Asymptotic profiles for Choquard equations with combined attractive nonlinearities

Abstract

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation $$-\Delta u+\epsilon u=(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u+|u{|}^{q-2}u\mathrm{in}{\mathbb{R}}^N,$$ where $N\ge 3$ is an integer, $p\in [\frac{N+\alpha }{N},\frac{N+\alpha }{N-2}]$, $q\in (2,\frac{2N}{N-2}]$, $I_{\alpha }$ is the Riesz potential and $\epsilon >0$ is a parameter. We show that as $\epsilon \to 0$ (resp. $\epsilon \to \infty$), after a suitable rescaling the ground state solutions of $(P_{\epsilon })$ converge in $H^1({\mathbb{R}}^N)$ to a particular solution of some limit equations. We also establish a sharp asymptotic characterisation of such a rescaling, and the exact asymptotic behaviours of $u_{\epsilon }(0),\Vert \nabla u_{\epsilon }{\Vert }_2^2,\Vert u_{\epsilon }{\Vert }_2^2,{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast |u_{\epsilon }{|}^p)|u_{\epsilon }{|}^p$ and $\Vert u_{\epsilon }{\Vert }_q^q$, which depend in a non-trivial way on the exponents $p,q$ and the space dimension $N$. We also discuss a connection of our results with an associated mass constrained problem with normalization constraint ${\int }_{{\mathbb{R}}^N}|u{|}^2=c^2$. As a consequence of the main results, we obtain the existence, multiplicity and exact asymptotic behaviour of positive normalized solutions of such a problem as $c\to 0$ and $c\to \infty$.

107pp