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Supercritical elliptic problems involving a Cordes like operator

The first author is supported by NSERC Discovery Grant

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  • 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\qquad\ \mbox{ on } \partial B_1, \end{array}\right. \end{equation} \ \ \ \ \ \ \ \ \ \ \ (1) $

    where $ B_1 $ is the unit ball in $ {{\mathbb{R}}}^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 $.

    Mathematics Subject Classification: Primary: 35J15; Secondary: 35J61.


    \begin{equation} \\ \end{equation}
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