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On a class of nonlinear Schrödinger equations. (English) Zbl 0763.35087

This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations

iψ t =- 2 2mΔψ+V(x)ψ-γ|ψ| p-1 ψ,(1)

where x n , 1<p<(n+2)/(n-2). Making a standing wave ansatz the problem reduces to that of studying the semilinear elliptic equation. In fact, a more general semilinear elliptic PDE


is considered. Using “mountain pass” and comparison arguments, the author gets existence of nontrivial solutions uW 1,2 ( n ) for (2), under some additional assumptions on b(x) and f(x,u).

35Q55NLS-like (nonlinear Schrödinger) equations
35Q51Soliton-like equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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