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Zbl 1173.35679
D'Aprile, Teresa; Pistoia, Angela
Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 4, 1423-1451 (2009). ISSN 0294-1449

Summary: We study the existence and multiplicity of sign-changing solutions for the Dirichlet problem $$\cases -\varepsilon^2\Delta v+V(x)v=f(v) &\text{ in }\Omega,\\ v=0 &\text{ on }\partial\Omega, \endcases$$ where $\varepsilon$ is a small positive parameter, $\Omega$ is a smooth, possibly unbounded, domain, $f$ is a superlinear and subcritical nonlinearity, $V$ is a positive potential bounded away from zero. No symmetry on $V$ or on the domain $\Omega$ is assumed. It is known by {\it X. Kang} and {\it J. Wei} [Adv. Differ. Equ. 5, No. 7--9, 899--928 (2000; Zbl 1217.35065)] that this problem has positive clustered solutions with peaks approaching a local maximum of $V$. The aim of this paper is to show the existence of clustered solutions with mixed positive and negative peaks concentrating at a local minimum point, possibly degenerate, of $V$.

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
35B40 Asymptotic behavior of solutions of PDE
35J20 Second order elliptic equations, variational methods
35J60 Nonlinear elliptic equations
35J65 (Nonlinear) BVP for (non)linear elliptic equations

Keywords: Schrödinger equation; clusters; finite-dimensional reduction; max-min argument

Citations: Zbl 1217.35065

Cited in: Zbl 1228.35019

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