Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in \({\mathbb{R}}^ 2\)
Consider the radially symmetric semilinear Dirichlet problem \[ (1)\quad -\Delta u=f(| x|,u),\quad x\in B,\quad u|_{\partial B}=0, \] where B is the unit disc in \({\mathbb{R}}^ 2\) centred at the origin. Sufficient conditions on f are given for (1) to have a positive radial solution u in B. Theorems are obtained for superlinear nonlinearities f of subcritical, critical, and supercritical type. A constrained variational method is employed involving approximation of the energy functional by a functional satisfying the Palais-Smale condition.
scientific article; zbMATH DE number 4115120
scientific article