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

Abstract

We study the existence/nonexistence of positive solution of $${\Delta }^{2}u-\mu \frac{u}{|x{|}^4}=\frac{|u{|}^{q_{\beta }-2}u}{|x{|}^{\beta }}\text{in Ω,}$$ when $\Omega$ is a bounded domain and $N\ge 5$, $q_{\beta }=\frac{2(N-\beta )}{N-4}$, $0\le \beta <4$ and $0\le \mu <(\frac{N(N-4)}{4}{)}^2$. 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 ${\mu }_0<(\frac{N(N-4)}{4}{)}^2$ and $N\ge 8$. Different behavior of PS sequences have been obtained depending on $\beta =0$ or $\beta >0$.

21 pages

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