Multiple solutions for a nonlinear neumann problem involving

A class of nonlinear Neumann problems driven by -Laplacian with a nonsmooth locally Lipschitz potential hemivariational inequality was considered.

The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, Weierstrass theorem and Mountain Pass theorem are used to prove the existence of at least two nontrivial solutions. Recently, there are several papers on the research of the Neumann-type problems involving the -Laplacian. Of the existing works in the literature, the majority deal with problems in which the potential function is smooth i.

Problems with a nonsmooth potential, were studied by Dai [ 67 ], who for the case established the existence of three or infinitely many solutions for Neumann-type differential inclusion problems involving the -Laplacian, using the nonsmooth three-critical-points theorem and nonsmooth Ricceri type variational principle, respectively. Not long ago, Qian et al. The authors prove the existence of at least one nontrivial solution of 1 using the nonsmooth Mountain Pass theorem and Weierstrass theorem.

Ifthen problem 1 becomes problem 2 as follows: In this paper, our goal is to establish the existence of at least two nontrivial solutions for problem 2.

We emphasize that the operator is said to be -Laplacian, which becomes -Laplacian when a constant.

multiple solutions for a nonlinear neumann problem involving

The -Laplacian possesses more complicated nonlinearities than the -Laplacian; for example, it is inhomogeneous and in general, it has not the first eigenvalue. The study of various mathematical problems with variable exponent growth conditions has received considerable attention in recent years.

These problems are interesting in applications to modeling electrorheological fluids see [ 1011 ] and image restoration see [ 12 ]. This paper is divided into three sections: in the second section, we introduce some necessary knowledge on the nonsmooth analysis and basic properties of the generalized Lebesgue-space and the generalized Lebesgue-Sobolev space.

In this section, we first review some facts on variable exponent spaces and. For the details, see [ 13 — 18 ]. Firstly, we need to give some notations, which we will use through this paper:. Denote by the set of all measurable real functions defined on.

multiple solutions for a nonlinear neumann problem involving

Two functions in are considered to be one element ofwhen they are equal almost everywhere. Fordefine with the norm with the norm. Let be a Banach space and its topological dual space and we denote as the duality bracket for pair.

For a locally Lipschitz functionwe define.

multiple solutions for a nonlinear neumann problem involving

It is obvious that the function is sublinear and continuous and so is the support function of a nonempty, convex, and -compact setdefined by. The multifunction is called the generalized subdifferential of. If is also convex, then coincides with subdifferential in the sense of convex analysis, defined by Ifthen. A point is a critical point of if. It is easily seen that if is a local minimum ofthen.

A locally Lipschitz function satisfies the nonsmooth -condition at level the nonsmooth -condition for shortif for every sequencesuch that andasthere is a strongly convergent subsequence, where. If this condition is satisfied at every levelthen we say that satisfies the nonsmooth -condition.

Lemma 1 see [ 19 ]. Consider the following. Moreover, is uniform convex. Lemma 2 see [ 15 ]. The conjugate space of iswhere. For any andone has.

Lemma 3 see [ 15 ]. In this paper, we denote by ; the dual space and by the dual pair. Consider the following function: We know that see [ 20 ] and -Laplacian operator is the derivative operator of in the weak sense. We denote ; then for all.PL EN. Widoczny [Schowaj] Abstrakt. Adres strony. Opuscula Mathematica. Constant-sign solutions for a nonlinear Neumann problem involving the discrete p-Laplacian.

Candito, P. In this paper, we investigate the existence of constant-sign solutions for a nonlinear Neumann boundary value problem involving the discrete p-Laplacian. Our approach is based on an abstract local minimum theorem and truncation techniques.

Opis fizyczny. D'Agui, G. Agarwal, K. Perera, D. Agarwal, On multipoint boundary value problems for discrete equations, J. Anderson, I. Tisdell, Solvability of discrete Neumann boundary value probles, Adv.

Bereanu, J. Difference Equ. Bereanu, P. Jebelean, C. Value Probl. Bonanno, P.Dipartimento di Matematica e Applicazioni "R. Angelo, Via Cintia, Napoli, Italy. MercaldoWell-posed elliptic Neumann problems involving irregular data and domains, Ann. Google Scholar. B 65 DiazUniqueness and continuum of foliated solutions for a quasilinear elliptic equation with a non-Lipschitz nonlinearity, Comm.

Partial Differential Equations17 Scuola Norm. Pisa Cl. Differential Equations, PorzioExistence and uniqueness results for nonlinear elliptic problems with a lower order term and measure datum, C.

Paris, Betta, A. Mercaldo, F. Murat and M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower-order term and right-hand side a measure, J. Pures Appl. Corrected reprint of J. I Math. MichailleUniqueness results and monotonicity properties for strongly nonlinear elliptic variational inequalities, Ann. PrignetRenormalized solutions of elliptic equations with general measure data, Ann.

Pura Appl. Toulouse Math. DroniouSolving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by a duality method, Adv. Differential Equations5 Partial Differential Equations34 MercaldoA second order derivation formula for functions defined by integrals, C.

MercaldoNeumann problems and Steiner symmetrization, Comm. Partial Differential Equations30 MercaldoExistence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data, Potential Anal. MercaldoExistence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data, Trans.We investigate the existence of at least three solutions for a discrete nonlinear Neumann boundary value problem involving the -Laplacian.

Our approach is based on three critical points theorems. In these last years, the study of discrete problems subject to various boundary value conditions has been widely approached by using different abstract methods as fixed point theorems, lower and upper solutions, and Brower degree see, e. Recently, also the critical point theory has aroused the attention of many authors in the study of these problems [ 4 — 12 ].

The main aim of this paper is to investigate different sets of assumptions which guarantee the existence and multiplicity of solutions for the following nonlinear Neumann boundary value problem. In particular, for every lying in a suitable interval of parameters, at least three solutions are obtained under mutually independent conditions. First, we require that the primitive of is sublinear at infinity and satisfies appropriate local growth condition Theorem 3.

Next, we obtain at least three positive solutions uniformly bounded with respect tounder a suitable sign hypothesis onan appropriate growth conditions on in a bounded interval, and without assuming asymptotic condition at infinity on Theorem 3.

Moreover, the existence of at least two nontrivial solutions for problem is obtained assuming that is sublinear at zero and superlinear at infinity Theorem 3. It is worth noticing that it is the first time that this type of results are obtained for discrete problem with Neumann boundary conditions; instead of Dirichlet problem, similar results have been already given in [ 6913 ].

Moreover, in [ 14 ], the existence of multiple solutions to problem is obtained assuming different hypotheses with respect to our assumptions see Remark 3. Investigation on the relation between continuous and discrete problems are available in the papers [ 1516 ]. General references on difference equations and their applications in different fields of research are given in [ 1718 ].

While for an overview on variational methods, we refer the reader to the comprehensive monograph [ 19 ]. Let be a real Banach space, let be two functions of class onand let be a positive real parameter. In order to study problemour main tools are critical points theorems for functional of type which insure the existence at least three critical points for every belonging to well-defined open intervals.

These theorems have been obtained, respectively, in [ 62021 ]. Assume that there exist andwith such that.

Then, for eachthe functional has at least three distinct critical points in. Assume that there exist two positive constantsandwith such that.We improve some results on the existence and multiplicity of solutions for the -biharmonic system. Our main results are new. Our approach is based on general variational principle and the theory of the variable exponent Sobolev spaces.

In this paper, we consider the existence of solutions for the following system: forwhere is a bounded domain with smooth boundary.

In recent years, many authors considered the existence and multiplicity of solutions for some fourth order problems [ 1 — 10 ]. In [ 4 ], based on critical point theory, the existence of infinitely many solutions has been established for a class of nonlinear elliptic equations involving the -biharmonic operator and under Navier boundary value conditions.

The -Laplacian operator is more complicated nonlinearities than -Laplacian; it is inhomogeneous and usually it does not have the so-called first eigenvalue, since the infimum of its principle eigenvalue is zero.

In [ 11 ], based on variational methods, the authors established the existence of an unbounded sequence of weak solutions for a class of differential equations with -Laplacian. In [ 12 ], when the nonlinearity has the subcritical growth and via variational methods [ 13 ], the authors obtained the existence of at least one, two, or three weak solutions for a class of differential equations with -Laplacian whenever the parameter belongs to a precise positive interval.

Recently, the -biharmonic problems have attracted more and more attention; we refer the reader to [ 1114 — 21 ]. In [ 20 ], the authors established the existence of at least three solutions for elliptic systems involving the -biharmonic operator. In [ 15 ], Allaoui et al. However, there are rare results on -biharmonic problem. Inspired by the aforementioned papers, our objective is to prove the existence and multiplicity solutions for problem 1 ; we study problem 1 by using the results as follows.

Theorem A see [ 1322 ]. Then, for each compact intervalthere exists with the following property: for everythe equation has at least three solutions in whose norms are less than.

Theorem B see [ 23 ]. For everylet one put Then, one has the following: a For every and everythe restriction of the functional to admits a global minimum, which is a critical point local minimum of in.

This paper is organized as follows. In Section 2we recall some basic facts about the variable exponent Lebesgue and Sobolev spaces, some important properties of the -biharmonic operator. In Section 3we establish the main results. In order to deal with the -biharmonic problem, we need some theories on spacesand introduce some notations used in the following. We introduce a norm on : Then becomes a Banach space; we call it a generalized Lebesgue space.

Proposition 1 see [ 24 ]. The conjugate space of iswhere.National Center for Theoretical Sciences, No. ZhangExistence and non-existence of solutions to elliptic equations related to the Caffarelli-Kohn-Nirenberg inequalities, Calc. Partial Diff. Google Scholar.

KangHardy-Sobolev critical elliptic equations with boundary singularities, Ann. Poincare Anal. Non Lineaire21 Ghoussoub and F. Robert, The effect of curvature on the best constant in the Hardy-Sobolev inequalities, Geom. HashizumeAsymptotic behavior of the least-energy solutions of a semilinear elliptic equation with the Hardy-Sobolev critical exponent, J. Differential Equations, LinA nonlinear elliptic pde with two Sobolev-Hardy critical exponents, Arch. Anal, LionsThe concentration-compactness principle in the calculus of variations.

The limit case. Iberoamericana1 MusinaGround state solutions of a critical problem involving cylindrical weights, Nonlinear Anal68 WangNeumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations93 Zhong and W. Yimei LiJiguang Bao. Semilinear elliptic system with boundary singularity. Jinhui ChenHaitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities.

Lei WeiZhaosheng Feng. Isolated singularity for semilinear elliptic equations. On the uniqueness of singular solutions for a Hardy-Sobolev equation.

Conference Publications, special : Xiaomei SunWenyi Chen.Metrics details. We consider the existence of at least two or three distinct weak solutions for the nonlinear elliptic equations. To do this, we give some critical point theorems for continuously differentiable functions with the Cerami condition which are extensions of the recent results in Bonanno Adv.

Nonlinear Anal. In the present paper, we are concerned with multiple solutions for the nonlinear Neumann boundary value problem associated with p -Laplacian type.

Concerning elliptic equations with nonlinear boundary conditions, we refer to [ 3 — 7 ]. To obtain the existence of two distinct weak solutions for this problem, they assumed that the nonlinear term f satisfies the Ambrosetti and Rabinowitz condition the AR condition, for short in [ 15 ]:. Roughly speaking, we give this result under the Cerami condition which is another compactness condition of the Palais-Smale type introduced by Cerami [ 17 ]. First we show the existence of at least two weak solutions for P without assuming that f satisfies the AR condition.

In recent years, some authors in [ 18 — 23 ] have tried to drop the AR condition that is crucial to guarantee the boundedness of the Palais-Smale sequence of the Euler-Lagrange functional which plays a decisive role in applying the critical point theory. In this respect, we observe that the energy functional associated with P satisfies the Cerami condition when the nonlinear term f does not satisfies AR condition. This together with the best Sobolev trace constant given in [ 3 ] yields the existence of at least two weak solutions for P.

This paper is organized as follows. Hence, from the inequality 2. This is crucial to get the existence of at least two distinct weak solutions for the problem P in the next section. Thenfor each. The proof is essentially the same as in that of [ 2 ]. This plays an important role in obtaining the fact that the problem P admits at least three distinct weak solutions.

In this section, we first collect some preliminary properties that will be used later. The following assertion can be found in [ 31 ]; see also [ 32 ]. Assume that J1 - J4 hold. Next we need the following assumptions for f and g. Assume that F1 - F2 and G1 hold. Now we obtain the positivity of the infimum of all eigenvalues for the problem E.

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