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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.

MSC:

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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