We consider a semilinear fourth-order elliptic equation with a right-hand side nonlinearity which exhibits an asymmetric growth at +∞ and at −∞. Namely, it is linear at −∞ and superlinear at +∞. Combining variational methods with Morse theory, we show that the problem has at least two nontrivial solutions, one of which is negative.
Keywords:fourth-order elliptic boundary value problems; multiple solutions; critical groups; Morse theory
Consider the following Navier boundary value problem:
the first and the last inequality are strict on sets (not necessary the same) of positive measure, and
In view of the conditions (H3) and equation (3) in (H4), it is clear that for all , is linear at −∞ and superlinear at +∞. Clearly, is a trivial solution of problem (1). It follows from (H1) and (H2) that the functional
where . Under the condition (H2), the critical points of I are solutions of problem (1). Let be the eigenvalues of and be the eigenfunction corresponding to . In fact, . Let denote the eigenspace associated with . Throughout this article, we denoted by the norm and . The aim of this paper is to prove a multiplicity theorem for problem (1) when the nonlinearity term exhibits an asymmetric behavior as approaches +∞ and −∞. In the past, some authors studied the following elliptic problem:
with asymmetric nonlinearities by using the Fučík spectrum of the operator . This approach requires that exhibits linear growth at both +∞ and −∞ and that the limits exist and belong to ℝ. See the works of Các , Dancer and Zhang , Magalhães , de Paiva , Schechter  and the references therein. Equations with nonlinearities which are superlinear in one direction and linear in the other were investigated by Arcoya and Villegas  and Perera . They let the nonlinearity be line at −∞ and satisfy the Ambrosetti-Rabinowitz condition at +∞. Particularly, it is worth noticing paper . The authors relax several of the above restrictions on the nonlinearity . Their nonlinearity is only measurable in . The limit as of need not exist and the growth at −∞ can be linear or sublinear. Furthermore, their nonlinearity does not satisfy the famous AR-condition. They use the truncated skill of first order weak derivative to verify the (PS) condition and obtain multiple solutions for problem (1) by combining variational methods and Morse theory.
To the authors’ knowledge, there seem to be few results on problem (1) when is asymmetric nonlinearity at positive infinity and at negative infinity. However, the method in  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 [, Remark 2.1.10] and [10,11]). As far as this point is concerned, we will make use of the new methods to overcome it.
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 , there was a survey of results obtained in this direction. In , Micheletti and Pistoia showed that admits at least two solutions by a variation of linking if is sublinear. Chipot  proved that the problem has at least three solutions by a variational reduction method and a degree argument. In , Zhang and Li showed that admits at least two nontrivial solutions by Morse theory and local linking if is superlinear and subcritical on u.
In this article, under the guidance of , we consider multiple solutions of problem (1) with the asymmetric nonlinearity by using variational methods and Morse theory.
2 Main result and auxiliary lemmas
Let us now state the main result.
Lemma 2.2Under the assumptions of Theorem 2.1, thenIsatisfies the (PS) condition.
where is a positive constant and is a sequence which converges to zero. By a standard argument, in order to prove that has a convergence subsequence, we have to show that it is a bounded sequence. To do this, we argue by contradiction assuming that for a subsequence, denoted by , we have
Passing to the limit we deduce from equation (11) that
for all n. It follows from the weak compactness of the unit ball of W that there exists a subsequence, say , such that weakly converges to u in W. Now Sobolev’s embedding theorem suggests that converges to u in . From inequality (19) we obtain
Moreover one has
Hence we have
3 Computation of the critical groups
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  for more information on Morse theory.
Let H be a Hilbert space and be a functional satisfying the (PS) condition or (C) condition, and be the qth singular relative homology group with integer coefficients. Let be an isolated critical point of I with , , and U be a neighborhood of . The group
Let be the set of critical points of I and , the critical groups of I at infinity are formally defined by 
For the convenience of our proof, we first recall two interesting results and prove two important propositions.
Proposition 3.3If the assumptions of Theorem 2.1 hold, then
Proof Under the guidance of  and , we begin to prove this result. Let . Indeed, it follows from above Proposition 3.1 that I and have same critical set. Since is dense in E, invoking Proposition 16 of Palais , we have
From equations (20) and (21), we see that in order to prove the proposition, it suffices to show that
In order to prove equation (22), we proceed as follows. We define the sets
Moreover, by condition (H2), we have
By (H2) and formula (2) in condition (H4), we have
Then, from equation (27), we obtain
Using inequality (33), we have
By the choice of a, we have
Combining with equation (36) leads to equation (22), which completes the proof. □
Proposition 3.4If the assumptions of Theorem 2.1 hold, then
From inequalities (40) and (43), we know that I has a local linking at 0. Then invoking Proposition 2.3 of Bartsch and Li , we obtain . □
4 Proof of the main result
Proof of Theorem 2.1 We consider the following problem:
For s small enough, it follows from inequality (45) that
(see Proposition 3.4). Because of equations (47) and (48), using Proposition 3.2, we obtain
The authors declare that they have no competing interests.
The authors read and approved the final manuscript.
The authors 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. 11101319) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No. 1301038C).
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