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. © 2002 Elsevier Science.