Multiple Solutions for Biharmonic Equations with Asymptotically Linear Nonlinearities

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:

(1.1)

where is a bounded smooth domain in , and satisfies the following:

for all ,

uniformly for , where and are constants;

, where

In view of the condition , problem (1.1) is called asymptotically linear at both zero and infinity. Clearly, is a trivial solution of problem (1.1). It follows from and that the functional

(1.2)

is of on the space with the norm

(1.3)

Under the condition , the critical points of are solutions of problem (1.1). Let be the eigenvalues of and be the eigenfunction corresponding to . Let denote the eigenspace associated to . Throughout this paper, we denoted by the norm.

If in the above condition is an eigenvalue of 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 is the mountain pass theorem (see [1–3]). To apply this theorem to the functional in (1.2), usually we need the following condition [1], that is, for some and ,

(AR)

(1.4)

It is well known that the condition (AR) plays an important role in verifying that the functional has a "mountain pass" geometry and a related sequence is bounded in when one uses the mountain pass theorem.

If 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 is of critical growth (see, e.g., [4–7] and their references).

It follows from the condition (AR) that after a simple computation. That is, must be superlinear with respect to at infinity. Noticing our condition the nonlinear term is asymptotically linear, not superlinear, with respect to 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 , it is difficult to verify that the functional has a "mountain pass" structure and the sequence is bounded.

In [8], Zhou studied the following elliptic problem:

(1.5)

where the conditions on are similar to and He provided a valid method to verify the sequence of the variational functional, for the above problem is bounded in (see also [9, 10]).

To the author's knowledge, there seems few results on problem (1.1) when is asymptotically linear at infinity. However, the method in [8] cannot be applied directly to the biharmonic problems. For example, for the Laplacian problem, implies where We can use or as a test function, which is helpful in proving a solution nonnegative. While for the biharmonic problems, this trick fails completely since does not imply (see [11, Remark ]). 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 and with respect to the conditions and in [8]. In the course of proving existence of positive solution and negative solution, the monotonicity condition of [8] on the nonlinear term 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 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 and hold, , and for some ; then problem (1.1) has at least three nontrivial solutions.

Theorem 2.2.

Assume that conditions )–( hold, and for some ; then problem (1.1) has at least three nontrivial solutions.

Consider the following problem:

(2.1)

where

(2.2)

Define a functional by

(2.3)

where and then

Lemma 2.3.

satisfies the (PS) condition.

Proof.

Let be a sequence such that as Note that

(2.4)

for all Assume that is bounded, taking in (2.4). By , there exists such that a.e. So is bounded in . If as set , and then . Taking in (2.4), it follows that is bounded. Without loss of generality, we assume that in , and then in . Hence, a.e. in . Dividing both sides of (2.4) by , we get

(2.5)

Then for a.e. , we deduce that as where . In fact, when by we have

(2.6)

When , we have

(2.7)

When , we have

(2.8)

Since , by (2.5) and the Lebesgue dominated convergence theorem, we arrive at

(2.9)

Choosing , we deduce that

(2.10)

Notice that

(2.11)

where

Now we show that there is a contradiction in both cases of and

Case 1.

Suppose then a.e. in By we have Thus (2.11) implies that

(2.12)

which contradicts to

Case 2.

Suppose then and It follows from (2.11) that

(2.13)

which contradicts to if and contradicts to if

Lemma 2.4.

Let be the eigenfunction corresponding to with . If , then

(a) there exist such that for all with ;

(b) as .

Proof.

By and , if , for any , there exist and such that for all ,

(2.14)

(2.15)

where if

Choose such that By (2.14), the Poincaré inequality, and the Sobolev inequality, we get

(2.16)

So, part (a) holds if we choose small enough.

On the other hand, if take such that . By (2.15), we have

(2.17)

Since and , it is easy to see that

(2.18)

and part (b) is proved.

Lemma 2.5.

Let , where . If satisfies )–( then

(i) the functional is coercive on , that is,

(2.19)

and bounded from below on ;

(ii) the functional is anticoercive on .

Proof.

For , by , for any , there exists such that for all ,

(2.20)

So we have

(2.21)

Choose such that This proves (i).

• (ii) We firstly consider the case . Write . Then and imply that

  

  (2.22)

(2.23)

It follows from (2.22) that for every , there exists a constant such that

(2.24)

For we have

(2.25)

Integrating (2.25) over , we deduce that

(2.26)

Let and use (2.23); we see that for a.e. A similar argument shows that for a.e. . Hence

(2.27)

By (2.27), we get

(2.28)

for with , where

In the case of , we do not need the assumption and it is easy to see that the conclusion also holds.

Lemma 2.6.

If , then satisfies the (PS) condition.

Proof.

Let be a sequence such that . One has

(2.29)

for all If is bounded, we can take . By , there exists a constant such that a.e. So is bounded in . If , as set , and then . Taking in (2.29), it follows that is bounded. Without loss of generality, we assume in , and then in . Hence, a.e. in . Dividing both sides of (2.29) by , we get

(2.30)

Then for a.e. , we have as In fact, if by , we have

(2.31)

If , we have

(2.32)

Since , by (2.30) and the Lebesgue dominated convergence theorem, we arrive at

(2.33)

It is easy to see that . In fact, if , then contradicts to . Hence, is an eigenvalue of . This contradicts our assumption.

Lemma 2.7.

Suppose that and satisfies . Then the functional satisfies the (C) condition which is stated in [13].

Proof.

Suppose satisfies

(2.34)

In view of , it suffices to prove that is bounded in . Similar to the proof of Lemma 2.6, we have

(2.35)

Therefore is an eigenfunction of , then for a.e. . It follows from that

(2.36)

holds uniformly in , which implies that

(2.37)

On the other hand, (2.34) implies that

(2.38)

Thus

(2.39)

which contradicts to (2.37). Hence 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 be a Hilbert space, let be a functional satisfying the (PS) condition or (C) condition, let be the th singular relative homology group with integer coefficients. Let be an isolated critical point of with and let be a neighborhood of . The group

(2.40)

is said to be the th critical group of at , where

Let be the set of critical points of and ; the critical groups of at infinity are formally defined by (see [15])

(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 is bounded from below on and as with . Then

(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 has a critical point satisfying . Since , , and by the maximum principle, we get . Hence is a positive solution of the problem (1.1) and satisfies

(3.1)

Using the results in [14], we obtain

(3.2)

Similarly, we can obtain another negative critical point of satisfying

(3.3)

Since the zero function is a local minimizer of , and then

(3.4)

On the other hand, by Lemmas 2.52.6 and Proposition 2.8, we have

(3.5)

Hence has a critical point satisfying

(3.6)

Since , it follows from (3.2)–(3.6) that , , and 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.