Open Access

Infinitely Many Solutions of Strongly Indefinite Semilinear Elliptic Systems

Boundary Value Problems20092009:865408

DOI: 10.1155/2009/865408

Received: 16 December 2008

Accepted: 6 July 2009

Published: 17 August 2009

Abstract

We proved a multiplicity result for strongly indefinite semilinear elliptic systems https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq1_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq3_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq4_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq6_HTML.gif 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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ1_HTML.gif
(11)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq8_HTML.gif are positive numbers which are in the range we shall specify later. Let us consider that the exponents https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq10_HTML.gif are below the critical hyperbola
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ2_HTML.gif
(12)
so one of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq12_HTML.gif could be larger than https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq13_HTML.gif ; for that matter, the quadratic part of the energy functional
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ3_HTML.gif
(13)

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

Hence the energy functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq14_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq15_HTML.gif . And there is a lack of compactness due to the fact that we are working in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq16_HTML.gif .

In [3], Yang shows that under some assumptions on the functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq17_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq18_HTML.gif there exist infinitely many solutions of the semilinear elliptic systems
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ4_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq19_HTML.gif , which are obtained as the domains of fractional powers of the operator
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ5_HTML.gif
(21)
Namely, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq20_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq21_HTML.gif , and the corresponding operator is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq22_HTML.gif . The spaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq23_HTML.gif , the usual fractional Sobolev space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq24_HTML.gif , are Hilbert spaces with inner product
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ6_HTML.gif
(22)
and associates norm
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ7_HTML.gif
(23)

It is known that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq25_HTML.gif is an isomorphism, and so we denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq26_HTML.gif the inverse of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq27_HTML.gif .

Now let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq29_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq30_HTML.gif . We define the Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq31_HTML.gif and the bilinear form https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq32_HTML.gif by the formula
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ8_HTML.gif
(24)
Using the Cauchy-Schwarz inequality, then it is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq33_HTML.gif is continuous and symmetric. Hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq34_HTML.gif induces a self-adjoint bounded linear operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq35_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ9_HTML.gif
(25)
Here and in what follows https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq36_HTML.gif denotes the inner product in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq37_HTML.gif induced by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq39_HTML.gif on the product space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq40_HTML.gif in the usual way. It is easy to see that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ10_HTML.gif
(26)

We can then prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq41_HTML.gif has two eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq43_HTML.gif , whose corresponding eigenspaces are

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ11_HTML.gif
(27)
which give a natural splitting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq44_HTML.gif . The spaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq45_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq46_HTML.gif are orthogonal with respect to the bilinear form https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq47_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ12_HTML.gif
(28)
We can also define the quadratic form https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq48_HTML.gif associated to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq50_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ13_HTML.gif
(29)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq51_HTML.gif . It follows then that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ14_HTML.gif
(210)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq52_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq53_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq54_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq55_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq56_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ15_HTML.gif
(211)
Similarly
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ16_HTML.gif
(212)

for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq57_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq58_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq59_HTML.gif is a number satisfying the condition
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ17_HTML.gif
(213)
and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq60_HTML.gif , it follows by (2.13) that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq61_HTML.gif and by H https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq62_HTML.gif lder inequalities that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ18_HTML.gif
(214)

In the sequel https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq63_HTML.gif denotes the norm in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq64_HTML.gif , and we denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq65_HTML.gif the weighted function spaces with the norm defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq66_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq67_HTML.gif . According to the properties of interpolation space, we have the following embedding theorem.

Theorem 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq68_HTML.gif . one defines the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq69_HTML.gif as follows: for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq70_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq71_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ19_HTML.gif
(215)

Then the inclusion of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq72_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq73_HTML.gif is compact if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq74_HTML.gif .

Proof.

Observe that, by H https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq75_HTML.gif lder's inequality and (2.14), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ20_HTML.gif
(216)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq76_HTML.gif ; hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq77_HTML.gif is well defined.

Then we will claim that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq78_HTML.gif is compact. Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq79_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq80_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq81_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq82_HTML.gif . Now, suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq83_HTML.gif weakly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq84_HTML.gif . We estimate
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ21_HTML.gif
(217)
letting
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ22_HTML.gif
(218)
we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ23_HTML.gif
(219)

so that by H https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq85_HTML.gif lder's inequality, we observe that, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq86_HTML.gif , we can choose a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq87_HTML.gif so that the integral over ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq88_HTML.gif ) is smaller than https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq89_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq90_HTML.gif , while for this fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq91_HTML.gif , by strong convergence of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq92_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq93_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq94_HTML.gif on any bounded region, the integral over ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq95_HTML.gif ) is smaller than https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq96_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq97_HTML.gif large enough. We thus have proved that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq98_HTML.gif strongly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq99_HTML.gif ; that is, the inclusion of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq100_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq101_HTML.gif is compact if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq102_HTML.gif .

3. Main Theorem

We consider below the problem of finding multiple solutions of the semilinear elliptic systems
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ24_HTML.gif
(31)
Now if we choose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq103_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq104_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq105_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ25_HTML.gif
(32)

and we assume that

(H) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq106_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq108_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq109_HTML.gif are positive numbers such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ26_HTML.gif
(33)
We set
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ27_HTML.gif
(34)
and we let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ28_HTML.gif
(35)

so that, under assumption (H), Theorem 2.1 holds, respectively, with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq110_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq111_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq112_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq113_HTML.gif ; that is, the inclusion of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq114_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq115_HTML.gif and the inclusion of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq116_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq117_HTML.gif are compact.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq118_HTML.gif , we let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ29_HTML.gif
(36)

denote the energy of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq119_HTML.gif . It is well known that under assumption (H) the energy functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq120_HTML.gif is well defined and continuously differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq121_HTML.gif , and for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq122_HTML.gif we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ30_HTML.gif
(37)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ31_HTML.gif
(38)

and it is also well known that the critical points of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq123_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq124_HTML.gif .

Since the functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq125_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq126_HTML.gif be a real Hilbert space, and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq127_HTML.gif be a functional with the following properties:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq128_HTML.gif has the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ32_HTML.gif
(39)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq129_HTML.gif is an invertible bounded self-adjoint linear operator in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq130_HTML.gif and where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq131_HTML.gif is such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq132_HTML.gif and the gradient https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq133_HTML.gif is a compact operator;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq134_HTML.gif is even, that is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq135_HTML.gif ;

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq136_HTML.gif satisfies the Palais-Smale condition. Furthermore, let
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ33_HTML.gif
(310)
be an orthogonal splitting into https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq137_HTML.gif -invariant subspaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq138_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq139_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq140_HTML.gif . Then,
  1. (a)

    suppose that there is an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq141_HTML.gif -dimensional linear subspace https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq142_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq143_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq144_HTML.gif ) such that for the spaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq145_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq146_HTML.gif one has

     

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq147_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq148_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq149_HTML.gif ;

(v) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq150_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq151_HTML.gif .Then there exist at least https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq152_HTML.gif pairs https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq153_HTML.gif of critical points of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq154_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq155_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq156_HTML.gif );
  1. (b)

    a similar result holds when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq157_HTML.gif , and one takes https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq158_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq159_HTML.gif .

     
It is known from Section 2 that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq160_HTML.gif induced by the bilinear form https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq161_HTML.gif is an invertible bounded self-adjoint linear operator satisfying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq162_HTML.gif . We shall need some finite dimensional subspace of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq163_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq164_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq165_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq166_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq167_HTML.gif , be a complete orthogonal system in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq168_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq169_HTML.gif denote the finite dimensional subspaces of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq170_HTML.gif generated by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq171_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq173_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq174_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq175_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq176_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq177_HTML.gif are isomorphisms, we know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq178_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq179_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq180_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq181_HTML.gif , is a complete orthogonal system in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq182_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq183_HTML.gif denote the finite dimensional subspaces of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq184_HTML.gif generated by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq185_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq186_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq187_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq188_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq189_HTML.gif . For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq190_HTML.gif , we introduce the following subspaces of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq191_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq192_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ34_HTML.gif
(311)

Lemma 3.3.

The functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq193_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq194_HTML.gif . For condition (iv), by (2.11) and Theorem 2.1, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq195_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ35_HTML.gif
(312)

and since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq196_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq197_HTML.gif , we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq198_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq199_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq200_HTML.gif small.

Next, let us prove condition (v). Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq201_HTML.gif be fixed, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq202_HTML.gif , and write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq203_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq204_HTML.gif . We have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ36_HTML.gif
(313)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq205_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq206_HTML.gif . Then we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq207_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq208_HTML.gif . Furthermore, we may write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq209_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq210_HTML.gif is orthogonal to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq211_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq212_HTML.gif . We also have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq213_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq214_HTML.gif is orthogonal to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq215_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq216_HTML.gif . It is easy to see that either https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq217_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq218_HTML.gif is positive. Suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq219_HTML.gif . Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ37_HTML.gif
(314)
Using the fact that the norms in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq220_HTML.gif are equivalent we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ38_HTML.gif
(315)
with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq221_HTML.gif independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq222_HTML.gif . So from (3.13) and (2.11) we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ39_HTML.gif
(316)

The same arguments can be applied if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq223_HTML.gif . So the result follows from (3.16).

A sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq224_HTML.gif is said to be the Palais-Smale sequence for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq225_HTML.gif (PS)-sequence for short) if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq226_HTML.gif uniformly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq227_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq228_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq229_HTML.gif . We say that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq230_HTML.gif satisfies the Palais-Smale condition (PS)-condition for short) if every (PS)-sequence of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq231_HTML.gif is relatively compact in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq232_HTML.gif .

Lemma 3.4.

Under assumption (H), the functional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq233_HTML.gif satisfies the (PS)-condition.

Proof.

We first prove the boundedness of (PS)-sequences of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq234_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq235_HTML.gif be a (PS)-sequence of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq236_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ40_HTML.gif
(317)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ41_HTML.gif
(318)
Taking https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq237_HTML.gif in (3.18), it follows from (3.17), (3.18), that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ42_HTML.gif
(319)
Next, we estimate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq238_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq239_HTML.gif . From (3.18) with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq240_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ43_HTML.gif
(320)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq241_HTML.gif . Using H https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq242_HTML.gif lder's inequality and by (3.20), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ44_HTML.gif
(321)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq243_HTML.gif , which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ45_HTML.gif
(322)
Similarly, we prove that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ46_HTML.gif
(323)
Adding (3.22) and (3.23) we conclude that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ47_HTML.gif
(324)
Using this estimate in (3.19), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ48_HTML.gif
(325)

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq244_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq245_HTML.gif , we conclude that both https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq246_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq247_HTML.gif are bounded, and consequently https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq248_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq249_HTML.gif are also bounded in terms of (3.24).

Finally, we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq250_HTML.gif contains a strongly convergent subsequence. It follows from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq251_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq252_HTML.gif which are bounded and Theorem 2.1 that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq253_HTML.gif contains a subsequence, denoted again by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq254_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ49_HTML.gif
(326)
It follows from (3.18) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ50_HTML.gif
(327)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ51_HTML.gif
(328)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_Equ52_HTML.gif
(329)

and by Theorem 2.1, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq255_HTML.gif strongly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq256_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq257_HTML.gif strongly in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F865408/MediaObjects/13661_2008_Article_885_IEq258_HTML.gif .

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.

Authors’ Affiliations

(1)
Department of Applied Science, Naval Academy

References

  1. Benci V: On critical point theory for indefinite functionals in the presence of symmetries. Transactions of the American Mathematical Society 1982, 274(2):533-572. 10.1090/S0002-9947-1982-0675067-XMATHMathSciNetView ArticleGoogle Scholar
  2. Heinz H-P: Existence and gap-bifurcation of multiple solutions to certain nonlinear eigenvalue problems. Nonlinear Analysis: Theory, Methods & Applications 1993, 21(6):457-484. 10.1016/0362-546X(93)90128-FMATHMathSciNetView ArticleGoogle Scholar
  3. Yang J: Multiple solutions of semilinear elliptic systems. Commentationes Mathematicae Universitatis Carolinae 1998, 39(2):257-268.MATHMathSciNetGoogle Scholar
  4. de Figueiredo DG, Felmer PL: On superquadratic elliptic systems. Transactions of the American Mathematical Society 1994, 343(1):99-116. 10.2307/2154523MATHMathSciNetView ArticleGoogle Scholar
  5. de Figueiredo DG, Yang J: Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Analysis: Theory, Methods & Applications 1998, 33(3):211-234. 10.1016/S0362-546X(97)00548-8MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Kuan-Ju Chen. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.