Open Access

Multiple solutions for a fourth-order nonlinear elliptic problem

Boundary Value Problems20122012:39

DOI: 10.1186/1687-2770-2012-39

Received: 24 December 2011

Accepted: 5 April 2012

Published: 5 April 2012

Abstract

The existence of multiple solutions for a class of fourth-order elliptic equation with respect to the generalized asymptotically linear conditions is established by using the minimax method and Morse theory.

Keywords

fourth-order elliptic boundary value problems multiple solutions mountain pass theorem Morse theory.

1 Introduction

Consider the following Navier boundary value problem
Δ 2 u ( x ) + c Δ u = f ( x , u ) , in Ω ; u = Δ u = 0 on Ω ,
(1)

where Δ2 is the biharmonic operator, and Ω is a bounded smooth domain in N (N > 4), and c < λ 1 * the first eigenvalue of -Δ in H 0 1 ( Ω ) .

The conditions imposed on f (x, t) are as follows:

( H 1 ) f C 1 ( Ω ̄ × , ) , f ( x , 0 ) = 0 , f ( x , t ) t 0 for all x Ω, t ;

( H 2 ) f1(t) ≤ f (x, t) ≤ f2 (t) uniformly in x Ω, where fl, f2 C () and we denote
f 0 - = lim t 0 inf f 1 ( t ) t , f 0 + = lim t 0 sup f 2 ( t ) t ,
f - = lim t inf f 1 ( t ) t , f + = lim t sup f 2 ( t ) t .
In view of the condition ( H 2 ) , problem (1) is called generalized asymptotically linear at both zero and infinity. Clearly, u = 0 is a trivial solution of problem (1). It follows from ( H 1 ) and ( H 2 ) that the functional
I ( u ) = 1 2 Ω ( Δ u 2 - c u 2 ) d x - Ω F ( x , u ) d x
(2)
is of C2 on the space H 0 1 ( Ω ) H 2 ( Ω ) with the norm
u : = Ω ( Δ u 2 - c u 2 ) d x 1 2 ,

where F ( x , t ) = 0 t f ( x , s ) d s . Under the condition ( H 2 ) , the critical points of I are solutions of problem (1). Let 0 < λ1 < λ2 < ··· < λ k < ··· be the eigenvalues of ( Δ 2 + c Δ , H 2 ( Ω ) H 0 1 ( Ω ) ) and ϕl(x) > 0 be the eigenfunction corresponding to λl. In fact, λ 1 = λ 1 * ( λ 1 * - c ) . Let E λ k denote the eigenspace associated to λ k . Throughout this article, we denoted by | · | p the L p (Ω) norm.

If f - = f + in the above condition ( H 2 ) is an eigenvalue of ( Δ 2 + c Δ , H 2 ( Ω ) H 0 1 ( Ω ) ) , then the problem (1) is called resonance at infinity. Otherwise, we call it non-resonance. A main tool of seeking the critical points of functional I is the mountain pass theorem (see [13]). To apply this theorem to the functional I in (2), usually we need the following condition [1], i.e., for some θ > 2 and M > 0,
( AR ) 0 < θ F ( x , s ) f ( x , s ) s  for a .e x Ω  and  s > M .

It is well known that the condition (AR) plays an important role in verifying that the functional I has a "mountain-pass" geometry and a related (PS) c sequence is bounded in H 2 ( Ω ) H 0 1 ( Ω ) when one uses the mountain pass theorem.

If f(x, t) admits subcritical growth and satisfies (AR) condition by the standard argument of applying mountain pass theorem, we know that problem (1) has nontrivial solutions. Similarly, lase f(x, t) is of critical growth (see, for example, [47] and their references).

It follows from the condition (AR) that lim t F ( x , t ) t 2 = + after a simple computation. That is, f(x, t) must be superlinear with respect to t at infinity. Noticing our condition ( H 2 ) , the nonlinear term f(x, t) is generalized asymptotically linear, not superlinear, with respect to t 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 I, it is difficult to verify that the functional I has a "mountain pass" structure and the (PS)c sequence is bounded.

In [8], Zhou studied the following elliptic problem
- Δ u = f ( x , u ) u H 0 1 ( Ω ) ,

where the conditions on f(x, t) are similar to ( H 1 ) and ( H 2 ) . He provided a valid method to verify the (PS) sequence of the variational functional, for above problem is bounded in H 0 1 ( Ω ) (see also [9, 10]).

To the authors' knowledge, there seems few results on problem (1) when f(x, t) is generalized asymptotically linear at infinity. However, the method in [8] cannot be applied directly to the biharmonic problems. For example, for the Laplacian problem, u H 0 1 ( Ω ) implies |u|, u+, u - H 0 1 ( Ω ) , where u+ = max(u, 0), u- = max(-u, 0). We can use u+ or u- as a test function, which is helpful in proving a solution nonnegative. While for the biharmonic problems, this trick fails completely since u H 0 2 ( Ω ) does not imply u+, u - H 0 2 ( Ω ) (see [[11], Remark 2.1.10] and [12, 13]). As far as this point is concerned, we will make use of the new methods to discuss in the following Lemma 2.2.

This fourth-order semilinear elliptic problem can be considered as an analogue of a class of second-order problems which have been studied by many authors. In [14], there was a survey of results obtained in this direction. In [15], Micheletti and Pistoia showed that (P1) admits at least two solutions by a variation of linking if f(x, u) is sublinear. Chipot [16] proved that the problem (P1) has at least three solutions by a variational reduction method and a degree argument. In [17], Zhang and Li showed that (P1) admits at least two nontrivial solutions by Morse theory and local linking if f(x, u) is superlinear and subcritical on u.

In this article, we consider multiple solutions of problem (1) in the 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) under our more general conditions ( H 1 ) and ( H 2 ) with respect to the conditions (H1) and (H3) in [8]. In the course of proving existence of positive solution and negative solution, our conditions are general, but the proof of our compact condition is more simple than that in [8]. Furthermore, we can obtain a nontrivial solution when the nonlinear term f is non-resonance at the infinity by using Morse theory.

2 Main result and auxiliary lemmas

Let us now state the main result.

Theorem 2.1. Assume conditions ( H 1 ) and ( H 2 ) hold, f 0 + < λ 1 and λ k < f - f + < λ k + 1 for some k ≥ 2, then problem (1) has at least three nontrivial solutions.

Consider the following problem
Δ 2 u + c Δ u = f + ( x , u ) , x Ω , u | Ω = Δ u | Ω = 0 ,
where
f + ( x , t ) = f ( x , t ) , t > 0 , 0 , t 0 .
Define a functional I + : H 2 ( Ω ) H 0 1 ( Ω ) by
I + ( u ) = 1 2 Ω ( Δ u 2 - c u 2 ) d x - Ω F + ( x , u ) d x ,

where F + ( x , t ) = 0 t f + ( x , s ) d s , then I + C 2 ( H 2 ( Ω ) H 0 1 ( Ω ) , ) .

Lemma 2.2. I+satisfies the (PS) condition.

Proof. Let { u n } H 2 ( Ω ) H 0 1 ( Ω ) be a sequence such that I + ( u n ) c , < I + ( u n ) , ϕ > 0 as n → ∞. Note that
< I + ( u n ) , ϕ > = Ω ( Δ u n Δ ϕ - c u n ϕ ) d x - Ω f + ( x , u n ) ϕ d x = ( ϕ )
(3)
for all ϕ H 2 ( Ω ) H 0 1 ( Ω ) . Assume that |u n |2 is bounded, taking ϕ = u n in (3). By ( H 2 ) , there exists c > 0 such that |f+(x, u n (x))| ≤ c|u n (x)|, a.e. x Ω. So (u n ) is bounded in H 2 ( Ω ) H 0 1 ( Ω ) . If |u n |2 → +∞, as n → ∞, set v n = u n u n 2 , then |v n |2 = 1. Taking ϕ = v n in (3), it follows that ||v n || is bounded. Without loss of generality, we assume that v n v in H 2 ( Ω ) H 0 1 ( Ω ) , then v n → v in L2(Ω). Hence, v n → v a.e. in Ω. Dividing both sides of (3) by |u n |2, we get
Ω ( Δ v n Δ ϕ - c v n ϕ ) d x - Ω f + ( x , u n ) u n 2 ϕ d x = ϕ u n 2 , ϕ H 2 ( Ω ) H 0 1 ( Ω ) .
(4)

Let ϵ > 0 and f - - ε > λ 1 , and choose a constant M > 0 such that f ( x , t ) ( f - - ϵ ) t , t > M .

Let ϕ H 2 ( Ω ) H 0 1 ( Ω ) and ϕ ≥ 0. From (4), we have
Ω ( Δ v n Δ ϕ - c v n ϕ ) d x = Ω f + ( x , u n ) u n 2 ϕ d x + ( 1 ) = u n + 2 v n < M f ( x , u n ) u n 2 ϕ d x + u n + 2 v n M f ( x , u n ) u n + 2 ϕ d x + ( 1 ) ( f - - ϵ ) Ω v n ϕ d x - c 0 u n + 2 v n M v n ϕ d x + ( 1 ) .
Letting n → ∞, for all ϕ H 2 ( Ω ) H 0 1 ( Ω ) and ϕ ≥ 0 we have
Ω ( Δ v Δ ϕ - c v ϕ ) d x ( f - - ϵ ) Ω v + ϕ d x .
(5)
Then we have
Δ 2 v + c Δ v ( f - - ϵ ) v + , x Ω , v Ω = Δ v Ω = 0 .

While let -Δv = u, by the comparison maximum principle v ≥ 0. Since the definition of v n , we have v 0 and we arrive at a contradiction by choosing ϕ = ϕ1 in (5).

Since |u n |2 is bounded, from (3) we get the boundedness of ||u n ||. A standard argument shows that {u n } has a convergent subsequence in H 2 ( Ω ) H 0 1 ( Ω ) . Therefore, I+ satisfies (PS) condition.

Lemma 2.3. Let ϕ1be the eigenfunction corresponding to λ1with ||ϕ1|| = 1. If f 0 + < λ 1 < f - , then

(a) There exist ρ, β > 0 such that I+ (u) ≥ β for all u H 2 ( Ω ) H 0 1 ( Ω ) with ||u|| = ρ;

(b) I+(1) = -∞ as t → +∞.

Proof. By ( H 1 ) and ( H 2 ) , if f - ( λ 1 , + ) , for any ε > 0, there exist A = A(ε) ≥ 0 and B = B(ε) such that for all (x, s) Ω × ,
F + ( x , s ) 1 2 ( f 0 + + ε ) s 2 + A s p + 1 ,
(6)
F + ( x , s ) 1 2 ( f - - ε ) s 2 - B ,
(7)

where p 1 , N + 4 N - 4 if N > 4.

Choose ε > 0 such that f 0 + + ε < λ 1 . By (6), the Poincaré inequality and the Sobolev inequality, we get
I + ( u ) = 1 2 Ω ( Δ u 2 - c u 2 ) d x - Ω F + ( x , u ) d x 1 2 Ω ( Δ u 2 - c u 2 ) d x - 1 2 Ω [ ( f 0 + + ε ) u 2 + A u p + 1 ] d x 1 2 1 - f 0 + + ε λ 1 u 2 - c u p + 1 .

So, part (a) holds if we choose ||u|| = ρ > 0 small enough.

On the other hand, if f - ( λ 1 , + ) , take ε > 0 such that f - - ε > λ 1 . By (7), we have
I + ( u ) 1 2 u 2 - f - - ε 2 u 2 2 + B Ω .
Since f - - ε > λ 1 and ||ϕl|| = 1, it is easy to see that
I + ( t ϕ 1 ) 1 2 1 - f - - ε λ 1 t 2 + B Ω -  as  t +

and part (b) is proved.

Lemma 2.4. Let H 2 ( Ω ) H 0 1 ( Ω ) = V W , where V = E λ 1 E λ 2 E λ k . If f satisfies ( H 1 ) - ( H 3 ) , then

(i) the functional I is coercive on W, that is
I ( u ) + as u + , u W

and bounded from below on W,

(ii) the functional I is anti-coercive on V.

Proof. For u W, by ( H 2 ) , for any ε > 0, there exists Bl = Bl(ε) such that for all (x, s) Ω × ,
F ( x , s ) 1 2 ( f + + ε ) s 2 + B 1 .
(8)
So we have
I ( u ) = 1 2 Ω ( Δ u 2 - c u 2 ) d x - Ω F ( x , u ) d x 1 2 Ω ( Δ u 2 - c u 2 ) d x - 1 2 ( f + + ε ) u 2 2 - B 1 Ω 1 2 1 - f + + ε λ k + 1 u 2 - B 1 Ω .

Choose ε > 0 such that f + + ε < λ k + 1 . This proves (i).

(ii) When λ k < f - f + < λ k + 1 , it is easy to see that the conclusion holds.

Lemma 2.5. If λ k < f - f + < λ k + 1 , then I satisfies the (PS) condition.

Proof. Let { u n } H 2 ( Ω ) H 0 1 ( Ω ) be a sequence such that |I (u n) | ≤ c, < I'(u n) , ϕ > → 0.

Since
< I ( u n ) , ϕ > = Ω ( Δ u n Δ ϕ - c u n ϕ ) d x - Ω f ( x , u n ) ϕ d x = ( ϕ )
(9)
for all ϕ H 2 ( Ω ) H 0 1 ( Ω ) . If |u n |2 is bounded, we can take ϕ = u n . By ( H 2 ) , there exists a constant c > 0 such that |f (x, u n (x))| ≤ c|u n (x)|, a.e. x Ω. So (u n ) is bounded in H 2 ( Ω ) H 0 1 ( Ω ) . If |u n |2 → +∞, as n → ∞, set v n = u n u n 2 , then |v n |2 = 1. Taking ϕ = v n in (9), it follows that ||vn|| is bounded. Without loss of generality, we assume vn v in H 2 ( Ω ) H 0 1 ( Ω ) , then vn → v in L2(Ω). Hence, vn → v a.e. in Ω. Dividing both sides of (9) by |u n |2, for any ϕ H 2 ( Ω ) H 0 1 ( Ω ) , we get
Ω ( Δ v n Δ ϕ - c v n ϕ ) d x - Ω f ( x , u n ) u n 2 ϕ d x = ϕ u n 2 .
(10)
Then for a.e. x Ω and suitable ϵ, we have
( f - - ϵ ) v 2 f ( x , u n ) u n 2 v ( f + + ϵ ) v 2
as n → ∞. In fact, if v(x) ≠ 0, by ( H 2 ) , we have
u n ( x ) = v n ( x ) u n 2 +
and
( f - - ϵ ) v 2 f ( x , u n ) u n 2 v = f ( x , u n ) u n v n v ( f + + ϵ ) v 2 .
as n → ∞. If v(x) = 0, we have
f ( x , u n ) u n 2 c v n 0 .
Since f ( x , u n ) u n 2 c v n , by choosing ϕ = v in (10) and the Lebesgue dominated convergence theorem, we arrive at
λ k v 2 2 Ω ( Δ v Δ v - c v v ) d x λ k + 1 v 2 2 .

From Fourier series theory, it is easy to see that v ≡ 0. It contradicts to lim n v n 2 = v 2 = 1 .

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 [18] for more information on Morse theory.

Let H be a Hilbert space and I Cl(H, ) be a functional satisfying the (PS) condition or (C) condition, and H q (X, Y) be the q th singular relative homology group with integer coefficients. Let u0 be an isolated critical point of I with I(u0) = c, c , and U be a neighborhood of u0. The group
C q ( I , u 0 ) : = H q ( I c U , I c U \ { u 0 } ) , q Z

is said to be the q th critical group of I at u0, where I c = {u H: I (u) ≤ c}.

Let K: = {u H: I'(u) = 0} be the set of critical points of I and a < inf I (K), the critical groups of I at infinity are formally defined by (see [19])
C q ( I , ) : = H q ( H , I a ) , q Z .

The following result comes from [18, 19] and will be used to prove the result in this article.

Proposition 2.6[19]. Assume that H = H + H - , I is bounded from below on H + and I(u) → -as ||u|| with u H - . Then
C k ( I , ) 0 , i f k = dim H - < .
(11)

3 Proof of the main result

Proof of Theorem 2.1. By Lemmas 2.2 and 2.3 and the mountain pass theorem, the functional I+ has a critical point ul satisfying I+(ul) ≥ β. Since I+ (0) = 0, ul ≠ 0 and by the maximum principle, we get ul > 0. Hence ul is a positive solution of the problem (1) and satisfies
C 1 ( I + , u 1 ) 0 , u 1 > 0 .
(12)
Using the results in [18], we obtain
C q ( I , u 1 ) = C q ( I C 0 1 ( Ω ) , u 1 ) = C q ( I + | C 0 1 ( Ω ) , u 1 ) = C q ( I + , u 1 ) = δ q 1 Z .
(13)
Similarly, we can obtain another negative critical point u2 of I satisfying
C q ( I , u 2 ) = δ q , 1 Z .
(14)
Since f 0 + < λ 1 , the zero function is a local minimizer of I, then
C q ( I , 0 ) = δ q , 0 Z .
(15)
On the other hand, by Lemmas 2.4, 2.5 and the Proposition 2.6, we have
C k ( I , ) 0 .
(16)
Hence I has a critical point u3 satisfying
C k ( I , u 3 ) 0 .
(17)

Since k ≥ 2, it follows from (13)-(17) that ul, u2, and u3 are three different nontrivial solutions of the problem (1).

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Tianshui Normal University

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© Pei; licensee Springer. 2012

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