Some existence results for superlinear elliptic boundary value problems involving critical exponents
This paper deals with the superlinear elliptic boundary value problem \[ - \Delta u-\lambda u=u | u|^{2^*-2}\quad in\quad \Omega;\quad u|_{\partial \Omega}=0, \] where \(\Omega\) is a smoothly bounded domain in \(R^ n\), \(n>2\), \(\lambda\in R\), \(2^*=2n/(n-2)\) and \(2^*\) is the limiting Sobolev exponent for the embedding \(H^ 1_ 0(\Omega)\to L^ p(\Omega)\). Solving this problem is equivalent to finding critical points in \(H^ 1_ 0(\Omega)\) of the energy functional \[ I_{\lambda}(u)=(1/2)\int_{\Omega}(| \nabla u|^ 2-\lambda | u|^ 2)dx-(1/2)\int_{\Omega}| u|^{2^*} dx. \] First, based on the global compactness theorem for the problem (1), the compactness question is discussed. Then some results about the existence and multiplicity of solutions to the above problem are obtained.
superlinear elliptic boundary value problem
scientific article; zbMATH DE number 3994107
scientific article