- Research Article
- Open Access
Multiple Nodal Solutions for Some Fourth-Order Boundary Value Problems via Admissible Invariant Sets
© Yang Yang et al. 2008
- Received: 6 May 2008
- Accepted: 16 September 2008
- Published: 9 October 2008
Existence and multiplicity results for nodal solutions are obtained for the fourth-order boundary value problem (BVP) , , , where is continuous. The critical point theory and admissible invariant sets are employed to discuss this problem.
- Banach Space
- Real Banach Space
- Order Preserve
- Critical Point Theory
- Solid Cone
where is continuous.
Owning to the importance of higher-order differential equations in physics, the existence and multiplicity of the solutions to such problems have been studied by many authors. They obtained the existence of solutions by the cone expansion or compression fixed point theorem [1–6]; sub-sup solution method [7–9]; critical point theory [10–13]; Morse theory [14, 15]; and eta [16, 17]. There are also papers which study nodal solutions for elliptic equations [18, 19]. In particular, in , Han and Li obtained multiple positive, negative, and sign-changing solutions by combining the critical point theory and the method of sub-sup solutions for the (BVP) (1.2). The main result is as follows:
there exist a strict subsolution and a strict supersolution of (BVP) (1.2) with , , and ;
is strictly increasing in ;
is locally Lipschitz continuous in ;
there exist and such that for all and
Theorem 1.1 (see ).
Assume that hold. Then, (BVP) (1.2) has at least four solutions.
Motivated by their ideas, we cannot help wondering if there are no strict subsolution and supersolution of (BVP) (1.2), can we still get the nodal solutions just by critical point theory? In this paper, we will use the admissible invariant sets and critical point theory to settle this problem. But we should point out that in all theorems of our paper, the nonlinearity is assumed to be odd in , while no such symmetry is required in .
The paper is organized as follows: in Section 2, we give some preliminaries, including the critical point theorems which will be used in our main results and some concepts concerning the partially ordered Banach space. The main results and proofs are established in Section 3.
Let be a Hilbert space and a Banach space densely embedded in . Assume that has a closed convex cone and that has interior points in , that is, with the interior and the boundary of in .
Let and for . We use the following notation: , , , for . Let and denote the norms in and , respectively.
Lemma 2.1 (see ).
Assume is a Hilbert space, and is a closed convex set of , , and . Then, there exists a pseudogradient vector field for , and . Furthermore, if is even, , then is odd.
We see that is odd in , if is odd in . Since for and the Brezis-Martin theorem  implies that for .
With the flow , a subset is called an invariant set if for .
Let us assume that
(F) , for , is continuous.
Under condition , we have for and is continuous in .
Definition 2.3 (see ).
Let be an invariant set under . is said to be an admissible invariant set for if (a) is the closure of an open set in , that is, ; (b) if for some and in as for some , then in ; (c) if such that in , then in ; (d) for any , for .
Lemma 2.4 (see ).
Let and hold. Assume is even, bounded from below, and satisfies (PS) condition. Assume that the positive cone is an admissible invariant set for and for all . Suppose there is a linear subspace with , such that for some , where Then, has at least pairs of critical points with negative critical values. More precisely,
(i)if , has at least one pair of critical points in and at least pairs of critical points in
(ii)if has at least one pair of critical points in and at least pairs of critical points in
Lemma 2.5 (see ).
Let and hold. Assume is even, , and satisfies (PS) condition. Assume that the positive cone is an admissible invariant set for and for all . Suppose there exist linear subspaces and with , ( , resp.), , such that for some , and . Then, has at least ( , resp.) pairs of critical points in with negative critical values.
Lemma 2.6 (see ).
Let and hold. Assume is even, and satisfies (PS) condition. Assume that the positive cone is an admissible invariant set for and for all . Suppose there exist linear subspaces and with , , , such that for some , and . Then for ( , resp.), has at least ( , resp.) pairs of critical points in with positive critical values.
Assume is even, , satisfies and condition for . Assume that is an admissible invariant set for , for all . , where are finite-dimensional subspaces of , and for each , let and Assume for each there exist such that , where , as . Then, has a sequence of critical points such that as , provided for large .
Next, we need some basic concepts of ordered Banach spaces.
By , we denote the usual real Hilbert space with the inner product for all
is nonnegative continuous;
is bounded and continuous.
Lemma 3.2 (see ).
is a linear completely continuous operator and also a linear completely continuous operator from In addition, is strongly order-preserving.
Lemma 3.3 (see ).
has a solution in .
The uniqueness of the solution for these two above equations is also equivalent.
From the proof of Lemma 3.3 , it is very clear if is a solution for (3.11), then is a solution for (3.7). Furthermore, if is a solution for (3.11), then is a solution for (3.7) with the same sign, which follows from Lemma 3.2.
Lemma 3.5 (see ).
Let , . Then,
(i) is Fréchet differentiable on and for all
(ii) is Fréchet differentiable on and for all
Hence, Lemma 3.3 implies that the operator equation has a solution in if and only if the functional has a critical point in . Thus, (BVP) (1.2) has been transformed into a variational problem.
We refer the following assumption:
is continuous and increasing in .
Under , is satisfied, and is strongly order-preserving.
The proof is similar to , and we omit it here.
Under , is an admissible invariant set for .
We know that is strongly order-preserving, so does given in Lemma 2.1. The Brezis-Martin theory implies that and are invariant sets under the negative pesudogradient flow of . Requirement (a) is satisfied automatically. For (d), we note that for all , we have , similar to the proof in , . To prove (b), let for some , so , let be a sequence such that in for some , then in . For (c), if , then , if in , for , then and , so in , and the proof is completed.
Lemma 3.8 (see ).
Any bounded sequence such that as has a convergent subsequence.
Next, we make more assumptions:
(f 2 ) uniformly for ;
(f 3 ) , uniformly for and some ;
(f 4 ) is odd in .
Theorem 3.9 (sublinear nonlinearity).
Under , (BVP) (1.2) has at least one pair of one-sign solutions , , and at least pairs of nodal solutions for .
So is coercive, bounded from below, and satisfies (PS) condition.
so for small. Result follows from Lemma 2.4.
Next, we consider an asymptotically linear problem:
(f 5 ) uniformly for ;
(f 6 ) , uniformly for .
Theorem 3.10 (asymptotically linear case).
Under , , , and , (BVP) (1.2) has at least pairs of nodal solutions provided or . Here, , if ; and , if .
So is bounded, where . Then, satisfies the (PS) condition.
If , let , and , then , and .
So, choose , then .
Choose large enough such that , and , result follows from Lemma 2.6.
which means , then result follows from Lemma 2.5.
Next, we consider a superlinear problem. Assume that
there is such that for large;
there are , such that for large.
Theorem 3.11 (superlinear nonlinearity).
Under , , , and , (BVP) (1.2) has infinitely many nodal solutions.
Result follows from Lemma 2.7.
If there exist no strict supsolution and supersolution required in , just only using the functional to get the critical point [10, 11], then we just know that (BVP) (1.2) has solutions, even we can know the sign of the critical point of the functional because is not strongly order-preserving in . In our paper, using admissible invariant sets in , we can settle the problem.
The authors are grateful to the referees for their useful suggestions which have improved the writing of the paper. Jihui Zhang thanks Z. Zhang and the members of AMSS very much for their hospitality and invitation to visit the Academy of Mathematics and Systems Sciences (AMSS), Academia Sinica, in January 2008. The authors also would like to thank Professor D. Cao, Professor S. Li, Professor Y. Ding, and Professor H. Yin for their help and many valuable discussions. This research was supported by the NNSF of China (Grant no.10871096), Foundation of Major Project of Science and Technology of Chinese Education Ministry, SRFDP of Higher Education, and NSF of Education Committee of Jiangsu Province. Zhitao Zhang was supported by NNSF of China (Grant no.10671195).
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