Existence for critical Hénon type equations. (English) Zbl 1265.35061

Summary: This paper is concerned with the existence of a nontrivial solution for \(-\Delta u=\lambda u+| x| ^{\alpha }| u| ^{2^{*}-2}u\) in \(\Omega \), \(u=0\) on \(\partial \Omega \), where \(\lambda >0\) and \(\Omega \subset \mathbb {R}^n\) is a smooth bounded domain. Let \(\lambda _k\), \(k=1,2,\dots \) be the eigenvalues of the operator \(-\Delta \); we show for \(\lambda _{k}<\lambda <\lambda _{k+1}\) that the above problem possesses at least a solution and each \(\lambda _{k}\) is a bifurcation point.


35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J60 Nonlinear elliptic equations
35J61 Semilinear elliptic equations