<p>In this paper, we investigate the following fractional nonlinear Choquard equation: <Equation ID="Equa"> <EquationSource Format="TEX">\(\begin{aligned} \left\{ \begin{array}{ll} {{\,\mathrm{\varepsilon }\,}}^{2s} (-\Delta )^{s} v +V(x) v= {{\,\mathrm{\varepsilon }\,}}^{-\alpha } (I_{\alpha }*F(v)) F'(v) \text{ in } \mathbb {R}^{N},\\ v\in H^{s}(\mathbb {R}^{N}), \,\, v&gt;0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}\)</EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({{\,\mathrm{\varepsilon }\,}}&gt;0\)</EquationSource> </InlineEquation> is a small parameter, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(s\in (0, 1)\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(N\ge 2\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((-\Delta )^{s}\)</EquationSource> </InlineEquation> denotes the fractional Laplacian, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(I_{\alpha }\)</EquationSource> </InlineEquation> is the Riesz potential of order <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\alpha \in ((N-4s)_{+}, N)\)</EquationSource> </InlineEquation>. The potential <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(V\in C^{0}(\mathbb {R}^N, (0, +\infty ))\)</EquationSource> </InlineEquation> satisfies <Equation ID="Equb"> <EquationSource Format="TEX">\(\begin{aligned} m_{0}:=\inf _{\Omega }V&lt;\min _{\partial \Omega }V, \end{aligned}\)</EquationSource> </Equation>for some bounded open set <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\Omega \subset \mathbb {R}^N\)</EquationSource> </InlineEquation>. The function <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(F\in C^{1}(\mathbb {R})\)</EquationSource> </InlineEquation> is a nonlinearity of Berestycki–Lions type. By employing suitable variational methods, we establish the existence of at least <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\textrm{cupl}(K)+1\)</EquationSource> </InlineEquation> solutions concentrating around the set <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(K:=\{x\in \Omega : V(x)=m_{0} \}\)</EquationSource> </InlineEquation> as <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\({{\,\mathrm{\varepsilon }\,}}\rightarrow 0^{+}.\)</EquationSource> </InlineEquation></p>

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Semiclassical Analysis for Fractional Choquard Equations with General Nonlinearities: Multiplicity and Concentration

  • Vincenzo Ambrosio

摘要

In this paper, we investigate the following fractional nonlinear Choquard equation: \(\begin{aligned} \left\{ \begin{array}{ll} {{\,\mathrm{\varepsilon }\,}}^{2s} (-\Delta )^{s} v +V(x) v= {{\,\mathrm{\varepsilon }\,}}^{-\alpha } (I_{\alpha }*F(v)) F'(v) \text{ in } \mathbb {R}^{N},\\ v\in H^{s}(\mathbb {R}^{N}), \,\, v>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}\) where \({{\,\mathrm{\varepsilon }\,}}>0\) is a small parameter, \(s\in (0, 1)\) , \(N\ge 2\) , \((-\Delta )^{s}\) denotes the fractional Laplacian, and \(I_{\alpha }\) is the Riesz potential of order \(\alpha \in ((N-4s)_{+}, N)\) . The potential \(V\in C^{0}(\mathbb {R}^N, (0, +\infty ))\) satisfies \(\begin{aligned} m_{0}:=\inf _{\Omega }V<\min _{\partial \Omega }V, \end{aligned}\) for some bounded open set \(\Omega \subset \mathbb {R}^N\) . The function \(F\in C^{1}(\mathbb {R})\) is a nonlinearity of Berestycki–Lions type. By employing suitable variational methods, we establish the existence of at least \(\textrm{cupl}(K)+1\) solutions concentrating around the set \(K:=\{x\in \Omega : V(x)=m_{0} \}\) as \({{\,\mathrm{\varepsilon }\,}}\rightarrow 0^{+}.\)