Nonsmooth Critical Point Theory Research Articles

Existence of positive and of multiple solutions for nonlinear periodic problems

In this paper we consider a scalar periodic problem driven by the ordinary p -Laplacian differential operator and having a nonsmooth potential. Using a variational method based on nonsmooth critical point theory, first we prove the existence of a strictly positive solution. Then by strengthening our hypotheses on the nonsmooth potential, we prove the existence of a second periodic solution.

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.

Positive Periodic and Homoclinic Solutions for Nonlinear Differential Equations with Nonsmooth Potential

We study the existence of positive solutions and of positive homoclinic (to zero) solutions for a class of periodic problems driven by the scalar ordinary p-Laplacian and having a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory and our results extend the recent works of Korman–Lazer (Electronic JDE (1994)) and of Grossinho–Minhos–Tersian (J. Math. Anal. Appl. 240 (1999)).

Nontrivial Solutions for Resonant Hemivariational Inequalities

We study a resonant semilinear elliptic hemivariational inequality. Under some assumptions of strong resonance on the Clarke subdifferential of the superpotential, and using nonsmooth critical point theory, the existence of a nontrivial solution of the problem is shown.

Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems

The aim of this monograph is to present a comprehensive survey of results about existence, multiplicity, perturbation from symmetry and concentration phenomena for a class of quasi-linear elliptic equations coming from functionals of the calculus of variations which turn out to be merely continuous. Some tools of non-smooth critical point theory will be employed.

“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.

SEMILINEAR HEMIVARIATIONAL INEQUALITIES WITH STRONG RESONANCE AT INFINITY

A semilinear elliptic equation with strong resonance at infinity and with a nonsmooth potential is studied. Using nonsmooth critical point theory and developing some abstract minimax principles which complement and extend results in the literature, two results on existence are obtained.

MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH A DISCONTINUOUS NONLINEARITY

We consider a nonlinear elliptic equation driven by the p-Laplacian with a discontinuous nonlinearity. Such problems have a "multivalued" and a "single-valued" interpretation. We are interested in the latter and we prove the existence of at least two distinct solutions, both smooth and one strictly positive. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions, coupled with penalization and truncation techniques.

Pairs of positive solutions for nonlinear elliptic equations with the p-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 C 1 (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 C 1 0-minimizers versus the local H 1 0-minimizers of a C 1-functional.

Minimax methods for singular elliptic equations with an application to a jumping problem

A jumping problem for a class of singular semilinear elliptic equations is considered. Minimax methods in the framework of nonsmooth critical point theory are applied.

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.

Multiple solutions for strongly resonant nonlinear elliptic problems with discontinuities

We examine a nonlinear strongly resonant elliptic problem driven by the $p$-Laplacian and with a discontinuous nonlinearity. We assume that the discontinuity points are countable and at them the nonlinearity has an upward jump discontinuity. We show that the problem has at least two nontrivial solutions without using a multivalued interpretation of the problem as it is often the case in the literature. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions.

Multiple critical points for nondifferentiable functionals involving Hardy potentials

In this paper we study general functionals of the calculus of variations with the presence of a Hardy potential. We will improve several results obtained in the semilinear framework. We will first prove a general weak lower semicontinuity result, which will imply the existence of a minimum point whenever the functional is coercive. Then we will demonstrate existence and multiplicity results of critical points, even if our functional is not differentiable. We will apply a nonsmooth critical point theory developed in Corvellec et al. (Nonlinear Anal. 1 (1993) 151) and Degiovanni and Marzocchi (Ann. Mat. Pura Appl. 167 (1994) 73).

Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities

In this paper we develop a critical point theory for nonsmooth locally Lipschitz functionals defined on a closed, convex set extending this way the work of Struwe (Variational Methods, Springer, Berlin, 1990). Through a deformation result, we obtain minimax principles producing critical points. Then we use the theory to obtain positive and negative solutions of nonlinear and semilinear hemivariational inequalities. In this context we improve a result on positive solutions for semilinear elliptic problems due to Nirenberg (Variational methods in nonlinear problems, in: Topics in Calculus of Variations, Lecture Notes in Mathematics, vol. 1365, Springer, Berlin, 1987).

A multiplicity result for semilinear resonant elliptic problems with nonsmooth potential

We study a semilinear elliptic problem with a nonsmooth potential and resonance at the origin between two successive eigenvalues of ( - Δ , H 0 1 ( Z ) ) . Our approach is variational and is based on the nonsmooth critical point theory for locally Lipschitz functions. We prove the existence of at least two nontrivial solutions.

On the Existence of Positive Solutions for Hemivariational Inequalities Driven by the p-Laplacian

We study nonlinear elliptic problems driven by the p-Laplacian and with a nonsmooth locally Lipschitz potential (hemivariational inequality). We do not assume that the nonsmooth potential satisfies the Ambrosetti--Rabinowitz condition. Using a variational approach based on the nonsmooth critical point theory, we establish the existence of at least one smooth positive solution.

Periodic solutions for nonautonomous systems with nonsmooth quadratic or superquadratic potential

We study a semilinear nonautonomous second order periodic system with a nonsmooth potential function which exhibits a quadratic or superquadratic growth. We establish the existence of a solution, using minimax methods of the nonsmooth critical point theory.

Non-linear second-order periodic systems with non-smooth potential

In this paper we study second order non-linear periodic systems driven by the ordinary vectorp-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by thep-Laplacian. In the last section of the paper we examine the scalar non-linear and semilinear problem. Our approach uses a generalized Landesman-Lazer type condition which generalizes previous ones used in the literature. Also for the semilinear case the problem is at resonance at any eigenvalue.

Periodic problems with asymmetric nonlinearities and nonsmooth potentials

We consider a nonlinear periodic boundary value problem driven by the ordinary scalar p-Laplacian and with a nonsmooth locally Lipschitz potential, which exhibits an asymmetric behaviour at +∞ and at −∞. Using the Fučik spectrum and a variational approach based on the nonsmooth critical point theory, we obtain a solution for the problem using a general minimax principle concerning linking sets.

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)).