JournalsrmiVol. 35, No. 6pp. 1885–1924

Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well

  • Silvia Cingolani

    Università degli Studi di Bari Aldo Moro, Italy
  • Kazunaga Tanaka

    Waseda University, Tokyo, Japan
Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well cover

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Abstract

We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation

ε2Δv+V(x)v=1εα(IαF(v))f(v)in RN,-\varepsilon^2\Delta v+V(x) v = \frac{1}{\varepsilon^\alpha}\,(I_\alpha*F(v))f(v) \quad \hbox{in } \mathbb{R}^N,

where N3N\geq 3, α(0,N)\alpha\in (0,N), Iα(x)=Aα/xNαI_\alpha(x)={A_\alpha/ |x|^{N-\alpha}} is the Riesz potential, FC1(R,R)F\in C^1(\mathbb{R},\mathbb{R}), F(s)=f(s)F'(s) = f(s) and ε>0\varepsilon>0 is a small parameter.

We develop a new variational approach and we show the existence of a family of solutions concentrating, as ε0\varepsilon\to 0, to a local minima of V(x)V(x) under general conditions on F(s)F(s). Our result is new also for f(s)=sp2sf(s)=|s|^{p-2}s and applicable for p(N+αN,N+αN2)p\in ({N+\alpha\over N}, {N+\alpha\over N-2}). Especially, we can give the existence result for locally sublinear case p(N+αN,2)p\in ({N+\alpha\over 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 cupl(K)+1{\rm cupl}(K)+1 solutions concentrating around~KK as ε0\varepsilon\to 0, where KΩK\subset \Omega is the set of minima of V(x)V(x) in a bounded potential well Ω\Omega, that is, m0infxΩV(x)<infxΩV(x)m_0 \equiv \inf_{x\in \Omega} V(x) < \inf_{x\in \partial\Omega}V(x) and K={xΩ;K=\{x\in\Omega; V(x)=m0}\, V(x)=m_0\}.

Cite this article

Silvia Cingolani, Kazunaga Tanaka, Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well. Rev. Mat. Iberoam. 35 (2019), no. 6, pp. 1885–1924

DOI 10.4171/RMI/1105