Sign-Changing Solutions to Equations of Elliptic TypeView this Special Issue
Review Article | Open Access
Infinitely Many Solutions for a Robin Boundary Value Problem
By combining the embedding arguments and the variational methods, we obtain infinitely many solutions for a class of superlinear elliptic problems with the Robin boundary value under weaker conditions.
In this paper, we consider the following equation:
where is a bounded domain in with smooth boundary and . Denote
and let be the eigenvalues of with the Robin boundary conditions. We assume that the following hold:(), such that
where , . If , let ;(), uniformly for .()there exist s.t.
(), ., such that
Obviously, can be deduced form (AR). Under (AR), the (PS) sequence can be deduced bounded. However, it is easy to see that the example 
Definition 1.1. Assume that is a Banach space, we say that satisfies Cerami condition (C), if for all :(i)any bounded sequence satisfying , possesses a convergent subsequence;(ii)there exist s.t. for any with , .
In the work in [2, 7], the Fountain theorem was obtained under the condition (PS). Though condition (C) is weaker than (PS), the well-known deformation theorem is still true under condition (C) (see ). There is the following Fountain theorem under condition (C).
Assume , where are finite dimensional subspace of . For each , let
Proposition 1.2. Assume that satisfies condition (C), and . For each , there exist such that(i), ,(ii).Then has a sequence of critical points , such that as .
As a particular linking theorem, Fountain theorem is a version of the symmetric Mountain-Pass theorem. Using the aforementioned theorem, the author in  proved multiple solutions for the problem (1.1) with Neumann boundary value condition; the author in  proved multiple solutions for the problem (1.1) with Dirichlet boundary value condition. In the present paper, we also use the theorem to give infinitely many solutions for problem (1.1). The main results are follows.
Theorem 1.3. Under assumptions ()–(), problem (1.1) has infinitely many solutions.
Remark 1.6. In the work in , they also obtained infinitely many solutions for problem (1.1) with Dirichlet boundary value condition under stronger conditions than the aforementioned and above. Furthermore, function (1.6) does not satisfy all conditions in . Therefore, Theorem 1.3 applied to Dirichlet boundary value problem improves those results in [1, 2, 8, 9].
Let the Sobolev space . Denote
to be the norm of in , and the norm of in . Consider the functional :
Then by , is and
The critical point of is just the weak solution of problem (1.1).
Since we do not assume condition (AR), we have to prove that the functional satisfies condition (C) instead of condition (PS).
Lemma 2.1. Under ()–(), satisfies condition (C).
Proof. For all , we assume that is bounded and
Going, if necessary, to a subsequence, we can assume that in , then
Since the Sobolev imbedding is compact, we have the right-hand side of (2.6) converges to 0. While , we have . It follows that in and , that is, condition (i) of Definition 1.1 holds.
Next, we prove condition (ii) of Definition 1.1, if not, there exist and satisfying, as then we have
Denote , then , that is, is bounded in , thus for some , we get
If , define a sequence as in  If for some , there is a number of satisfying (2.10), we choose one of them. For all , let , it follows by a.e. that Then for large enough, by (2.9), (2.11), and , we have That is, . Since and , then . Thus We see that From the aforementioned, we infer that which contradicts (2.8).
If , by (2.7) That is, Since exists, and by in (the weakly convergent sequence is bounded), we get where is the constant of Sobolev Trace imbedding from , see . We have For , we get . Then by By using Fatou lemma, since the Lebesgue measure , On the other hand, by there exists , such that for . Moreover, Now, there is s.t. Together with (2.19) and (2.21), (2.23), it is a contradiction.
This proves that satisfies condition (C).
3. Proof of Theorem 1.3
then . It shows that by and satisfies condition (C) by Lemma 2.1.(i)After integrating, we obtain from that there exist such that
Notice that and , we infer that
we can deduce that is the equivalent norm of in . Since and all norms are equivalent in the finite-dimensional space, there exists , for all , we get
Next by , there is such that for . Take := , , then for all , we obtain
Therefore, we get that for large enough (),
Remark 3.2. In the next paper, we wish to consider the sign-changing solutions for problem (1.1).
We thank the referee for useful comments. C. Li is supported by NSFC (10601058, 10471098, 10571096). This work was supported by the Chinese National Science Foundation (10726003), the National Science Foundation of Shandong (Q2008A03), and the Foundation of Qufu Normal University.
- P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Washington, DC, USA, 1986.
- M. Willem, Minimax Theorems, vol. 24 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Boston, Mass, USA, 1996.
- S. B. Liu and S. J. Li, “Infinitely many solutions for a superlinear elliptic equation,” Acta Mathematica Sinica. Chinese Series, vol. 46, no. 4, pp. 625–630, 2003 (Chinese).
- L. Jeanjean, “On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 129, no. 4, pp. 787–809, 1999.
- P. Bartolo, V. Benci, and D. Fortunato, “Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 7, no. 9, pp. 981–1012, 1983.
- A. X. Qian, “Existence of infinitely many nodal solutions for a Neumann boundary value problem,” Boundary Value Problem, vol. 3, pp. 329–335, 2005.
- S. Li and Z.-Q. Wang, “Ljusternik-Schnirelman theory in partially ordered Hilbert spaces,” Transactions of the American Mathematical Society, vol. 354, no. 8, pp. 3207–3227, 2002.
- S. Shi and S. Li, “Existence of solutions for a class of semilinear elliptic equations with the Robin boundary value condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3292–3298, 2009.
- W. Zou, “Variant fountain theorems and their applications,” Manuscripta Mathematica, vol. 104, no. 3, pp. 343–358, 2001.
- L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 1998.
Copyright © 2010 Aixia Qian and Chong Li. 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.