A Variational Approach to Discontinuous Problems with Critical Sobolev Exponents

Abstract

We employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form −Δu=λh(x)H(u−a)uq+u2*−1 in RN, where λ, a>0 are parameters, h(x) is both nonnegative and integrable on RN, H is the Heaviside function, 2* is the critical Sobolev exponent, and 0≤q<2*−1. We obtain existence, multiplicity and regularity of solutions by distinguishing the cases 0≤q≤1 and 1<q<2*−1.