Open Access

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq3_HTML.gif . We suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq4_HTML.gif , the nonlinearity https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq6_HTML.gif , the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq7_HTML.gif is a Caratheodory function and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq10_HTML.gif satisfies the growth condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq12_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq15_HTML.gif where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq18_HTML.gif and there exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq19_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq20_HTML.gif An interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq21_HTML.gif is centered in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq22_HTML.gif with the radius https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq23_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq25_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq26_HTML.gif is a solution to the problem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq27_HTML.gif with https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq30_HTML.gif has zero mean value
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ2_HTML.gif
(1.2)
the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq31_HTML.gif is bounded by a https://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])
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ3_HTML.gif
(1.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq35_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq36_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq38_HTML.gif of functions whose https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq40_HTML.gif of measurable real-valued functions whose https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq43_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq44_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq45_HTML.gif with the norm https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq47_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq48_HTML.gif is absolutely continuous, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq50_HTML.gif .

We denote https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq52_HTML.gif , which is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ4_HTML.gif
(2.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ5_HTML.gif
(2.2)
We say that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq53_HTML.gif is a critical point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq54_HTML.gif , if
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq55_HTML.gif of the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq56_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ7_HTML.gif
(2.4)

for all https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq60_HTML.gif in (2.4) we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ8_HTML.gif
(2.5)
We denote
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ10_HTML.gif
(2.7)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq62_HTML.gif Hence there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq63_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ11_HTML.gif
(2.8)
on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq64_HTML.gif . The condition (1.2) implies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq65_HTML.gif and from (2.8) we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq66_HTML.gif Using https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq68_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ12_HTML.gif
(2.9)

Thus https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq71_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq72_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq74_HTML.gif (as https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq77_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq78_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq79_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq80_HTML.gif be a functional such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq82_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq83_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq84_HTML.gif and a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq85_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq86_HTML.gif ,

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

(c) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq90_HTML.gif has a critical point in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq91_HTML.gif .

3. Main Result

We define
https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq92_HTML.gif such that
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq94_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq95_HTML.gif has a critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq96_HTML.gif and due to Lemma 2.1   https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq98_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq99_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq100_HTML.gif and https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq102_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq103_HTML.gif and a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq104_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ17_HTML.gif
(3.5)
From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq105_HTML.gif and from (3.5) it follows
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq106_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq107_HTML.gif and functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq108_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq109_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq110_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq111_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq112_HTML.gif , https://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 https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq116_HTML.gif is bounded from below on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq117_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq118_HTML.gif , we have
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq119_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq120_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ22_HTML.gif
(3.10)

Since the function https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq123_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq124_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq125_HTML.gif , a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq126_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq127_HTML.gif , and there is a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq128_HTML.gif such that https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq132_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ23_HTML.gif
(3.11)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ24_HTML.gif
(3.12)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq133_HTML.gif be an arbitrary sequence bounded in https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ26_HTML.gif
(3.14)
Put https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq136_HTML.gif then (3.13), (3.14) imply
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ27_HTML.gif
(3.15)
Due to compact imbedding https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq137_HTML.gif and (3.15) we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq138_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq139_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq140_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq141_HTML.gif and set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq142_HTML.gif in (3.13), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ28_HTML.gif
(3.16)
Because the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq143_HTML.gif is bounded (assumption (3.3)) and https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq145_HTML.gif . We get
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ31_HTML.gif
(3.19)
Because https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq146_HTML.gif , https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq148_HTML.gif similarly. This implies that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq149_HTML.gif is bounded. Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq150_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq151_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq152_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq153_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq154_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ33_HTML.gif
(3.21)
The strong convergence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq156_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq157_HTML.gif and the assumption (3.3) imply
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_Equ34_HTML.gif
(3.22)
If we set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq158_HTML.gif , https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq160_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F586971/MediaObjects/13661_2010_Article_942_IEq161_HTML.gif . This shows that https://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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