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:
where is the biharmonic operator, and Ω is a bounded smooth domain in ( ), and the first eigenvalue of −△ in .
The conditions imposed on are as follows:
(H1) , for all and for all and all ;
(H2) there exist and such that for all , and , where , if ;
(H3) uniformly for , where l is a nonnegative constant;
(H4) there exist such that for , we have
(H5) there exist and an integer such that
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
is on the space with the norm
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.
Theorem 2.1Assume conditions (H1)-(H5) hold. If , then problem (1) has at least two nontrivial solutions.
Lemma 2.2Under the assumptions of Theorem 2.1, thenIsatisfies the (PS) condition.
Proof Let be a sequence such that for every ,
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
Without loss of generality we can assume for all and define . Obviously, and then it is possible to extract a subsequence (denoted also by ) such that
where and . Dividing both sides of inequality (10) by , we obtain
Passing to the limit we deduce from equation (11) that
for all .
Now we claim that a.e. . To verify this, let us observe that by choosing in equation (15) we have
where . But, on the other hand, from (H3) and equation (3) in (H4), we have
for some positive constant . Moreover, using a.e. , equation (13) and the superlinearity of f, we also deduce
Therefore, if we will obtain by Fatou’s lemma that
which contradicts inequality (16). Thus and the claim is proved.
Clearly, , by (H3), there exists such that for a.e. . By using Lebesgue dominated convergence theorem in equation (15), we have
for all . This contradicts . □
Lemma 2.3Let , where . If is an integer, , a.e. on Ω and the inequality is strict on a set of positive measure, then there exists such that
for all .
Proof We claim that there exists a constant such that
for all . In fact, if not, there exists a sequence such that
for all , which implies for all n. By the homogeneity of the above inequality, we may assume that and
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
which implies that and on a positive measure subset. It contradicts the unique continuation property of the eigenfunction. □
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
is said to be the qth critical group of I at , where .
Let be the set of critical points of I and , the critical groups of I at infinity are formally defined by 
From the deformation theorem, we see that the above definition is independent of the particular choice of . If then
For the convenience of our proof, we first recall two interesting results and prove two important propositions.
Under (H2), if is an isolated critical point ofI, then .
If and for some integer we have and , then either or .
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
Consider the map defined by
Clearly, is a continuous homotopy and for all . Therefore, is contractible in itself.
By equation (3) in (H4), given any , we can find such that
Similarly, from condition (H3), and by choosing even bigger if necessary, we observe that there is a number such that
Moreover, by condition (H2), we have
for some .
Let . By inequalities (23), (24), and (25), for all we have
Recalling that is arbitrary, from (26), we have
Using formula (4) in condition (H4), we see that there exist constants and such that
By (H2) and formula (2) in condition (H4), we have
for some . By inequalities (28) and (29), for any we have
where C is a positive constant. Let be the continuous embedding map. Let denote the duality brackets for the pair . We let , and so
Then, from equation (27), we obtain
From conditions (H2) and (H3), we see that given , we can find such that
Using inequality (33), we have
for , where is defined as
and is a positive constant. So is coercive, thus we find such that . We pick
Then inequality (32) implies that we can find such that
Moreover, the implicit function theorem implies that .
By the choice of a, we have
We define the set . The map defined by
is a continuous deformation of , and for all (see equations (34) and (35)). Therefore, is a strong deformation retract of . Hence we have
Recalling that in the first part of the proof, we established that is contractible. This yields
Combining with equation (36) leads to equation (22), which completes the proof. □
Proposition 3.4If the assumptions of Theorem 2.1 hold, then
where (Vbeing defined in Lemma 2.3).
Proof By condition (H5), given , we can find such that
Since V is finite dimensional, all norms are equivalent. Thus we can find small such that
for all . Taking inequalities (37) and (38) into account, for all with we have
Similar to the proof of Lemma 2.3, there exists such that
for all and .
On the other hand, for given , it follows from (H2) and (H5) that
for all and . By (41) and Lemma 2.3, we have
for all . From inequality (42), we infer that for ρ small enough we have
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:
Define a functional by
where , then . Obviously, by conditions (H1) and (H3), we know that is coercive and boundedness from below. Thus we can find such that
Next, we claim that . By condition (H5), given , there exists such that
For s small enough, it follows from inequality (45) that
and thus, by equation (44), , so . From condition (H1) and strong maximum principle, we have and
Since is an interior point of , from Proposition 3.1, we know
Let , be such that . We consider the sublevel sets
Suppose that 0 and are the only critical points of I. Otherwise, we have a second nontrivial smooth solution and so we are done. By Proposition 3.3, we have
We know that I satisfies the (PS) condition (see Lemma 2.2). Hence choosing small enough, we have
(see Proposition 3.4). Because of equations (47) and (48), using Proposition 3.2, we obtain
If , then there is a critical point of I such that
If , then there is a critical point of I such that
Since , from equations (46) and (49), we see that . It is obvious that . Therefore and are two solutions of problem (1). □
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).
Các, NP: On nontrivial solutions of a Dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue. J. Differ. Equ.. 80, 379–404 (1989). Publisher Full Text
Dancer, EN, Zhang, Z: Fučík spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity. J. Math. Anal. Appl.. 250, 449–464 (2000). Publisher Full Text
Schechter, M: The Fučík spectrum. Indiana Univ. Math. J.. 43, 1139–1157 (1994). Publisher Full Text
Arcoya, D, Villegas, S: Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at −∞ and superlinear at +∞. Math. Z.. 219, 499–513 (1995). Publisher Full Text
Perera, K: Existence and multiplicity results for a Sturm-Liouville equation asymptotically linear at −∞ and superlinear at +∞. Nonlinear Anal.. 39, 669–684 (2000). Publisher Full Text
Liu, Y, Wang, ZP: Biharmonic equations with asymptotically linear nonlinearities. Acta Math. Sci.. 27, 549–560 (2007). Publisher Full Text
Lazer, AC, Mckenna, PJ: Large amplitude periodic oscillation in suspension bridges: some new connections with nonlinear analysis. SIAM Rev.. 32, 537–578 (1990). Publisher Full Text
Micheletti, AM, Pistoia, A: Multiplicity solutions for a fourth order semilinear elliptic problems. Nonlinear Anal. TMA. 31, 895–908 (1998). Publisher Full Text
Bartsch, T, Li, SJ: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. TMA. 28, 419–441 (1997). Publisher Full Text
Qian, AX: Multiple solutions for a fourth-order asymptotically linear elliptic problem. Acta Math. Sin.. 22, 1121–1126 (2006). Publisher Full Text
Perera, K: Critical groups of critical points produced by local linking with applications. Abstr. Appl. Anal.. 3, 437–446 (1998). Publisher Full Text
Palais, RS: Homotopy theory of infinite dimensional manifolds. Topology. 5, 1–16 (1966). Publisher Full Text