Open Access

Existence of positive solutions of elliptic mixed boundary value problem

Boundary Value Problems20122012:91

DOI: 10.1186/1687-2770-2012-91

Received: 19 January 2012

Accepted: 6 August 2012

Published: 16 August 2012

Abstract

In this paper, we use variational methods to prove two existence of positive solutions of the following mixed boundary value problem:

{ Δ u = f ( x , u ) , x Ω , u = 0 , x σ , u ν = g ( x , u ) , x Γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equa_HTML.gif

One deals with the asymptotic behaviors of f ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq1_HTML.gif near zero and infinity and the other deals with superlinear of f ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq1_HTML.gif at infinity.

MSC:35M12, 35D30.

Keywords

elliptic mixed boundary value problem positive solutions mountain pass theorem Sobolev embedding theorem

1 Introduction and preliminaries

This paper is concerned with the existence of positive solutions of the following elliptic mixed boundary value problem:
{ Δ u = f ( x , u ) , x Ω , u = 0 , x σ , u ν = g ( x , u ) , x Γ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ1_HTML.gif
(1)

where Ω is a bounded domain in R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq2_HTML.gif with Lipschitz boundary Ω, σ Γ = Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq3_HTML.gif, σ Γ = Ø https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq4_HTML.gif, Γ is a sufficiently smooth ( n 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq5_HTML.gif-dimensional manifold, and ν is the outward normal vector on Ω. We assume f : Ω × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq6_HTML.gif, g : Γ × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq7_HTML.gif are continuous and satisfy

(S1) f ( x , t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq8_HTML.gif, t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq9_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif, f ( x , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq11_HTML.gif. f ( x , t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq12_HTML.gif, t < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq13_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif.

(S2) For almost every x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif, f ( x , t ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq14_HTML.gif is nondecreasing with respect to t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq15_HTML.gif.

(S3) lim t 0 f ( x , t ) t = p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq16_HTML.gif, lim t + f ( x , t ) t = q ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq17_HTML.gif uniformly in a.e. x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif, where p ( x ) < λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq18_HTML.gif, λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq19_HTML.gif is the first eigenvalue of (2), 0 p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq20_HTML.gif, q ( x ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq21_HTML.gif.

(S4) There exists c 1 , c 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq22_HTML.gif such that | f ( x , t ) | c 1 + c 2 | t | p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq23_HTML.gif for some p ( 2 , 2 n n 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq24_HTML.gif as n 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq25_HTML.gif and p ( 2 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq26_HTML.gif as n = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq27_HTML.gif.

The eigenvalue problem of (1) is studied by Liu and Su in [1]
{ Δ u = λ u in  Ω , u = 0 on  σ , u ν = λ u on  Γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ2_HTML.gif
(2)

There exists a set of eigenvalues { λ k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq28_HTML.gif and corresponding eigenfunctions { u k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq29_HTML.gif which solve problem (2), where 0 λ 1 λ 2 λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq30_HTML.gif, λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq31_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq32_HTML.gif, λ 1 = inf 0 u V Ω | u | 2 d x Ω | u | 2 d x + Γ | u | 2 d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq33_HTML.gif.

There have been many papers concerned with similar problems at resonance under the boundary condition; see [210]. Moreover, some multiplicity theorems are obtained by the topological degree technique and variational methods; interested readers can see [1117]. Problem (1) is different from the classical ones, such as those with Dirichlet, Neuman, Robin, No-flux, or Steklov boundary conditions.

In this paper, we assume V : = { v H 1 ( Ω ) : v | σ = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq34_HTML.gif is a closed subspace of H 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq35_HTML.gif. We define the norm in V as u 2 = Ω | u | 2 d x + Γ | γ u | 2 d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq36_HTML.gif, L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq37_HTML.gif is the L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq38_HTML.gif norm, L p ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq39_HTML.gif is the L p ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq40_HTML.gif norm, γ : V L 2 ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq41_HTML.gif is the trace operator with γ u = u Γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq42_HTML.gif for all u H 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq43_HTML.gif, that is continuous and compact (see [18]). Furthermore, we define g = γ f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq44_HTML.gif, 0 g ( x , t ) | γ f ( x , t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq45_HTML.gif for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq15_HTML.gif (see [1]). Then, by (S3), we obtain
lim t + g ( x , t ) t lim t + | γ f ( x , t ) | t = q ( x ) 0 , a.e.  x Ω ¯ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ3_HTML.gif
(3)
Let Ω be a bounded domain with a Lipschitz boundary; there is a continuous embedding V L y ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq46_HTML.gif for y [ 2 , 2 n n 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq47_HTML.gif when n 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq25_HTML.gif, and y [ 2 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq48_HTML.gif when n = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq49_HTML.gif. Then there exists γ y > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq50_HTML.gif, such that
u L y ( Ω ) γ y u , u V . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ4_HTML.gif
(4)
Moreover, there is a continuous boundary trace embedding V L z ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq51_HTML.gif for z [ 2 , 2 ( n 1 ) n 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq52_HTML.gif when n 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq25_HTML.gif, and z [ 2 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq53_HTML.gif when n = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq27_HTML.gif. Then there exists k z > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq54_HTML.gif, such that
u L z ( Γ ) k z u , u V . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ5_HTML.gif
(5)
It is well known that to seek a nontrivial weak solution of problem (1) is equivalent to finding a nonzero critical value of the C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq55_HTML.gif functional
J ( u ) = 1 2 Ω | u | 2 d x Ω F ( x , u ) d x Γ G ( s , u ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ6_HTML.gif
(6)
where u V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq56_HTML.gif, F ( x , u ) = 0 u f ( x , t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq57_HTML.gif, G ( x , u ) = 0 u g ( x , t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq58_HTML.gif. Moreover, by (S1) and the Strong maximum principle, a nonzero critical point of J is in fact a positive solution of (1). In order to find critical points of the functional (6), one often requires the technique condition, that is, for some μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq59_HTML.gif, | u | M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq60_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif,
0 < μ F ( x , u ) u f ( x , u ) , F ( x , u ) = 0 u f ( x , t ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ7_HTML.gif
(AR)

It is easy to see that the condition (AR) implies that lim u + F ( x , u ) u 2 = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq61_HTML.gif, that is, f ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq62_HTML.gif must be superlinear with respect to u at infinity. In the present paper, motivated by [19] and [20], we study the existence and nonexistence of positive solutions for problem (1) with the asymptotic behavior assumptions (S3) of f at zero and infinity. Moreover, we also study superlinear of f at infinity with q ( x ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq63_HTML.gif in (S3), which is weaker than the (AR) condition, that is the (AR) condition does not hold.

In order to get our conclusion, we define the minimization problem
Λ = inf { Ω | u | 2 d x : u V , Ω q ( x ) u 2 d x + Γ q ( s ) u 2 d s = 1 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ8_HTML.gif
(7)

then Λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq64_HTML.gif, which is achieved by some φ Λ V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq65_HTML.gif with φ Λ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq66_HTML.gif a.e. in Ω; see Lemma 1.

We denote by c, c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq67_HTML.gif, c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq68_HTML.gif universal constants unless specified otherwise. Our main results are as follows.

Theorem 1 Let conditions (S 1) to (S 3) hold, then:
  1. (i)

    If Λ > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq69_HTML.gif, then the problem (1) has no any positive solution in V.

     
  2. (ii)

    If Λ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq70_HTML.gif, then the problem (1) has at least one positive solution in V.

     
  3. (iii)

    If Λ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq71_HTML.gif, then the problem (1) has one positive solution u ( x ) V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq72_HTML.gif if and only if there exists a constant c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq73_HTML.gif such that u ( x ) = c φ Λ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq74_HTML.gif and f ( x , u ) = q ( x ) u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq75_HTML.gif, g ( x , u ) = q ( x ) u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq76_HTML.gif a.e. x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif, where φ Λ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq66_HTML.gif is the function which achieves Λ.

     
Corollary 2 Let conditions (S 1) to (S 3) with q ( x ) l > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq77_HTML.gif hold, then:
  1. (i)

    If l < λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq78_HTML.gif, then the problem (1) has no any positive solution in V.

     
  2. (ii)

    If λ 1 < l < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq79_HTML.gif, then the problem (1) has at least one positive solution in V.

     
  3. (iii)

    If l = λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq80_HTML.gif, then the problem (1) has one positive solution u ( x ) V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq72_HTML.gif if and only if there exists a constant c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq73_HTML.gif such that u ( x ) = c φ 1 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq81_HTML.gif and f ( x , u ) = λ 1 u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq82_HTML.gif, g ( x , u ) = λ 1 u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq83_HTML.gif a.e. x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif, where φ 1 ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq84_HTML.gif is the eigenfunction of the λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq19_HTML.gif.

     

Theorem 3 Let conditions (S 1) to (S 4) with q ( x ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq63_HTML.gif hold, then the problem (1) has at least one positive solution in V.

2 Some lemmas

We need the following lemmas.

Lemma 1 If q ( x ) L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq21_HTML.gif, q ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq85_HTML.gif, q ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq86_HTML.gif, then Λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq64_HTML.gif and there exists φ Λ ( x ) V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq87_HTML.gif such that Λ = Ω | φ Λ | 2 d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq88_HTML.gif and Ω q ( x ) φ Λ 2 d x + Γ q ( s ) φ Λ 2 d s = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq89_HTML.gif. Moreover, φ Λ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq66_HTML.gif a.e. in V.

Proof By the Sobolev embedding function V L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq90_HTML.gif and Fatou’s lemma, it is easy to know that Λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq64_HTML.gif and there exists φ Λ ( x ) V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq87_HTML.gif, which satisfies Λ, that is, Ω q ( x ) φ Λ 2 d x + Γ q ( s ) φ Λ 2 d s = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq91_HTML.gif. Furthermore, we assume φ Λ ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq92_HTML.gif, then φ Λ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq93_HTML.gif could replace by | φ Λ ( x ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq94_HTML.gif. By the Strong maximum principle, we know φ Λ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq66_HTML.gif a.e. in V. □

Lemma 2 If conditions (S 1) to (S 3) hold, then there exists β , ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq95_HTML.gif such that J | B ρ ( 0 ) β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq96_HTML.gif, u V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq97_HTML.gif, u = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq98_HTML.gif.

Proof By condition (S3), there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq99_HTML.gif, ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq100_HTML.gif such that f ( x , u ) u λ 1 ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq101_HTML.gif, g ( x , u ) u γ f ( x , u ) u λ 1 ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq102_HTML.gif as 0 < | u | δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq103_HTML.gif. Which implies that F ( x , u ) 1 2 ( λ 1 ε ) u 2 + c | u | y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq104_HTML.gif, G ( x , u ) 1 2 ( λ 1 ε ) u 2 + c | u | z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq105_HTML.gif.

By (4) and (5), we obtain
J ( u ) = 1 2 u L 2 ( Ω ) 2 Ω F ( x , u ) d x Γ G ( s , u ) d s 1 2 u L 2 ( Ω ) 2 + 1 2 γ u L 2 ( Γ ) 2 1 2 γ u L 2 ( Γ ) 2 1 2 ( λ 1 ε ) u L 2 ( Ω ) 2 c u L y ( Ω ) y 1 2 ( λ 1 ε ) u L 2 ( Γ ) 2 c u L z ( Γ ) z 1 2 u 2 1 2 ( λ 1 ε ) 1 λ 1 u 2 c γ y y u y 1 2 ( λ 1 ε + 1 ) 1 λ 1 + 1 u 2 c k z z u z = [ ε ( 2 λ 1 + 1 ) 2 λ 1 ( λ 1 + 1 ) 1 2 ] u 2 c γ y y u y c k z z u z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equb_HTML.gif

Hence, y , z > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq106_HTML.gif; we take ε which satisfies ε ( 2 λ 1 + 1 ) 2 λ 1 ( λ 1 + 1 ) 1 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq107_HTML.gif, that is, ε > λ 1 ( λ 1 + 1 ) 2 λ 1 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq108_HTML.gif. Then we take a positive constant β such that J | B ρ ( 0 ) β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq96_HTML.gif as u = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq98_HTML.gif, and is small enough. □

Lemma 3 If conditions (S 1) to (S 3) hold, Λ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq70_HTML.gif, φ Λ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq66_HTML.gif is defined by Lemma 1, then J ( t φ Λ ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq109_HTML.gif as t + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq110_HTML.gif.

Proof If Λ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq70_HTML.gif, φ Λ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq66_HTML.gif is defined by Lemma 1, by Fatou’s lemma, and (S3), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equc_HTML.gif

So, J ( t φ Λ ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq109_HTML.gif as t + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq110_HTML.gif. □

Lemma 4 Let conditions (S 1) and (S 2) hold. If a sequence { u n } V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq111_HTML.gif satisfies J ( u n ) , u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq112_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq113_HTML.gif, then there exists a subsequence of { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq114_HTML.gif, still denoted by { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq114_HTML.gif such that J ( t u n ) 1 + t 2 2 n + J ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq115_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq15_HTML.gif, n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq116_HTML.gif.

Proof Since J ( u n ) , u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq117_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq113_HTML.gif, for a subsequence, we may assume that
1 n < J ( u n ) , u n = u n L 2 ( Ω ) 2 Ω f ( x , u n ) u n d x Γ g ( s , u n ) u n d s < 1 n , n 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ9_HTML.gif
(8)
For any fixed x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif and n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq116_HTML.gif, set
ψ 1 ( t ) = t 2 2 f ( x , u n ) u n F ( x , t u n ) , ψ 2 ( t ) = t 2 2 g ( s , u n ) u n G ( s , t u n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equd_HTML.gif
Then (S2) implies that
ψ 1 ( t ) = t f ( x , u n ) u n f ( x , t u n ) u n = t u n [ f ( x , u n ) f ( x , t u n ) t ] = { 0 , 0 < t 1 ; 0 , t > 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Eque_HTML.gif

It implies that ψ 1 ( t ) ψ 1 ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq118_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq119_HTML.gif. Following the same procedures, we obtain ψ 2 ( t ) ψ 2 ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq120_HTML.gif, t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq119_HTML.gif.

For all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq15_HTML.gif and positive integer n, by (8), we have
J ( t u n ) = t 2 2 u n L 2 ( Ω ) 2 Ω F ( x , t u n ) d x Γ G ( s , t u n ) d s t 2 2 [ 1 n + Ω f ( x , u n ) u n d x + Γ g ( s , u n ) u n d s ] Ω F ( x , t u n ) d x Γ G ( s , t u n ) d s t 2 2 n + Ω [ 1 2 f ( x , u n ) u n F ( x , u n ) ] d x + Γ [ 1 2 g ( s , u n ) u n G ( s , u n ) ] d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ10_HTML.gif
(9)
On the other hand, by (8), one has
J ( u n ) = 1 2 u n L 2 ( Ω ) 2 Ω F ( x , u n ) d x Γ G ( s , u n ) d s 1 2 [ 1 n + Ω f ( x , u n ) u n d x + Γ g ( s , u n ) u n d s ] Ω F ( x , u n ) d x Γ G ( s , u n ) d s = 1 2 n + Ω [ 1 2 f ( x , u n ) u n F ( x , u n ) ] d x + Γ [ 1 2 g ( s , u n ) u n G ( s , u n ) ] d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equf_HTML.gif
One has
Ω [ 1 2 f ( x , u n ) u n F ( x , u n ) ] d x + Γ [ 1 2 g ( s , u n ) u n G ( s , u n ) ] d s J ( u n ) + 1 2 n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ11_HTML.gif
(10)

Combining (9) and (10), we have J ( t u n ) 1 + t 2 2 n + J ( u n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq121_HTML.gif. □

Lemma 5 (see [21])

Suppose E is a real Banach space, J C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq122_HTML.gif satisfies the following geometrical conditions:
  1. (i)

    J ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq123_HTML.gif; there exists ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq124_HTML.gif such that J | B ρ ( 0 ) r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq125_HTML.gif;

     
  2. (ii)
    There exists e E B ρ ( 0 ) ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq126_HTML.gif such that J ( e ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq127_HTML.gif. Let Γ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq128_HTML.gif be the set of all continuous paths joining 0 and e:
    Γ 1 = { h C ( [ 0 , 1 ] , E ) | h ( 0 ) = 0 , h ( 1 ) = e } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equg_HTML.gif
     
and
c = inf h Γ 1 max t [ 0 , 1 ] J ( h ( t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equh_HTML.gif

Then there exists a sequence { u n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq129_HTML.gif such that J ( u n ) c β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq130_HTML.gif and ( 1 + u n ) × J ( u n ) E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq131_HTML.gif.

3 Proofs of main results

Proof of Theorem 1 (i) If u V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq56_HTML.gif is one positive solution of problem (1), by (3), one has
0 = J ( u ) , u = Ω | u | 2 d x Ω f ( x , u ) u d x Γ g ( s , u ) u d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equi_HTML.gif
That is,
Ω | u | 2 d x = Ω f ( x , u ) u d x + Γ g ( s , u ) u d s Ω q ( x ) u 2 d x + Γ q ( s ) u 2 d s = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equj_HTML.gif
It implies that Λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq132_HTML.gif. This completes the proof of Theorem 1(i).
  1. (ii)
    By Lemma 2, there exists β , ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq133_HTML.gif such that J | B ρ ( 0 ) β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq96_HTML.gif with u = ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq98_HTML.gif. By Lemma 3, we obtain J ( t 0 φ Λ ( x ) ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq134_HTML.gif as t 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq135_HTML.gif. Define
    https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ12_HTML.gif
    (11)
     
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ13_HTML.gif
(12)
where φ Λ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq66_HTML.gif is given by Lemma 1. Then c β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq136_HTML.gif and by Lemma 3, there exists { u n } V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq111_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ14_HTML.gif
(13)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ15_HTML.gif
(14)
  1. (14)
    implies that
    J ( u n ) , u n = u n L 2 ( Ω ) 2 Ω f ( x , u n ) u n d x Γ g ( s , u n ) u n d s = o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ16_HTML.gif
    (15)
     

Here, in what follows, we use o ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq137_HTML.gif to denote any quantity which tends to zero as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq113_HTML.gif.

If { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq114_HTML.gif is bounded in V, when Ω is bounded and f ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq62_HTML.gif, g ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq138_HTML.gif are subcritical, we can get { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq114_HTML.gif has a subsequence strong convergence to a critical value of J, and our proof is complete. So, to prove the theorem, we only need show that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq114_HTML.gif is bounded in V. Supposing that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq139_HTML.gif is unbounded, that is, u n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq140_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq113_HTML.gif. We order
t n = 2 c u n , w n = t n u n = 2 c u n u n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ17_HTML.gif
(16)

Then { w n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq141_HTML.gif is bounded in V. By extracting a subsequence, we suppose w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq142_HTML.gif is a strong convergence in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq143_HTML.gif, w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq142_HTML.gif is a convergence a.e. x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif, w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq144_HTML.gif is a weak convergence in V.

We claim that w 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq145_HTML.gif. In fact, by (S1) and (S3), we know x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq146_HTML.gif, u n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq147_HTML.gif, and there exists M 1 , M 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq148_HTML.gif such that | f ( x , u n ) u n | M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq149_HTML.gif, | g ( x , u n ) u n | M 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq150_HTML.gif. If w = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq151_HTML.gif, w n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq152_HTML.gif is a strong convergence in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq143_HTML.gif, and by (15) and (16) we know
4 c = t n 2 u n 2 = t n 2 ( u n L 2 ( Ω ) 2 + γ u n L 2 ( Γ ) 2 ) = t n 2 Ω f ( x , u n ) u n d x + t n 2 Γ g ( s , u n ) u n d s + t n 2 γ u n L 2 ( Γ ) 2 + o ( 1 ) = Ω f ( x , u n ) u n w n 2 d x + Γ g ( s , u n ) u n w n 2 d s + t n 2 u n L 2 ( Γ ) 2 + o ( 1 ) M 1 Ω w n 2 d x + M 2 Γ w n 2 d s + w n L 2 ( Γ ) 2 + o ( 1 ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equk_HTML.gif

It is contradiction with c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq73_HTML.gif, so w 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq145_HTML.gif.

As follows, we prove w 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq145_HTML.gif satisfies
Ω φ ( x ) w ( x ) d x Ω q 1 ( x ) φ ( x ) w ( x ) d x Γ q 2 ( s ) φ ( s ) w ( s ) d s = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equl_HTML.gif
We order
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equm_HTML.gif

By (S1) and (S3), there exists M 3 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq153_HTML.gif such that 0 p n ( x ) M 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq154_HTML.gif, 0 q n ( x ) M 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq155_HTML.gif, x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq156_HTML.gif. We select a suitable subsequence and there exists h 1 ( x ) L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq157_HTML.gif, h 2 ( x ) L 2 ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq158_HTML.gif such that p n ( x ) h 1 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq159_HTML.gif is a strong convergence in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq143_HTML.gif, q n ( x ) h 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq160_HTML.gif is a strong convergence in L 2 ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq161_HTML.gif, and 0 h 1 ( x ) M 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq162_HTML.gif, 0 h 2 ( x ) M 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq163_HTML.gif, x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq156_HTML.gif.

It follows from w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq142_HTML.gif is a strong convergence in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq143_HTML.gif that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equn_HTML.gif

Hence, { p n ( x ) w n ( x ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq164_HTML.gif is bounded in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq143_HTML.gif, p n ( x ) w n ( x ) h 1 ( x ) w + ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq165_HTML.gif in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq143_HTML.gif; { q n ( x ) w n ( x ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq166_HTML.gif is bounded in L 2 ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq161_HTML.gif, q n ( x ) w n ( x ) h 2 ( x ) w + ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq167_HTML.gif in L 2 ( Γ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq161_HTML.gif.

By (16), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equo_HTML.gif
Since w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq144_HTML.gif is a weak convergence in V, we obtain
Ω φ ( x ) w ( x ) d x Ω h 1 ( x ) φ ( x ) w + ( x ) d x Γ h 2 ( s ) φ ( s ) w + ( s ) d s = 0 , φ V . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equp_HTML.gif
We order φ = w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq168_HTML.gif; this yields w 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq169_HTML.gif, so w = w + 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq170_HTML.gif. By the Strong maximum principle, we know w > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq171_HTML.gif a.e. in Ω, so u n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq172_HTML.gif a.e. in Ω. Combining (S3) and (3), we obtain
Ω φ ( x ) w ( x ) d x Ω q ( x ) φ ( x ) w ( x ) d x Γ q ( s ) φ ( s ) w ( s ) d s = 0 , φ V . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equq_HTML.gif
This is a contradiction with Λ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq70_HTML.gif. This completes the proof of Theorem 1(ii).
  1. (iii)
    If Λ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq71_HTML.gif, by Lemma 1, there exists some φ Λ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq66_HTML.gif, such that
    Ω v ( x ) φ Λ ( x ) d x = Ω q ( x ) v ( x ) φ Λ ( x ) d x + Γ q ( s ) v ( s ) φ Λ ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ18_HTML.gif
    (17)
     
If u is a positive solution of (1), for the above φ Λ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq93_HTML.gif, we have
Ω u ( x ) φ Λ ( x ) d x = Ω f ( x , u ( x ) ) φ Λ ( x ) d x + Γ g ( s , u ( s ) ) φ Λ ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ19_HTML.gif
(18)
We order v = u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq173_HTML.gif in (17), and it follows from (18) that
Ω u ( x ) φ Λ ( x ) d x = Ω q ( x ) u ( x ) φ Λ ( x ) d x + Γ q ( s ) u ( s ) φ Λ ( s ) d s = Ω f ( x , u ( x ) ) φ Λ ( x ) d x + Γ g ( s , u ( s ) ) φ Λ ( s ) d s Ω q ( x ) u ( x ) φ Λ ( x ) d x + Γ q ( s ) u ( s ) φ Λ ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equr_HTML.gif

which implies that Ω ( f ( x , u ) q ( x ) u ( x ) ) φ Λ ( x ) d x + Γ ( g ( s , u ) q ( s ) u ( s ) ) φ Λ ( s ) d s = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq174_HTML.gif.

When φ Λ ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq66_HTML.gif a.e. in Ω, combining (S2), (S3), and (3), we obtain
f ( x , u ) q ( x ) u ( x ) , g ( x , u ) q ( x ) u ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equs_HTML.gif

Then we must have f ( x , u ) = q ( x ) u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq75_HTML.gif, g ( x , u ) = q ( x ) u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq175_HTML.gif a.e. in Ω, u ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq176_HTML.gif also achieves Λ (=1). When u = c φ Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq177_HTML.gif, c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq73_HTML.gif, we have Ω | φ Λ | 2 d x = Ω q ( x ) φ Λ 2 d x + Γ q ( s ) φ Λ 2 d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq178_HTML.gif, which achieves Λ.

On the other hand, if for some c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq73_HTML.gif, u ( x ) = c φ Λ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq74_HTML.gif and f ( x , c φ Λ ( x ) ) = c q ( x ) φ Λ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq179_HTML.gif, g ( x , u ) = c q ( x ) φ Λ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq180_HTML.gif a.e. x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif, since c φ Λ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq181_HTML.gif also achieves Λ. This means u ( x ) = c φ Λ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq74_HTML.gif is a solution of problem (1) as Λ = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq71_HTML.gif. This completes the proof of Theorem 1(iii). □

Proof of Corollary 2 Note that when q ( x ) l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq182_HTML.gif, then Λ = λ 1 l https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq183_HTML.gif. The conclusion follows from Theorem 1. □

Proof of Theorem 3 When q ( x ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq63_HTML.gif, we can replace φ Λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq184_HTML.gif by φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq185_HTML.gif in (11) and define c as in (12), then following the same procedures as in the proof of Theorem 1(ii), we need to show only that { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq114_HTML.gif is bounded in V. For this purpose, let { w n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq141_HTML.gif be defined as in (16). If { w n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq141_HTML.gif is bounded in V, we know w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq142_HTML.gif is a strong convergence in L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq143_HTML.gif, w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq142_HTML.gif is convergence a.e. x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq10_HTML.gif, w n w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq144_HTML.gif is a weak convergence in V, and w V https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq186_HTML.gif.

If u n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq140_HTML.gif, then t n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq187_HTML.gif and w ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq188_HTML.gif. We set Ω 1 = { x Ω : w ( x ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq189_HTML.gif, Ω 2 = { x Ω : w ( x ) 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq190_HTML.gif. Obviously, by (16), | u n | + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq191_HTML.gif a.e. in Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq192_HTML.gif. When q ( x ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq63_HTML.gif in (S3), there exists K 1 , K 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq193_HTML.gif and n large enough we have | f ( x , u n ) u n | K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq194_HTML.gif, | g ( x , u n ) u n | K 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq195_HTML.gif uniformly in x Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq196_HTML.gif. Hence, by (15) and (16), we obtain
4 c = lim n + t n 2 u n 2 = lim n + t n 2 ( u n L 2 ( Ω ) 2 + γ u n L 2 ( Γ ) 2 ) = lim n + t n 2 ( Ω f ( x , u n ) u n d x + Γ g ( s , u n ) u n d s + γ u n L 2 ( Γ ) 2 ) = lim n + ( Ω f ( x , u n ) u n w n 2 d x + Γ g ( s , u n ) u n w n 2 d s + t n 2 γ u n L 2 ( Γ ) 2 ) K 1 Ω w 2 d x + K 2 Γ w 2 d s + w L 2 ( Γ ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equt_HTML.gif

Noticing that w ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq197_HTML.gif in Ω 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq192_HTML.gif and K 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq198_HTML.gif, K 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq199_HTML.gif can be chosen large enough, so m Ω 2 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq200_HTML.gif and then w ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq188_HTML.gif in Ω.

Then we know lim n + Ω F ( x , w n ) d x + lim n + Γ G ( s , w n ) d s = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq201_HTML.gif, and consequently,
J ( w n ) = 1 2 w n L 2 ( Ω ) 2 + o ( 1 ) = 1 2 w n 2 1 2 w n L 2 ( Γ ) 2 + o ( 1 ) 1 2 ( 1 1 λ 1 + 1 ) w n 2 + o ( 1 ) = 2 c ( 1 1 λ 1 + 1 ) + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ20_HTML.gif
(19)
By u n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq140_HTML.gif, t n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq187_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq113_HTML.gif, then it follows Lemma 4 and (13), we obtain
J ( w n ) = J ( t n u n ) 1 + t n 2 2 n c . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ21_HTML.gif
(20)

Obviously, (19) and (20) are contradictory. So { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq114_HTML.gif is bounded in V. This completes the proof of Theorem 3. □

4 Example

In this section, we give two examples on f ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq62_HTML.gif: One satisfies (S1) to (S3) with q ( x ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq63_HTML.gif, but does not satisfy the (AR) condition; the other illustrates how the assumptions on the boundary are not trivial and compatible with the inner assumptions in Ω.

Example 1 Set:
f ( x , t ) = { 0 , t 0 ; t l n ( 1 + t ) , t > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equu_HTML.gif
Then it is easy to verify that f ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq202_HTML.gif satisfies (S1) to (S3) with p ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq203_HTML.gif as t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq204_HTML.gif and q ( x ) = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq205_HTML.gif as t + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq110_HTML.gif. In addition,
F ( x , t ) = 1 2 t 2 ln ( 1 + t ) 1 4 t 2 + 1 2 t 1 2 ln ( 1 + t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equv_HTML.gif

So, for some μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq59_HTML.gif, μ F ( x , t ) = t 2 ln ( 1 + t ) ( μ 2 μ 4 ln ( 1 + t ) + μ 2 t l n ( 1 + t ) μ 2 t 2 ) > t 2 ln ( 1 + t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq206_HTML.gif, for all t large.

This means f ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq202_HTML.gif does not satisfy the (AR) condition.

Example 2 Consider the following problem:
{ u ( x ) = α u ( x ) , 0 < x < l , u ( 0 ) = 0 , u ( l ) = α u ( l ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_Equ22_HTML.gif
(21)

where α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq207_HTML.gif is a constant. It is obvious that g = γ f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq44_HTML.gif as f ( x , u ) = α u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq208_HTML.gif. Problem (21) is a case of (1); we can obtain the nontrivial solution: u ( x ) = C ˜ sin α x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq209_HTML.gif, C ˜ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-91/MediaObjects/13661_2012_Article_202_IEq210_HTML.gif.

Author’s contributions

Li G carried out all studies in this article.

Declarations

Acknowledgements

The author would like to thank the referees for carefully reading this article and making valuable comments and suggestions.

Authors’ Affiliations

(1)
Department of Mathematics and Information Science, Qujing Normal University

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© Li; licensee Springer 2012

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