\(W^{1,p}\) versus \(C^1\) local minimizers and multiplicity results for quasilinear elliptic equations.
The paper considers the existence and multiplicity of positive (negative) solutions for the equation \[ \begin{gathered} \text{div}(|\nabla u(x)|^{p-2}\nabla u(x))+ \lambda| u(x)|^{q-1} u(x)+ g(u(x))= 0\quad\text{in }\Omega,\\ u\in W^{1,p}_0(\Omega),\end{gathered}\tag{1} \] where \(\Omega\subset\mathbb{R}^n\), \(n\geq 3\), is a bounded smooth domain, \(p> 1\), \(0< q< p=1\), \(g(u)u\geq 0\) and \(g\) has a growth larger than or equal to \(p-2\). Most results are obtained for \(p> 2\) and a typical one is that there exists \(\Lambda> 0\) such that for \(\lambda>\Lambda\) the equation (1) has no positive solutions, but for \(0<\lambda<\Lambda\) there are at least two positive solutions.
scientific article; zbMATH DE number 2003886
scientific article