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

Abstract

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

$$\begin{aligned} -\Delta u - \frac{1}{2}(x \cdot \nabla u) = \lambda h(x) + H(u-a)|u|^{2^{*}-2}u \ \text{ in } \ \mathbb {R}^{N}, \end{aligned}$$

where $a>0,$ $2^*:=2N/(N-2);$ $N\ge 3$, $h:\mathbb {R^{N}}\rightarrow \mathbb {R}$ 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 $x \in \mathbb {R}^{N}$ such that $u_{i}(x) >a$ has positive measure and the set of $x \in \mathbb {R}^{N}$ such that $u_{i}(x) =a$ has zero measure. In addition, we will study the asymptotic behavior of such solutions as $a\rightarrow 0$.