Open Access

Existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient

Boundary Value Problems20132013:173

DOI: 10.1186/1687-2770-2013-173

Received: 15 May 2013

Accepted: 10 July 2013

Published: 24 July 2013

Abstract

We provide the existence of a positive solution for the quasilinear elliptic equation

div ( a ( x , | u | ) u ) = f ( x , u , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equa_HTML.gif

in Ω under the Dirichlet boundary condition. As a special case ( a ( x , t ) = t p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq1_HTML.gif), our equation coincides with the usual p-Laplace equation. The solution is established as the limit of a sequence of positive solutions of approximate equations. The positivity of our solution follows from the behavior of f ( x , t ξ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq2_HTML.gif as t is small. In this paper, we do not impose the sign condition to the nonlinear term f.

MSC:35J92, 35P30.

Keywords

nonhomogeneous elliptic operator positive solution the first eigenvalue with weight approximation

1 Introduction

In this paper, we consider the existence of a positive solution for the following quasilinear elliptic equation:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equb_HTML.gif

where Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq3_HTML.gif is a bounded domain with C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq4_HTML.gif boundary Ω. Here, A : Ω ¯ × R N R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq5_HTML.gif is a map which is strictly monotone in the second variable and satisfies certain regularity conditions (see the following assumption (A)). Equation (P) contains the corresponding p-Laplacian problem as a special case. However, in general, we do not suppose that this operator is ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq6_HTML.gif-homogeneous in the second variable.

Throughout this paper, we assume that the map A and the nonlinear term f satisfy the following assumptions (A) and (f), respectively.
  1. (A)

    A ( x , y ) = a ( x , | y | ) y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq7_HTML.gif, where a ( x , t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq8_HTML.gif for all ( x , t ) Ω ¯ × ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq9_HTML.gif, and there exist positive constants C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq10_HTML.gif, C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq11_HTML.gif, C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq12_HTML.gif, C 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq13_HTML.gif, 0 < t 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq14_HTML.gif and 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq15_HTML.gif such that

     
  2. (i)

    A C 0 ( Ω ¯ × R N , R N ) C 1 ( Ω ¯ × ( R N { 0 } ) , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq16_HTML.gif;

     
  3. (ii)

    | D y A ( x , y ) | C 1 | y | p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq17_HTML.gif for every x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq18_HTML.gif, and y R N { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq19_HTML.gif;

     
  4. (iii)

    D y A ( x , y ) ξ ξ C 0 | y | p 2 | ξ | 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq20_HTML.gif for every x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq18_HTML.gif, y R N { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq19_HTML.gif and ξ R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq21_HTML.gif;

     
  5. (iv)

    | D x A ( x , y ) | C 2 ( 1 + | y | p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq22_HTML.gif for every x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq23_HTML.gif, y R N { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq19_HTML.gif;

     
  6. (v)

    | D x A ( x , y ) | C 3 | y | p 1 ( log | y | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq24_HTML.gif for every x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq18_HTML.gif, y R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq25_HTML.gif with 0 < | y | < t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq26_HTML.gif.

     
  1. (f)
    f is a continuous function on Ω × [ 0 , ) × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq27_HTML.gif satisfying f ( x , 0 , ξ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq28_HTML.gif for every ( x , ξ ) Ω × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq29_HTML.gif and the following growth condition: there exist 1 < q < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq30_HTML.gif, b 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq31_HTML.gif and a continuous function f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq32_HTML.gif on Ω × [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq33_HTML.gif such that
    b 1 ( 1 + t q 1 ) f 0 ( x , t ) f ( x , t , ξ ) b 1 ( 1 + t q 1 + | ξ | q 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ1_HTML.gif
    (1)
     

for every ( x , t , ξ ) Ω × [ 0 , ) × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq34_HTML.gif.

In this paper, we say that u W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq35_HTML.gif is a (weak) solution of (P) if
Ω A ( x , u ) φ d x = Ω f ( x , u , u ) φ d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equc_HTML.gif

for all φ W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq36_HTML.gif.

A similar hypothesis to (A) is considered in the study of quasilinear elliptic problems (see [[1], Example 2.2.], [25] and also refer to [6, 7] for the generalized p-Laplace operators). From now on, we assume that C 0 p 1 C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq37_HTML.gif, which is without any loss of generality as can be seen from assumptions (A)(ii), (iii).

In particular, for A ( x , y ) = | y | p 2 y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq38_HTML.gif, that is, div A ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq39_HTML.gif stands for the usual p-Laplacian Δ p u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq40_HTML.gif, we can take C 0 = C 1 = p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq41_HTML.gif in (A). Conversely, in the case where C 0 = C 1 = p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq41_HTML.gif holds in (A), by the inequalities in Remark 3(ii) and (iii), we see that a ( x , t ) = | t | p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq42_HTML.gif whence A ( x , y ) = | y | p 2 y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq38_HTML.gif. Hence, our equation contains the p-Laplace equation as a special case.

In the case where f does not depend on the gradient of u, there are many existence results because our equation has the variational structure (cf. [1, 4, 8]). Although there are a few results for our equation (P) with f including u, we can refer to [7, 9] and [10] for the existence of a positive solution in the case of the ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq43_HTML.gif-Laplacian or m-Laplacian ( 1 < m < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq44_HTML.gif). In particular, in [9] and [7], the nonlinear term f is imposed to be nonnegative. The results in [7] and [10] are applied to the m-Laplace equation with an ( m 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq45_HTML.gif-superlinear term f w.r.t. u. Here, we mention the result in [9] for the p-Laplacian. Faria, Miyagaki and Motreanu considered the case where f is ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq6_HTML.gif-sublinear w.r.t. u and u, and they supposed that f ( x , u , u ) c u r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq46_HTML.gif for some c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq47_HTML.gif and 0 < r < p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq48_HTML.gif. The purpose of this paper is to remove the sign condition and to admit the condition like f ( x , u , u ) λ u p 1 + o ( u p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq49_HTML.gif for large λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq50_HTML.gif as u 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq51_HTML.gif. Concerning the condition for f as | u | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq52_HTML.gif, Zou in [10] imposed that there exists an L > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq53_HTML.gif satisfying f ( x , u , u ) = L u m 1 + o ( | u | m 1 + | u | m 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq54_HTML.gif as | u | , | u | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq55_HTML.gif for the m-Laplace problem. Hence, we cannot apply the result of [10] and [9] to the case of f ( x , u , u ) = λ m ( x ) u p 1 + ( 1 u p 1 ) | u | r 1 + o ( u p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq56_HTML.gif as u 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq51_HTML.gif for 1 < r < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq57_HTML.gif and m L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq58_HTML.gif (admitting sign changes), but we can do our result if λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq50_HTML.gif is large.

In [9], the positivity of a solution is proved by the comparison principle. However, since we are not able to do it for our operator in general, after we provide a non-negative and non-trivial solution as a limit of positive approximate solutions (in Section 2), we obtain the positivity of it due to the strong maximum principle for our operator.

1.1 Statements

To state our first result, we define a positive constant A p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq59_HTML.gif by
A p : = C 1 p 1 ( C 1 C 0 ) p 1 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ2_HTML.gif
(2)

which is equal to 1 in the case of A ( x , y ) = | y | p 2 y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq38_HTML.gif (i.e., the case of the p-Laplacian) because we can choose C 0 = C 1 = p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq41_HTML.gif. Then, we introduce the hypothesis (f1) to the function f 0 ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq60_HTML.gif in (f) as t is small.

(f1) There exist m L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq58_HTML.gif and b 0 > μ 1 ( m ) A p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq61_HTML.gif such that the Lebesgue measure of { x Ω ; m ( x ) > 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq62_HTML.gif is positive and
lim inf t 0 + f 0 ( x , t ) t p 1 b 0 m ( x ) uniformly in  x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ3_HTML.gif
(3)
where f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq32_HTML.gif is the continuous function in (f) and μ 1 ( m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq63_HTML.gif is the first positive eigenvalue of the p-Laplacian with the weight function m obtained by
μ 1 ( m ) : = inf { Ω | u | p d x ; u W 0 1 , p ( Ω )  and  Ω m | u | p d x = 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ4_HTML.gif
(4)
Theorem 1 Assume (f1). Then equation (P) has a positive solution u int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq64_HTML.gif, where
P : = { u C 0 1 ( Ω ¯ ) ; u ( x ) 0 in Ω } , int P : = { u C 0 1 ( Ω ¯ ) ; u ( x ) > 0 in Ω and u / ν < 0 on Ω } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equd_HTML.gif

and ν denotes the outward unit normal vector on Ω.

Next, we consider the case where A is asymptotically ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq6_HTML.gif-homogeneous near zero in the following sense:

(AH0) There exist a positive function a 0 C ( Ω ¯ , ( 0 , + ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq65_HTML.gif and a ˜ 0 ( x , t ) C ( Ω ¯ × [ 0 , + ) , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq66_HTML.gif such that
A ( x , y ) = a 0 ( x ) | y | p 2 y + a ˜ 0 ( x , | y | ) y for every  x Ω , y R N and https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ5_HTML.gif
(5)
lim t 0 + a ˜ 0 ( x , t ) t p 2 = 0 uniformly in  x Ω ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ6_HTML.gif
(6)

Under (AH0), we can replace the hypothesis (f1) with the following (f2):

(f2) There exist m L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq58_HTML.gif and b 0 > λ 1 ( m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq67_HTML.gif such that (3) and the Lebesgue measure of { x Ω ; m ( x ) > 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq62_HTML.gif is positive, where λ 1 ( m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq68_HTML.gif is the first positive eigenvalue of div ( a 0 ( x ) | u | p 2 u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq69_HTML.gif with a weight function m obtained by
λ 1 ( m ) : = inf { Ω a 0 ( x ) | u | p d x ; u W 0 1 , p ( Ω )  and  Ω m | u | p d x = 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ7_HTML.gif
(7)

Theorem 2 Assume (AH0) and (f2). Then equation (P) has a positive solution u int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq64_HTML.gif.

Throughout this paper, we may assume that f ( x , t , ξ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq70_HTML.gif for every t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq71_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq72_HTML.gif and ξ R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq21_HTML.gif because we consider the existence of a positive solution only. In what follows, the norm on W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq73_HTML.gif is given by u : = u p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq74_HTML.gif, where u q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq75_HTML.gif denotes the usual norm of L q ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq76_HTML.gif for u L q ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq77_HTML.gif ( 1 q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq78_HTML.gif). Moreover, we denote u ± : = max { ± u , 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq79_HTML.gif.

1.2 Properties of the map A

Remark 3 The following assertions hold under condition (A):
  1. (i)

    for all x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq18_HTML.gif, A ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq80_HTML.gif is maximal monotone and strictly monotone in y;

     
  2. (ii)

    | A ( x , y ) | C 1 p 1 | y | p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq81_HTML.gif for every ( x , y ) Ω ¯ × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq82_HTML.gif;

     
  3. (iii)

    A ( x , y ) y C 0 p 1 | y | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq83_HTML.gif for every ( x , y ) Ω ¯ × R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq82_HTML.gif,

     

where C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq10_HTML.gif and C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq11_HTML.gif are the positive constants in (A).

Proposition 4 ([[3], Proposition 1])

Let A : W 0 1 , p ( Ω ) W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq84_HTML.gif be a map defined by
A ( u ) , v = Ω A ( x , u ) v d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Eque_HTML.gif

for u , v W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq85_HTML.gif. Then A is maximal monotone, strictly monotone and has ( S ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq86_HTML.gif property, that is, any sequence { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq87_HTML.gif weakly convergent to u with lim sup n A ( u n ) , u n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq88_HTML.gif strongly converges to u.

2 Constructing approximate solutions

Choose a function ψ P { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq89_HTML.gif. In this section, for such ψ and ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq90_HTML.gif, we consider the following elliptic equation:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equf_HTML.gif

In [7], the case ψ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq91_HTML.gif in the above equation is considered.

Lemma 5 Suppose (f1) or (f2). Then there exists λ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq92_HTML.gif such that f ( x , t , ξ ) t + λ 0 t p 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq93_HTML.gif for every x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq72_HTML.gif, t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq94_HTML.gif and ξ R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq21_HTML.gif.

Proof From the growth condition of f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq32_HTML.gif and (3), it follows that
f 0 ( x , t ) t b 0 m t p b 1 t p for every  ( x , t ) Ω × [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equg_HTML.gif

holds, where b 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq95_HTML.gif is a positive constant independent of ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq96_HTML.gif. Therefore, for λ 0 b 0 m + b 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq97_HTML.gif, we easily see that f ( x , t , ξ ) t + λ 0 t p f 0 ( x , t ) t + λ 0 t p 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq98_HTML.gif for every x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq72_HTML.gif, t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq94_HTML.gif and ξ R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq21_HTML.gif holds. □

Proposition 6 If u ε W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq99_HTML.gif is a non-negative solution of ( P ; ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq100_HTML.gif) for ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq101_HTML.gif, then u ε L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq102_HTML.gif. Moreover, for any ε 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq103_HTML.gif, there exists a positive constant D > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq104_HTML.gif such that u ε D max { 1 , u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq105_HTML.gif holds for every ε [ 0 , ε 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq106_HTML.gif.

Proof Set p ¯ = N p / ( N p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq107_HTML.gif if N > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq108_HTML.gif, and in the case of N p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq109_HTML.gif, p ¯ > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq110_HTML.gif is an arbitrarily fixed constant. Let u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq111_HTML.gif be a non-negative solution of ( P ; ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq100_HTML.gif) with 0 ε ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq112_HTML.gif (some ε 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq103_HTML.gif). For r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq113_HTML.gif, choose a smooth increasing function η ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq114_HTML.gif such that η ( t ) = t r + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq115_HTML.gif if 0 t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq116_HTML.gif, η ( t ) = d 0 t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq117_HTML.gif if t d 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq118_HTML.gif and η ( t ) d 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq119_HTML.gif if 1 t d 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq120_HTML.gif for some 0 < d 2 < 1 < d 0 , d 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq121_HTML.gif. Define ξ M ( u ) : = M r + 1 η ( u / M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq122_HTML.gif for M > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq123_HTML.gif.

If u ε L r + p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq124_HTML.gif, then by taking ξ M ( u ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq125_HTML.gif as a test function (note that η https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq126_HTML.gif is bounded), we have
C 0 p 1 Ω | u ε | p ξ M ( u ε ) d x Ω A ( x , u ε ) u ε ξ M ( u ε ) d x = Ω ( f ( x , u ε , u ε ) + ε ψ ) ξ M ( u ε ) d x b 1 Ω ( 1 + u ε q 1 + ε 0 ψ ) M r + 1 η ( u ε / M ) d x + b 1 Ω | u ε | q 1 ξ M ( u ε ) d x d 0 d 1 ( 2 b 1 + ε 0 ψ ) ( u ε r + q r + q + u ε r + 1 r + 1 ) + b 1 Ω | u ε | q 1 ξ M ( u ε ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ8_HTML.gif
(8)
due to Remark 3(iii) and M r + 1 η ( t / M ) d 0 d 1 t r + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq127_HTML.gif. Putting β : = p / ( p q + 1 ) < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq128_HTML.gif, we see that ( ξ M ( u ε ) ) / ( ξ M ( u ε ) ) ( q 1 ) / p = u ε r + 1 / ( ( r + 1 ) u ε r ) ( q 1 ) / p u ε 1 + r / β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq129_HTML.gif provided 0 < u ε < M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq130_HTML.gif (note r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq113_HTML.gif). Similarly, if M u ε d 1 M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq131_HTML.gif, then ( ξ M ( u ε ) ) / ( ξ M ( u ε ) ) ( q 1 ) / p d 0 d 1 M r + 1 / ( d 2 M r ) ( q 1 ) / p = d 0 d 1 d 2 ( 1 q ) / p M 1 + r / β d 0 d 1 d 2 ( 1 q ) / p u ε 1 + r / β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq132_HTML.gif, and if u ε > d 1 M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq133_HTML.gif, then ( ξ M ( u ε ) ) / ( ξ M ( u ε ) ) ( q 1 ) / p = d 0 1 / β M r / β u ε d 0 1 / β u ε 1 + r / β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq134_HTML.gif (note d 1 > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq135_HTML.gif). Thus, according to Young’s inequality, for every δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq136_HTML.gif, there exists C δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq137_HTML.gif such that
Ω | u ε | q 1 ξ M ( u ε ) d x δ Ω | u ε | p ξ M ( u ε ) d x + C δ u ε > 0 ( ξ M ( u ε ) ) β ( ξ M ( u ε ) ) ( q 1 ) β / p d x δ Ω | u ε | p ξ M ( u ε ) d x + C δ d 3 Ω u ε r + β d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ9_HTML.gif
(9)
where β : = p / ( p q + 1 ) < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq128_HTML.gif and d 3 = max { d 0 d 1 d 2 ( 1 q ) / p , d 0 1 / β } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq138_HTML.gif (>1). As a result, because of r + p > r + q , r + β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq139_HTML.gif, according to Hölder’s inequality and the monotonicity of t r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq140_HTML.gif with respect to r on [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq141_HTML.gif, taking a 0 < δ < C 0 / b 1 ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq142_HTML.gif and setting u ε M ( x ) : = min { u ε ( x ) , M } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq143_HTML.gif, we obtain
b 4 ( r ) p max { 1 , u ε r + p r + p } ( r ) p Ω | u ε | p ξ M ( u ε ) d x ( r ) p Ω | u ε M | p ( u ε M ) r d x = ( u ε M ) r p C ( u ε M ) r p ¯ p = C u ε M p ¯ r r + p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ10_HTML.gif
(10)

provided u ε L r + p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq124_HTML.gif by (8) and (9), where r = 1 + r / p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq144_HTML.gif, C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq145_HTML.gif comes from the continuous embedding of W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq146_HTML.gif into L p ¯ ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq147_HTML.gif and d 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq148_HTML.gif is a positive constant independent of u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq111_HTML.gif, ε and r. Consequently, Moser’s iteration process implies our conclusion. In fact, we define a sequence { r m } m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq149_HTML.gif by r 0 : = p ¯ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq150_HTML.gif and r m + 1 : = p ¯ ( p + r m ) / p p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq151_HTML.gif. Then, we see that u ε L p ¯ ( p + r m ) / p ( Ω ) = L p + r m + 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq152_HTML.gif holds if u ε L p + r m ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq153_HTML.gif by applying Fatou’s lemma to (10) and letting M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq154_HTML.gif. Here, we also see r m + 1 = p ¯ r m / p + p ¯ p ( p ¯ / p ) m + 1 r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq155_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq156_HTML.gif. Therefore, by the same argument as in Theorem C in [4], we can obtain u ε L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq102_HTML.gif and u ε D max { 1 , u ε } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq105_HTML.gif for some positive constant D independent of u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq111_HTML.gif and ε. □

Lemma 7 Suppose (f1) or (f2). If u ε W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq99_HTML.gif is a solution of ( P ; ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq100_HTML.gif) for ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq90_HTML.gif, then u ε int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq157_HTML.gif.

Proof Taking ( u ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq158_HTML.gif as a test function in ( P ; ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq100_HTML.gif), we have
C 0 p 1 ( u ε ) p p Ω A ( x , u ε ) ( ( u ε ) ) d x = ε Ω ψ ( u ε ) d x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equh_HTML.gif

because of f ( x , t , ξ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq70_HTML.gif if t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq71_HTML.gif and by Remark 3(iii). Hence, u ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq159_HTML.gif follows. Because Proposition 6 guarantees that u ε L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq102_HTML.gif, we have u ε C 0 1 , α ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq160_HTML.gif (for some 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq161_HTML.gif) by the regularity result in [11]. Note that u ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq162_HTML.gif because of ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq90_HTML.gif and ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq163_HTML.gif. In addition, Lemma 5 implies the existence of λ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq92_HTML.gif such that div A ( x , u ε ) + λ 0 u ε p 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq164_HTML.gif in the distribution sense. Therefore, according to Theorem A and Theorem B in [4], u ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq165_HTML.gif in Ω and u ε / ν < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq166_HTML.gif on Ω, namely, u ε int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq157_HTML.gif. □

The following result can be shown by the same argument as in [[9], Theorem 3.1].

Proposition 8 Suppose (f1) or (f2). Then, for every ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq90_HTML.gif, ( P ; ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq100_HTML.gif) has a positive solution u ε int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq157_HTML.gif.

Proof Fix any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq90_HTML.gif and let { e 1 , , e m , } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq167_HTML.gif be a Schauder basis of W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq73_HTML.gif (refer to [12] for the existence). For each m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq168_HTML.gif, we define the m-dimensional subspace V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq169_HTML.gif of W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq73_HTML.gif by V m : = lin.sp. { e 1 , , e m } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq170_HTML.gif. Moreover, set a linear isomorphism T m : R m V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq171_HTML.gif by T m ( ξ 1 , , ξ m ) : = i = 1 m ξ i e i V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq172_HTML.gif, and let T m : V m ( R m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq173_HTML.gif be a dual map of T m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq174_HTML.gif. By identifying R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq175_HTML.gif and ( R m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq176_HTML.gif, we may consider that T m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq177_HTML.gif maps from V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq178_HTML.gif to R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq175_HTML.gif. Define maps A m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq179_HTML.gif and B m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq180_HTML.gif from V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq169_HTML.gif to V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq178_HTML.gif as follows:
A m ( u ) , v : = Ω A ( x , u ) v d x and B m ( u ) , v : = Ω f ( x , u , u ) v d x + ε Ω ψ v d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equi_HTML.gif
for u, v V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq181_HTML.gif. We claim that for every m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq168_HTML.gif, there exists u m V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq182_HTML.gif such that A m ( u m ) B m ( u m ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq183_HTML.gif in V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq178_HTML.gif. Indeed, by the growth condition of f, Remark 3(iii) and Hölder’s inequality, we easily have
A m ( u ) B m ( u ) , u C 0 p 1 u p b 1 ( u 1 + u q q + u p q 1 u β ) ε ψ u 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ11_HTML.gif
(11)
for every u V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq184_HTML.gif, where β = p / ( p q + 1 ) < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq185_HTML.gif. This implies that A m B m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq186_HTML.gif is coercive on V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq169_HTML.gif by q < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq187_HTML.gif. Set a homotopy H m ( t , y ) : = t y + ( 1 t ) T m ( A m ( T m ( y ) ) B m ( T m ( y ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq188_HTML.gif for t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq189_HTML.gif and y R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq190_HTML.gif. By recalling that A m B m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq186_HTML.gif is coercive on V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq169_HTML.gif, we see that there exists an R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq191_HTML.gif such that ( H m ( t , y ) , y ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq192_HTML.gif for every t [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq189_HTML.gif and | y | R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq193_HTML.gif because https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq194_HTML.gif and the norm of R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq175_HTML.gif are equivalent on V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq169_HTML.gif. Therefore, we have
1 = deg ( I m , B R ( 0 ) , 0 ) = deg ( H m ( 1 , ) , B R ( 0 ) , 0 ) = deg ( H m ( 0 , ) , B R ( 0 ) , 0 ) = deg ( T m ( A m B m ) T m , B R ( 0 ) , 0 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equj_HTML.gif

where I m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq195_HTML.gif is the identity map on R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq175_HTML.gif, B R ( 0 ) : = { y R m ; | y | < R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq196_HTML.gif and deg ( g , B , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq197_HTML.gif denotes the degree on R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq175_HTML.gif for a continuous map g : B R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq198_HTML.gif (cf. [13]). Hence, this yields the existence of y m R m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq199_HTML.gif such that ( T m ( A m B m ) T m ) ( y m ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq200_HTML.gif, and so the desired u m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq201_HTML.gif is obtained by setting u m = T m ( y m ) V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq202_HTML.gif since T m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq177_HTML.gif is injective.

Because (11) with u = u m W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq203_HTML.gif leads to the boundedness of u m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq204_HTML.gif by q < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq187_HTML.gif, we may assume, by choosing a subsequence, that u m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq201_HTML.gif converges to some u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq205_HTML.gif weakly in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq73_HTML.gif and strongly in L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq206_HTML.gif. Let P m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq207_HTML.gif be a natural projection onto V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq169_HTML.gif, that is, P m u = i = 1 m ξ i e i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq208_HTML.gif for u = i = 1 ξ i e i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq209_HTML.gif. Since u m , P m u 0 V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq210_HTML.gif and A m ( u m ) B m ( u m ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq183_HTML.gif in V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq178_HTML.gif, by noting that A m = A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq211_HTML.gif on V m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq169_HTML.gif for a map A defined in Proposition 4, we obtain
A ( u m ) , u m u 0 + A ( u m ) , u 0 P m u 0 = A m ( u m ) , u m P m u 0 = B m ( u m ) , u m P m u 0 = Ω ( f ( x , u m , u m ) + ε ψ ) ( u m u 0 ) d x + Ω ( f ( x , u m , u m ) + ε ψ ) ( u 0 P m u 0 ) d x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equk_HTML.gif

as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq156_HTML.gif, where we use the boundedness of u m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq204_HTML.gif, the growth condition of f and u m u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq212_HTML.gif in L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq206_HTML.gif. In addition, since A ( u m ) W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq213_HTML.gif is bounded, by the boundedness of u m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq204_HTML.gif, we see that A ( u m ) , u 0 P m u 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq214_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq156_HTML.gif, whence A ( u m ) , u m u 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq215_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq156_HTML.gif holds. As a result, it follows from the ( S ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq86_HTML.gif property of A that u m u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq212_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq73_HTML.gif as m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq156_HTML.gif.

Finally, we shall prove that u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq205_HTML.gif is a solution of ( P ; ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq100_HTML.gif). Fix any l N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq216_HTML.gif and φ V l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq217_HTML.gif. For each m l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq218_HTML.gif, by letting m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq156_HTML.gif in A m ( u m ) , φ = B m ( u m ) , φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq219_HTML.gif, we have
Ω A ( x , u 0 ) φ d x = Ω f ( x , u 0 , u 0 ) φ d x + ε Ω ψ φ d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ12_HTML.gif
(12)

Since l is arbitrary, (12) holds for every φ l 1 V l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq220_HTML.gif. Moreover, the density of l 1 V l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq221_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq73_HTML.gif guarantees that (12) holds for every φ W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq36_HTML.gif. This means that u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq205_HTML.gif is a solution of ( P ; ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq100_HTML.gif). Consequently, our conclusion u 0 int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq222_HTML.gif follows from Lemma 7. □

3 Proof of theorems

Lemma 9 Let φ , u int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq223_HTML.gif. Then
Ω A ( x , u ) ( φ p u p 1 ) d x A p φ p p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equl_HTML.gif

holds, where A p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq59_HTML.gif is the positive constant defined by (2).

Proof Because of φ , u int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq223_HTML.gif, there exist δ 1 > δ 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq224_HTML.gif such that δ 1 u φ δ 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq225_HTML.gif in Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq226_HTML.gif. Thus, δ 1 φ / u δ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq227_HTML.gif and 1 / δ 2 u / φ 1 / δ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq228_HTML.gif in Ω. Hence, u / φ , φ / u L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq229_HTML.gif hold. Therefore, we have
A ( x , u ) ( φ p u p 1 ) = p ( φ u ) p 1 A ( x , u ) φ ( p 1 ) ( φ u ) p A ( x , u ) u p C 1 p 1 ( φ u ) p 1 | u | p 1 | φ | C 0 ( φ u ) p | u | p = { ( p C 0 p 1 ) 1 / p φ u | u | } p 1 ( p p 1 ) 1 / p C 1 C 0 ( 1 p ) / p | φ | C 0 ( φ u ) p | u | p A p | φ | p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ13_HTML.gif
(13)

in Ω by (ii) and (iii) in Remark 3 and Young’s inequality. □

Lemma 10 Assume that a 0 C ( Ω ¯ , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq230_HTML.gif and let φ , u int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq223_HTML.gif. Then
Ω a 0 ( x ) | φ | p 2 φ ( φ p u p φ p 1 ) d x Ω a 0 ( x ) | u | p 2 u ( φ p u p u p 1 ) d x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equm_HTML.gif

holds.

Proof First, we note that u / φ , φ / u L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq229_HTML.gif hold by the same reason as in Lemma 9. Applying Young’s inequality to the second term of the right-hand side in (14) (refer to (13) with C 0 = C 1 = p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq41_HTML.gif), we obtain
a 0 ( x ) | φ | p 2 φ ( φ p u p φ p 1 ) a 0 ( x ) ( | φ | p p ( u φ ) p 1 | φ | p 1 | u | + ( p 1 ) ( u φ ) p | φ | p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ14_HTML.gif
(14)
a 0 ( x ) ( | φ | p | u | p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ15_HTML.gif
(15)
in Ω. Similarly, we also have
a 0 ( x ) | u | p 2 u ( φ p u p u p 1 ) a 0 ( x ) ( | φ | p | u | p ) in  Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ16_HTML.gif
(16)

The conclusion follows from (15) and (16). □

Under (f1) or (f2), we denote a solution u ε int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq157_HTML.gif of ( P ; ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq100_HTML.gif) for each ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq90_HTML.gif obtained by Proposition 8.

Lemma 11 Assume (f1) or (f2). Let I : = ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq231_HTML.gif. Then { u ε } ε I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq232_HTML.gif is bounded in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq73_HTML.gif.

Proof Taking u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq111_HTML.gif as a test function in ( P ; ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq100_HTML.gif), we have
C 0 p 1 u ε p p Ω A ( x , u ε ) u ε d x = Ω f ( x , u ε , u ε ) u ε d x + ε Ω ψ u ε d x b 1 ( u ε 1 + u ε q q + u ε p q 1 u ε β ) + ψ u ε 1 b 1 ( u ε + u ε q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equn_HTML.gif

by Remark 3(iii), the growth condition of f, Hölder’s inequality and the continuity of the embedding of W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq73_HTML.gif into L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq206_HTML.gif, where β = p / ( p q + 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq233_HTML.gif (<p) and b 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq95_HTML.gif is a positive constant independent of u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq111_HTML.gif. Because of q < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq187_HTML.gif, this yields the boundedness of u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq234_HTML.gif ( = u ε p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq235_HTML.gif). □

Lemma 12 Assume (f1) or (f2). Then | u ε | / u ε L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq236_HTML.gif and | u ε | / u ε p p λ 0 | Ω | / C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq237_HTML.gif hold for every ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq90_HTML.gif, where | Ω | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq238_HTML.gif denotes the Lebesgue measure of Ω, and where C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq10_HTML.gif and λ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq239_HTML.gif are positive constants as in (A) and Lemma 5, respectively.

Proof Fix any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq90_HTML.gif and choose any ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq240_HTML.gif. By taking ( u ε + ρ ) 1 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq241_HTML.gif as a test function, we obtain
( 1 p ) Ω A ( x , u ε ) u ε ( u ε + ρ ) p d x = Ω f ( x , u ε , u ε ) + ε ψ ( u ε + ρ ) p 1 d x λ 0 Ω u ε p 1 ( u ε + ρ ) p 1 d x λ 0 | Ω | , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ17_HTML.gif
(17)
by Lemma 5 and ε ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq242_HTML.gif. On the other hand, by Remark 3(iii) and 1 p < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq243_HTML.gif, we have
( 1 p ) Ω A ( x , u ε ) u ε ( u ε + ρ ) p d x C 0 Ω | u ε | p ( u ε + ρ ) p d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ18_HTML.gif
(18)

Therefore, (17) and (18) imply the inequality Ω | u ε | p / ( u ε + ρ ) p d x λ 0 | Ω | / C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq244_HTML.gif for every ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq240_HTML.gif. As a result, by letting ρ 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq245_HTML.gif, our conclusion is shown. □

Lemma 13 Assume (f2) and (AH0). Let φ int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq246_HTML.gif. If u ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq247_HTML.gif in C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq248_HTML.gif as ε 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq249_HTML.gif, then
lim ε 0 + | Ω a ˜ 0 ( x , | u ε | ) u ε ( φ p u ε p u ε p 1 ) d x | = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equo_HTML.gif

holds, where a ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq250_HTML.gif is a continuous function as in (AH0).

Proof Note that u ε / φ , φ / u ε L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq251_HTML.gif hold (as in the proof of Lemma 9). Because we easily see that | Ω a ˜ 0 ( x , | u | ) | u | 2 d x | C u p p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq252_HTML.gif for every u W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq35_HTML.gif with some C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq253_HTML.gif independent of u (see (6)), it is sufficient to show | Ω a ˜ 0 ( x , | u ε | ) u ε ( φ p / u ε p 1 ) d x | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq254_HTML.gif as ε 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq249_HTML.gif. Here, we fix any δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq136_HTML.gif. By the property of a ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq250_HTML.gif (see (6)) and because we are assuming that u ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq247_HTML.gif in C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq248_HTML.gif as ε 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq249_HTML.gif, we have | a ˜ 0 ( x , | u ε | ) | δ | u ε | p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq255_HTML.gif for every x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq72_HTML.gif provided sufficiently small ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq90_HTML.gif. Therefore, for such sufficiently small ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq90_HTML.gif, we obtain
| Ω a ˜ 0 ( x , | u ε | ) u ε ( φ p u ε p 1 ) d x | p Ω | a ˜ 0 ( x , | u ε | ) | | u ε | | φ | φ p 1 u ε p 1 d x + ( p 1 ) Ω | a ˜ 0 ( x , | u ε | ) | | u ε | 2 φ p u ε p d x δ φ C 0 1 ( Ω ¯ ) p { p Ω ( | u ε | u ε ) p 1 d x + ( p 1 ) Ω ( | u ε | u ε ) p d x } δ φ C 0 1 ( Ω ¯ ) p | Ω | ( p ( λ 0 / C 0 ) 1 1 / p + ( p 1 ) ( λ 0 / C 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equp_HTML.gif

because of | u ε | / u ε L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq236_HTML.gif by Lemma 12. Since δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq136_HTML.gif is arbitrary, our conclusion is shown. □

3.1 Proof of main results

Proof of Theorems

Let ε ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq256_HTML.gif. Due to Proposition 6 and Lemma 11, we have u ε M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq257_HTML.gif for some M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq258_HTML.gif independent of ε ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq256_HTML.gif. Hence, there exist M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq259_HTML.gif and 0 < α < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq161_HTML.gif such that u ε C 0 1 , α ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq260_HTML.gif and u ε C 0 1 , α ( Ω ¯ ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq261_HTML.gif for every ε ( 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq256_HTML.gif by the regularity result in [11]. Because the embedding of C 0 1 , α ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq262_HTML.gif into C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq248_HTML.gif is compact and by u ε int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq157_HTML.gif, there exists a sequence { ε n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq263_HTML.gif and u 0 P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq264_HTML.gif such that ε n 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq265_HTML.gif and u n : = u ε n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq266_HTML.gif in C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq248_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq267_HTML.gif. If u 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq268_HTML.gif occurs, then u 0 int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq222_HTML.gif by the same reason as in Lemma 7, and hence our conclusion is proved. Now, we shall prove u 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq268_HTML.gif by contradiction for each theorem. So, we suppose that u 0 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq269_HTML.gif, whence u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq270_HTML.gif in C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq248_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq267_HTML.gif.

Proof of Theorem 1 Let φ int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq246_HTML.gif be an eigenfunction corresponding to the first positive eigenvalue μ 1 ( m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq63_HTML.gif (cf. [14, 15], it is well known that we can obtain φ as the minimizer of (4)), namely, φ is a positive solution of Δ p u = μ 1 ( m ) m ( x ) | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq271_HTML.gif in Ω and u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq272_HTML.gif on Ω. Since p-Laplacian is ( p 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq6_HTML.gif-homogeneous, we may assume that φ satisfies Ω m ( x ) φ p d x = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq273_HTML.gif, and hence φ p p = μ 1 ( m ) Ω m ( x ) φ p d x = μ 1 ( m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq274_HTML.gif holds by taking φ as a test function. Choose ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq240_HTML.gif satisfying b 0 A p μ 1 ( m ) > ρ φ p p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq275_HTML.gif (note that b 0 A p μ 1 ( m ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq276_HTML.gif as in (f1)). Due to (f1), there exists a δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq136_HTML.gif such that f 0 ( x , t ) ( b 0 m ( x ) ρ ) t p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq277_HTML.gif for every 0 t δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq278_HTML.gif and x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq72_HTML.gif. Since we are assuming u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq270_HTML.gif in C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq248_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq267_HTML.gif, u n δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq279_HTML.gif occurs for sufficiently large n. Then, for such sufficiently large n, according to Lemma 9, (1) and ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq280_HTML.gif, we obtain
A p μ 1 ( m ) = A p φ p p Ω A ( x , u n ) ( φ p u n p 1 ) d x = Ω f ( x , u n , u n ) + ε ψ u n p 1 φ p d x Ω f 0 ( x , u n ) u n p 1 φ p d x b 0 Ω m ( x ) φ p d x ρ φ p p = b 0 ρ φ p p > A p μ 1 ( m ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equq_HTML.gif

This is a contradiction. □

Proof of Theorem 2 Since > sup x Ω a 0 ( x ) inf x Ω a 0 ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq281_HTML.gif holds, by the standard argument as in the p-Laplacian, we see that λ 1 ( m ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq282_HTML.gif and it is the first positive eigenvalue of div ( a 0 ( x ) | u | p 2 u ) = λ m ( x ) | u | p 2 u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq283_HTML.gif in Ω and u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq272_HTML.gif on Ω. Therefore, by the well-known argument, there exists a positive eigenfunction φ 1 int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq284_HTML.gif corresponding to λ 1 ( m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq68_HTML.gif (we can obtain φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq285_HTML.gif as the minimizer of (7)). Hence, by taking φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq285_HTML.gif as a test function, we have 0 < Ω a 0 ( x ) | φ 1 | p d x = λ 1 ( m ) Ω m ( x ) φ 1 p d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq286_HTML.gif. Thus, Ω m ( x ) φ 1 p d x > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq287_HTML.gif follows. Because u n int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq288_HTML.gif is a solution of ( P ; ε n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq289_HTML.gif and φ 1 int P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq284_HTML.gif is an eigenfunction corresponding to λ 1 ( m ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq68_HTML.gif, according to Lemma 11 and Lemma 13 (note A ( x , y ) = a 0 | y | p 2 y + a ˜ 0 ( x , | y | ) y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq290_HTML.gif as in (AH0)), we obtain
0 Ω a 0 ( x ) | φ 1 | p 2 φ 1 ( φ 1 p u n p φ 1 p 1 ) d x Ω a 0 ( x ) | u n | p 2 u n ( φ 1 p u n p u n p 1 ) d x λ 1 ( m ) Ω m ( φ 1 p u n p ) d x Ω f 0 ( x , u n ) u n p 1 φ 1 p d x + Ω a ˜ 0 ( x , | u n | ) u n ( φ 1 p u n p u n p 1 ) d x + Ω f ( x , u n , u n ) u n d x + ε n Ω ψ u n d x = Ω ( f 0 ( x , u n ) u n p 1 b 0 m ( x ) ) φ 1 p d x ( b 0 λ 1 ( m ) ) m ( x ) φ 1 p d x + o ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equ19_HTML.gif
(19)
as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq267_HTML.gif since we are assuming u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq270_HTML.gif in C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq248_HTML.gif, where we use the facts that ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq280_HTML.gif and φ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq291_HTML.gif in Ω. Furthermore, by Fatou’s lemma and (3), we have
lim inf n Ω ( f 0 ( x , u n ) u n p 1 b 0 m ( x ) ) φ 1 p d x 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_Equr_HTML.gif

As a result, by taking a limit superior with respect to n in (19), we have 0 ( b 0 λ 1 ( m ) ) m ( x ) φ 1 p d x < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-173/MediaObjects/13661_2013_Article_425_IEq292_HTML.gif. This is a contradiction. □

Declarations

Acknowledgements

The author would like to express her sincere thanks to Professor Shizuo Miyajima for helpful comments and encouragement. The author thanks Professor Dumitru Motreanu for giving her the opportunity of this work. The author thanks referees for their helpful comments.

Authors’ Affiliations

(1)
Department of Mathematics, Tokyo University of Science

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