Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well. (English) Zbl 1431.35169
Summary: We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation \[-\varepsilon^2\Delta v+V(x) v = \frac{1}{\varepsilon^\alpha}\,(I_\alpha*F(v))f(v) \quad \text{in } \mathbb{R}^N,\] where \(N\geq 3, \alpha\in (0,N), I_\alpha(x)={A_\alpha/ |x|^{N-\alpha}}\) is the Riesz potential, \(F\in C^1(\mathbb{R},\mathbb{R}), F'(s) = f(s)\) and \(\varepsilon>0\) is a small parameter.
We develop a new variational approach and we show the existence of a family of solutions concentrating, as \(\varepsilon\to 0\), to a local minima of \(V(x)\) under general conditions on \(F(s)\). Our result is new also for \(f(s)=|s|^{p-2}s\) and applicable for \(p\in (\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2})\). Especially, we can give the existence result for locally sublinear case \(p\in (\frac{N+\alpha}{N},2)\), which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen.
We also study the multiplicity of positive single-peak solutions and we show the existence of at least \(\mathrm{cupl}(K)+1\) solutions concentrating around \(K\) as \(\varepsilon\to 0\), where \(K\subset \Omega\) is the set of minima of \(V(x)\) in a bounded potential well \(\Omega \), that is, \(m_0 \equiv \inf_{x\in \Omega} V(x) < \inf_{x\in \partial\Omega}V(x)\) and \(K=\{x\in\Omega; \, V(x)=m_0\}\).
We develop a new variational approach and we show the existence of a family of solutions concentrating, as \(\varepsilon\to 0\), to a local minima of \(V(x)\) under general conditions on \(F(s)\). Our result is new also for \(f(s)=|s|^{p-2}s\) and applicable for \(p\in (\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2})\). Especially, we can give the existence result for locally sublinear case \(p\in (\frac{N+\alpha}{N},2)\), which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen.
We also study the multiplicity of positive single-peak solutions and we show the existence of at least \(\mathrm{cupl}(K)+1\) solutions concentrating around \(K\) as \(\varepsilon\to 0\), where \(K\subset \Omega\) is the set of minima of \(V(x)\) in a bounded potential well \(\Omega \), that is, \(m_0 \equiv \inf_{x\in \Omega} V(x) < \inf_{x\in \partial\Omega}V(x)\) and \(K=\{x\in\Omega; \, V(x)=m_0\}\).
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q40 | PDEs in connection with quantum mechanics |
| 35J20 | Variational methods for second-order elliptic equations |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 35B09 | Positive solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
Keywords:
nonlinear Choquard equation; semiclassical states; non-local nonlinearities; positive solutions; potential well; relative cup-lengthReferences:
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