Calculus of Variations and Partial Differential Equations | 2019
Existence and uniqueness of solutions for Choquard equation involving Hardy–Littlewood–Sobolev critical exponent
Abstract
In this paper, we first prove that each positive solution of $$\\begin{aligned} -\\Delta u=\\big (I_{\\alpha }*|u|^{2_{\\alpha }^{*}}\\big )|u|^{2_{\\alpha }^*-2}u, \\quad u\\in \\mathcal {D}^{1,2}({\\mathbb {R}}^{N}) \\end{aligned}$$\n is radially symmetric, monotone decreasing about some point and has the form $$\\begin{aligned} c_\\alpha \\left( \\frac{t}{t^2+|x-x_0|^2}\\right) ^{\\frac{N-2}{2}}, \\end{aligned}$$\n where $$0<\\alpha <N$$\n if $$N=3$$\n or 4, and $$N-4\\le \\alpha <N$$\n if $$N\\ge 5$$\n , $${2_{\\alpha }^{*}}:=\\frac{N+\\alpha }{N-2}$$\n is the upper Hardy–Littlewood–Sobolev critical exponent, $$t>0$$\n is a constant and $$c_\\alpha >0$$\n depends only on $$\\alpha $$\n and N. Based on this uniqueness result, we then study the following nonlinear Choquard equation $$\\begin{aligned} -\\Delta u+V(x)u=\\left( I_{\\alpha }*|u|^{2_{\\alpha }^{*}}\\right) |u|^{2_{\\alpha }^*-2}u, \\quad u\\in \\mathcal {D}^{1,2}({\\mathbb {R}}^{N}). \\end{aligned}$$\n By using Lions’ Concentration-Compactness Principle, we obtain a global compactness result, i.e. we give a complete description for the Palais–Smale sequences of the corresponding energy functional. Adopting this description, we are succeed in proving the existence of at least one positive solution if $$\\Vert V(x)\\Vert _{L^\\frac{N}{2}}$$\n is suitable small. This result generalizes the result for semilinear Schrodinger equation by Benci and Cerami (J Funct Anal 88:90–117, 1990) to Choquard equation.