We consider Dirichlet problems of the form $-|x|^\alpha\Delta u=\lambda u+g(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\alpha,\lambda\in\mathbb{R}$, $g\in C(\mathbb{R})$ is a superlinear and subcritical function, and $\Omega$ is a domain in $\mathbb{R}^2$. We study the existence of positive solutions with respect to the values of the parameters $\alpha$ and $\lambda$, and according that $0\in\Omega$ or $0\in\partial\Omega$, and that $\Omega$ is an exterior domain or not.
On a class of two-dimensional singular elliptic problems
CALDIROLI, Paolo;
2001-01-01
Abstract
We consider Dirichlet problems of the form $-|x|^\alpha\Delta u=\lambda u+g(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\alpha,\lambda\in\mathbb{R}$, $g\in C(\mathbb{R})$ is a superlinear and subcritical function, and $\Omega$ is a domain in $\mathbb{R}^2$. We study the existence of positive solutions with respect to the values of the parameters $\alpha$ and $\lambda$, and according that $0\in\Omega$ or $0\in\partial\Omega$, and that $\Omega$ is an exterior domain or not.
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