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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ1_HTML.gif
(1.1)

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

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

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

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

In view of the condition http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq19_HTML.gif that the functional
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ2_HTML.gif
(1.2)
is of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq20_HTML.gif on the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq21_HTML.gif with the norm
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ3_HTML.gif
(1.3)

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

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq32_HTML.gif in the above condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq33_HTML.gif is an eigenvalue of http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq37_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq38_HTML.gif ,

(AR)
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq40_HTML.gif sequence is bounded in http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq44_HTML.gif after a simple computation. That is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq45_HTML.gif must be superlinear with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq46_HTML.gif at infinity. Noticing our condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq47_HTML.gif the nonlinear term http://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 http://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 http://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 http://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 http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ5_HTML.gif
(1.5)

where the conditions on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq53_HTML.gif are similar to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq54_HTML.gif and http://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 http://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 http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq59_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq60_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq61_HTML.gif We can use http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq62_HTML.gif or http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq64_HTML.gif does not imply http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq65_HTML.gif (see [11, Remark http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq67_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq68_HTML.gif with respect to the conditions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq69_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq71_HTML.gif of [8] on the nonlinear term http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq75_HTML.gif hold, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq76_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq77_HTML.gif for some http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq79_HTML.gif )–( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq80_HTML.gif hold, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq81_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq82_HTML.gif for some http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ6_HTML.gif
(2.1)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ7_HTML.gif
(2.2)
Define a functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq84_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ8_HTML.gif
(2.3)

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

Lemma 2.3.

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

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq88_HTML.gif be a sequence such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq89_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq90_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq91_HTML.gif Note that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ9_HTML.gif
(2.4)
for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq92_HTML.gif Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq93_HTML.gif is bounded, taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq94_HTML.gif in (2.4). By http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq95_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq96_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq97_HTML.gif a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq98_HTML.gif So http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq99_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq100_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq101_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq102_HTML.gif set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq103_HTML.gif , and then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq104_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq105_HTML.gif in (2.4), it follows that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq107_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq108_HTML.gif , and then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq109_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq110_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq111_HTML.gif a.e. in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq113_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ10_HTML.gif
(2.5)
Then for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq114_HTML.gif , we deduce that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq115_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq116_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq117_HTML.gif . In fact, when http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq118_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq119_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ11_HTML.gif
(2.6)
When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq120_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ12_HTML.gif
(2.7)
When http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq121_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ13_HTML.gif
(2.8)
Since http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ14_HTML.gif
(2.9)
Choosing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq123_HTML.gif , we deduce that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ15_HTML.gif
(2.10)
Notice that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ16_HTML.gif
(2.11)

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq126_HTML.gif

Case 1.

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

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

Case 2.

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

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

Lemma 2.4.

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

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

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

Proof.

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

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

Choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq159_HTML.gif such that http://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
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq162_HTML.gif take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq163_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq164_HTML.gif . By (2.15), we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ22_HTML.gif
(2.17)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq165_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq166_HTML.gif , it is easy to see that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq167_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq168_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq169_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq170_HTML.gif )–( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq171_HTML.gif then

(i) the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq172_HTML.gif is coercive on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq173_HTML.gif , that is,
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq174_HTML.gif ;

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

Proof.

For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq177_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq178_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq179_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq180_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq181_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ25_HTML.gif
(2.20)
So we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ26_HTML.gif
(2.21)
Choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq182_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq184_HTML.gif . Write http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq185_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq186_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq187_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq188_HTML.gif imply that
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ27_HTML.gif
    (2.22)
     
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq189_HTML.gif , there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq190_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ29_HTML.gif
(2.24)
For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq191_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ30_HTML.gif
(2.25)
Integrating (2.25) over http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq192_HTML.gif , we deduce that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ31_HTML.gif
(2.26)
Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq194_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq195_HTML.gif a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq196_HTML.gif A similar argument shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq197_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq198_HTML.gif a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq199_HTML.gif . Hence
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ33_HTML.gif
(2.28)

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

In the case of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq203_HTML.gif , we do not need the assumption http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq205_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq206_HTML.gif satisfies the (PS) condition.

Proof.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq207_HTML.gif be a sequence such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq208_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq209_HTML.gif . One has
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ34_HTML.gif
(2.29)
for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq210_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq211_HTML.gif is bounded, we can take http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq212_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq213_HTML.gif , there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq214_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq215_HTML.gif a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq216_HTML.gif So http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq217_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq218_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq219_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq220_HTML.gif set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq221_HTML.gif , and then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq222_HTML.gif . Taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq223_HTML.gif in (2.29), it follows that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq225_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq226_HTML.gif , and then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq227_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq228_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq229_HTML.gif a.e. in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq231_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ35_HTML.gif
(2.30)
Then for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq232_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq233_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq234_HTML.gif In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq235_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq236_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ36_HTML.gif
(2.31)
If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq237_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ37_HTML.gif
(2.32)
Since http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq239_HTML.gif . In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq240_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq241_HTML.gif contradicts to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq242_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq243_HTML.gif is an eigenvalue of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq245_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq246_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq247_HTML.gif . Then the functional http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq249_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ39_HTML.gif
(2.34)
In view of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq250_HTML.gif , it suffices to prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq251_HTML.gif is bounded in http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ40_HTML.gif
(2.35)
Therefore http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq253_HTML.gif is an eigenfunction of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq254_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq255_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq256_HTML.gif . It follows from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq257_HTML.gif that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ41_HTML.gif
(2.36)
holds uniformly in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq258_HTML.gif , which implies that
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ43_HTML.gif
(2.38)
Thus
http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq260_HTML.gif be a Hilbert space, let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq262_HTML.gif be the http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq264_HTML.gif be an isolated critical point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq265_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq266_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq267_HTML.gif be a neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq268_HTML.gif . The group
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq269_HTML.gif th critical group of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq270_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq271_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq272_HTML.gif

Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq274_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq275_HTML.gif ; the critical groups of http://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])
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq277_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq278_HTML.gif is bounded from below on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq279_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq280_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq281_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq282_HTML.gif . Then
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq283_HTML.gif has a critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq284_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq285_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq286_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq288_HTML.gif . Hence http://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
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq290_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq291_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ50_HTML.gif
(3.3)
Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq293_HTML.gif , and then
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ52_HTML.gif
(3.5)
Hence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq294_HTML.gif has a critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq295_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_Equ53_HTML.gif
(3.6)

Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq297_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq298_HTML.gif , and http://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

References

  1. Ambrosetti A, Rabinowitz P: Dual variational methods in critical point theory and applications. Journal of Functional Analysis 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7MathSciNetView Article
  2. Brézis H, Nirenberg L: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Communications on Pure and Applied Mathematics 1983,36(4):437-477. 10.1002/cpa.3160360405MathSciNetView Article
  3. Rabinowitz PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMs Regional Conference Series in Mathematics, no. 65. American Mathematical Society, Providence, RI, USA; 1986.
  4. Bernis F, García-Azorero J, Peral I: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Advances in Differential Equations 1996,1(2):219-240.MathSciNet
  5. Deng YB, Wang GS: On inhomogeneous biharmonic equations involving critical exponents. Proceedings of the Royal Society of Edinburgh 1999,129(5):925-946. 10.1017/S0308210500031012MathSciNetView Article
  6. Gazzola F, Grunau H-C, Squassina M: Existence and nonexistence results for critical growth biharmonic elliptic equations. Calculus of Variations and Partial Differential Equations 2003,18(2):117-143. 10.1007/s00526-002-0182-9MathSciNetView Article
  7. Noussair ES, Swanson CA, Yang J:Critical semilinear biharmonic equations in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq300_HTML.gif . Proceedings of the Royal Society of Edinburgh 1992,121(1-2):139-148. 10.1017/S0308210500014189MathSciNetView Article
  8. Zhou H-S: Existence of asymptotically linear Dirichlet problem. Nonlinear Analysis: Theory, Methods & Applications 2001, 44: 909-918. 10.1016/S0362-546X(99)00314-4View Article
  9. Stuart CA, Zhou HS:Applying the mountain pass theorem to an asymptotically linear elliptic equation on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq301_HTML.gif . Communications in Partial Differential Equations 1999,24(9-10):1731-1758. 10.1080/03605309908821481MathSciNetView Article
  10. Li GB, Zhou H-S:Multiple solutions to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq302_HTML.gif -Laplacian problems with asymptotic nonlinearity as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F241518/MediaObjects/13661_2010_Article_907_IEq303_HTML.gif at infinity. Journal of the London Mathematical Society 2002,65(1):123-138. 10.1112/S0024610701002708MathSciNetView Article
  11. Ziemer WP: Weakly Differentiable Functions, Graduate Texts in Mathematics. Volume 120. Springer, New York, NY, USA; 1989:xvi+308.View Article
  12. Liu Y, Wang ZP: Biharmonic equations with asymptotically linear nonlinearities. Acta Mathematica Scientia 2007,27(3):549-560. 10.1016/S0252-9602(07)60055-1View Article
  13. Su JB, Zhao LG: An elliptic resonance problem with multiple solutions. Journal of Mathematical Analysis and Applications 2006,319(2):604-616. 10.1016/j.jmaa.2005.10.059MathSciNetView Article
  14. Chang K-C: Infinite-Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston, Mass, USA; 1993:x+312.View Article
  15. Bartsch T, Li SJ: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Analysis: Theory, Methods & Applications 1997,28(3):419-441. 10.1016/0362-546X(95)00167-TMathSciNetView Article

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.