Multi-peak bound states for nonlinear Schrödinger equations. (English) Zbl 0901.35023

The authors consider the equation \[ \varepsilon^2 \Delta u-V(x)u +f(u)=0 \] in a smooth, possibly unbounded domain \(\Omega \subset \mathbb{R}^n\). Here \(\varepsilon\) is a small parameter, \(f\) a superlinear function, and the potential \(V\) is positive, locally Hölder continuous, and bounded away from zero. This equation arises from studying standing wave solutions of nonlinear Schrödinger equations. Assume that there are \(K\) disjoint bounded domains \(\Lambda_i \Subset \Omega\) such that \(\inf_{\Lambda_i} V<\inf_{\partial \Lambda_i} V\). If the nonlinearity \(f\) is appropriate, then for all sufficiently small \(\varepsilon>0\) the authors prove the existence of a positive solution \(u_\varepsilon \in H^1_0 (\Omega)\) possessing exactly \(K\) local maxima \(x_{\varepsilon,i} \in \Lambda_i\). Moreover, for certain positive constants \(\alpha,\beta\), \[ u_\varepsilon (x)\leq \alpha \exp \left(-{\beta \over \varepsilon} | x-x_{\varepsilon,i} |\right) \text{ in } \Omega \setminus \bigcup_{j\neq i} \Lambda_j \] and \(V(x_{\varepsilon,i}) \to \inf_{\Lambda_i} V\).


35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
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