Showing papers by "Bin Ge published in 2010"
TL;DR: In this article, the variational method for locally Lipschitz functions was used to prove the existence of at least two nontrivial solutions of constant sign for the nonlinear elliptic problem.
Abstract: In this paper we study the nonlinear elliptic problem driven by p ( x ) -Laplacian with a nonsmooth locally Lipschitz potential (hemivariational inequality), that is (P) { − div ( ‖ ∇ u ( x ) ‖ R N p ( x ) − 2 ∇ u ( x ) ) ∈ ∂ j ( x , u ( x ) ) , a.a. x ∈ Ω , u = 0 , on ∂ Ω , where Ω ⊂ R N is a bounded domain and p : Ω ¯ → R is a continuous function satisfying some given assumptions. 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. Finally, we obtain the existence of at least two nontrivial solutions of constant sign.
16 citations
TL;DR: In this paper, the variational method for locally Lipschitz functions was used to prove that there exist at least two nontrivial solutions when α + p −, when α − > p +, and when α ≥ p +.
Abstract: In this paper we consider a differential inclusion in R N involving a p ( x ) -Laplacian of the type (P) { − Δ p ( x ) u + e ( x ) | u | p ( x ) − 2 u ∈ ∂ j ( x , u ( x ) ) , in R N , u ∈ W 1 , p ( x ) ( R N ) , where p : R N → R is a continuous function satisfying some given assumptions. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, on the basis of the Weierstrass Theorem and the Mountain Pass Theorem, we prove that there exist at least two nontrivial solutions, when α + p − . Finally, we obtain the existence of at least one nontrivial solution, when α − > p + .
11 citations