## 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 $$V(x_{\varepsilon,i}) \to \inf_{\Lambda_i} V$$.

### MSC:

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

### Keywords:

multibump solution; positive solution; local maxima
Full Text:

### References:

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