Existence of positive solutions for a class of elliptic problems with fast increasing weights and critical exponent discontinuous nonlinearity

In this paper, using variational methods, we show the existence of at least two nonnegative solutions to a class of elliptic problems with fast increasing weights given by - Δ u - 1 2 ( x · ∇ u ) = λ h ( x ) + H ( u - a ) | u | 2 ∗ - 2 u in R N , where a > 0 , 2 ∗ : = 2 N / ( N - 2 ) ; N ≥ 3 , h : R N → R is a nonnegative function and H is the Heaviside function. For small a > 0 , we will obtain two nonnegative solutions u i , i = 1 , 2 for this equation. The first solution will be obtained using a nonsmooth version of the Mountain Pass Theorem and the second solution will be obtained by a local application of the Ekeland Variational Principle. We will also show that the set of x ∈ R N such that u i ( x ) > a has positive measure and the set of x ∈ R N such that u i ( x ) = a has zero measure. In addition, we will study the asymptotic behavior of such solutions as a → 0 .