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Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity. (English) Zbl 1142.35082

The authors consider the nonlinear Schrödinger problem -ε 2 Δu+V(x)u=K(x)u p , u>0, posed in n , where ε>0, 1<p<(n+2)/(n-2). Here V and K are smooth bounded and positive potentials. The authors assume that A 0 /(1+|x| α )V(x)A 1 and 0K(x)k/(1+|x| β ) on n with some hypotheses on α and β. They are looking for generalized solutions of this problem which belong to W 1,2 ( n ). They introduce the function Q(x)=V θ (x)K 2/(p-1) (x), with θ=(p+1)/(p-1)-n/2.

In the main result, the authors prove that if x 0 is an isolated stable stationary point of Q, for ε small enough, then, the Schrödinger problem has a finite energy solution which concentrates at x 0 . For the proof, the authors quote the previous result [see A. Ambrosetti, V. Felli and A. Malchiodi, J. Eur. Math. Soc. 7, No. 1, 117–144 (2005; Zbl 1064.35175)], where the existence was proved for the problem, assuming other hypotheses. The authors introduce some change of variables and a truncated nonlinearity which lead to an energy functional I ε defined on a Hilbert space E. The authors then use functional analysis arguments and intricate computations.

35Q55NLS-like (nonlinear Schrödinger) equations
35D05Existence of generalized solutions of PDE (MSC2000)
58E05Abstract critical point theory
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