Ni-Serrin type equations arising from capillarity phenomena with non-standard growth

Boundary Value Problems20132013:55

DOI: 10.1186/1687-2770-2013-55

Received: 28 December 2012

Accepted: 28 February 2013

Published: 18 March 2013

Abstract

In the present paper, in view of the variational approach, we discuss a Ni-Serrin type equation involving non-standard growth condition and arising from the capillarity phenomena. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.

MSC:35D05, 35J60, 35J70.

Keywords

p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq1_HTML.gif-Laplacian variable exponent Sobolev space mountain pass theorem genus theory variational method capillarity phenomena

1 Introduction

We study the existence and multiplicity of solutions for a Ni-Serrin type equation involving non-standard growth condition and arising from capillarity phenomena of the following type:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equa_HTML.gif

where Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq2_HTML.gif is a bounded domain with smooth boundary Ω, p C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq3_HTML.gif such that 1 < p ( x ) < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq4_HTML.gif for any x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif and L ( u ) : = Ω | u | p ( x ) + 1 + | u | 2 p ( x ) p ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq6_HTML.gif.

Capillarity can be briefly explained by considering the effects of two opposing forces: adhesion, i.e., the attractive (or repulsive) force between the molecules of the liquid and those of the container; and cohesion, i.e., the attractive force between the molecules of the liquid. The study of capillary phenomena has gained some attention recently. This increasing interest is motivated not only by fascination in naturally-occurring phenomena such as motion of drops, bubbles and waves but also its importance in applied fields ranging from industrial and biomedical and pharmaceutical to microfluidic systems.

The study of ground states for equations of the form
div ( u 1 + | u | 2 ) = f ( u ) in  R N , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ1_HTML.gif
(1.1)

where G ( u ) = u 1 + | u | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq7_HTML.gif is the Kirchhoff stress term and the source term f was very general, was initiated by Ni and Serrin [1, 2]. Moreover, radial solutions of the problem (1.1) have been studied in the context of the analysis of capillarity surfaces for a function of the form f ( u ) = k u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq8_HTML.gif, k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq9_HTML.gif (see [35]). Recently, in [6] Rodrigues studied a version of the problem (P) for the case M ( L ( u ) ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq10_HTML.gif and f ( x , u ) λ f ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq11_HTML.gif, λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq12_HTML.gif.

We note that if we choose the functional L ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq13_HTML.gif as Ω | u | p ( x ) p ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq14_HTML.gif in (P), then we get the problem
{ M ( Ω | u | p ( x ) p ( x ) d x ) div ( | u | p ( x ) 2 u ) = f ( x , u ) in  Ω , u = 0 on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ2_HTML.gif
(1.2)
which is called the p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq1_HTML.gif-Kirchhoff type equation [79]. In this case, the problem (1.2) indicates a generalization of a model, the so-called Kirchhoff equation, introduced by Kirchhoff in [10]. To be more precise, Kirchhoff established a model given by the equation
ρ 2 u t 2 ( P 0 h + E 2 l 0 l | u x | 2 d x ) 2 u x 2 = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ3_HTML.gif
(1.3)

where ρ, P 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq15_HTML.gif, h, E, l are constants, which extends the classical D’Alambert wave equation by considering the effects of the changes in the length of the strings during the vibrations. A distinguishing feature of Kirchhoff equation (1.3) is that the equation contains a nonlocal coefficient P 0 h + E 2 l 0 l | u x | 2 d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq16_HTML.gif which depends on the average E 2 l 0 l | u x | 2 d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq17_HTML.gif of the kinetic energy 1 2 | u x | 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq18_HTML.gif on [ 0 , l ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq19_HTML.gif, and hence the equation is no longer a pointwise identity.

The nonlinear problems involving the p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq1_HTML.gif-Laplacian operator, that is, div ( | u | p ( x ) 2 u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq20_HTML.gif, are extremely attractive because they can be used to model dynamical phenomena which arise from the study of electrorheological fluids or elastic mechanics, in the modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [1115]. The detailed application backgrounds of the p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq1_HTML.gif-Laplacian can be found in [1620] and references therein.

2 Abstract framework and preliminary results

We state some basic properties of the variable exponent Lebesgue-Sobolev spaces L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq21_HTML.gif and W 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq22_HTML.gif, where Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq2_HTML.gif is a bounded domain (for details, see [2124]).

Set
C + ( Ω ¯ ) = { p ; p C ( Ω ¯ ) , inf p ( x ) > 1  for all  x Ω ¯ } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equb_HTML.gif
Let p C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq23_HTML.gif and denote
p : = inf x Ω ¯ p ( x ) and p + : = sup x Ω ¯ p ( x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equc_HTML.gif
For any p C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq23_HTML.gif, we define the variable exponent Lebesgue space by
L p ( x ) ( Ω ) = { u u : Ω R  is measurable , Ω | u ( x ) | p ( x ) d x < } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equd_HTML.gif
then L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq21_HTML.gif endowed with the norm
| u | p ( x ) = inf { μ > 0 : Ω | u ( x ) μ | p ( x ) d x 1 } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Eque_HTML.gif

becomes a Banach space.

Proposition 1 [22, 24]

For any u L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq24_HTML.gif and v L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq25_HTML.gif, we have
| Ω u v d x | ( 1 p + 1 ( p ) ) | u | p ( x ) | v | p ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equf_HTML.gif

where L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq26_HTML.gif is a conjugate space of L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq27_HTML.gif such that 1 p ( x ) + 1 p ( x ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq28_HTML.gif.

The modular of L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq21_HTML.gif, which is the mapping ρ : L p ( x ) ( Ω ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq29_HTML.gif, is defined by
ρ ( u ) = Ω | u ( x ) | p ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equg_HTML.gif

for all u L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq30_HTML.gif.

Proposition 2 [22, 24]

If u , u n L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq31_HTML.gif ( n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq32_HTML.gif), then the following statements are equivalent:
  1. (i)

    lim n | u n u | p ( x ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq33_HTML.gif;

     
  2. (ii)

    lim n ρ ( u n u ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq34_HTML.gif;

     
  3. (iii)

    u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq35_HTML.gif in measure in Ω and lim n ρ ( u n ) = ρ ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq36_HTML.gif.

     

Proposition 3 [22, 24]

If u , u n L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq31_HTML.gif ( n = 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq32_HTML.gif), we have
  1. (i)

    | u | p ( x ) < 1 ( = 1 ; > 1 ) ρ ( u ) < 1 ( = 1 ; > 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq37_HTML.gif;

     
  2. (ii)

    | u | p ( x ) > 1 | u | p ( x ) p ρ ( u ) | u | p ( x ) p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq38_HTML.gif; | u | p ( x ) < 1 | u | p ( x ) p + ρ ( u ) | u | p ( x ) p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq39_HTML.gif;

     
  3. (iii)

    lim n | u n | p ( x ) = 0 lim n ρ ( u n ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq40_HTML.gif; lim n | u n | p ( x ) = lim n ρ ( u n ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq41_HTML.gif.

     
The variable exponent Sobolev space W 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq42_HTML.gif is defined by
W 1 , p ( x ) ( Ω ) = { u L p ( x ) ( Ω ) : | u | L p ( x ) ( Ω ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equh_HTML.gif
with the norm
u 1 , p ( x ) = | u | p ( x ) + | u | p ( x ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equi_HTML.gif

for all u W 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq43_HTML.gif.

The space W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq44_HTML.gif is defined as the closure of C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq45_HTML.gif in W 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq42_HTML.gif with respect to the norm u 1 , p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq46_HTML.gif. For u W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq47_HTML.gif, we can define an equivalent norm
u = | u | p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equj_HTML.gif
since the Poincaré inequality holds, i.e., there exists a positive constant C 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq48_HTML.gif such that
| u | p ( x ) C 1 | u | p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equk_HTML.gif

for all u W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq47_HTML.gif [18, 24].

Proposition 4 [22, 24]

If 1 < p p + < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq49_HTML.gif, then the spaces L p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq21_HTML.gif, W 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq42_HTML.gif and W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq44_HTML.gif are separable and reflexive Banach spaces.

Proposition 5 [22, 24]

Let q C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq50_HTML.gif. If q ( x ) < p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq51_HTML.gif for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq52_HTML.gif, then the embedding W 1 , p ( x ) ( Ω ) L q ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq53_HTML.gif is compact and continuous, where p ( x ) = N p ( x ) N p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq54_HTML.gif if p ( x ) < N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq55_HTML.gif and p ( x ) = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq56_HTML.gif if p ( x ) N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq57_HTML.gif.

Proposition 6 [18]

Let X be a Banach space and let define the functional Λ = Ω | u | p ( x ) p ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq58_HTML.gif. Then Λ : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq59_HTML.gif is convex. The mapping Λ : X X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq60_HTML.gif is a strictly monotone, bounded homeomorphism of ( S + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq61_HTML.gif type, namely
u n u in X and lim ¯ n Λ ( u n ) , u n u 0 implies u n u in  X . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equl_HTML.gif

Definition 7 Let X be a Banach space and J : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq62_HTML.gif be a C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq63_HTML.gif-functional. We say that a functional J satisfies the Palais-Smale condition ((PS) for short) if any sequence { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq64_HTML.gif in X, such that { J ( u n ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq65_HTML.gif is bounded and J ( u n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq66_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq67_HTML.gif, admits a convergent subsequence.

We say that u W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq47_HTML.gif is a weak solution of (P) if
M ( L ( u ) ) Ω ( | u | p ( x ) 2 u + | u | 2 p ( x ) 2 u 1 + | u | 2 p ( x ) ) v d x = Ω f ( x , u ) v d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equm_HTML.gif
for any v W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq68_HTML.gif. The energy functional I : W 0 1 , p ( x ) ( Ω ) R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq69_HTML.gif corresponding to the problem (P) is
I ( u ) = M ( L ( u ) ) Ω F ( x , u ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equn_HTML.gif

where M ( t ) = 0 t M ( ξ ) d ξ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq70_HTML.gif and F ( x , u ) = 0 u f ( x , φ ) d φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq71_HTML.gif.

Thanks to the conditions (M0) and (f0) (see below), the functional I is well defined and of class C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq63_HTML.gif. Since the problem (P) is in the variational setting, the critical points of I are weak solutions of (P). Moreover, the derivative of I is the mapping I : W 0 1 , p ( x ) ( Ω ) ( W 0 1 , p ( x ) ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq72_HTML.gif given by the formula
I ( u ) , v = M ( L ( u ) ) Ω ( | u | p ( x ) 2 u + | u | 2 p ( x ) 2 u 1 + | u | 2 p ( x ) ) v d x Ω f ( x , u ) v d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equo_HTML.gif
for any u , v W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq73_HTML.gif, where
Ω ( | u | p ( x ) 2 u + | u | 2 p ( x ) 2 u 1 + | u | 2 p ( x ) ) v d x : = L ( u ) , v = L ( u ) v . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equp_HTML.gif

3 Main results

Theorem 8 Assume the following conditions hold:

(M0) M : R + R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq74_HTML.gif is a continuous function and satisfies the condition
m 0 t α 1 M ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equq_HTML.gif

for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq75_HTML.gif, where m 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq76_HTML.gif and α > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq77_HTML.gif are positive real numbers;

(f0) f : Ω ¯ × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq78_HTML.gif satisfies the Carathéodory condition and there exist positive constants C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq79_HTML.gif and C 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq80_HTML.gif such that
| f ( x , t ) | C 2 + C 3 | t | q ( x ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equr_HTML.gif

for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif and t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq81_HTML.gif, where p , q C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq82_HTML.gif such that q + < α p < p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq83_HTML.gif. Then (P) has a weak solution.

Proof By the assumptions (M0) and (f0), we have
I ( u ) = M ( L ( u ) ) Ω F ( x , u ) d x m 0 0 L ( u ) ξ α 1 d ξ Ω F ( x , u ) d x m 0 α ( L ( u ) ) α C 4 q Ω | u | q ( x ) d x C 4 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equs_HTML.gif
Therefore, by Proposition 3 and Proposition 5, it follows
I ( u ) m 0 α ( p + ) α ( u p + 1 + u 2 p ) α C 4 q u q + C 4 2 α m 0 α ( p + ) α u α p C 4 q u q + C 4 + as  u . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ4_HTML.gif
(3.1)

By the assumption q + < α p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq84_HTML.gif, I is coercive. Since I is weakly lower semicontinuous, I has a minimum point u in W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq44_HTML.gif and u is a weak solution of (P). □

Theorem 9 Assume the following conditions hold:

(M1) M : R + R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq85_HTML.gif is a continuous function and satisfies the condition
m 1 t α 1 M ( t ) m 2 t α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equt_HTML.gif

for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq75_HTML.gif, where m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq86_HTML.gif, m 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq87_HTML.gif and α real numbers such that 0 < m 1 m 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq88_HTML.gif and α > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq77_HTML.gif;

(M2) M satisfies
M ( t ) M ( t ) t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equu_HTML.gif

for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq75_HTML.gif;

(f1) f : Ω ¯ × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq78_HTML.gif satisfies the Carathéodory condition and there exist positive constants C 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq89_HTML.gif and C 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq90_HTML.gif such that
| f ( x , t ) | C 5 + C 6 | t | β ( x ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equv_HTML.gif

for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif and t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq81_HTML.gif, where β C + ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq91_HTML.gif such that β ( x ) < p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq92_HTML.gif for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq93_HTML.gif and α p + < β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq94_HTML.gif;

(f2) f ( x , t ) = o ( | t | α p + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq95_HTML.gif, t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq96_HTML.gif uniformly for x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq52_HTML.gif;

(f3) There exists t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq97_HTML.gif such that F ( x , t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq98_HTML.gif for x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif and all t t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq99_HTML.gif;

(AR) Ambrosetti-Rabinowitz’s condition holds, i.e., t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq100_HTML.gif, θ > m 2 m 1 α p + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq101_HTML.gif such that
0 θ F ( x , t ) f ( x , t ) t , | t | t a.e. x Ω ¯ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equw_HTML.gif

Then (P) has at least one nontrivial weak solution.

To obtain the result of Theorem 9, we need to show that Lemma 10 and Lemma 11 hold.

Lemma 10 Suppose (M1), (M2), (AR) and (f1) hold. Then I satisfies the (PS) condition.

Proof Let us assume that there exists a sequence { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq102_HTML.gif in W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq44_HTML.gif such that
I ( u n ) c and I ( u n ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ5_HTML.gif
(3.2)
Then
c + u n I ( u n ) 1 θ I ( u n ) u n = M ( L ( u n ) ) 1 θ M ( L ( u n ) ) L ( u n ) u n + Ω ( 1 θ f ( x , u n ) u n F ( x , u n ) ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equx_HTML.gif
Since 1 + | u | 2 p ( x ) | u | p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq103_HTML.gif, we have L ( u n ) 2 p + Ω | u n | p ( x ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq104_HTML.gif. Therefore,
L ( u n ) u n = Ω ( | u n | p ( x ) 2 u n + | u n | 2 p ( x ) 2 u n 1 + | u n | 2 p ( x ) ) u n d x 2 Ω | u n | p ( x ) d x p + L ( u n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equy_HTML.gif
By the above inequalities and assumptions (M1), (M2) and (AR), we get
c + u n M ( L ( u n ) ) L ( u n ) p + θ M ( L ( u n ) ) L ( u n ) c ( 1 p + θ ) M ( L ( u n ) ) L ( u n ) c ( θ p + θ ) L ( u n ) α 1 L ( u n ) c 2 α ( θ p + ) θ ( p + ) α u n α p c . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equz_HTML.gif
This implies that { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq105_HTML.gif is bounded in W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq44_HTML.gif. Passing to a subsequence if necessary, there exists u W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq47_HTML.gif such that u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq106_HTML.gif. Therefore, by Proposition 5, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ6_HTML.gif
(3.3)
By (3.2), we have I ( u n ) , u n u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq107_HTML.gif. Thus
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equaa_HTML.gif
From (f1) and Proposition 1, it follows
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equab_HTML.gif
If we consider the relations given in (3.3), we get
Ω f ( x , u n ) ( u n u ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equac_HTML.gif
Hence,
M ( L ( u n ) ) Ω ( | u n | p ( x ) 2 u n + | u n | 2 p ( x ) 2 u n 1 + | u n | 2 p ( x ) ) ( u n u ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equad_HTML.gif
From (M1), we get
Ω ( | u n | p ( x ) 2 u n + | u n | 2 p ( x ) 2 u n 1 + | u n | 2 p ( x ) ) ( u n u ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ7_HTML.gif
(3.4)

Since the functional (3.4) is of type ( S + ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq108_HTML.gif (see Proposition 3.1 in [6]), we get u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq109_HTML.gif in W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq110_HTML.gif. We are done. □

Lemma 11 Suppose (M1), (AR) and (f1)-(f3) hold. Then the following statements hold:
  1. (i)

    There exist two positive real numbers γ and a such that I ( u ) a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq111_HTML.gif, u W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq47_HTML.gif with u = γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq112_HTML.gif;

     
  2. (ii)

    There exists u W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq113_HTML.gif such that u > γ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq114_HTML.gif, I ( u ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq115_HTML.gif.

     
Proof (i) Let u < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq116_HTML.gif. Then by (M1) and Proposition 3, we have
I ( u ) 2 α m 1 α ( p + ) α u α p + Ω F ( x , u ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equae_HTML.gif
Since α p + < β < p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq117_HTML.gif, by Proposition 5 we have the continuous embeddings W 0 1 , p ( x ) ( Ω ) L α p + ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq118_HTML.gif and W 0 1 , p ( x ) ( Ω ) L β + ( Ω ) L β ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq119_HTML.gif, and also there are positive constants C 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq120_HTML.gif, C 8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq121_HTML.gif and C 9 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq122_HTML.gif such that
| u | α p + C 7 u , u W 0 1 , p ( x ) ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ8_HTML.gif
(3.5)
and
| u | β C 8 u , | u | β + C 9 u , u W 0 1 , p ( x ) ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ9_HTML.gif
(3.6)
From (f1) and (f2), we get F ( x , t ) ε | t | α p + + C ε | t | β ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq123_HTML.gif for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif and t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq81_HTML.gif, where ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq124_HTML.gif is small enough and C ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq125_HTML.gif. Therefore, by (M1), Proposition 3 and (3.5), (3.6), it follows
I ( u ) 2 α m 1 α ( p + ) α u α p + ε Ω | u | α p + d x C ε Ω | u | β ( x ) d x 2 α m 1 α ( p + ) α u α p + ( ε | u | α p + α p + + C ε β + | u | β + β + + C ε β | u | β β ) 2 α m 1 α ( p + ) α u α p + ε C 10 α p + u α p + C 8 β u β C 9 β + u β + ( 2 α m 1 α ( p + ) α ε C 10 α p + ) u α p + max { C 8 β , C 9 β + } u β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equaf_HTML.gif

providing that ε C 10 α p + < m 1 2 α ( p + ) α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq126_HTML.gif. Since u < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq116_HTML.gif and α p + < β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq94_HTML.gif, there exist two positive real numbers γ and a such that I ( u ) a > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq111_HTML.gif, u W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq113_HTML.gif with u = γ ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq127_HTML.gif.

(ii) From (AR) and (f3), one easily deduces
F ( x , t ) F ( x , t ) t θ t θ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equag_HTML.gif
for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif and t t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq99_HTML.gif. Therefore, for δ > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq128_HTML.gif and nonnegative u W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq47_HTML.gif such that { x Ω ¯ : u ( x ) t } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq129_HTML.gif, we get
Ω F ( x , δ u ) d x { δ u t } F ( x , δ u ) d x δ θ t θ { δ u t } F ( x , t ) u θ d x δ θ t θ { u t } F ( x , t ) u θ d x δ θ { u t } F ( x , t ) d x > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equah_HTML.gif
(recall that F 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq130_HTML.gif and F ( , t ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq131_HTML.gif almost everywhere). On the other hand, when t > t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq132_HTML.gif, from (M1) we obtain that
M ( t ) m 2 α t α m 2 α t m 2 m 1 α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equai_HTML.gif
Since t > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq133_HTML.gif, it is obvious L ( t ω ) t p + L ( ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq134_HTML.gif. Hence, for ω W 0 1 , p ( x ) ( Ω ) { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq135_HTML.gif, we have
I ( t ω ) = M ( L ( t ω ) ) Ω F ( x , t ω ) d x m 2 α ( L ( t ω ) ) m 2 m 1 α Ω F ( x , t ω ) d x m 2 α t m 2 m 1 α p + ( L ( ω ) ) m 2 m 1 α t θ { ω t } F ( x , ω ) d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equaj_HTML.gif

From the assumption on θ (see (AR)), we conclude I ( t ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq136_HTML.gif as t + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq137_HTML.gif. □

Proof of Theorem 9 From Lemma 10, Lemma 11 and the fact that I ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq138_HTML.gif, I satisfies the mountain pass theorem (see [25, 26]). Therefore, I has at least one nontrivial weak solution. The proof of Theorem 9 is completed. □

In the sequel, using Krasnoselskii’s genus theory (see [25, 27]), we show the existence of infinitely many solutions of the problem (P). So, we recall some basic notations of Krasnoselskii’s genus.

Let X be a real Banach space and set
R = { E X { 0 } : E  is compact and  E = E } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equak_HTML.gif
Definition 12 Let E R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq139_HTML.gif and X = R k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq140_HTML.gif. The genus γ ( E ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq141_HTML.gif of E is defined by
γ ( E ) = min { k 1 ;  there exists an odd continuous mapping  ϕ : E R k { 0 } } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equal_HTML.gif

If such a mapping does not exist for any k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq9_HTML.gif, we set γ ( E ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq142_HTML.gif. Note also that if E is a subset which consists of finitely many pairs of points, then γ ( E ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq143_HTML.gif. Moreover, from the definition, γ ( ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq144_HTML.gif. A typical example of a set of genus k is a set which is homeomorphic to a ( k 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq145_HTML.gif dimensional sphere via an odd map.

Now, we will give some results of Krasnoselskii’s genus which are necessary throughout the present paper.

Theorem 13 Let X = R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq146_HTML.gif and Ω be the boundary of an open, symmetric and bounded subset Ω R N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq2_HTML.gif with 0 Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq147_HTML.gif. Then γ ( Ω ) = N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq148_HTML.gif.

Corollary 14 γ ( S N 1 ) = N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq149_HTML.gif.

Remark 15 If X is of an infinite dimension and separable and S is the unit sphere in X, then γ ( S ) = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq150_HTML.gif.

Theorem 16 Suppose that M and f satisfy the following conditions:

(M3) M : R + R + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq151_HTML.gif is a continuous function and satisfies the condition
m 3 t δ 1 M ( t ) m 4 t α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equam_HTML.gif

for all t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq75_HTML.gif, where m 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq152_HTML.gif, m 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq153_HTML.gif, δ and α are real numbers such that 0 < m 3 m 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq154_HTML.gif and 1 < δ α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq155_HTML.gif;

(f4) f : Ω ¯ × R R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq78_HTML.gif is a continuous function and there exist positive constants C 11 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq156_HTML.gif, C 12 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq157_HTML.gif, C 13 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq158_HTML.gif and C 14 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq159_HTML.gif such that
C 11 + C 12 | t | s ( x ) 1 f ( x , t ) C 13 + C 14 | t | q ( x ) 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equan_HTML.gif

for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif and t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq160_HTML.gif, where s , q C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq161_HTML.gif such that 1 < s ( x ) < q ( x ) < p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq162_HTML.gif for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif;

(f5) f is an odd function according to t, that is,
f ( x , t ) = f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equao_HTML.gif

for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif and t R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq163_HTML.gif.

If p ( x ) < q ( x ) < p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq164_HTML.gif for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif and q + < δ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq165_HTML.gif, then the problem (P) has infinitely many solutions.

The following result obtained by Clarke in [28] is the main idea which we use in the proof of Theorem 16.

Theorem 17 Let J C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq166_HTML.gif be a functional satisfying the (PS) condition. Furthermore, let us suppose that:
  1. (i)

    J is bounded from below and even;

     
  2. (ii)

    There is a compact set K R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq167_HTML.gif such that γ ( K ) = k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq168_HTML.gif and sup x K J ( x ) < J ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq169_HTML.gif.

     

Then J possesses at least k pairs of distinct critical points and their corresponding critical values are less than J ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq170_HTML.gif.

Lemma 18 Suppose (M3), (f4) and the inequality q + < δ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq165_HTML.gif hold.
  1. (i)

    I is bounded from below;

     
  2. (ii)

    I satisfies the (PS) condition.

     
Proof (i) By the assumptions (M3) and (f4), we have
I ( u ) = M ( L ( u ) ) Ω F ( x , u ) d x m 3 0 L ( u ) ξ δ 1 d ξ C 14 q Ω | u | q ( x ) d x C 13 2 δ m 3 δ ( p + ) δ u δ p + C 14 q Ω | u | q ( x ) d x C 13 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equap_HTML.gif
By Proposition 3 and Proposition 5, we get
I ( u ) 2 δ m 3 δ ( p + ) δ u δ p C 14 q max { | u | q ( x ) q , | u | q ( x ) q + } C 13 2 δ m 3 δ ( p + ) δ u δ p C 14 q max { C q u q , C q + u q + } C 13 2 δ m 3 δ ( p + ) δ u δ p C 14 q C q + u q + C 13 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ10_HTML.gif
(3.7)

for u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq171_HTML.gif large enough. Hence, I is bounded from below.

(ii) Let us assume that there exists a sequence { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq172_HTML.gif in W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq44_HTML.gif such that
I ( u n ) c and I ( u n ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ11_HTML.gif
(3.8)
From (3.8) we have | I ( u n ) | C 16 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq173_HTML.gif. This fact combined with (3.7) implies that
C 16 I ( u n ) 2 δ m 3 δ ( p + ) δ u δ p C 15 u q + C 17 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equaq_HTML.gif

where u n > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq174_HTML.gif. Since q + < δ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq165_HTML.gif, we obtain that { u n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq175_HTML.gif is bounded in W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq44_HTML.gif.

Hence, we may extract a subsequence { u n } W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq176_HTML.gif and u W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq47_HTML.gif such that u n u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq106_HTML.gif in W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq44_HTML.gif. In the rest of the proof, if we consider similar relations given in (3.3) and growth conditions assumed on f and apply the same processes which we used in the proof of Lemma 10, we can see that I satisfies the (PS) condition. □

Proof of Theorem 16 Set (see [7, 25])
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equar_HTML.gif
then we have
< c 1 c 2 c k c k + 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equas_HTML.gif

Now, we will show that c k < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq177_HTML.gif for every k N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq178_HTML.gif. Since W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq179_HTML.gif is a reflexive and separable Banach space, for any k N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq178_HTML.gif, we can choose a k-dimensional linear subspace X k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq180_HTML.gif of W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq179_HTML.gif such that X k C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq181_HTML.gif. As the norms on X k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq180_HTML.gif are equivalent, there exists r k ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq182_HTML.gif such that u X k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq183_HTML.gif with u r k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq184_HTML.gif implies | u | L δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq185_HTML.gif.

Set S r k ( k ) = { u X k : u = r k } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq186_HTML.gif. By the compactness of S r k ( k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq187_HTML.gif and the condition (f4), there exists a constant η k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq188_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ12_HTML.gif
(3.9)
for all u S r k ( k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq189_HTML.gif. If we consider (M3) and (f4), for u S r k ( k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq190_HTML.gif and t ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq191_HTML.gif, we have
I ( t u ) = M ( L ( t u ) ) Ω F ( x , t u ) d x m 4 α ( L ( t u ) ) α t s + η k C 18 m 4 α ( Ω | t u | p ( x ) + 1 + | t u | 2 p ( x ) p ( x ) d x ) α t s + η k C 18 m 4 α ( p ) α ( t p u p + 1 + t p u p ) α t s + η k C 18 m 4 α ( p ) α 2 α 1 ( ( 2 t p u p ) α + 1 ) t s + η k C 18 m 4 α ( p ) α 2 2 α 1 t α p r k α p t s + η k + m 4 α ( p ) α 2 α 1 C 18 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equ13_HTML.gif
(3.10)
providing that C 18 m 4 α ( p ) α 2 α 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq192_HTML.gif. Since s + < q q + < δ p α p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq193_HTML.gif, we can find t k ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq194_HTML.gif and ε k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq195_HTML.gif such that
I ( t k u ) ε k < 0 for all  u S r k ( k ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equat_HTML.gif
i.e.,
I ( u ) ε k < 0 for all  u S t k r k ( k ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equau_HTML.gif

It is clear that γ ( S t k r k ( k ) ) = k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq196_HTML.gif, so c k ε k < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq197_HTML.gif. Finally, by Lemma 18 above, we can apply Theorem 17 to obtain that the functional I admits at least k pairs of distinct critical points, and since k is arbitrary, we obtain infinitely many critical points of I. The proof is completed. □

Theorem 19 Suppose (M3), (f4) and (f5) hold. If q ( x ) < p ( x ) < p ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq198_HTML.gif for all x Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq5_HTML.gif, then the problem (P) has a sequence of solutions { ± u k : k = 1 , 2 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq199_HTML.gif such that I ( ± u k ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq200_HTML.gif.

Proof In the beginning, we will show that I is coercive. If we follow the same processes applied in the proof of Theorem 8 and consider the fact q + < p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq201_HTML.gif, it is easy to get the coerciveness of I. Since I is weak lower semi-continuous, I attains its minimum on W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq202_HTML.gif, i.e., (P) has a solution. By help of coerciveness, we know that I satisfies the (PS) condition on W 0 1 , p ( x ) ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq179_HTML.gif. Moreover, from the condition (f5), I is even.

In the rest of the proof, since we develop the same arguments which we used in the proof of Theorem 16, we omit the details. Therefore, if we follow similar steps to those in (3.9) and (3.10) and consider the inequalities s + < q q + < p < α p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq203_HTML.gif, we can find t k ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq194_HTML.gif and ε k > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq195_HTML.gif such that
I ( u ) ε k < 0 for all  u S t k r k ( k ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_Equav_HTML.gif

Obviously, γ ( S t k r k ( k ) ) = k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq196_HTML.gif, so c k ε k < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq204_HTML.gif. By Krasnoselskii’s genus, each c k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq205_HTML.gif is a critical value of I, hence there is a sequence of solutions { ± u k : k = 1 , 2 , } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq206_HTML.gif such that I ( ± u k ) < 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-55/MediaObjects/13661_2012_Article_404_IEq200_HTML.gif. □

Declarations

Acknowledgements

The author would like to thank the referee for some valuable comments and helpful suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Dicle University

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