Open Access

Multiple Solutions for Biharmonic Equations with Asymptotically Linear Nonlinearities

Boundary Value Problems20102010:241518

DOI: 10.1155/2010/241518

Received: 26 February 2010

Accepted: 22 April 2010

Published: 27 May 2010

Abstract

The existence of multiple solutions for a class of fourth elliptic equation with respect to the resonance and nonresonance conditions is established by using the minimax method and Morse theory.

1. Introduction

Consider the following Navier boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq1_HTML.gif is a bounded smooth domain in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq2_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq3_HTML.gif satisfies the following:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq5_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq7_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq9_HTML.gif uniformly for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq10_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq12_HTML.gif are constants;

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq14_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq15_HTML.gif

In view of the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq16_HTML.gif , problem (1.1) is called asymptotically linear at both zero and infinity. Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq17_HTML.gif is a trivial solution of problem (1.1). It follows from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq19_HTML.gif that the functional
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ2_HTML.gif
(1.2)
is of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq20_HTML.gif on the space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq21_HTML.gif with the norm
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ3_HTML.gif
(1.3)

Under the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq22_HTML.gif , the critical points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq23_HTML.gif are solutions of problem (1.1). Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq24_HTML.gif be the eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq25_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq26_HTML.gif be the eigenfunction corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq27_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq28_HTML.gif denote the eigenspace associated to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq29_HTML.gif . Throughout this paper, we denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq30_HTML.gif the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq31_HTML.gif norm.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq32_HTML.gif in the above condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq33_HTML.gif is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq34_HTML.gif then problem (1.1) is called resonance at infinity. Otherwise, we call it non-resonance. A main tool of seeking the critical points of functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq35_HTML.gif is the mountain pass theorem (see [13]). To apply this theorem to the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq36_HTML.gif in (1.2), usually we need the following condition [1], that is, for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq38_HTML.gif ,

(AR)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ4_HTML.gif
(1.4)

It is well known that the condition (AR) plays an important role in verifying that the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq39_HTML.gif has a "mountain pass" geometry and a related https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq40_HTML.gif sequence is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq41_HTML.gif when one uses the mountain pass theorem.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq42_HTML.gif admits subcritical growth and satisfies (AR) condition by the standard argument of applying mountain pass theorem, we known that problem (1.1) has nontrivial solutions. Similarly, lase https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq43_HTML.gif is of critical growth (see, e.g., [47] and their references).

It follows from the condition (AR) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq44_HTML.gif after a simple computation. That is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq45_HTML.gif must be superlinear with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq46_HTML.gif at infinity. Noticing our condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq47_HTML.gif the nonlinear term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq48_HTML.gif is asymptotically linear, not superlinear, with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq49_HTML.gif at infinity, which means that the usual condition (AR) cannot be assumed in our case. If the mountain pass theorem is used to seek the critical points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq50_HTML.gif , it is difficult to verify that the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq51_HTML.gif has a "mountain pass" structure and the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq52_HTML.gif sequence is bounded.

In [8], Zhou studied the following elliptic problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ5_HTML.gif
(1.5)

where the conditions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq53_HTML.gif are similar to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq55_HTML.gif He provided a valid method to verify the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq56_HTML.gif sequence of the variational functional, for the above problem is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq57_HTML.gif (see also [9, 10]).

To the author's knowledge, there seems few results on problem (1.1) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq58_HTML.gif is asymptotically linear at infinity. However, the method in [8] cannot be applied directly to the biharmonic problems. For example, for the Laplacian problem, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq59_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq60_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq61_HTML.gif We can use https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq62_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq63_HTML.gif as a test function, which is helpful in proving a solution nonnegative. While for the biharmonic problems, this trick fails completely since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq64_HTML.gif does not imply https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq65_HTML.gif (see [11, Remark https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq66_HTML.gif ]). As far as this point is concerned, we will make use of the methods in [12] to discuss in the following Lemma 2.3. In this paper we consider multiple solutions of problem (1.1) in the cases of resonance and non-resonance by using the mountain pass theorem and Morse theory. At first, we use the truncated skill and mountain pass theorem to obtain a positive solution and a negative solution of problem (1.1) under our more general condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq67_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq68_HTML.gif with respect to the conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq69_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq70_HTML.gif in [8]. In the course of proving existence of positive solution and negative solution, the monotonicity condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq71_HTML.gif of [8] on the nonlinear term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq72_HTML.gif is not necessary, this point is very important because we can directly prove existence of positive solution and negative solution by using Rabinowitz's mountain pass theorem. That is, the proof of our compact condition is more simple than that in [8]. Furthermore, we can obtain a nontrivial solution when the nonlinear term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq73_HTML.gif is resonance or non-resonance at the infinity by using Morse theory.

2. Main Results and Auxiliary Lemmas

Let us now state the main results.

Theorem 2.1.

Assume that conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq75_HTML.gif hold, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq76_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq77_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq78_HTML.gif ; then problem (1.1) has at least three nontrivial solutions.

Theorem 2.2.

Assume that conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq79_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq80_HTML.gif hold, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq82_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq83_HTML.gif ; then problem (1.1) has at least three nontrivial solutions.

Consider the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ6_HTML.gif
(2.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ7_HTML.gif
(2.2)
Define a functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq84_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ8_HTML.gif
(2.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq85_HTML.gif and then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq86_HTML.gif

Lemma 2.3.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq87_HTML.gif satisfies the (PS) condition.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq88_HTML.gif be a sequence such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq89_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq90_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq91_HTML.gif Note that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ9_HTML.gif
(2.4)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq92_HTML.gif Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq93_HTML.gif is bounded, taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq94_HTML.gif in (2.4). By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq95_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq96_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq97_HTML.gif a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq98_HTML.gif So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq99_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq100_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq101_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq102_HTML.gif set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq103_HTML.gif , and then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq104_HTML.gif . Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq105_HTML.gif in (2.4), it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq106_HTML.gif is bounded. Without loss of generality, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq107_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq108_HTML.gif , and then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq109_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq110_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq111_HTML.gif a.e. in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq112_HTML.gif . Dividing both sides of (2.4) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq113_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ10_HTML.gif
(2.5)
Then for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq114_HTML.gif , we deduce that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq115_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq116_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq117_HTML.gif . In fact, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq118_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq119_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ11_HTML.gif
(2.6)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq120_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ12_HTML.gif
(2.7)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq121_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ13_HTML.gif
(2.8)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq122_HTML.gif , by (2.5) and the Lebesgue dominated convergence theorem, we arrive at
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ14_HTML.gif
(2.9)
Choosing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq123_HTML.gif , we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ15_HTML.gif
(2.10)
Notice that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ16_HTML.gif
(2.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq124_HTML.gif

Now we show that there is a contradiction in both cases of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq126_HTML.gif

Case 1.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq127_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq128_HTML.gif a.e. in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq129_HTML.gif By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq130_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq131_HTML.gif Thus (2.11) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ17_HTML.gif
(2.12)

which contradicts to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq132_HTML.gif

Case 2.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq133_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq135_HTML.gif It follows from (2.11) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ18_HTML.gif
(2.13)

which contradicts to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq136_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq137_HTML.gif and contradicts to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq138_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq139_HTML.gif

Lemma 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq140_HTML.gif be the eigenfunction corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq141_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq142_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq143_HTML.gif , then

(a) there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq144_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq145_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq146_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq147_HTML.gif ;

(b) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq148_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq149_HTML.gif .

Proof.

By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq151_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq152_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq153_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq154_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq155_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq156_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ19_HTML.gif
(2.14)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ20_HTML.gif
(2.15)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq157_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq158_HTML.gif

Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq159_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq160_HTML.gif By (2.14), the Poincaré inequality, and the Sobolev inequality, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ21_HTML.gif
(2.16)

So, part (a) holds if we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq161_HTML.gif small enough.

On the other hand, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq162_HTML.gif take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq163_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq164_HTML.gif . By (2.15), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ22_HTML.gif
(2.17)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq165_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq166_HTML.gif , it is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ23_HTML.gif
(2.18)

and part (b) is proved.

Lemma 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq167_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq168_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq169_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq170_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq171_HTML.gif then

(i) the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq172_HTML.gif is coercive on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq173_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ24_HTML.gif
(2.19)

and bounded from below on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq174_HTML.gif ;

(ii) the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq175_HTML.gif is anticoercive on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq176_HTML.gif .

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq177_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq178_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq179_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq180_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq181_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ25_HTML.gif
(2.20)
So we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ26_HTML.gif
(2.21)
Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq182_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq183_HTML.gif This proves (i).
  1. (ii)
    We firstly consider the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq184_HTML.gif . Write https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq185_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq186_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq187_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq188_HTML.gif imply that
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ27_HTML.gif
    (2.22)
     
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ28_HTML.gif
(2.23)
It follows from (2.22) that for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq189_HTML.gif , there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq190_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ29_HTML.gif
(2.24)
For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq191_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ30_HTML.gif
(2.25)
Integrating (2.25) over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq192_HTML.gif , we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ31_HTML.gif
(2.26)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq193_HTML.gif and use (2.23); we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq194_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq195_HTML.gif a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq196_HTML.gif A similar argument shows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq197_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq198_HTML.gif a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq199_HTML.gif . Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ32_HTML.gif
(2.27)
By (2.27), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ33_HTML.gif
(2.28)

for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq200_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq201_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq202_HTML.gif

In the case of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq203_HTML.gif , we do not need the assumption https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq204_HTML.gif and it is easy to see that the conclusion also holds.

Lemma 2.6.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq205_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq206_HTML.gif satisfies the (PS) condition.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq207_HTML.gif be a sequence such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq208_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq209_HTML.gif . One has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ34_HTML.gif
(2.29)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq210_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq211_HTML.gif is bounded, we can take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq212_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq213_HTML.gif , there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq214_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq215_HTML.gif a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq216_HTML.gif So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq217_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq218_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq219_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq220_HTML.gif set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq221_HTML.gif , and then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq222_HTML.gif . Taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq223_HTML.gif in (2.29), it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq224_HTML.gif is bounded. Without loss of generality, we assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq225_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq226_HTML.gif , and then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq227_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq228_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq229_HTML.gif a.e. in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq230_HTML.gif . Dividing both sides of (2.29) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq231_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ35_HTML.gif
(2.30)
Then for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq232_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq233_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq234_HTML.gif In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq235_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq236_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ36_HTML.gif
(2.31)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq237_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ37_HTML.gif
(2.32)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq238_HTML.gif , by (2.30) and the Lebesgue dominated convergence theorem, we arrive at
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ38_HTML.gif
(2.33)

It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq239_HTML.gif . In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq240_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq241_HTML.gif contradicts to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq242_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq243_HTML.gif is an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq244_HTML.gif . This contradicts our assumption.

Lemma 2.7.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq245_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq246_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq247_HTML.gif . Then the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq248_HTML.gif satisfies the (C) condition which is stated in [13].

Proof.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq249_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ39_HTML.gif
(2.34)
In view of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq250_HTML.gif , it suffices to prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq251_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq252_HTML.gif . Similar to the proof of Lemma 2.6, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ40_HTML.gif
(2.35)
Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq253_HTML.gif is an eigenfunction of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq254_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq255_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq256_HTML.gif . It follows from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq257_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ41_HTML.gif
(2.36)
holds uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq258_HTML.gif , which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ42_HTML.gif
(2.37)
On the other hand, (2.34) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ43_HTML.gif
(2.38)
Thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ44_HTML.gif
(2.39)

which contradicts to (2.37). Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq259_HTML.gif is bounded.

It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equation. Let us recall some results which will be used later. We refer the readers to the book [14] for more information on Morse theory.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq260_HTML.gif be a Hilbert space, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq261_HTML.gif be a functional satisfying the (PS) condition or (C) condition, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq262_HTML.gif be the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq263_HTML.gif th singular relative homology group with integer coefficients. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq264_HTML.gif be an isolated critical point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq265_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq266_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq267_HTML.gif be a neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq268_HTML.gif . The group
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ45_HTML.gif
(2.40)

is said to be the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq269_HTML.gif th critical group of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq270_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq271_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq272_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq273_HTML.gif be the set of critical points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq274_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq275_HTML.gif ; the critical groups of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq276_HTML.gif at infinity are formally defined by (see [15])
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ46_HTML.gif
(2.41)

The following result comes from [14, 15] and will be used to prove the results in this paper.

Proposition 2.8 (see [15]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq277_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq278_HTML.gif is bounded from below on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq279_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq280_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq281_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq282_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ47_HTML.gif
(2.42)

3. Proof of the Main Results

Proof of Theorem 2.1.

By Lemmas 2.32.4 and the mountain pass theorem, the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq283_HTML.gif has a critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq284_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq285_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq286_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq287_HTML.gif , and by the maximum principle, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq288_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq289_HTML.gif is a positive solution of the problem (1.1) and satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ48_HTML.gif
(3.1)
Using the results in [14], we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ49_HTML.gif
(3.2)
Similarly, we can obtain another negative critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq290_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq291_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ50_HTML.gif
(3.3)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq292_HTML.gif the zero function is a local minimizer of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq293_HTML.gif , and then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ51_HTML.gif
(3.4)
On the other hand, by Lemmas 2.52.6 and Proposition 2.8, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ52_HTML.gif
(3.5)
Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq294_HTML.gif has a critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq295_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ53_HTML.gif
(3.6)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq296_HTML.gif , it follows from (3.2)–(3.6) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq297_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq298_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq299_HTML.gif are three different nontrivial solutions of problem (1.1).

Proof of Theorem 2.2.

By Lemmas 2.52.7 and the Proposition 2.8, we can prove the conclusion (3.5). The other proof is similar to that of Theorem 2.1.

Declarations

Acknowledgments

The author would like to thank the referees for valuable comments and suggestions for improving this paper. This work was supported by the National NSF (Grant no. 10671156) of China.

Authors’ Affiliations

(1)
Center for Nonlinear Studies, Northwest University
(2)
Department of Mathematics, Tianshui Normal University

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Copyright

© Ruichang Pei. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.