Supercritical elliptic problems involving a Cordes like operator

Abstract

In this work we obtain positive bounded solutions of various perturbations of \begin{equation} \left\{ \begin{array}{lcl} \hfill -\Delta u - \gamma \sum_{i,j=1}^N \frac{x_i x_j}{|x|^2} u_{x_i x_j} &=& u^p \qquad \mbox{ in } B_1, \\ \hfill u &=& 0 \hfill \mbox{ on } \partial B_1, \end{array}\right. \end{equation} where $B_1$ is the unit ball in $ \IR^N$ where $N \ge 3$, $ \gamma>0$ and $ 1<p<p_{N,\gamma}$ where \begin{equation*} p_{N,\gamma}:=\left\{ \begin{array}{lc} \frac{N+2+3 \gamma}{N-2-\gamma} & \qquad \mbox{ if } \gamma<N-2, \\ \infty & \qquad \mbox{ if } \gamma \ge N-2. \end{array}\right. \end{equation*} Note for $\gamma>0$ this allows for supercritical range of $p$.