On the number of positive symmetric solutions of a nonautonomous semilinear elliptic problem
The author considers the problem \[ -\Delta u+a(x)u=K(x)|u|^{p-2}u \quad \text{in }\Omega ,\qquad u>0\quad \text{in } \Omega ,\qquad u=0\quad \text{on } \partial \Omega , \] where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^n\), \(a\), \(K\) are continuous, \(a\geq 0\), \(K>0,\) and \(2<p\leq 2n/(n-2)\). Moreover, \(\Omega\), \(a\), \(K\) are supposed to be invariant under the action of a closed subgroup \(G\) of the orthogonal group of \(\mathbb{R}^n\). The lower estimate of the number of \(G\)-invariant solutions is given (for \(p\) close to \(2n/(n-2)\)) in terms of the Lusternik-Shnirelmann category of the set of \(G\)-maximal values of \(K\).
scientific article; zbMATH DE number 1512513