Existence and multiplicity of solutions for a class of sublinear Schrödinger-Maxwell equations

Boundary Value Problems20132013:177

DOI: 10.1186/1687-2770-2013-177

Received: 24 September 2012

Accepted: 5 July 2013

Published: 30 July 2013

Abstract

In this paper I consider a class of sublinear Schrödinger-Maxwell equations, and new results about the existence and multiplicity of solutions are obtained by using the minimizing theorem and the dual fountain theorem respectively.

Keywords

Schrödinger-Maxwell equations sublinear minimizing theorem dual fountain theorem

1 Introduction and main result

Consider the following semilinear Schrödinger-Maxwell equations:
{ Δ u + V ( x ) u + ϕ u = f ( x , u ) , in  R 3 , Δ ϕ = u 2 , lim | x | ϕ ( x ) = 0 , in  R 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ1_HTML.gif
(1)

Such a system, also known as the nonlinear Schrödinger-Poisson system, arises in an interesting physical context. Indeed, according to a classical model, the interaction of a charge particle with an electromagnetic field can be described by coupling the nonlinear Schrödinger and the Maxwell equations (we refer to [1, 2] for more details on the physical aspects and on the qualitative properties of the solutions). In particular, if we are looking for electrostatic-type solutions, we just have to solve (1).

In recent years, system (1), with V ( x ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq1_HTML.gif or being radially symmetric, has been widely studied under various conditions on f; see, for example, [311]. Since (1) is set on R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq2_HTML.gif, it is well known that the Sobolev embedding H 1 ( R 3 ) L s ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq3_HTML.gif ( 2 s 2 = 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq4_HTML.gif) is not compact, and then it is usually difficult to prove that a minimizing sequence or a sequence that satisfies the ( PS ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq5_HTML.gif condition, briefly a Palais-Smale sequence, is strongly convergent if we seek solutions of (1) by variational methods. If V ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq6_HTML.gif is radial (for example, V ( x ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq7_HTML.gif), we can avoid the lack of compactness of Sobolev embedding by looking for solutions of (1) in the subspace of radial functions of H 1 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq8_HTML.gif, which is usually denoted by H r 1 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq9_HTML.gif, since the embedding H r 1 ( R 3 ) L s ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq10_HTML.gif ( 2 < s < 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq11_HTML.gif) is compact. Specially, Ruiz [11] dealt with (1) under the assumption that V ( x ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq7_HTML.gif and f ( u ) = u p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq12_HTML.gif ( 1 < p < 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq13_HTML.gif) and got some general existence, nonexistence and multiplicity results.

Moreover, in [12] the authors considered system (1) with periodic potential V ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq6_HTML.gif, and the existence of infinitely many geometrically distinct solutions was proved by the nonlinear superposition principle established in [13].

There are also some papers treating the case with nonradial potential V ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq6_HTML.gif. More precisely, Wang and Zhou [14] got the existence and nonexistence results of (1) when f ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq14_HTML.gif is asymptotically linear at infinity. Chen and Tang [15] proved that (1) has infinitely many high energy solutions under the condition that f ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq15_HTML.gif is superlinear at infinity in u by the fountain theorem. Soon after, Li, Su and Wei [16] improved their results.

Up to now, there have been few works concerning the case that V ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq6_HTML.gif is nonradial potential and f ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq15_HTML.gif is sublinear at infinity in u. Very recently, Sun [17] treated the above case based on the variant fountain theorem established in Zou [18].

Theorem 1.1 [17]

Assume that the following conditions hold:

( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq16_HTML.gif) V C ( R 3 , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq17_HTML.gif satisfies inf x R 3 V ( x ) a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq18_HTML.gif, where a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq19_HTML.gif is a constant. For every M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq20_HTML.gif, meas { x R 3 : v ( x ) M } < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq21_HTML.gif.

( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq22_HTML.gif) F ( x , u ) = a ( x ) | u | r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq23_HTML.gif, where F ( x , u ) = 0 u f ( x , y ) d y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq24_HTML.gif, a : R 3 R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq25_HTML.gif is a positive function such that a L 2 2 r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq26_HTML.gif and 1 < r < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq27_HTML.gif.

Then problem (1) has infinitely many nontrivial solutions { ( u k , ϕ k ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq28_HTML.gif satisfying
1 2 R 3 ( | u k | 2 + V ( x ) u k 2 ) d x 1 4 R 3 | ϕ k | 2 d x + 1 2 R 3 ϕ k u k 2 d x R 3 F ( x , u k ) d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equa_HTML.gif

as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif.

In the present paper, based on the dual fountain theorem, we can prove the same result under a more generic condition, which generalizes the result in [17]. Our first result can be stated as follows.

Theorem 1.2 Assume that V satisfies

( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq30_HTML.gif) V C ( R 3 , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq17_HTML.gif and inf x R 3 V ( x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq31_HTML.gif;

and f satisfies the following conditions.

( W 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq32_HTML.gif) There exist constants δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq33_HTML.gif, r 1 ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq34_HTML.gif and a function a 1 L 2 2 r 1 ( R 3 , [ 0 , + ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq35_HTML.gif such that
| f ( x , u ) | a 1 ( x ) | u | r 1 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equb_HTML.gif

for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif and | u | δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq37_HTML.gif;

( W 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq38_HTML.gif) There exist constants M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq20_HTML.gif, r 2 ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq39_HTML.gif and a function a 2 L 2 2 r 2 ( R 3 , [ 0 , + ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq40_HTML.gif such that
| f ( x , u ) | a 2 ( x ) | u | r 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equc_HTML.gif

for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif and | u | M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq41_HTML.gif;

( W 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq42_HTML.gif) For every m > δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq43_HTML.gif, there exist a constant r 3 ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq44_HTML.gif and a function b m L 2 2 r 3 ( R 3 , [ 0 , + ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq45_HTML.gif such that
| f ( x , u ) | b m ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equd_HTML.gif

for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif and | u | m http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq46_HTML.gif;

( W 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq47_HTML.gif) There exist constants r 4 ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq48_HTML.gif, η > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq49_HTML.gif and ζ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq50_HTML.gif such that
F ( x , u ) η | u | r 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Eque_HTML.gif

for all x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq51_HTML.gif and | u | ζ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq52_HTML.gif, where meas { Ω } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq53_HTML.gif, F ( x , u ) : = 0 u f ( x , y ) d y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq54_HTML.gif;

( W 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq55_HTML.gif) F ( x , u ) = F ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq56_HTML.gif for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif and u R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq57_HTML.gif.

Then problem (1) has infinitely many nontrivial solutions { ( u k , ϕ k ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq28_HTML.gif satisfying
1 2 R 3 ( | u k | 2 + V ( x ) u k 2 ) d x 1 4 R 3 | ϕ k | 2 d x + 1 2 R 3 ϕ k u k 2 d x R 3 F ( x , u k ) d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equf_HTML.gif

as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif.

By Theorem 1.2, we obtain the following corollary.

Corollary 1.3 Assume that L satisfies ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq30_HTML.gif) and W satisfies

( W 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq58_HTML.gif) F ( x , u ) = a ( x ) | u | r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq23_HTML.gif, where F ( x , u ) = 0 u f ( x , y ) d y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq24_HTML.gif, 1 < r < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq27_HTML.gif is a constant and a : R 3 R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq59_HTML.gif is a function such that a L 2 2 r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq60_HTML.gif and a ( x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq61_HTML.gif for x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq51_HTML.gif, where meas { Ω } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq53_HTML.gif.

Then problem (1) has infinitely many nontrivial solutions { ( u k , ϕ k ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq28_HTML.gif satisfying
1 2 R 3 ( | u k | 2 + V ( x ) u k 2 ) d x 1 4 R 3 | ϕ k | 2 d x + 1 2 R 3 ϕ k u k 2 d x R 3 F ( x , u k ) d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equg_HTML.gif

as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif.

Remark 1.4 In Theorem 1.2, infinitely many solutions for problem (1) are obtained under the symmetry condition ( W 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq62_HTML.gif) by using the dual fountain theorem. As a special case of Theorem 1.2, Corollary 1.3 generalizes and improves Theorem 1.1. To show this, it suffices to compare ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq16_HTML.gif) and ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq30_HTML.gif), ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq22_HTML.gif) and ( W 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq58_HTML.gif). Firstly, it is clear that ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq30_HTML.gif) is really weaker than ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq16_HTML.gif). Secondly, in ( H 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq22_HTML.gif) a is assumed to be positive, while in ( W 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq58_HTML.gif) we assume that a is indefinite.

Moreover, under all the conditions of Theorem 1.2 except ( W 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq55_HTML.gif) we obtain an existence result.

Theorem 1.5 Assume that L satisfies ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq30_HTML.gif) and W satisfies ( W 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq32_HTML.gif), ( W 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq38_HTML.gif), ( W 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq42_HTML.gif), ( W 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq47_HTML.gif). Then problem (1) possesses a nontrivial solution.

Remark 1.6 In Theorem 1.5 we obtain the existence of solutions for problem (1) under the assumption that f ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq63_HTML.gif is indefinite and without any coercive assumptions respect to V such as ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq16_HTML.gif). There are functions V and f which satisfy Theorem 1.5, but do not satisfy the corresponding results in [216]. For example,
V ( x ) 1 , f ( x , u ) = a ˜ ( x ) | u | 3 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ2_HTML.gif
(2)
and
a ˜ ( x ) = { ( 1 ) n n 3 ( | x | n ) for  n | x | n + 1 n 2 , 0 else , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ3_HTML.gif
(3)
in which n 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq64_HTML.gif. It is clear that a ˜ C ( R 3 , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq65_HTML.gif is indefinite. Denoting by π the area of the unit ball in R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq2_HTML.gif, we obtain
R 3 a ˜ 4 ( x ) d x = n = 3 ( n n + 1 n 2 n 12 r 2 ( r n ) 4 d r + n + 1 n 2 n + 2 n 2 n 12 r 2 ( n + 2 n 2 r ) 4 d r ) π = π n = 3 2 n 12 0 1 n 2 r 6 d x = 2 π 7 n = 3 n 2 < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ4_HTML.gif
(4)

which means that a ˜ L 2 2 3 2 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq66_HTML.gif. So, (2) satisfies our results, but does not satisfy the results in [317].

2 Preliminary results

In order to establish our results via critical point theory, we firstly describe some properties of the space H 1 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq8_HTML.gif, on which the variational functional associated with problem (1) is defined. Define the function space
H 1 ( R 3 ) : = { u L 2 ( R 3 ) : u ( L 2 ( R 3 ) ) 3 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equh_HTML.gif
equipped with the norm
u H 1 : = ( R 3 ( | u | 2 + u 2 ) d x ) 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equi_HTML.gif
and the function space
D 1 , 2 ( R 3 ) : = { u L 2 : u ( L 2 ( R 3 ) ) 3 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equj_HTML.gif
with the norm
u D 1 , 2 = ( R 3 | u | 2 d x ) 1 / 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equk_HTML.gif
Let
E : = { u H 1 ( R 3 ) : R 3 V ( x ) u 2 d x < + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equl_HTML.gif
equipped with the inner product
( u , v ) = R 3 ( u v + V ( x ) u v ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equm_HTML.gif
and the corresponding norm
u 2 = ( u , u ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equn_HTML.gif
Note that the following embeddings
E L s ( R 3 ) , 2 s 2 , D 1 , 2 ( R 3 ) L 2 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equo_HTML.gif
are continuous, where 2 = 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq67_HTML.gif is the critical exponent for the Sobolev embeddings in dimension 3. Therefore, there exist constants C p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq68_HTML.gif and C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq69_HTML.gif such that
u L p C p u , u L 2 C u D 1 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ5_HTML.gif
(5)
for all u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq70_HTML.gif. Here L p ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq71_HTML.gif ( 2 p 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq72_HTML.gif) denotes the Banach spaces of a function on R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq2_HTML.gif with values in R under the norm
u L p = ( R 3 | u ( x ) | p d x ) 1 / p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equp_HTML.gif
Let
L a r ( R 3 ) : = { u : R 3 R : R 3 a ( x ) | u | r d x < + } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equq_HTML.gif
where a ( x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq61_HTML.gif for a.e. x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif. Then L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq73_HTML.gif is a Banach space with the norm
u L a r = ( R 3 a ( x ) | u | r d x ) 1 / r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equr_HTML.gif

Lemma 2.1 Suppose that assumption ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq30_HTML.gif) holds. Then the embedding of E in L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq73_HTML.gif is compact, where r ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq74_HTML.gif, a L 2 2 r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq60_HTML.gif is positive for a.e. x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif.

Proof For any bounded set K E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq75_HTML.gif, there exists a positive constant M 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq76_HTML.gif such that u M 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq77_HTML.gif for all u K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq78_HTML.gif. We claim that K is precompact in L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq73_HTML.gif. In fact, since a L 2 2 r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq60_HTML.gif, for any ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq79_HTML.gif, there exists T ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq80_HTML.gif such that
( | x | T ε a ( x ) 2 2 r d x ) ( 2 r ) / 2 < ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equs_HTML.gif
For any u , v K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq81_HTML.gif, applying the Hölder inequality for r such that r 2 + 2 r 2 = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq82_HTML.gif and the first inequality in (5), we have
| x | T ε a ( x ) | u v | r d x ( | x | T ε a ( x ) 2 2 r d x ) ( 2 r ) / 2 ( | x | T ε | u v | 2 d x ) r / 2 u v L 2 r ( | x | T ε a ( x ) 2 2 r d x ) ( 2 r ) / 2 C 2 r u v r ε 2 C 2 r M 0 r ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ6_HTML.gif
(6)
Besides, since E ( B T ε ( 0 ) ) H 1 ( B T ε ( 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq83_HTML.gif is compactly embedded in L a r ( B T ε ( 0 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq84_HTML.gif, where B T ε ( 0 ) = { x R 3 : | x | T ε } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq85_HTML.gif, there are u 1 , u 2 , , u m K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq86_HTML.gif such that for any u K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq78_HTML.gif,
| x | T ε a ( x ) | u u i | r d x < ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ7_HTML.gif
(7)

Now it follows from (6) and (7) that K is precompact in L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq73_HTML.gif. Obviously, we have E is compact embedded in L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq73_HTML.gif, where r ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq87_HTML.gif, a L 2 2 r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq60_HTML.gif is positive for a.e. x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif. □

Lemma 2.2 Assume that assumptions ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq30_HTML.gif), ( W 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq32_HTML.gif), ( W 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq38_HTML.gif) and ( W 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq42_HTML.gif) hold and u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq88_HTML.gif in E. Then
f ( x , u n ) f ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equt_HTML.gif

in L 2 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq89_HTML.gif.

Proof Assume that u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq88_HTML.gif in E. Then, by Lemma 2.1,
u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equu_HTML.gif
in L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq90_HTML.gif, where r ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq74_HTML.gif, a L 2 2 r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq60_HTML.gif is positive for a.e. x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif. Passing to a subsequence if necessary, it can be assumed that
n = 1 u n u L a r < + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equv_HTML.gif
It is clear that
h k ( x ) : = n = 1 k | u n ( x ) u ( x ) | L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ8_HTML.gif
(8)
and
h g h l L a r n = l g u n u L a r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ9_HTML.gif
(9)
for all g > l N + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq91_HTML.gif. Since { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq92_HTML.gif is a Cauchy sequence in L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq73_HTML.gif, so by (9) we know that { h k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq93_HTML.gif is also a Cauchy sequence in L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq73_HTML.gif. Therefore, by the completeness of L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq73_HTML.gif, there exists h L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq94_HTML.gif such that h k h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq95_HTML.gif in L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq73_HTML.gif. Now we show that
h k ( x ) h ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ10_HTML.gif
(10)
for all k N + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq96_HTML.gif and almost every x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif. If not, there exist k 0 N + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq97_HTML.gif and S R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq98_HTML.gif, with meas { S } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq99_HTML.gif, such that
h k 0 ( x ) > h ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equw_HTML.gif
for all x S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq100_HTML.gif. Then there exist a constant c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq101_HTML.gif and S 0 S http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq102_HTML.gif, with meas { S 0 } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq103_HTML.gif, such that
h k 0 ( x ) h ( x ) + c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equx_HTML.gif
for all x S 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq104_HTML.gif. By the definition of h k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq105_HTML.gif, we have
h k ( x ) h k 0 ( x ) h ( x ) + c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equy_HTML.gif
for all k k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq106_HTML.gif and x S 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq104_HTML.gif. Therefore, one has
R 3 a ( x ) | h k h | r d x S 0 a ( x ) | h k h | r d x c r S 0 a ( x ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equz_HTML.gif
Letting k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif, we get
0 c r S 0 a ( x ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equaa_HTML.gif
which contradicts the fact that a ( x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq61_HTML.gif for a.e. x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif. Now we have proved (10). It follows from ( W 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq38_HTML.gif) that there exists M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq20_HTML.gif such that
| f ( x , u ) | a 2 ( x ) | u | r 2 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ11_HTML.gif
(11)
for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif and | u | M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq41_HTML.gif. By ( W 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq32_HTML.gif), there exists δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq33_HTML.gif such that
| f ( x , u ) | a 1 ( x ) | u | r 1 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ12_HTML.gif
(12)
for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif and | u | δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq37_HTML.gif, which together with ( W 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq42_HTML.gif) shows there exists b M L 2 2 r 3 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq107_HTML.gif such that
| f ( x , u ) | a 1 ( x ) | u | r 1 1 + b M ( x ) δ r 3 1 | u | r 3 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ13_HTML.gif
(13)
for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif and | u | M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq108_HTML.gif. Combining (11) and (13), we have
| f ( x , u ) | a 1 ( x ) | u | r 1 1 + a 2 ( x ) | u | r 2 1 + b M δ r 3 1 | u | r 3 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ14_HTML.gif
(14)
for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif and u R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq57_HTML.gif. Hence, by (10) one has
| f ( x , u n ) f ( x , u ) | a 1 ( x ) ( | u n | r 1 1 + | u | r 1 1 ) + a 2 ( x ) ( | u n | r 2 1 + | u | r 2 1 ) + b M ( x ) δ r 3 1 ( | u n | r 3 1 + | u | r 3 1 ) a 1 ( x ) ( | u n u | r 1 1 + 2 | u ( x ) | r 1 1 ) + a 2 ( x ) ( | u n u | r 2 1 + 2 | u | r 2 1 ) + b M ( x ) δ r 3 1 ( | u n u | r 3 1 + 2 | u | r 3 1 ) a 1 ( x ) ( | h | r 1 1 + 2 | u | r 1 1 ) + a 2 ( x ) ( | h | r 2 1 + 2 | u | r 2 1 ) + b M ( x ) δ r 3 1 ( | h | r 3 1 + 2 | u | r 3 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equab_HTML.gif
for all n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq109_HTML.gif and x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif. It follows that
| f ( x , u n ) f ( x , u ) | 2 d x 6 a 1 2 ( x ) ( | h | 2 ( r 1 1 ) + 4 | u | 2 ( r 1 1 ) ) d x + 6 a 2 2 ( x ) ( | h | 2 ( r 2 1 ) + 4 | u | 2 ( r 2 1 ) ) d x + 6 b M 2 ( x ) δ 2 ( r 3 1 ) ( | h | 2 ( r 3 1 ) + 4 | u | 2 ( r 3 1 ) ) d x = : ϱ ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ15_HTML.gif
(15)
for all n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq109_HTML.gif. By the Hölder inequality, we have
R 3 a 1 2 ( x ) | h | 2 ( r 1 1 ) d x ( R 3 a 1 ( x ) 2 2 r 1 d x ) 2 r 1 r 1 ( R 3 a 1 ( x ) | h | r 1 d x ) 2 ( r 1 1 ) r 1 = a 1 L 2 2 r 1 2 r 1 h L a 1 r 1 2 ( r 1 1 ) < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ16_HTML.gif
(16)
Similarly, we can prove
R 3 a 1 2 ( x ) | u | 2 ( r 1 1 ) d x < , R 3 a 2 2 ( x ) | h | 2 ( r 2 1 ) d x < , R 3 a 2 2 ( x ) | u | 2 ( r 2 1 ) d x < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ17_HTML.gif
(17)
also
R 3 b M 2 ( x ) | h | 2 ( r 3 1 ) d x < , R 3 b M 2 ( x ) | u | 2 ( r 3 1 ) d x < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ18_HTML.gif
(18)
It follows from (15), (16), (17) and (18) that
ϱ L 1 ( R 3 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equac_HTML.gif
which together with Lebesgue’s convergence theorem shows
R 3 | f ( x , u n ) f ( x , u ) | 2 d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ19_HTML.gif
(19)

as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq110_HTML.gif. Now we have proved the lemma. □

In the proof of Theorem 1.2, the following lemma is needed.

Lemma 2.3 Assume that G R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq111_HTML.gif is an open set. Then, for any closed set H G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq112_HTML.gif, there exists a function φ C 0 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq113_HTML.gif such that φ ( x ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq114_HTML.gif for all x R 3 G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq115_HTML.gif, φ ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq116_HTML.gif for all x H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq117_HTML.gif and 0 ϕ ( x ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq118_HTML.gif for all x G H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq119_HTML.gif.

Proof Letting
α ˜ ( x ) = { e 1 | x | 2 1 , | x | < 1 , 0 , | x | 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equad_HTML.gif
then α ˜ C 0 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq120_HTML.gif and supp α ˜ = B 1 ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq121_HTML.gif. For any given ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq79_HTML.gif, defining α and α ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq122_HTML.gif as follows,
α ( x ) = α ˜ ( x ) R 3 α ˜ ( x ) d x , α ε ( x ) = 1 ε 3 α ( x ε ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equae_HTML.gif
one has α ε C 0 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq123_HTML.gif, supp α ε = { x : | x | ε } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq124_HTML.gif and R 3 α ε ( x ) d x = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq125_HTML.gif. Denoting
d 0 = inf x H , y G d ( x , y ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equaf_HTML.gif
and
G θ : = { x G , d ( x , G ) θ } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equag_HTML.gif
it is clear that d 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq126_HTML.gif and H G d 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq127_HTML.gif. Lastly, we define
ψ ( x ) = { 1 , x G d 0 2 , 0 , x R G d 0 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equah_HTML.gif
and
φ ( x ) = R 3 ψ ( x y ) α d 0 4 ( y ) d y , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equai_HTML.gif

then φ ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq116_HTML.gif for all x H http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq117_HTML.gif and φ ( x ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq114_HTML.gif for all x G d 0 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq128_HTML.gif. Moreover, by the definition of α ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq122_HTML.gif, we have φ C 0 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq113_HTML.gif and 0 φ ( x ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq129_HTML.gif. □

Since E is a Hilbert space, then there exists a basis { v n } X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq130_HTML.gif such that X = j 1 X j ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq131_HTML.gif, where X j = span { v j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq132_HTML.gif. Letting Y k = j = 1 k X j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq133_HTML.gif, Z k = j k X j ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq134_HTML.gif, now we show the following lemma, which will be used in the proof of Theorem 1.2.

Lemma 2.4 Suppose r ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq87_HTML.gif and a L 2 2 r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq26_HTML.gif, then we have
β k ( a , r ) : = sup u Z k , u = 1 u L a r 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equaj_HTML.gif

as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif.

Proof It is clear that 0 < β k + 1 ( a , r ) β k ( a , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq135_HTML.gif, so there exists β ( a , r ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq136_HTML.gif such that
β k ( a , r ) β ( a , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ20_HTML.gif
(20)
as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif. By the definition of β k ( a , r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq137_HTML.gif, there exists u k Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq138_HTML.gif with u k = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq139_HTML.gif such that
u k L a r > β k ( a , r ) 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ21_HTML.gif
(21)
Since { u k } k N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq140_HTML.gif is bounded, then there exists u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq70_HTML.gif such that
u k u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equak_HTML.gif
as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif. Now, since { v j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq141_HTML.gif is a basis of E, it follows that for all j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq142_HTML.gif,
0 = ( u k , v j ) k > j ( u , v j ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equal_HTML.gif
as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq143_HTML.gif, which shows that u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq144_HTML.gif. By Lemma 2.1 we have
u k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equam_HTML.gif

in L a r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq90_HTML.gif for all r ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq87_HTML.gif and a L 2 2 r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq60_HTML.gif, which together with (20) and (21) implies that β ( a , r ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq145_HTML.gif for all r ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq87_HTML.gif and a L 2 2 r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq60_HTML.gif. □

We obtain the existence of a solution for problem (1) by using the following standard minimizing argument.

Lemma 2.5 [19]

Let E be a real Banach space and Φ C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq146_HTML.gif satisfying the ( PS ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq5_HTML.gif condition. If Φ is bounded from below,
c : = inf E Φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equan_HTML.gif

is a critical value of Φ.

In order to prove the multiplicity of solutions, we will use the dual fountain theorem. Firstly, we introduce the definition of the ( PS ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq147_HTML.gif condition.

Definition 2.6 Let Φ C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq148_HTML.gif and c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq149_HTML.gif. The function Φ satisfies the ( PS ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq147_HTML.gif condition if any sequence { u n j } E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq150_HTML.gif, such that
Φ ( u n j ) c , Φ | Y n j ( u n j ) 0 as  n j , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equao_HTML.gif

contains a subsequence converging to a critical point of Φ.

Now we show the following dual fountain theorem.

Lemma 2.7 [20]

If Φ ( u ) = Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq151_HTML.gif and for every k k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq106_HTML.gif, there exists ρ k > γ k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq152_HTML.gif such that
  1. (i)

    a k : = inf u Z k , u = ρ k Φ ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq153_HTML.gif,

     
  2. (ii)

    b k : = max u Y k , u = γ k Φ ( u ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq154_HTML.gif,

     
  3. (iii)

    d k : = inf u Z k , u = ρ k Φ ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq155_HTML.gif as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif.

     

Moreover, if Φ C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq156_HTML.gif satisfies the ( PS ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq147_HTML.gif condition for all c [ d k 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq157_HTML.gif, then Φ has a sequence of critical points { u k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq158_HTML.gif such that Φ ( u k ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq159_HTML.gif as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif.

3 Proof of theorems

Define the functional I : E × D 1 , 2 ( R 3 ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq160_HTML.gif by
I ( u , ϕ ) = 1 2 u 2 1 4 R 3 | ϕ | 2 d x + 1 2 R 3 ϕ u 2 d x R 3 F ( x , u ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ22_HTML.gif
(22)

It is easy to know that I exhibits a strong indefiniteness, namely it is unbounded both from below and from above on an infinitely dimensional subspace. This indefiniteness can be removed using the reduction method described in [1], by which we are led to study a variable functional that does not present such a strong indefinite nature.

Now we recall this method. For any u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq70_HTML.gif, consider the linear functional T u : D 1 , 2 ( R 3 ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq161_HTML.gif defined as
T u ( v ) = R 3 u 2 v d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equap_HTML.gif
By the Hölder inequality and using the second inequality in (5), we have
R 3 u 2 v d x u 2 L 6 / 5 v L 6 u L 12 / 5 v L 6 C 12 / 5 C u 2 v D 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equaq_HTML.gif
So, T u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq162_HTML.gif is continuous on D 1 , 2 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq163_HTML.gif. Set
μ ( u , v ) = R 3 u v d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equar_HTML.gif
for all u , v D 1 , 2 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq164_HTML.gif. Obviously, μ ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq165_HTML.gif is bilinear, bounded and coercive. Hence, the Lax-Milgram theorem implies that for every u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq70_HTML.gif, there exists a unique ϕ u D 1 , 2 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq166_HTML.gif such that
T u ( v ) = μ ( ϕ u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equas_HTML.gif
for any v D 1 , 2 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq167_HTML.gif, that is,
R 3 u 2 v d x = R 3 ϕ u v d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equat_HTML.gif
for any v D 1 , 2 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq167_HTML.gif. Using integration by parts, we get
R 3 ϕ u v d x = R 3 v Δ ϕ u d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equau_HTML.gif
for any v D 1 , 2 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq167_HTML.gif, therefore
Δ ϕ u = u 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ23_HTML.gif
(23)
in a weak sense. We can write an integral expression for ϕ u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq168_HTML.gif in the form
ϕ u = 1 4 π R 3 u 2 ( y ) | x y | d y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equav_HTML.gif

for any u C 0 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq169_HTML.gif (see [21], Theorem 1); by density it can be extended for any u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq70_HTML.gif (see Lemma 2.1 of [22]). Clearly, ϕ u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq170_HTML.gif and ϕ u = ϕ u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq171_HTML.gif for all u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq70_HTML.gif.

It follows from (23) that
R 3 ϕ u u 2 d x = R 3 ϕ u ( Δ ϕ u ) d x = R 3 | ϕ u | 2 d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ24_HTML.gif
(24)
and by the Hölder inequality, we have
ϕ u D 1 , 2 2 = R 3 ϕ u u 2 d x ( R 3 ϕ u 6 d x ) 1 / 6 ( R 3 | u | 12 5 ) 5 / 6 = C ϕ u D 1 , 2 u L 12 / 5 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equaw_HTML.gif
and it follows that
ϕ u D 1 , 2 C u L 12 / 5 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ25_HTML.gif
(25)
Hence,
R 3 ϕ u u 2 d x C 2 u L 12 / 5 4 C 2 C 12 / 5 4 u 4 : = C u 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ26_HTML.gif
(26)
So, we can consider the functional Φ : E R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq172_HTML.gif defined by Φ ( u ) = I ( u , ϕ u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq173_HTML.gif. By (24), the reduced functional takes the form
Φ ( u ) = 1 2 u 2 + 1 4 R 3 ϕ u u 2 d x R 3 F ( x , u ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ27_HTML.gif
(27)
By (12), we have
| F ( x , u ) | a 1 ( x ) r 1 | u | r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ28_HTML.gif
(28)
for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif and | u | δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq37_HTML.gif, where r 1 ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq34_HTML.gif and a 1 L 2 2 r 1 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq174_HTML.gif. Let u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq70_HTML.gif, then u C 0 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq175_HTML.gif, the space of continuous function u on R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq2_HTML.gif, such that u ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq176_HTML.gif as | x | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq177_HTML.gif. Therefore there exists T 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq178_HTML.gif such that
| u ( x ) | δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ29_HTML.gif
(29)
for all | x | > T 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq179_HTML.gif. Hence, one has
| x | > T 1 | F ( x , u ) | d x | x | > T 1 a 1 ( x ) r 1 | u ( x ) | r 1 d x 1 r 1 ( | x | T 1 a 1 ( x ) 2 2 r 1 d x ) ( 2 r 1 ) / 2 ( | x | T 1 | u ( x ) | 2 d x ) r 1 / 2 1 r 1 ( | x | T 1 a 1 ( x ) 2 2 r 1 d x ) ( 2 r 1 ) / 2 u L 2 r 1 1 r 1 C 2 r 1 u r 1 a 1 L 2 2 r 1 < , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equax_HTML.gif
which together with (26) shows that Φ is well defined. Furthermore, it is well known that Φ is a C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq180_HTML.gif functional with derivative given by
Φ ( u ) , v = R 3 [ ( u v ) + V ( x ) u v + ϕ u u v f ( x , u ) v ] d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equay_HTML.gif

It can be proved that ( u , ϕ ) E × D 1 , 2 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq181_HTML.gif is a solution of problem (1) if and only if u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq70_HTML.gif is a critical point of the functional Φ and ϕ = ϕ u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq182_HTML.gif; see, for instance, [1].

Lemma 3.1 Under conditions ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq30_HTML.gif), ( W 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq32_HTML.gif), ( W 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq38_HTML.gif), ( W 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq42_HTML.gif), Φ satisfies the ( PS ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq147_HTML.gif condition.

Proof Assume that { u n j } E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq183_HTML.gif is a sequence such that
Φ ( u n j ) c , Φ | Y n j ( u n j ) 0 as  n j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equaz_HTML.gif
Then there exists σ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq184_HTML.gif such that
| Φ ( u n j ) | σ , Φ | Y n j ( u n j ) E σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equba_HTML.gif

for all n j N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq185_HTML.gif.

Firstly, we show that { u n j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq186_HTML.gif is bounded. By (14), we have
| F ( x , u ) | a 1 ( x ) r 1 | u | r 1 + a 2 ( x ) r 2 | u | r 2 + b M ( x ) r 3 δ r 3 1 | u | r 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ30_HTML.gif
(30)
for all u R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq57_HTML.gif and x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif, which together with R 3 ϕ u n j u n j 2 d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq187_HTML.gif implies
u n j 2 = 2 Φ ( u n j ) 1 2 R 3 ϕ u n j u n j 2 d x + 2 R 3 F ( x , u n j ) d x 2 σ + 2 r 1 R 3 a 1 ( x ) | u n j | r 1 d x + 2 r 2 R 3 a 2 ( x ) | u n j | r 2 d x + 2 r 3 δ r 3 1 R 3 b M ( x ) | u n j | r 3 d x 2 σ + 2 r 1 ( R 3 a 1 ( x ) 2 2 r 1 d x ) ( 2 r 1 ) / 2 ( R 3 | u n j | 2 d x ) r 1 / 2 + 2 r 2 ( R 3 a 2 ( x ) 2 2 r 2 d x ) ( 2 r 2 ) / 2 ( R 3 | u n j | 2 d x ) r 2 / 2 + 2 r 3 δ r 3 1 ( R 3 b M ( x ) 2 2 r 3 d x ) ( 2 r 3 ) / 2 ( R 3 | u n j | 2 d x ) r 3 / 2 2 σ + 2 r 1 C 2 r 1 a 1 L 2 2 r 1 u n j r 1 + 2 r 2 C 2 r 2 a 2 L 2 2 r 2 u n j r 2 + 2 r 3 δ r 3 1 C 2 r 3 b M L 2 2 r 3 u n j r 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ31_HTML.gif
(31)

Noting that r i < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq188_HTML.gif for all i = 1 , 2 , 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq189_HTML.gif, so u n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq190_HTML.gif is bounded.

By the fact that { u n j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq186_HTML.gif is bounded in E, there exists u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq70_HTML.gif and a constant d > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq191_HTML.gif such that
sup n j N u n j d , u d http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ32_HTML.gif
(32)
and
u n j u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbb_HTML.gif
in E as n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq192_HTML.gif. It is obvious that
Φ ( u n j ) Φ ( u ) , u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ33_HTML.gif
(33)
and
ϕ u u ( u n j u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ34_HTML.gif
(34)
as n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq192_HTML.gif. On the other hand, by ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq30_HTML.gif), (32) and Lemma 2.2, one has
| R 3 ( f ( x , u n j ) f ( x , u ) ) u n j d x | f ( x , u n j ) f ( x , u ) L 2 u n j L 2 C 2 f ( x , u n j ) f ( x , u ) L 2 u n j C 2 d f ( x , u n j ) f ( x , u ) L 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ35_HTML.gif
(35)
as n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq192_HTML.gif, which implies
Φ ( u n j ) Φ ( u ) , u n j 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ36_HTML.gif
(36)
as n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq192_HTML.gif. Summing up (33) and (36), we have
Φ ( u n j ) Φ ( u ) , u n j u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ37_HTML.gif
(37)
as n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq192_HTML.gif. By the Hölder inequality and (25), one gets
R 3 ϕ u n j u n j ( u n j u ) d x ϕ u n j u n j L 2 u n j u L 2 ϕ u n j L 6 u n j L 3 u n j u L 2 C ϕ u n j D 1 , 2 u n j L 3 u n j u L 2 C 2 u n j L 12 / 5 2 u n j L 3 u n j u L 2 C 2 C 12 / 5 2 C 3 C 2 u n j 3 u n j u 2 C 2 C 12 / 5 2 C 3 C 2 d 4 < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbc_HTML.gif
Then by Lebesgue’s convergence theorem, we have
R 3 ϕ u n j u n j ( u n j u ) d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbd_HTML.gif
as n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq192_HTML.gif, which together with (34) implies
R 3 ( ϕ u n j u n j ϕ u u ) ( u n j u ) d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ38_HTML.gif
(38)
as n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq192_HTML.gif. By Lemma 2.2 and (32), we get
| R 3 ( f ( x , u n j ) f ( x , u ) ) ( u n j u ) d x | f ( x , u n j ) f ( x , u ( x ) ) L 2 u n j u L 2 C 2 f ( x , u n j ) f ( x , u ) L 2 u n j u 2 C 2 d f ( x , u n j ) f ( x , u ) L 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Eqube_HTML.gif
as n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq192_HTML.gif. Moreover, an easy computation shows that
Φ ( u n j ) Φ ( u ) , u n j u = u n j u 2 + R 3 ( ϕ u n j u n j ϕ u u ) ( u n j u ) d x R 3 ( f ( x , u n j ) f ( x , u ) ) ( u n j u ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbf_HTML.gif

Consequently, u n j u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq193_HTML.gif as n j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq192_HTML.gif. Φ satisfies the ( PS ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq147_HTML.gif condition. □

Remark 3.2 Under conditions ( V 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq30_HTML.gif), ( W 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq32_HTML.gif), ( W 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq38_HTML.gif), ( W 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq42_HTML.gif), Φ satisfies the ( PS ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq5_HTML.gif condition. Assume that { u n } E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq194_HTML.gif is a sequence such that I ( u n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq195_HTML.gif is bounded and
I ( u n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbg_HTML.gif
as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq110_HTML.gif. Then there exists σ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq184_HTML.gif such that
| I ( u n ) | σ , I ( u n ) E σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbh_HTML.gif

for all n N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq109_HTML.gif. The rest of the proof is the same as that of Lemma 3.1.

Proof of Theorem 1.2 For any k N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq196_HTML.gif, we take k disjoint open sets { Ω i | i = 1 , , k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq197_HTML.gif such that
i = 1 k Ω i Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbi_HTML.gif
For any ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq79_HTML.gif and Ω i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq198_HTML.gif, there exist a closed set H i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq199_HTML.gif and an open set G i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq200_HTML.gif such that H i Ω i G i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq201_HTML.gif and
meas { G i Ω i } < ε , meas { Ω i H i } < ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbj_HTML.gif

For every G i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq200_HTML.gif ( i = 1 , , k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq202_HTML.gif), by Lemma 2.3 there exists φ i C 0 ( G i , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq203_HTML.gif such that φ i | H i = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq204_HTML.gif and 0 φ i 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq205_HTML.gif. Letting v i = φ i φ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq206_HTML.gif, can be extended to be a basis { v n } X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq130_HTML.gif. Therefore X = j 1 X j ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq131_HTML.gif, where X j = span { v j } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq132_HTML.gif. Now we define Y k : = j = 1 k X j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq207_HTML.gif, Z k : = j k X j ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq208_HTML.gif.

By Lemma 3.1, Φ C 1 ( E , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq148_HTML.gif satisfies the ( PS ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq209_HTML.gif condition and Φ ( u ) = Φ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq210_HTML.gif. Hence, to prove Theorem 1.2, we should just show that Φ has the geometric property (i), (ii) and (iii) in Lemma 2.7.
  1. (i)
    By Lemma 2.4
    β k ( a , r ) = sup u Z k , u = 1 u L a r 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbk_HTML.gif
     
as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif for r ( 1 , 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq87_HTML.gif and a L 2 2 r ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq60_HTML.gif. In view of (30) and the fact that R 3 ϕ u u 2 d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq211_HTML.gif, we have
Φ ( u ) = 1 2 u 2 + 1 4 R 3 ϕ u u 2 d x R 3 F ( x , u ) d x 1 2 | u | 2 R 3 F ( x , u ) d x 1 2 u 2 2 r 1 R 3 a 1 ( x ) | u | r 1 d x 2 r 2 R 3 a 2 ( x ) | u | r 2 d x 2 r 3 δ r 3 1 R 3 b M ( x ) | u | r 3 d x 1 2 u 2 2 u L a 1 r 1 r 1 r 1 2 u L a 2 r 2 r 2 r 2 2 u L a 3 r 3 r 3 r 3 δ r 3 1 1 2 u 2 2 β k ( a 1 , r 1 ) r 1 r 1 u r 1 2 β k ( a 2 , r 2 ) r 2 r 2 u r 2 2 β k ( b M , r 3 ) r 3 r 3 δ r 3 1 u r 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ39_HTML.gif
(39)
Let r : = min { r 1 , r 2 , r 3 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq212_HTML.gif, β k : = max { β k ( a 1 , r 1 ) , β k ( a 2 , r 2 ) , β k ( b M , r 3 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq213_HTML.gif, C : = max { 2 r 1 , 2 r 2 , 2 r 3 δ r 3 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq214_HTML.gif, then β k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq215_HTML.gif as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif. Hence, we have
Φ ( u ) 1 2 u 2 3 C β k r u r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ40_HTML.gif
(40)
when u 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq216_HTML.gif and β k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq217_HTML.gif. Now we can choose ρ k = ( 12 β k r C ) 1 / ( 2 r ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq218_HTML.gif, then ρ k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq219_HTML.gif as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif. When k is large enough, we have ρ k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq220_HTML.gif, β k 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq217_HTML.gif, which together with (40) shows
a k : = inf u Z k , u = ρ k Φ ( u ) 1 4 ρ k 2 > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbl_HTML.gif
  1. (ii)
    For any u Y k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq221_HTML.gif, there exists λ i = 1 , 2 , , k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq222_HTML.gif such that
    u = i = 1 k λ i v i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbm_HTML.gif
     
Then we have
u L r 4 r 4 = R 3 | u ( x ) | r 4 d x = i = 1 k | λ i | r 4 Ω i | v i ( x ) | r 4 d x + i = 1 k | λ i | r 4 G i Ω i | v i ( x ) | r 4 d x = i = 1 k | λ i | r 4 Ω i | v i ( x ) | r 4 d x + i = 1 k | λ i | r 4 G i Ω i | φ i ( x ) | r 4 φ i r 4 d x i = 1 k | λ i | r 4 Ω i | v i ( x ) | r 4 d x + i = 1 k | λ i | r 4 φ i r 4 meas { G i Ω i } i = 1 k | λ i | r 4 Ω i | v i ( x ) | r 4 + i = 1 k | λ i | r 4 φ i r 4 ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ41_HTML.gif
(41)
and also
u 2 = R 3 [ | u | 2 + V ( x ) u 2 ] d x = i = 1 k λ i 2 G i [ | v i | 2 + V ( x ) v i 2 ] d x = i = 1 k λ i 2 v i 2 = i = 1 k λ i 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ42_HTML.gif
(42)
Since all the norms of a finite dimensional space are equivalent, there is a constant C ˜ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq223_HTML.gif such that
C ˜ u u L r 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbn_HTML.gif
for all u Y k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq221_HTML.gif. By (30), one has
F ( x , λ i v i ) a 1 ( x ) r 1 | λ i v i | r 1 a 2 ( x ) r 2 | λ i v i | r 2 b M ( x ) r 3 δ r 3 1 | λ i v i | r 3 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbo_HTML.gif
Therefore, we have
i = 1 k G i Ω i F ( x , λ i v i ) d x i = 1 k G i Ω i | λ i | r 1 r 1 a 1 ( x ) | v i | r 1 d x i = 1 k G i Ω i | λ i | r 2 r 2 a 2 ( x ) | v i | r 2 d x i = 1 k G i Ω i | λ i | r 3 r 3 δ r 3 1 b M ( x ) | v i | r 3 d x i = 1 k | λ i | r 1 r 1 a 1 L 2 2 r 1 ( G i Ω i | v i | 2 d x ) r 1 / 2 i = 1 k | λ i | r 2 r 2 a 2 L 2 2 r 2 ( G i Ω i | v i | 2 d x ) r 2 / 2 i = 1 k | λ i | r 3 r 3 δ r 3 1 b M L 2 2 r 3 ( G i Ω i | v i | 2 d x ) r 3 / 2 i = 1 k | λ i | r 1 r 1 a 1 L 2 2 r 1 ( G i Ω i | φ i | 2 φ i 2 d x ) r 1 / 2 i = 1 k | λ i | r 2 r 2 a 2 L 2 2 r 2 ( G i Ω i | φ i | 2 φ i 2 d x ) r 2 / 2 i = 1 k | λ i | r 3 r 3 δ r 3 1 b M L 2 2 r 3 ( G i Ω i | φ i | 2 φ i 2 d x ) r 3 / 2 = 1 r 1 a 1 L 2 2 r 1 i = 1 k | λ i | r 1 φ i r 1 ( meas { G i Ω i } ) r 1 / 2 1 r 2 a 2 L 2 2 r 2 i = 1 k | λ i | r 2 φ i r 2 ( meas { G i Ω i } ) r 2 / 2 1 r 3 δ r 3 1 b M L 2 2 r 3 i = 1 k | λ i | r 3 φ i r 3 ( meas { G i Ω i } ) r 3 / 2 1 r 1 a 1 L 2 2 r 1 i = 1 k | λ i | r 1 φ i r 1 ε r 1 / 2 1 r 2 a 2 L 2 2 r 2 i = 1 k | λ i | r 2 φ i r 2 ε r 2 / 2 1 r 3 δ r 3 1 b M L 2 2 r 3 i = 1 k | λ i | r 3 φ i r 3 ε r 3 / 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ43_HTML.gif
(43)
For any u Y k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq221_HTML.gif with u = i = 1 k λ i 2 = γ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq224_HTML.gif, we can choose γ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq225_HTML.gif small enough such that | λ i v i ( x ) | < ζ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq226_HTML.gif for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif and i = 1 , , k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq202_HTML.gif, which together with ( W 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq227_HTML.gif) implies
F ( x , λ i v i ) η | λ i v i | r 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ44_HTML.gif
(44)
for all x Ω i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq228_HTML.gif and i = 1 , , k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq202_HTML.gif. Combining (24), (41), (42), (43) and (44), we have
Φ ( u ) = 1 2 u 2 + 1 4 R 3 ϕ u u 2 d x R 3 F ( x , u ) d x = 1 2 u 2 + C 4 u 4 i = 1 k G i F ( x , λ i v i ) d x 1 2 u 2 i = 1 k [ G i Ω i F ( x , λ i v i ) d x + Ω i F ( x , λ i v i ) d x ] 1 2 u 2 + C 4 u 4 + 1 r 1 a 1 L 2 2 r 1 i = 1 k | λ i | r 1 φ i r 1 ε r 1 / 2 + 1 r 2 a 2 L 2 2 r 2 i = 1 k | λ i | r 2 φ i r 2 ε r 2 / 2 + 1 r 3 δ r 3 1 b M L 2 2 r 3 i = 1 k | λ i | r 3 φ i r 3 ε r 3 / 2 η i = 1 k | λ i | r 4 Ω i | v i | r 4 d x = 1 2 u 2 + C 4 u 4 + 1 r 1 a 1 L 2 2 r 1 i = 1 k | λ i | r 1 φ i r 1 ε r 1 / 2 + 1 r 2 a 2 L 2 2 r 2 i = 1 k | λ i | r 2 φ i r 2 ε r 2 / 2 + 1 r 3 δ r 3 1 b M L 2 2 r 3 i = 1 k | λ i | r 3 φ i r 3 ε r 3 / 2 η ( u L r 4 r 4 i = 1 k | λ i | r 4 φ i r 4 ε ) 1 2 u 2 + C 4 u 4 η C ˜ r 4 u r 4 + 1 r 1 a 1 L 2 2 r 1 i = 1 k | λ i | r 1 φ i r 1 ε r 1 / 2 + 1 r 2 a 2 L 2 2 r 2 i = 1 k | λ i | r 2 φ i r 2 ε r 2 / 2 + 1 r 3 δ r 3 1 b M L 2 2 r 3 i = 1 k | λ i | r 3 φ i r 3 ε r 3 / 2 + η i = 1 k | λ i | r 4 φ i r 4 ε = 1 2 i = 1 k λ i 2 + C 4 ( i = 1 k λ i 2 ) 2 η C ˜ r 4 ( i = 1 k λ i 2 ) r 4 / 2 + 1 r 1 a 1 L 2 2 r 1 i = 1 k | λ i | r 1 φ i r 1 ε r 1 / 2 + 1 r 2 a 2 L 2 2 r 2 i = 1 k | λ i | r 2 φ i r 2 ε r 2 / 2 + 1 r 3 δ r 3 1 b M L 2 2 r 3 i = 1 k | λ i | r 3 φ i r 3 ε r 3 / 2 + η i = 1 k | λ i | r 4 φ i r 4 ε = 1 2 γ k 2 + C 4 γ k 4 η ( C ˜ γ k ) r 4 + 1 r 1 a 1 L 2 2 r 1 i = 1 k | λ i | r 1 φ i r 1 ε r 1 / 2 + 1 r 2 a 2 L 2 2 r 2 i = 1 k | λ i | r 2 φ i r 2 ε r 2 / 2 + 1 r 3 δ r 3 1 b M L 2 2 r 3 i = 1 k | λ i | r 3 φ i r 3 ε r 3 / 2 + η i = 1 k | λ i | r 4 φ i r 4 ε γ k 2 + C 4 γ k 4 η ( C ˜ γ k ) r 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbp_HTML.gif
for all u Y k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq221_HTML.gif with u = γ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq229_HTML.gif, when ε and γ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq230_HTML.gif are both small enough. Since r 4 < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq231_HTML.gif, we can choose γ k < ρ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq232_HTML.gif small enough such that
b k : = max u Y k , u = γ k Φ ( u ) < 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbq_HTML.gif
  1. (iii)
    By (40), for any u Z k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq233_HTML.gif with u = ρ k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq234_HTML.gif, we have
    Φ ( u ) 3 C β k r u r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbr_HTML.gif
     
Therefore
0 inf u Z k , u ρ k Φ ( u ) 3 C β k r ρ k r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbs_HTML.gif
Since β k , ρ k 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq235_HTML.gif as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif, we have
d k : = inf u Z k , u ρ k Φ ( u ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbt_HTML.gif

as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif.

Hence, by Lemma 2.7, we obtain that problem (1) has infinitely many solutions { ( u k , ϕ k ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq236_HTML.gif satisfying
1 2 R 3 ( | u k | 2 + V ( x ) u k 2 ) d x 1 4 R 3 | ϕ k | 2 d x + 1 2 R 3 ϕ k u k 2 d x R 3 F ( x , u k ) d x 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbu_HTML.gif

as k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq29_HTML.gif. □

Proof of Theorem 1.5 Similar to (31), there exist constants k i > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq237_HTML.gif, i = 1 , 2 , 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq189_HTML.gif, such that
Φ ( u ) 1 2 u 2 i = 1 3 k i u r i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equ45_HTML.gif
(45)
for all u E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq70_HTML.gif. Since 1 < r i < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq238_HTML.gif, it follows from (45) that the functional Φ is bounded from below. By Lemma 2.5 and Remark 3.2, Φ possesses a critical point u satisfying
Φ ( u ) = inf E Φ , Φ ( u ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbv_HTML.gif
It remains to show that u is nontrivial. For every ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq79_HTML.gif, there exist an open set G and a closed set H such that H Ω G http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq239_HTML.gif and
meas { G Ω } < ε , meas { Ω H } < ε . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbw_HTML.gif
By Lemma 2.3, there exists a function φ C 0 ( R 3 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq113_HTML.gif such that 0 φ ( x ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq129_HTML.gif and φ | H ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq240_HTML.gif, φ | R G ( x ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq241_HTML.gif, then φ E http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq242_HTML.gif. Choosing 0 < λ < min { δ , ζ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq243_HTML.gif, then | λ φ ( x ) | < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq244_HTML.gif for all x R 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_IEq36_HTML.gif, which together with (28) shows
F ( x , λ φ ( x ) ) a 1 ( x ) r 1 | λ φ ( x ) | r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-177/MediaObjects/13661_2012_Article_430_Equbx_HTML.gif