Infinitely Many Solutions of Strongly Indefinite Semilinear Elliptic Systems

Abstract

We proved a multiplicity result for strongly indefinite semilinear elliptic systems in , in where and are positive numbers which are in the range we shall specify later.

1. Introduction

In this paper, we shall study the existence of multiple solutions of the semilinear elliptic systems

(11)

where and are positive numbers which are in the range we shall specify later. Let us consider that the exponents , are below the critical hyperbola

(12)

so one of and could be larger than ; for that matter, the quadratic part of the energy functional

(13)

has to be redefined, and we then need fractional Sobolev spaces.

Hence the energy functional is strongly indefinite, and we shall use the generalized critical point theorem of Benci [1] in a version due to Heinz [2] to find critical points of . And there is a lack of compactness due to the fact that we are working in .

In [3], Yang shows that under some assumptions on the functions and there exist infinitely many solutions of the semilinear elliptic systems

(14)

We shall propose herein a result similar to [3] for problem (1.1).

2. Abstract Framework and Fractional Sobolev Spaces

We recall some abstract results developed in [4] or [5].

We shall work with space , which are obtained as the domains of fractional powers of the operator

(21)

Namely, for , and the corresponding operator is denoted by . The spaces , the usual fractional Sobolev space , are Hilbert spaces with inner product

(22)

and associates norm

(23)

It is known that is an isomorphism, and so we denote by the inverse of .

Now let , with . We define the Hilbert space and the bilinear form by the formula

(24)

Using the Cauchy-Schwarz inequality, then it is easy to see that is continuous and symmetric. Hence induces a self-adjoint bounded linear operator such that

(25)

Here and in what follows denotes the inner product in induced by and on the product space in the usual way. It is easy to see that

(26)

We can then prove that has two eigenvalues and , whose corresponding eigenspaces are

(27)

which give a natural splitting . The spaces and are orthogonal with respect to the bilinear form , that is,

(28)

We can also define the quadratic form associated to and as

(29)

for all . It follows then that

(210)

where , , . If , that is, , we have

(211)

Similarly

(212)

for .

If where is a number satisfying the condition

(213)

and , it follows by (2.13) that and by Hlder inequalities that

(214)

In the sequel denotes the norm in , and we denote by the weighted function spaces with the norm defined on by . According to the properties of interpolation space, we have the following embedding theorem.

Theorem 2.1.

Let . one defines the operator as follows: for , ,

(215)

Then the inclusion of into is compact if .

Proof.

Observe that, by Hlder's inequality and (2.14), we have

(216)

where ; hence is well defined.

Then we will claim that is compact. Since , for any , there exists , such that . Now, suppose weakly in . We estimate

(217)

letting

(218)

we have

(219)

so that by Hlder's inequality, we observe that, for any , we can choose a so that the integral over () is smaller than for all , while for this fixed , by strong convergence of to in on any bounded region, the integral over () is smaller than for large enough. We thus have proved that strongly in ; that is, the inclusion of into is compact if .

3. Main Theorem

We consider below the problem of finding multiple solutions of the semilinear elliptic systems

(31)

Now if we choose , , , such that

(32)

and we assume that

(H) , and and are positive numbers such that

(33)

We set

(34)

and we let

(35)

so that, under assumption (H), Theorem 2.1 holds, respectively, with and , and and ; that is, the inclusion of into and the inclusion of into are compact.

If , we let

(36)

denote the energy of . It is well known that under assumption (H) the energy functional is well defined and continuously differentiable on , and for all we have

(37)

(38)

and it is also well known that the critical points of are weak solutions of problem (3.1). The main theorem is the following.

Theorem 3.1.

Under assumption (H), problem (3.1) possesses infinitely many solutions .

Since the functional are strongly indefinite, a modified multiplicity critical points theorem Heinz [2] which is the generalized critical point theorem of Benci [1] will be used. For completeness, we state the result from here.

Theorem 3.2.

(see [2]) Let be a real Hilbert space, and let be a functional with the following properties:

(i) has the form

(39)

where is an invertible bounded self-adjoint linear operator in and where is such that and the gradient is a compact operator;

(ii) is even, that is ;

(iii) satisfies the Palais-Smale condition. Furthermore, let

(310)

be an orthogonal splitting into -invariant subspaces , such that . Then,

• (a) suppose that there is an -dimensional linear subspace of () such that for the spaces , one has

(iv) such that , ;

(v) such that .Then there exist at least pairs of critical points of such that ();

• (b) a similar result holds when , and one takes , .

It is known from Section 2 that the operator induced by the bilinear form is an invertible bounded self-adjoint linear operator satisfying . We shall need some finite dimensional subspace of . Let , , ,, be a complete orthogonal system in . Let denote the finite dimensional subspaces of generated by , , ,,. Since and are isomorphisms, we know that , , ,, is a complete orthogonal system in . Let denote the finite dimensional subspaces of generated by , , ,,. For each , we introduce the following subspaces of and

(311)

Lemma 3.3.

The functional defined in (3.6) satisfies conditions (ii), (iv), and (v) of Theorem 3.2.

Proof.

Condition (ii) is an immediate consequence of the definition of . For condition (iv), by (2.11) and Theorem 2.1, for ,

(312)

and since , , we conclude that for with small.

Next, let us prove condition (v). Let be fixed, let , and write and . We have

(313)

Let and . Then we have and . Furthermore, we may write , where is orthogonal to in . We also have , where is orthogonal to in . It is easy to see that either or is positive. Suppose . Then we have

(314)

Using the fact that the norms in are equivalent we obtain

(315)

with constant independent of . So from (3.13) and (2.11) we obtain

(316)

The same arguments can be applied if . So the result follows from (3.16).

A sequence is said to be the Palais-Smale sequence for (PS)-sequence for short) if uniformly in and in . We say that satisfies the Palais-Smale condition (PS)-condition for short) if every (PS)-sequence of is relatively compact in .

Lemma 3.4.

Under assumption (H), the functional satisfies the (PS)-condition.

Proof.

We first prove the boundedness of (PS)-sequences of . Let be a (PS)-sequence of such that

(317)

(318)

Taking in (3.18), it follows from (3.17), (3.18), that

(319)

Next, we estimate and . From (3.18) with , we have

(320)

for all . Using Hlder's inequality and by (3.20), we obtain

(321)

for all , which implies that

(322)

Similarly, we prove that

(323)

Adding (3.22) and (3.23) we conclude that

(324)

Using this estimate in (3.19), we get

(325)

Since and , we conclude that both and are bounded, and consequently and are also bounded in terms of (3.24).

Finally, we show that contains a strongly convergent subsequence. It follows from and which are bounded and Theorem 2.1 that contains a subsequence, denoted again by , such that

(326)

It follows from (3.18) that

(327)

Therefore,

(328)

(329)

and by Theorem 2.1, we conclude that strongly in and strongly in .

Proof of Theorem 3.1.

Applying Lemmas 3.3 and 3.4 and Theorem 3.2, we can obtain the conclusion of Theorem 3.1.