On some nonlinear equations involving the \(p\)-Laplacian with critical Sobolev growth
The author considers nonlinear elliptic equations \[ -\text{div}(|\nabla|^{p- 2}\nabla u)+ a(x)|u|^{p-2} u= f(x)|u|^{p^*- 2}\quad\text{in }\Omega, \] \[ u\not\equiv 0,\quad u= 0\quad\text{on }\partial\Omega \] in which \(p^*= np/(n- p)\) is the critical Sobolev exponent for the embedding of \(W^{1,p}\) in \(L^p\). Conditions are found to be satisfied by the domain \(\Omega\) and function \(f(x)\) in order that there exist a positive solution \(u\in W^{1,p}_0(\Omega)\cap C^{1,\alpha}(\overline\Omega)\cap C^{1,\alpha}(\Omega)\) to the equation (Theorem 1) as well as conditions under which a modal solution \(u\in W^{1,p}(\Omega)\) exists (Theorem 2). In the proof of both theorems modified veriational techniques are used.
scientific article; zbMATH DE number 1449026