Sign-Changing Solutions for a Fourth-Order Elliptic Equation with Hardy Singular Terms
The existence and multiplicity of sign-changing solutions for a class of fourth elliptic equations with Hardy singular terms are established by using the minimax methods.
Consider the following Navier boundary value problem: where is a bounded smooth domain in , .
The conditions imposed on are as follows: there exists such that where ; for all , ;, uniformly for , where and are constants; uniformly for , where ; there exist and such that is odd in .
In recent years, 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 (4) admits at least two solutions by a variation of linking if is sublinear. And in , the authors proved that the problem (4) has at least three solutions by a variational reduction method and a degree argument. In , Zhang and Li showed that (4) admits at least two nontrivial solutions by Morse theory and local linking if is superlinear and subcritical on .
To the authors’ knowledge, there seem few results about the sign-changing solutions on problem (1) with hardy singular terms. In this paper, motivated by [5–8], the existence and multiplicity of sign-changing solutions for problem (1) are obtained by introducing a compact embedding theorem and a maximum principle. Our results are new.
2. Preliminaries and Auxiliary Lemmas
We introduce the new working space which is obtained by the completion of with respect to the norm (see ) associated with the inner product
Throughout this paper, we denoted by the norm.
At first, we here give two important lemmas.
Lemma 1. (see ).
Lemma 2 (see [6, Corollary 4.1]). Assume , , and . Let us suppose that the operator is coercive on . Let such that . Let be a solution of Then in .
Now, we consider the following eigenvalue problem:
The first eigenvalue of this problem is given by
By Lemma 1, for . The minimizing sequence is compact in . By standard argument, we may assume that the first eigenfunction is positive in (see [9, page 167]). The second eigenvalue is given by which possesses a sign-changing eigenfunction . Similarly, we can characterize the th eigenvalue with a sign-changing eigenfunction. By standard elliptic theory, as .
It follows from that the functional is of on the space . Under the condition , the critical points of are solutions of problem (1).
If in the above condition is an eigenvalue of , then the problem (1) is called resonance at infinity. Otherwise, we call it nonresonance.
For looking for sign-changing solutions of problem (1), we recall a very useful result.
Proposition 3 (see [10, Theorem 3.2]). Let be a Hilbert space and be a functional defined on . Assume that satisfies the (PS) condition on and has the expression for . Assume that and are open convex subset of with the properties that , , and . If there exists a path such that then f has at least four critical points, one in , one in , one in , and one in .
Remark 4. If satisfies the condition, then this proposition still holds (see ).
3. Main Results
Let us now state the main results.
Theorem 5. Assume conditions and hold. If and for some , then problem (1) has a positive solution, a negative solution, and a sign-changing solution.
Theorem 7. Assume conditions – hold. If and for some , then problem (1) has a positive solution, a negative solution, and a sign-changing solution.
Remark 8. When , the case is called resonance and not considered by . This result is completely new.
Theorem 9. Assume conditions , , and hold. If , then problem (1) has infinitely many sign-changing solutions.
Lemma 10. Under the assumptions of Theorem 5, if , then satisfies the (PS) condition.
Proof. Let be a sequence such that , , and . Since
for all . If is bounded, we can take . By , there exists a constant such that , a.e. . So is bounded in . If , as , set , then . Taking in (14), it follows that is bounded. Without loss of generality, we assume in , and then in . Hence, a.e. in and (). Dividing both sides of (14) by , for all , we get
Then for a.e. , we have as . In fact, if , by , we have
If , we have
Since , by (15) and the Lebesgue dominated convergence theorem, we arrive at
It is easy to see that . In fact, if , then contradicts . Hence, is an eigenvalue of . This contradicts our assumption. Thus is bounded. By standard argument (see the proof of our Lemma 12 below), in . The lemma is proved.
Proof. Suppose satisfies
In view of , it suffices to prove that is bounded in . Similar to the proof of Lemma 10, we have
Therefore is an eigenfunction of , and then for a.e. . It follows from that holds uniformly in , which implies that
On the other hand, (19) implies that
Thus which contradicts (22). Hence is bounded.
Lemma 12. Assume and hold. Then satisfies the (PS) condition.
Proof. Assume that is a (PS) sequence; and is bounded. A routine argument implies that is bounded. By [6, Theorem A.2], we have
where . For given in , , and we may choose such that , . By the Sobolev embedding theorem, we have
We infer from (26) that is compact in . By ,
This completes the proof of this lemma.
For the aim of using Proposition 3 that proves our main results, we prove an important lemma below.
From previous Section 1, we know that is functional and its gradient at is given by
Then for all . We consider the convex cones and ; moreover, for , assume
Note that and are open convex subsets of and contains only sign-changing functions.
Lemma 13. Assume and hold. Then, there exists such that for there holds
Moreover, if is a nontrivial solution of problem (1), then is positive (negative) in the sense that in .
Proof. Indeed, if and , , then
For every , there exists such that
Choose such that . Using (32), the Hölder inequality, the Poincaré inequality, and the Sobolev embedding theorem, we have where are constants. Hence where . Take such that . Now if , then we have
Thus for every , by (35) we have thus . Hence . In a similar way, . If , and (resp.,—) is a nontrivial solution of problem (1), then . By (35) we have ; that is, (resp., ). By Lemma 2, we imply that in .
Lemma 14. Assume , , and hold. Then, there exists such that for there holds
Proof. The proof is quite similar to that of Lemma 4.2 in . We omit it here.
Lemma 15. Assume holds. Then where the definition of introduced in our proof of Theorem 9.
Proof. Because , then by , as . This lemma follows immediately.
Lemma 16. Assume and hold. Let , and then there exists such that .
Proof. By the conditions and , we know that, for any , there exists , such that
Using (40) and the Sobolev embedding theorem, we have
By (32) we have for every . So there exists such that
Hence this lemma is proved.
4. Proof of the Main Results
Proof of Theorem 5 and Theorem 7. Motivated by the Proof of Theorem 4.2 in , we still define a path as
Obviously, and . By the Fatou’s lemma, the condition with and a direct computation shows that
So, it yields that there exists such that . Hence we obtain By using Lemmas 10, 11, and 13, Proposition 3, and Lemma 16, we can find a critical point in which is a positive solution, a critical point in which is a negative solution, and a critical point in which is a sign-changing solution.
Before beginning our proof of Theorem 9, we need the following important proposition.
Proposition 17 (see [9, Theorem 5.6]). Assume is a Hilbert space with inner product , and the corresponding norm , and , , where . denotes a positive closed convex cone of . Assume that , where , , and .Assume that, for any , there is a constant such thatwhere denotes another norm of such that for all .Assume that . If the even functional satisfies condition at level for each , then for all small, where ( are two subspaces of with , , , and , where , and are fixed constants.
Now, we give an outline proof for our Theorem 9.
Proof of Theorem 9. Let denote the eigenspace of . We fix and let . Consider another norm of , . Write .
Let where are fixed constants. By our assumptions, we may find a constant such that where . For any , there is a constant such that By Lemma 15, where . Then and are satisfied. By Lemma 14, the condition holds.
Now, we define Let By Lemma 12, satisfies the (PS) condition. Thus, by Proposition 17 and the Proof of Theorem 5.7 in , we know that the functional posses a sequence sign-changing solution .
Conflict of Interests
The authors declare that they have no competing interests.
The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant no. 10671156) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant no. 1301038C).