UNIVERSITATEA DE VEST DIN TIMISOARA

Scoala Doctorala de Matematica

TEZA DE DOCTORAT

Conducator stiintific:

Prof. Dr. Petru JEBELEAN

Doctorand:

Calin - Constantin SERBAN

Timisoara

2013

WEST UNIVERSITY OF TIMISOARA

Doctoral School of Mathematics

Doctoral Thesis

Existence and Multiplicity Results

for Discrete p()-Laplacian and forSingular -Laplacian

Scientific Advisor:

Prof. Dr. Petru JEBELEAN

Ph.D. Student:

Calin - Constantin SERBAN

Timisoara, 2013

UNIVERSITATEA DE VEST DIN TIMISOARA

Scoala Doctorala de Matematica

Teza de Doctorat

Rezultate de Existenta si Multiplicitate

pentru p()-Laplacian Discret si-Laplacian Singular

Conducator stiintific:

Prof. Dr. Petru JEBELEAN

Doctorand:

Calin - Constantin SERBAN

Timisoara, 2013

This work was supported by the strategic grant

POSDRU/CPP107/DMI1.5/S/78421,

Project ID 78421 (2010), co-financed by the

European Social Fund Investing in People,

within the Sectoral Operational Programme

Human Resources Development 2007-2013.

Contents

Introduction 1

1 Preliminaries 12

2 Existence and multiplicity of solutions for discrete

p()-Laplacian 212.1 Existence results for problems with a general

potential boundary condition . . . . . . . . . . . . . . . . . 23

2.2 Existence of solutions for periodic and

Neumann problems . . . . . . . . . . . . . . . . . . . . . . . 42

2.3 Multiplicity of solutions for periodic and

Neumann problems . . . . . . . . . . . . . . . . . . . . . . . 64

3 Existence and multiplicity results for singular

-Laplacian 84

3.1 Numerical extremal solutions for a problem

with mixed boundary conditions . . . . . . . . . . . . . . . . 86

3.2 Nontrivial solutions for a class of

one-parameter mixed problems . . . . . . . . . . . . . . . . . 102

3.3 Existence of solutions for Neumann problems

with singular nonlinearities . . . . . . . . . . . . . . . . . . . 108

Bibliography 118

i

Introduction

This thesis is concerned with existence and multiplicity of solutions for

boundary value problems involving the discrete p()-Laplacian operator andthe singular -Laplacian operator. Our approach mainly relies on the direct

method in calculus of variations, Saddle Point Theorem, Mountain Pass The-

orem, some abstract three critical points results, monotone iterative tech-

niques, the method of lower and upper solutions, as well on Leray-Schauder

degree type arguments.

The work is divided in three chapters. The first one consists of four

paragraphs, while the second and the third are each structured in three

sections.

Some preliminary notions and results which are needed throughout the

thesis are introduced in Chapter 1. In the first paragraph of the chapter some

basic properties of lower semi-continuous, coercive and convex functions are

recalled. Then, in the following two paragraphs, we make a short overview

on some results related to the classical and Szulkins critical point theory. In

the final paragraph we provide two abstract three critical points theorems.

In Chapter 2 we use the critical point theory to establish the existence

and multiplicity of solutions for discrete p()-Laplacian equations subjectedto periodic, Neumann and general potential type boundary conditions. As

usually, the idea is to transfer the problem of the existence of solutions into

a problem of finding critical points for the corresponding Euler-Lagrange

functional. So, for given s (1,), hs will be the homeomorphism definedby hs(x) = |x|s2x, for all x R. If a, b N with a < b, then Z[a, b]will denote the discrete interval {a, a + 1, . . . , b}. Also, given T a positiveinteger and a function p : Z[0, T ] (1,), p() will stand for the discretep()-Laplacian operator; recall, that is

p(k1)x(k 1) : = (hp(k1)(x(k 1)))= hp(k)(x(k)) hp(k1)(x(k 1)),

where x(k) = x(k + 1) x(k) is the forward difference operator.

1

Introduction 2

Various problems in applied mathematics lead to consideration of dif-

ference equations (see e.g., R.P. Agarwal [1], W.G. Kelly and A.C. Peterson

[89] and the references therein). Recently, much attention has been paid

to their study. Certainly this is largely due to the dynamic development

of computer technology. For this reason, the research concerning boundary

value problems involving the discrete p()-Laplacian operator has gained anincreasing popularity in very recent time. According to most of the papers

in this field, it seems that the study was initiated in 2009 by M. Mihailescu

et al. in [110], where some eigenvalue problems for discrete p()-Laplacianwere investigated. After that, various existence and multiplicity results for

related discrete anisotropic boundary value problems were obtained. Most

of them concern with homogeneous Dirichlet problems. In this direction

we refer the reader to [66]-[69], [90], [100]. But, the study of discrete p()-Laplacian equations subjected to other boundary conditions seems lagging

behind. To the best of our knowledge, only A. Guiro et al. [72] and B. Kone

and S. Ouaro [91] proved existence of solutions for some discrete anisotropic

Neumann and mixed boundary value problems. Hence, the present chapter

aims to partly fill the gap in this area.

Concerning continuous anisotropic problems, it is worth to point out

that these were intensively studied in the last decade by many authors. We

refer the reader e.g. to [51], [53], [56]-[62], [64], [65], [74], [85], [86], [95], [107]-

[109], [127]-[130] for existence and multiplicity results of the p(x)-Laplacian

operator, using the theory of variable exponent Lebesque and Sobolev spaces

see X.L. Fan and D. Zhao [63], O. Kovacik and J. Rakosnk [92] and the

recent monograph of L. Diening et al. [52].

In Section 2.1 we deal with difference equations of type

p(k1)x(k 1) = f(k, x(k)), () k Z[1, T ], (1)

subjected to the general potential boundary condition

(hp(0)(x(0)),hp(T )(x(T ))) j(x(0), x(T + 1)), (2)

where f : Z[1, T ]R R is a continuous function, j : RR (,+]is convex, proper, lower semi-continuous and j denotes the subdifferential

of j. First, we emphasize that the potential boundary condition (2) recovers

the classical ones and hence a unified approach of all the classical boundary

conditions is provided (see Remark 2.1.15 for details).

Szulkins critical point theory [121] is employed when dealing with prob-

lem (1), (2). So, the associated Euler-Lagrange functional is defined on the

Introduction 3

finite dimensional space

XPO := {x : Z[0, T + 1] R}

(here, the index PO comes from potential) by

I(x) = (x)Tk=1

F (k, x(k)), () x XPO,

where : XPO (,+] is given by

(x) =T+1k=1

1

p(k 1)|x(k 1)|p(k1) + j(x(0), x(T + 1)), () x XPO

and F : Z[1, T ] R R is the primitive of f with respect to the secondvariable, i.e.,

F (k, t) =

t0

f(k, )d, () k Z[1, T ], t R. (3)

First of all, we show in Proposition 2.1.4 that if x XPO is a critical pointof the functional I in the sense of Szulkin, then x is a solution of problem(1), (2).

There are two main results in this section, namely Theorem 2.1.5 and

Theorem 2.1.12. In the first one we use a minimization argument in order

to prove the existence of at least one solution. Then, we obtain in Theorem

2.1.12 the existence of at least one nontrivial solution for problemp(k1)x(k 1) + r(k)hp(k)(x(k)) = f(k, x(k)), () k Z[1, T ],

(hp(0)(x(0)),hp(T )(x(T ))) j(x(0), x(T + 1)),

where r : Z[1, T ] [0,) is a given function. The main tool used here willbe the Mountain Pass Theorem. The results from this section are obtained

in [16]. Also, it is worth to point out that Theorem 3.1 and Theorem 4.4 in

[122] obtained for p = constant, are immediate consequences of the above

mentioned Theorem 2.1.5, respectively Theorem 2.1.12.

In Section 2.2 we are concerned with the existence of solutions for the

periodic problemp(k1)x(k 1) = f(k, x(k)), () k Z[1, T ],

x(0) x(T + 1) = 0 = x(0)x(T )(4)

Introduction 4

and for the Neumann problemp(k1)x(k 1) = f(k, x(k)), () k Z[1, T ],

x(0) = 0 = x(T ),

(5)

where as above f : Z[1, T ]R R is a continuous function whose primitiveF is defined in (3). To establish the existence results for problems (4) and

(5) we shall use the classical critical point theory of Ambrosetti-Rabinowitz

[5]. The solutions which we obtain appear either as minimizers or as saddle

points of the corresponding energy functional

EX(x) =T+1k=1

1

p(k 1)|x(k 1)|p(k1)

Tk=1

F (k, x(k)), () x X,

where

X = XP := {x : Z[0, T + 1] R | x(0) = x(T + 1)} ,

when we refer to the periodic problem (4), respectively

X = XN := {x : Z[0, T + 1] R} (= XPO) ,

in the case of the Neumann problem (5).

In Proposition 2.2.2 (resp. Proposition 2.2.4) we show that a function

x XP (resp. x XN) is a solution of problem (4) (resp. (5)) if and onlyif it is a critical point of EXP (resp. EXN ).

First, we deal with the case when no convexity assumptions are made

on the nonlinearity f . So, if f is bounded and one of the following Ahmad-

Lazer-Paul [4] type conditions:

Tk=1

F (k, t) , as |t| (6)

orTk=1

F (k, t) +, as |t| (7)

holds true,