Periodic Problem with a Potential Landesman Lazer Condition

Boundary Value Problems20102010:586971

DOI: 10.1155/2010/586971

Received: 6 January 2010

Accepted: 22 September 2010

Published: 26 September 2010

Abstract

We prove the existence of a solution to the periodic nonlinear second-order ordinary differential equation with damping http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq3_HTML.gif . We suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq4_HTML.gif , the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq5_HTML.gif satisfies the potential Landesman Lazer condition and prove that a critical point of a corresponding energy functional is a solution to this problem.

1. Introduction

Let us consider the nonlinear problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq6_HTML.gif , the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq7_HTML.gif is a Caratheodory function and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq8_HTML.gif .

To state an existence result to (1.1) Amster [1] assumes that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq9_HTML.gif is a nondecreasing function (see also [2]). He supposes that the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq10_HTML.gif satisfies the growth condition http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq12_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq15_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq16_HTML.gif is the first eigenvalue of the problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq18_HTML.gif and there exist http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq19_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq20_HTML.gif An interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq21_HTML.gif is centered in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq22_HTML.gif with the radius http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq23_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq24_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq26_HTML.gif is a solution to the problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq27_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq28_HTML.gif .

In [3, 4] authors studied (1.1) with a constant friction term http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq29_HTML.gif and results with repulsive singularities were obtained in [5, 6].

In this paper we present new assumptions, we suppose that the friction term http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq30_HTML.gif has zero mean value
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ2_HTML.gif
(1.2)
the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq31_HTML.gif is bounded by a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq32_HTML.gif function and satisfies the following potential Landesman-Lazer condition (see also [7, 8])
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ3_HTML.gif
(1.3)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq33_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq34_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq35_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq37_HTML.gif

To obtain our result we use variational approach even if the linearization of the periodic problem (1.1) is a non-self-adjoint operator.

2. Preliminaries

Notation 2.

We will use the classical space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq38_HTML.gif of functions whose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq39_HTML.gif th derivative is continuous and the space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq40_HTML.gif of measurable real-valued functions whose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq41_HTML.gif th power of the absolute value is Lebesgue integrable. We denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq42_HTML.gif the Sobolev space of absolutely continuous functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq43_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq45_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq46_HTML.gif . By a solution to (1.1) we mean a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq47_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq48_HTML.gif is absolutely continuous, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq49_HTML.gif satisfies the boundary conditions and (1.1) is satisfied a.e. in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq50_HTML.gif .

We denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq51_HTML.gif and we study (1.1) by using variational methods. We investigate the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq52_HTML.gif , which is defined by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ4_HTML.gif
(2.1)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ5_HTML.gif
(2.2)
We say that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq53_HTML.gif is a critical point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq54_HTML.gif , if
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ6_HTML.gif
(2.3)
We see that every critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq55_HTML.gif of the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq56_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ7_HTML.gif
(2.4)

for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq57_HTML.gif .

Now we prove that any critical point of the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq58_HTML.gif is a solution to (1.1) mentioned above.

Lemma 2.1.

Let the condition (1.2) be satisfied. Then any critical point of the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq59_HTML.gif is a solution to (1.1).

Proof.

Setting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq60_HTML.gif in (2.4) we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ8_HTML.gif
(2.5)
We denote
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ9_HTML.gif
(2.6)
then previous equality (2.5) implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq61_HTML.gif and by parts in (2.4) we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ10_HTML.gif
(2.7)
for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq62_HTML.gif Hence there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq63_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ11_HTML.gif
(2.8)
on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq64_HTML.gif . The condition (1.2) implies http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq65_HTML.gif and from (2.8) we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq66_HTML.gif Using http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq67_HTML.gif and differentiating equality (2.8) with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq68_HTML.gif we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ12_HTML.gif
(2.9)

Thus http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq69_HTML.gif is a solution to (1.1).

We say that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq70_HTML.gif satisfies the Palais-Smale condition (PS) if every sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq71_HTML.gif for which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq72_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq73_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq74_HTML.gif (as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq75_HTML.gif possesses a convergent subsequence.

To prove the existence of a critical point of the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq76_HTML.gif we use the Saddle Point Theorem which is proved in Rabinowitz [9] (see also [10]).

Theorem 2.2 (Saddle Point Theorem).

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq77_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq78_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq79_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq80_HTML.gif be a functional such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq81_HTML.gif and

(a) there exists a bounded neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq82_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq83_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq84_HTML.gif and a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq85_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq86_HTML.gif ,

(b) there is a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq87_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq88_HTML.gif ,

(c) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq89_HTML.gif satisfies the Palais-Smale condition (PS).

Then, the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq90_HTML.gif has a critical point in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq91_HTML.gif .

3. Main Result

We define
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ13_HTML.gif
(3.1)
Assume that the following potential Landesman-Lazer type condition holds:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ14_HTML.gif
(3.2)
We also suppose that there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq92_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ15_HTML.gif
(3.3)

Theorem 3.1.

Under the assumptions (1.2), (3.2), (3.3), problem (1.1) has at least one solution.

Proof.

We verify that the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq93_HTML.gif satisfies assumptions of the Saddle Point Theorem 2.2 on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq94_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq95_HTML.gif has a critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq96_HTML.gif and due to Lemma 2.1   http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq97_HTML.gif is the solution to (1.1).

It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq98_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq99_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq100_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq101_HTML.gif .

In order to check assumption (a), we prove
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ16_HTML.gif
(3.4)
by contradiction. Then, assume on the contrary there is a sequence of numbers http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq102_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq103_HTML.gif and a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq104_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ17_HTML.gif
(3.5)
From the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq105_HTML.gif and from (3.5) it follows
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ18_HTML.gif
(3.6)
We note that from (3.2) it follows there exist constants http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq106_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq107_HTML.gif and functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq108_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq109_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq110_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq111_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq112_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq113_HTML.gif , respectively. We suppose that for this moment http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq114_HTML.gif . Using (3.6) and Fatou's lemma we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ19_HTML.gif
(3.7)

a contradiction to (3.2). We proceed for the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq115_HTML.gif Then assumption (a) of Theorem 2.2 is verified.

(b) Now we prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq116_HTML.gif is bounded from below on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq117_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq118_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ20_HTML.gif
(3.8)
and assumption (3.3) implies
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ21_HTML.gif
(3.9)
Hence and due to compact imbedding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq119_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq120_HTML.gif we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ22_HTML.gif
(3.10)

Since the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq121_HTML.gif is strictly positive equality (3.10) implies that the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq122_HTML.gif is bounded from below.

Using (3.4), (3.10) we see that there exists a bounded neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq123_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq124_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq125_HTML.gif , a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq126_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq127_HTML.gif , and there is a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq128_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq129_HTML.gif .

In order to check assumption (c), we show that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq130_HTML.gif satisfies the Palais-Smale condition. First, we suppose that the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq131_HTML.gif is unbounded and there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq132_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ23_HTML.gif
(3.11)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ24_HTML.gif
(3.12)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq133_HTML.gif be an arbitrary sequence bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq134_HTML.gif . It follows from (3.12) and the Schwarz inequality that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ25_HTML.gif
(3.13)
From (3.3) we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ26_HTML.gif
(3.14)
Put http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq135_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq136_HTML.gif then (3.13), (3.14) imply
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ27_HTML.gif
(3.15)
Due to compact imbedding http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq137_HTML.gif and (3.15) we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq138_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq139_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq140_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq141_HTML.gif and set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq142_HTML.gif in (3.13), we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ28_HTML.gif
(3.16)
Because the nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq143_HTML.gif is bounded (assumption (3.3)) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq144_HTML.gif the second integral in previous equality (3.16) converges to zero. Therefore
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ29_HTML.gif
(3.17)
Now we divide (3.11) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq145_HTML.gif . We get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ30_HTML.gif
(3.18)
Equalities (3.17), (3.18) imply
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ31_HTML.gif
(3.19)
Because http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq146_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq147_HTML.gif . Using Fatou's lemma and (3.19) we conclude
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ32_HTML.gif
(3.20)
a contradiction to (3.2). We proceed for the case http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq148_HTML.gif similarly. This implies that the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq149_HTML.gif is bounded. Then there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq150_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq151_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq152_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq153_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq155_HTML.gif (taking a subsequence if it is necessary). It follows from equality (3.13) that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ33_HTML.gif
(3.21)
The strong convergence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq156_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq157_HTML.gif and the assumption (3.3) imply
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ34_HTML.gif
(3.22)
If we set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq158_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq159_HTML.gif in (3.21) and subtract these equalities, then using (3.22) we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ35_HTML.gif
(3.23)

Hence we obtain the strong convergence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq160_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq161_HTML.gif . This shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq162_HTML.gif satisfies the Palais-Smale condition and the proof of Theorem 3.1 is complete.

Declarations

Acknowledgment

This work was supported by Research Plan MSM 4977751301.

Authors’ Affiliations

(1)
Department of Mathematics, University of West Bohemia

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Copyright

© Petr Tomiczek. 2010

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