## 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

### Keywords:

positive solutions; semilinear elliptic equation
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