On a min-max procedure for the existence of a positive solution for certain scalar field equations in \({\mathbb{R}}^ N\). (English) Zbl 0731.35036

The problem of the existence of positive solutions \(u\in H^ 1({\mathbb{R}}^ n)\) of the semilinear elliptic equation \[ -\Delta u+u- q(x)| u|^{p-1}u=0 \] is considered. Under suitable conditions on p,n and \(q(x)\in L^{\infty}({\mathbb{R}}^ n)\) a min-max procedure for the corresponding variational functional J which is based on topological methods shows the existence of at least one positive solution.


35J65 Nonlinear boundary value problems for linear elliptic equations
47H11 Degree theory for nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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