A Variational Approach to Discontinuous Problems with Critical Sobolev Exponents1

We employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form − u = 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