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

The authors consider the nonlinear Schrödinger problem \(-\varepsilon ^{2}\Delta u+V(x)u=K(x)u^{p}\), \(u>0\), posed in \(\mathbb{R}^{n}\), where \( \varepsilon >0\), \(1<p<(n+2)/(n-2)\). Here \(V\) and \(K\) are smooth bounded and positive potentials. The authors assume that \(A_{0}/(1+| x| ^{\alpha })\leq V(x)\leq A_{1}\) and \(0\leq K(x)\leq k/(1+| x| ^{\beta })\) on \(\mathbb{R}^{n}\) with some hypotheses on \(\alpha \) and \(\beta \). They are looking for generalized solutions of this problem which belong to \(W^{1,2}(\mathbb{R}^{n})\). They introduce the function \(Q(x)=V^{\theta }(x)K^{2/(p-1)}(x)\), with \(\theta =(p+1)/(p-1)-n/2\).
In the main result, the authors prove that if \(x_{0}\) is an isolated stable stationary point of \(Q\), for \(\varepsilon \) 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 \( I_{\varepsilon }\) defined on a Hilbert space \(E\). The authors then use functional analysis arguments and intricate computations.


35Q55 NLS equations (nonlinear Schrödinger equations)
35D05 Existence of generalized solutions of PDE (MSC2000)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces


Zbl 1064.35175
Full Text: DOI


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