High energy positive solution for mixed and Neumann elliptic problems with critical nonlinearity
The authors consider the problem \[ -\Delta u+ a(x)u= u^\gamma,\quad u>0\quad\text{in }\Omega,\quad u= 0\quad\text{on }\Gamma_0,\quad {\partial u\over\partial n}= 0\quad\text{on }\Gamma_1, \] where \(\Omega\subset\mathbb{R}^N\), \(\gamma= 2N/(N-2)\), \(\Gamma_1\neq\emptyset\), \(a\in L^{N/2}(\Omega)\) is a nonnegative function. They prove existence and multiplicity results for a special class of functions \(a(x)\).
scientific article; zbMATH DE number 1038436
scientific article