Multiple bound states for the Schrödinger-Poisson problem. (English) Zbl 1188.35171
Summary: We study the problem
\[ -\Delta u+u+V(x)u= u^p, \qquad -\Delta V=\lambda u^2, \quad \lim_{|x|\to+\infty} V(x)=0, \]
where \(u,V:\mathbb R^3\to\mathbb R\) are radial functions, \(\lambda>0\) and \(1<p<5\). We give multiplicity results, depending on \(p\) and on the parameter \(\lambda\).
\[ -\Delta u+u+V(x)u= u^p, \qquad -\Delta V=\lambda u^2, \quad \lim_{|x|\to+\infty} V(x)=0, \]
where \(u,V:\mathbb R^3\to\mathbb R\) are radial functions, \(\lambda>0\) and \(1<p<5\). We give multiplicity results, depending on \(p\) and on the parameter \(\lambda\).
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35J60 | Nonlinear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
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