EXISTENCE OF PERIODIC SOLUTIONS FOR A DIFFERENTIAL INCLUSION SYSTEMS INVOLVING THE p（t）-LAPLACIAN

We study a nonlinear periodic problem driven by the p(t)-Laplacian and having a nonsmooth potential (hemivariational inequalities). Using a variational method based on nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of at least two nontrivial solutions under the generalized subquadratic and then establish the existence of at least one nontrivial solution under the generalized superquadratic.     

Multiple solutions to a class of inclusion problem with the p(x)-Laplacian

In this article, we obtain the existence of at least two nontrivial solutions for a nonlinear elliptic problem involving p(x)-Laplacian type operator and nonsmooth potentials. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.     

Positive solutions for nonlinear hemivariational inequalities

In this paper we study the existence of positive solutions for nonlinear problems driven by the p-Laplacian or more generally, by multivalued p-Laplacian-like operators. Both problems have a nonsmooth locally Lipschitz potential (hemivariational inequalities). Using variational methods based on the nonsmooth critical point theory, we prove two existence results with the p-Laplacian and multivalued p-Laplacian-like operators.     

“Two Nontrivial Critical Points for Nonsmooth Functionals via Local Linking and Applications”

In this paper, we extend to nonsmooth locally Lipschitz functionals the multiplicity result of Brezis–Nirenberg (Communication Pure Applied Mathematics and 44 (1991)) based on a local linking condition. Our approach is based on the nonsmooth critical point theory for locally Lipschitz functions which uses the Clarke subdifferential. We present two applications. This first concerns periodic systems driven by the ordinary vector p-Laplacian. The second concerns elliptic equations at resonance driven by the partial p-Laplacian with Dirichlet boundary condition. In both cases the potential function is nonsmooth, locally Lipschitz.     

Multiple solutions for a Robin‐type differential inclusion problem involving the p(x)‐Laplacian

In this paper, we obtain the existence of at least two nontrivial solutions for a Robin‐type differential inclusion problem involving p(x)‐Laplacian type operator and nonsmooth potentials. Our approach is variational, and it is based on the nonsmooth critical point theory for locally Lipschitz functions. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence and multiplicity of solutions for Neumann problems

In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro-Lazer-Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of (−Δp,W1,p(Z)). Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p=2) corresponds to the super-sub quadratic situation.     

Multiplicity of nontrivial solutions for elliptic equations with nonsmooth potential and resonance at higher eigenvalues

We consider a semilinear elliptic equation with a nonsmooth, locally Lipschitz potential function (hemivariational inequality). Our hypotheses permit double resonance at infinity and at zero (double-double resonance situation). Our approach is based on the nonsmooth critical point theory for locally Lipschitz functionals and uses an abstract multiplicity result under local linking and an extension of the Castro-Lazer-Thews reduction method to a nonsmooth setting, which we develop here using tools from nonsmooth analysis.     

Multiple Solutions for a Differential Inclusion Problem with Nonhomogeneous Boundary Conditions

In this paper, we use a mountain pass theorem with Cerami type conditions for locally Lipschitz functions to investigate the existence of at least one nontrivial solution for a differential inclusion problem involving the p-Laplacian and with nonlinear and nonsmooth boundary conditions. Moreover, by a symmetric version of the mountain pass theorem, we prove the existence of infinitely many solutions.     

Multiple solutions for a class of elliptic hemivariational inequalities

The existence of multiple solutions to elliptic hemivariational inequality problems in bounded domains is investigated via a suitable nonsmooth version of a classical technique due to Struwe and a recent saddle point theorem for locally Lipschitz continuous functionals.     

Multiple solutions of constant sign for nonlinear nonsmooth eigenvalue problems near resonance

In this paper we study a class of nonlinear elliptic eigenvalue problems driven by the p-Laplacian and having a nonsmooth locally Lipschitz potential. We show that as the parameter approaches (= the principal eigenvalue of ) from the right, the problem has three nontrivial solutions of constant sign. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions. In the process of the proof we also establish a generalization of a recent result of Brezis and Nirenberg for C₀¹ versus W₀^1,p minimizers of a locally Lipschitz functional. In addition we prove a result of independent interest on the existence of an additional critical point in the presence of a local minimizer of constant sign. Finally by restricting further the asymptotic behavior of the potential at infinity, we show that for all the problem has two solutions one strictly positive and the other strictly negative.Received: 7 January 2003, Accepted: 12 May 2003, Published online: 4 September 2003Mathematics Subject Classification (2000): 35J20, 35J85, 35R70     

Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential

In this paper, we study a nonlinear elliptic problem at resonance driven by the p-Laplacian and with a nonsmooth potential (hemivariational inequality). Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions due to Chang. We prove a theorem guaranteeing the existence of one solution which is smooth and strictly positive. Then by strengthening the assumptions, we establish a multiplicity result providing the existence of at least two distinct solutions. Our hypotheses permit resonance and asymmetric behavior at +∞ and −∞. As a byproduct of our analysis we obtain an nonlinear and nonsmooth generalization of a result of Brézis–Nirenberg about H₀¹ versus C₀¹ minimizers of a smooth functional.     

Existence of Solutions and of Multiple Solutions for Nonlinear Nonsmooth Periodic Systems

In this paper we examine nonlinear periodic systems driven by the vectorial p-Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the sublinear problem. For the semilinear problem (i.e. p = 2) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem for the superlinear problem. Our work generalizes some recent results of Tang (PAMS 126(1998)).     

An obstacle problem for nonlinear hemivariational inequalities at resonance

In this paper we examine an obstacle problem for a nonlinear hemivariational inequality at resonance driven by the p-Laplacian. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functionals defined on a closed, convex set, we prove two existence theorems. In the second theorem we have a pointwise interpretation of the obstacle problem, assuming in addition that the obstacle is also a kind of lower solution for the nonlinear elliptic differential inclusion.     

Pairs of positive solutions for nonlinear elliptic equations with the <Emphasis Type="Italic">p</Emphasis>-Laplacian and a nonsmooth potential

In this paper we examine a nonlinear elliptic problem driven by the p-Laplacian differential operator and with a potential function which is only locally Lipschitz, not necessarily C¹ (hemivariational inequality). Using the nonsmooth critical point theory of Chang, we obtain two strictly positive solutions. One solution is obtained by minimization of a suitable modification of the energy functional. The second solution is obtained by generalizing a result of Brezis-Nirenberg about the local C¹₀-minimizers versus the local H¹₀-minimizers of a C¹-functional. Mathematics Subject Classification (2000) 35J50, 35J85, 35R70     

Nonlinear eigenvalue Neumann problems with discontinuities

In this paper we study nonlinear eigenvalue problems with Neumann boundary conditions and discontinuous terms. First we consider a nonlinear problem involving the p-Laplacian and we prove the existence of a solution for the multivalued approximation of it, then we pass to semilinear problems and we prove the existence of multiple solutions. The approach is based on the critical point theory for nonsmooth locally Lipschitz functionals.     

Existence and multiplicity of solutions for second order periodic systems with the p-Laplacian and a nonsmooth potential

In this paper we study nonlinear periodic systems driven by the ordinary p-Laplacian with a nonsmooth potential. We prove an existence theorem using a nonsmooth variant of the reduction method. We also prove two multiplicity results. The first is for scalar problems and uses the nonsmooth second deformation lemma. The second is for systems and it is based on the nonsmooth local linking theorem.     

Semilinear hemivariational inequalities at resonance

In this paper we examine a semilinear hemivariational inequality at resonance in the first eigenvalue λ₁ of (−Δ,H ₀ ¹ (Z)). We prove two existence theorems for such problems. Our approach is variational and is based on the nonsmooth critical point theory of Chang, which uses the subdifferential calculus of Clarke for locally Lipschitz functions.     

Existence and subharmonicity of solutions for nonlinear non-smooth periodic systems with a -Laplacian

In the present paper, we consider the nonlinear periodic systems driven by a vectorial p-Laplacian with a locally Lipschitz nonlinearity. Some existence, multiplicity and subharmonic results are obtained by using non-smooth critical point theory.     

Existence and multiplicity results for quasilinear hemivariational inequalities at resonance

In this paper we consider quasilinear hemivariational inequality at resonance. We prove existence results for strongly resonant quasilinear problem, resonant problem under a Tang‐type condition as well as two multiplicity results. The method of the proofs is based on the nonsmooth critical point theory for locally Lipschitz functions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and multiplicity results for some elliptic systems with discontinuous nonlinearities

In this paper we complete two tasks. We first extend the critical point theorem obtained by Li and Willem [S.J. Li, M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl. 189 (1995), 6-32] to the nonsmooth case in which the energy functional is locally Lipschitz and satisfies the weaker nonsmooth Cerami condition. Then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results.