Abstract
We consider a nonlinear elliptic equation driven by the -Laplacian with a nonsmooth potential (hemivariational inequality) and Dirichlet boundary condition. Using a variational approach based on nonsmooth critical point theory together with the method of upper and lower solutions, we prove the existence of at least three nontrivial smooth solutions: one positive, the second negative, and the third sign changing (nodal solution). Our hypotheses on the nonsmooth potential incorporate in our framework of analysis the so-called asymptotically -linear problems.
Highlights
The aim of this work is to prove the existence of multiple solutions of constant sign and of nodal solutions sign changing solutions for nonlinear elliptic equations driven by the pLaplacian and having a nonsmooth potential hemivariational inequalities
Hemivariational inequalities are a new type of variational expressions, which arise in applications if one considers more realistic mechanical laws of multivalued and nonmonotone nature
Various engineering applications of hemivariational inequalities can be found in the book of Naniewicz-Panagiotopoulos 2
Summary
The aim of this work is to prove the existence of multiple solutions of constant sign and of nodal solutions sign changing solutions for nonlinear elliptic equations driven by the pLaplacian and having a nonsmooth potential hemivariational inequalities. In contrast Zhang-Li 7 use invariance properties of the negative gradient flow of the corresponding equation in C01 Z These methods were extended to “smooth” problems driven by the p-Laplacian differential operator. In this paper using techniques from nonsmooth critical point theory in conjunction with the method of upper and lower solutions, we are able to extend the works of DancerDu 6 and Carl-Perera 8 to hemivariational inequalities. Helpful in this respect is the nonsmooth second deformation lemma of Corvellec 17. Sign-changing solutions for problems with discontinuous nonlinearities were obtained by Averna et al 18 , but in contrast to our work they deal with p-superlinear problems
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call