Caffarelli-Kohn-Nirenberg type equations of fourth order with the critical exponent and Rellich potential

TL;DR

This paper investigates the existence and nonexistence of positive solutions for a class of fourth-order elliptic equations with critical exponents and Rellich potential, highlighting the influence of domain topology and parameters.

Contribution

It establishes nonexistence results in star-shaped domains and existence results in bounded domains with nontrivial topology for certain parameter ranges.

Findings

Nonexistence of positive solutions in star-shaped domains.

Existence of positive solutions in bounded domains with nontrivial topology.

Different behaviors of Palais-Smale sequences depending on eta=0 or eta>0.

Abstract

We study the existence/nonexistence of positive solution of $$ {\Delta^2u-\mu\frac{u}{|x|^4}=\frac{|u|^{q_{\beta}-2}u}{|x|^{\beta}}\quad\textrm{in $\Omega$,}} $$ when $\Omega$ is a bounded domain and $N\geq 5$, $q_{\beta}=\frac{2(N-\beta)}{N-4}$, $0\leq \beta<4$ and $0\leq\mu<\big(\frac{N(N-4)}{4}\big)^2$. We prove the nonexistence result when $\Omega$ is an open subset of $\mathbf R^N$ which is star shaped with respect to the origin. We also study the existence of positive solution in $\Omega$ when $\Omega$ is a bounded domain with non trivial topology and $\beta=0$, $\mu\in(0,\mu_0)$, for certain $\mu_0<\big(\frac{N(N-4)}{4}\big)^2$ and $N\geq 8$. Different behavior of PS sequences have been obtained depending on $\beta=0$ or $\beta>0$.