Existence of a nontrival solution for Dirichlet problem involving p(x)-Laplacian

In this paper we study the nonlinear Dirichlet problem involving p(x)-Laplacian (hemivariational inequality) with nonsmooth potential. By using nonsmooth critical point theory for locally Lipschitz functionals due to Chang and the properties of variational Sobolev spaces, we establish conditions which ensure the existence of solution for our problem.Comment: arXiv admin note: substantial text overlap with arXiv:1212.368

Existence result for differential inclusion with p(x)-Laplacian

In this paper we study the nonlinear elliptic problem with p(x)-Laplacian (hemivariational inequality). We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to Chang

Resonant nonlinear periodic problems with the scalar p-Laplacian and a nonsmooth potential

We study periodic problems driven by the scalar p-Laplacian with a nonsmooth potential. Using the nonsmooth critical point theory for locally Lipsctiz functions,we prove two existence theorems under conditions of resonance at infinity with respect to the first two eigenvalues of the negative scalar p-Laplacian with periodic boundary conditions.Universidade de Aveir

Subharmonic solutions of sublinear second order systems with impacts

AbstractWe mainly consider subharmonic bouncing solutions of sublinear second order systems with an obstacle based on variational method

Existence result for differential inclusion with p(x)-Laplacian

In this paper we study the nonlinear elliptic problem with p(x)-Laplacian (hemivariational inequality). We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to ChangIn this paper we study the nonlinear elliptic problem with p(x)-Laplacian (hemivariational inequality). We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to Chan

Existence of three solutions for Kirchhoff nonlocal operators of elliptic type

In this paper we prove the existence of at least three solutions to the following Kirchhoff nonlocal fractional equation: begin{equation*} begin{cases} M left (int_{mathbb{R}^ntimes mathbb{R}^n} |u (x) - u (y)|^2 K (x - y) d x d y - int_Omega |u (x)|^2 d x right) ((- Delta)^s u - lambda u) \ hspace{2cm} in theta (partial j (x, u (x)) + mu partial k (x, u (x))), & textrm{in};; Omega,\ u = 0, & textrm{in};; mathbb{R}^n setminus Omega, end{cases} end{equation*} where $(- Delta)^s$ is the fractional Laplace operator, $s in (0, 1)$ is a fix, $lambda, theta, mu$ are real parameters and $Omega$ an open bounded subset of $mathbb{R}^n$, $n > 2 s$, with Lipschitz boundary. The approach is fully based on a recent three critical points theorem of Teng [K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. (RWA) 14 (2013) 867-874]

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 \hbox{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.Comment: 23 page

Nonlinear elliptic equations with nonsmooth potential : variational and topological methods

Doutoramento em MatemáticaNesta tese de doutoramento, estudamos a existência e a multiplicidade de soluções para algumas classes de equações elípticas não lineares com potencial não suave. Os resultados originais foram obtidos, utilizando métodos variacionais e da teoria de grau. A nossa abordagem variacional é baseada em descobertas recentes na teoria não suave (nonsmooth) dos pontos críticos. A teoria de grau é aplicada a determinadas perturbações multívocas de operadores de tipo monótono (operadores do tipo (S)+ ). O primeiro problema que consideramos é um problema de valor próprio semi-linear com potencial não suave (ver Capítulo 3). O resultado de existência obtido estende para uma versão não suave, e sob hipóteses de crescimento mais fracas, um resultado obtido por Rabinowitz para potenciais suaves. Mais, sob condições no potencial que permitem ressonância, quer em zero, quer no infinito, provamos um resultado de multiplicidade. Para um problema elíptico não linear derivado do p-Laplaciano e com um potencial não suave (ver Capítulo 4), estabelecemos a existência de, pelo menos, três soluções suaves, não triviais e distintas, sendo duas delas de sinal constante (uma positiva e uma negativa). Problemas semi-lineares de Neumann, que são duplamente ressonantes na origem, relativamente a qualquer intervalo espectral [λk,λk+1], são estudados no Capítulo 5. O resultado de multiplicidade obtido para um potencial não suave estende resultados existentes para o caso do potencial suave, nos quais a ressonância é completa relativamente a λk, mas incompleta relativamente a λk+1. Respondemos afirmativamente à questão aberta em relação à validade do resultado de multiplicidade, quando ocorre, também, ressonância completa relativamente a λk+1 (situação de dupla ressonância). A última parte da tese (Capítulo 6) é dedicada ao estudo de uma classe de problemas de Neumann, em que o operador diferencial não é homogéneo, nem variacional. Portanto, os métodos mini-max da teoria dos pontos críticos (suave e não-suave) não podem ser utilizados. Usando o espectro do operador diferencial assimptótico, juntamente com métodos da teoria de grau, estabelecemos a existência de soluções suaves não triviais.In this Ph.D. thesis, we study the existence and the multiplicity of solutions to some classes of nonlinear elliptic equations with a nonsmooth potential. Our new results were obtained by using variational and degree theoretic methods. The variational approach we used is based on recent developments in nonsmooth critical point theory. The degree theory we used concerns certain multivalued perturbations of a class of monotone type operators (the (S)+ type operators). The first problem we consider is a semilinear eigenvalue problem with a nonsmooth potential (see Chapter 3). The existence result we obtained extends to nonsmooth setting and under weaker growth assumptions, a result obtained by Rabinowitz for smooth potentials. Moreover, under conditions on the potential which allow resonance both at zero and at infinity, we prove a multiplicity result. For a nonlinear elliptic problem driven by the p-Laplacian and with a nonsmooth potential (see Chapter 4), we establish the existence of at least three distinct nontrivial smooth solutions, two of them with constant sign (one positive and one negative). Semilinear Neumann problems which are doubly resonant at the origin with respect to any spectral interval [λk,λk+1] were studied in Chapter 5. The multiplicity result we obtained for nonsmooth potential, extend results known for the case of smooth potential, where the resonance is complete with respect to λk, but incomplete (nonuniform nonresonance) with respect to λk+1. We give a positive answer to an open question asking whether the multiplicity result also holds when complete resonance occurs also with respect to λk+1 (double resonance situation). The last part of the thesis (Chapter 6) is devoted to the study of a class of Neumann problems where the differential operator driving the problem is neither homogeneous, nor variational. So the minimax methods of critical point theory (smooth and nonsmooth alike) fail. Using the spectrum of the asymptotic differential operator together with degree theoretic methods, we establish the existence of nontrivial smooth solutions

Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems

We consider quasilinear strongly resonant problems with discontinuous right-hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and a generalized version of the Ekeland variational principle. At the end of the paper we show that the nonsmooth Palais-Smale (PS)-condition implies the coercivity of the functional, extending this way a well-known result of the “smooth” case