Multiple Nodal Solutions for Some Fourth-Order Boundary Value Problems via Admissible Invariant Sets

Abstract

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.

1. Introduction

In this paper, we consider the existence of nodal solutions to the semilinear fourth-order equation:

(1.1)

(1.2)

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 [20], 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 [20]).

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 [20].

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.

2. Preliminaries

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 [21]).

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.

Consider the pseudogradient flow on associated with the vector field ,

(2.1)

We see that is odd in , if is odd in . Since for and the Brezis-Martin theorem [22] implies that for .

Definition 2.2 (see [21, 23]).

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 [21]).

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 [24]).

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 inand at leastpairs of critical points in

(ii)ifhas at least one pair of critical points inand at leastpairs of critical points in

Lemma 2.5 (see [21]).

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 [21]).

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.

Lemma 2.7 (see [21, 25]).

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.

Definition 2.8.

An ordered real Banach space is a pair , where is a real Banach space and a closed convex subset of such that and . The partial order on is given by the cone . For , we write

(2.2)

If has nonempty interior, then it is called a solid cone. If every ordered interval is bounded, then is called a normal cone. An operator is called order preserving (in the literature sometimes increasing) if

(2.3)

strictly order preserving if

(2.4)

and strongly order preserving if

(2.5)

3. Main Results

In this section, we will employ the abstract results in Section 2 to establish some existence theorems on sign-changing solutions of (BVP) (1.2). Firstly, we give some lemmas to change (BVP) (1.2) to a variational problem. Let be the usual real Banach space with the norm for all . We can easily verify that

(3.1)

is also a Banach space with respect to . Let

(3.2)

then is a normal solid cone in and

(3.3)

By , we denote the usual real Hilbert space with the inner product for all

It is well known that the solution of (BVP) (1.2) in is equivalent to the solution of the following integral equation in :

(3.4)

where is the Green's function of the linear boundary value problem for all subject to that is,

(3.5)

Define operators by

(3.6)

Since , (3.4) is equivalent to the following operator equation in :

(3.7)

Remark 3.1.

It is easy to see that

• (i) is nonnegative continuous;

• (ii) ;

• (iii) is bounded and continuous.

Lemma 3.2 (see [20]).

is a linear completely continuous operator and also a linear completely continuous operator from In addition, is strongly order-preserving.

From the definition of , we can obtain that for all with . Therefore, for all with . It is well known that all eigenvalues of are

(3.8)

which have the corresponding orthonormal eigenfunctions

(3.9)

and .

Lemma 3.3 (see [10]).

The operator equation

(3.10)

has a solution in if and only if the operator equation

(3.11)

has a solution in .

The uniqueness of the solution for these two above equations is also equivalent.

Remark 3.4.

From the proof of Lemma 3.3 [10], 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 [10]).

Let , . Then,

(i)is Fréchet differentiable onandfor all

(ii)is Fréchet differentiable onandfor all

Choose and to be our Hilbert space and Banach space, respectively. Define a functional :

(3.12)

Then, according to Lemma 3.5, we have

(3.13)

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 .

Lemma 3.6.

Under , is satisfied, and is strongly order-preserving.

Proof.

The proof is similar to [20], and we omit it here.

Lemma 3.7.

Under , is an admissible invariant set for .

Proof.

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 [23], . 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 [15]).

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 .

Proof.

It is easy to see that and holds. is an admissible invariant set for , and for . Also, is even, . By , there exist , such that for all , then

(3.14)

So is coercive, bounded from below, and satisfies (PS) condition.

Take ; from , there exist , such that , , choose , then , and

(3.15)

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 .

Proof.

Take and such that for , . Now let be a (PS) sequence for . Writing with , , and taking inner product of and , we see that

(3.16)

So is bounded, where . Then, satisfies the (PS) condition.

If , let , and , then , and .

From , we know that there exist and such that

(3.17)

Then, for , , we can obtain, when ,

(3.18)

So, choose , then .

From , we can get there exist , such that

(3.19)

Then, when , we have

(3.20)

Choose large enough such that , and , result follows from Lemma 2.6.

If , let , , then , . From (3.17), when ,

(3.21)

When , we know from (3.19),

(3.22)

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.

Proof.

From condition by the standard argument, satisfies condition for every . Let . From , we obtain for all . Define , it is very clear and , so and . So if ,

(3.23)

Choosing , we obtain, if and ,

(3.24)

Let . From , after integrating, we obtain the existence of such that for . Hence, we have for and is constant. Therefore, when ,

(3.25)

Noting , choose large enough, such that , and

(3.26)

Result follows from Lemma 2.7.

Remark 3.12.

If there exist no strict supsolution and supersolution required in [20], 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.