Multiple positive solutions for a semilinear equation with prescribed singularity
The author deals with the study of the existence and multiplicity of positive solutions for the following problem \[ -\Delta u-\mu V(x)u= K(x)|u|^{2^*-2} u+\theta h(x),\quad u\in H^1_0(\Omega),\tag{1} \] where \(\Omega\subset\mathbb{R}^N\) \((N\geq 3)\) is an open bounded domain with smooth boundary \(\partial\Omega\), \(0\leq\mu< \widetilde\mu= {(N-2)^2\over 4}\) and \(\widetilde\mu\) is the best constant in the Hardy inequality. \(2^*= {2N\over N-2}\) is the so-called critical Sobolev exponent. The weight \(V(x)\) has finitely many singular points. The author adresses to the following question: If \(V(x)\) has finitely many singular points, say \(m\) in \(\Omega\), does it produce at least \(m\) really positive solutions to (1)? The result of the author shows that the answer is affirmative. To this end, the author uses a variational method.
scientific article; zbMATH DE number 2159780