Global C1 , α regularity and existence of multiple solutions for singular p(x)-Laplacian equations

Abstract

In this paper we study the following singular p(x)-Laplacian problem { -div (vertical bar del u vertical bar(p(x)-2)del u) = lambda/u(beta(x)) + u(q(x)), in Omega, u > 0, in Omega, u = 0, on partial derivative Omega, where Omega is a bounded domain in R-N, N >= 2, with smooth boundary partial derivative Omega, beta is an element of C-1((Omega) over bar) with 0 < beta(x) < 1, p is an element of C-1((Omega) over bar), q is an element of C((Omega) over bar) with p(x) > 1, p(x) < q(x)+ 1 < p*(x) for x is an element of(Omega) over bar, where p*(x) = Np(x)/N-p(x) for p(x) < N and p*(x) = infinity for p(x) >= N. We establish C1, a regularity of weak solutions of the problem and strong comparison principle. Based on these two results, we prove the existence of multiple (at least two) positive solutions for a certain range of lambda.olutions for a certain range of λ.