Existence of positive solutions of the equation \(-\Delta u+a(x)u=u^{(N+2)/(N-2)}\) in \({\mathbb{R}}^ N\)
Consider the following problem: find u such that \[ (*)\;u(x)>0,\;-\Delta u+a(x)u=| u|^{2^*-2}u\text{ on } \Omega,\;u\in {\mathcal D}^{1,2}(\Omega), \] where \(\Omega \subseteq {\mathbb{R}}^ N\), \(N\geq 3\), \(2^*=2N/(n-2)\), \(a(x)\geq 0\) and \({\mathcal D}^{1,2}(\Omega)\) is a closure of \(C^{\infty}_ 0(\Omega)\) with respect to the standard Sobolev norm of \(H^ 1(\Omega)\). Then, they prove the following Theorem. If \(\Omega ={\mathbb{R}}^ N\) and \(a(x)\geq 0\forall x\in {\mathbb{R}}^ N\) and \(a(x)\geq \nu >0\) in a neighborhood of a point \(\bar x,\) \(\exists p_ 1<N/2\) and \(p_ 2>N/2\) and for \(N=3\), \(p_ 2<3\), such that \(a(x)\in L^ p\forall p\in [p_ 1,p_ 2]\), \(| a|_{L^{N/2}}<S(2^{2/N}-1)\), where \[ S=\inf \{\int_{{\mathbb{R}}^ N}| \nabla u|^ 2 dx/(\int_{{\mathbb{R}}^ N}| u|^{2^*} dx)^{2/2^*};\;u\in {\mathcal D}^{1,2}({\mathbb{R}}^ N)\}, \] then the problem (*) has at least one positive solution.
scientific article; zbMATH DE number 4157046
scientific article