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Kazune Takahashi
Electronic journal of differential equations, 11/2018, Letnik: 2018, Številka: 194Journal Article
We investigate the Henon type equation involving the critical Sobolev exponent with Dirichret boundary condition $$ - \Delta u = \lambda \Psi u + | x |^\alpha u^{2^*-1} $$ in $\Omega$ included in a unit ball, under several conditions. Here, $\Psi$ is a non-trivial given function with $0 \leq \Psi \leq 1$ which may vanish on $\partial \Omega$. Let $\lambda_1$ be the first eigenvalue of the Dirichret eigenvalue problem $-\Delta \phi = \lambda \Psi \phi$ in $\Omega$. We show that if the dimension $N \geq 4$ and $0 < \lambda < \lambda_1$, there exists a positive solution for small $\alpha > 0$. Our methods include the mountain pass theorem and the Talenti function.
