Electron. J. Differential Equations, Vol. 2018 (2018), No. 194, pp. 1-17.

Positive solution for Henon type equations with critical Sobolev growth

Kazune Takahashi

Abstract:
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.

Submitted April 2, 2018. Published November 28, 2018.
Math Subject Classifications: 35J20, 35J60, 35J61, 35J91.
Key Words: Critical Sobolev exponent; Henon equation; mountain pass theorem; Talenti function.

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Kazune Takahashi
Graduate School of Mathematical Sciences
The University of Tokyo
3-8-1 Komaba Meguroku Tokyo 153-8914, Japan
email: kazunetakahashi@gmail.com

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