| Abstract: |
| We consider the problem $ -\Delta u=\lambda{(1+|\nabla u|^q)\over (1- u)^p},$ in $B$, with $ u=0$ on $\partial B,$ where $B$ is the unit ball in $\mathbb{R}^N$, $p>0,$ $q\geq 0 $ and $\lambda\geq 0.$ The problem with $q=0$ is well known. In fact, Joseph \& Lundgren found that for a bounded range of dimensions there are infinitely many solutions for some $\lambda=\lambda_*>0$. On the other hand, they also found that for large dimensions there exists $\lambda^*$ such that there exists a unique solution for $\lambda\in (0,\lambda^*).$ In this talk, we present results of existence of solutions for this problem when $q>0$. In this case, we found a rich structure of solutions depending on $p$, $q$ and the dimension $N.$ In addition, we study numerically the behaviour of solutions for related problems with a gradient term in the nonlinearity. |
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