| Abstract: |
| In this talk, we will discuss the existence of positive solutions of a semi-linear elliptic system defined in a cylinder $\Omega=\Omega`\times(0,a)\subset\mathbb{R}^n$, where $\Omega`\subset\mathbb{R}^{n-1}$ is a bounded and smooth domain. The system couples a superlinear equation defined in the whole cylinder $\Omega$ with another superlinear (or linear) equation defined at the bottom of the cylinder $\Omega`\times\{0\}$. We provide a priori $L^\infty$ bounds for all positive solutions of the system when the nonlinear terms satisfy certain growth conditions. Using the a priori bounds and topological arguments, we prove the existence of positive solutions for these particular semi-linear elliptic systems. |
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