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On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions. (English) Zbl 0879.35053

The authors consider nonlinear Schrödinger equations \[ -h^2\Delta u+V(x)u- K(x)|u|^{p-1}u=0,\quad x\in\mathbb{R}^n.\tag{\(*\)} \] Here \(1<p<(n+ 2)/(n-2)\), \(n>2\), and \(V\) and \(K\) are positive smooth functions with \(K\) bounded and \(V\) bounded from below by a positive constant \(\overline V>0\). They are interested in the existence and concentration of positive ground states (i.e. solutions of \((*)\) with minimal energy among all nontrivial solutions) in the semiclassical limit \(h\to 0\).
In the case of constant \(K(x)\equiv 1\), P. H. Rabinowitz [Z. Angew. Math. Phys. 43, No. 2, 270-291 (1992; Zbl 0763.35087)] proved for small \(h\) existence of ground states of \((*)\), provided \(\liminf_{x\to\infty}V(x)> \inf_{x\in\mathbb{R}^n}V(x)\geq\overline V>0\). In a previous paper, the first author [Commun. Math. Phys. 153, No. 2, 229-244 (1993; Zbl 0795.35118)] showed that in this situation for any sequence of ground states of \((*)\) there is a subsequence which concentrates at a global minimum point of \(V\) as \(h\to 0\). It is mentioned that in the complementing case, where \(V\) is constant and \(K\) is variable, concentration occurs at global maximum points of \(K\).
In the present paper, the authors address the question what happens in the case of nonconstant competing potentials \(V\) and \(K\). They give a sufficient condition for the existence of ground states of \((*)\) for small \(h\), which generalizes Rabinowitz’s condition mentioned above. They find further a function in terms of \(V\) and \(K\), the minimum points of which are the concentration points for the bound states, as \(h\to 0\). Moreover, also more general equations and concentration results for positive bound states (solutions of \((*)\), not necessarily with minimal energy) are investigated.

MSC:

35J60 Nonlinear elliptic equations
35B25 Singular perturbations in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
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