- Open Access
Multiple solutions for a fourth-order nonlinear elliptic problem which is superlinear at +∞ and linear at −∞
© Pei and Zhang; licensee Springer. 2014
- Received: 14 August 2013
- Accepted: 16 December 2013
- Published: 10 January 2014
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.
- fourth-order elliptic boundary value problems
- multiple solutions
- critical groups
- Morse theory
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;
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.
Let us now state the main result.
Theorem 2.1 Assume conditions (H1)-(H5) hold. If , then problem (1) has at least two nontrivial solutions.
Lemma 2.2 Under the assumptions of Theorem 2.1, then I satisfies the (PS) condition.
for all .
which contradicts inequality (16). Thus and the claim is proved.
for all . This contradicts . □
for all .
which implies that and on a positive measure subset. It contradicts the unique continuation property of the eigenfunction. □
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.
is said to be the q th critical group of I at , where .
For the convenience of our proof, we first recall two interesting results and prove two important propositions.
Proposition 3.1 
Under (H2), if is an isolated critical point of I, then .
Proposition 3.2 
If and for some integer we have and , then either or .
Clearly, is a continuous homotopy and for all . Therefore, is contractible in itself.
for some .
Moreover, the implicit function theorem implies that .
where (V being defined in Lemma 2.3).
for all and .
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 . □
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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