Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation. (English) Zbl 1173.35679
Summary: We study the existence and multiplicity of sign-changing solutions for the Dirichlet problem
\[ \begin{cases} -\varepsilon^2\Delta v+V(x)v=f(v) &\text{ in }\Omega,\\ v=0 &\text{ on }\partial\Omega, \end{cases} \]
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 X. Kang and 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 \(V\).
\[ \begin{cases} -\varepsilon^2\Delta v+V(x)v=f(v) &\text{ in }\Omega,\\ v=0 &\text{ on }\partial\Omega, \end{cases} \]
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 X. Kang and 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:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
Citations:
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