Existence and multiplicity of solutions for nonlocal p(x)-Laplacian problems in ℝN

Abstract

In this paper, we study the nonlocal p(x)-Laplacian problem of the following form {M(∫ℝN1/p(x)(|▽u| p(x)+|u|p(x))dx)(-div(|▽u|p(x)-2▽u) +|u|p(x)-2u) { = f(x,u) in ℝN, uεW 1,p(+)(ℝN). By using the method of weight function and the theory of the variable exponent Sobolev space, under appropriate assumptions on f and M, we obtain some results on the existence multiplicity of solutions of this problem. Moreover, we get much better results with f in a Special form. © 2012 Guo and Zhao licensee Springer.