Symmetric positive solutions to nonlinear Choquard equations with potentials

Abstract

Existence results for a class of Choquard equations with potentials are established. The potential has a limit at infinity and it is taken invariant under the action of a closed subgroup of linear isometries of $\mathbb {R}^N$. As a consequence, the positive solution found will be invariant under the same action. Power nonlinearities with exponent greater or equal than two or less than two will be handled. Our results include the physical case.

Appendix A. Technical Lemma

Lemma A.1

(Lemma 4.1 in [8]) Let $u, v : \mathbb {R}^N \rightarrow \mathbb {R}$ be two continuous functions such that

$$\begin{aligned} u(x) \le C(1+\left| x\right| )^a, \quad v(x) \le C(1+\left| x\right| )^{a'} \end{aligned}$$

as $\left| x\right| \rightarrow \infty$, where $a, a'<0$ such that $a+a'<-N$. Let $\xi \in \mathbb {R}^N$ such that $\left| \xi \right| \rightarrow \infty$. We denote $u_{\xi }(x)=u(x-\xi )$. Then the following asymptotic estimate holds:

$$\begin{aligned} \int _{\mathbb {R}^N} u_{\xi } v\le C \left| \xi \right| ^{\tau } \end{aligned}$$

where $\tau = \max \{ a, a', a+a' +N\}<0$.

Lemma A.2

(Lemma 3.7 in [3]) Let $u, v : \mathbb {R}^N \rightarrow \mathbb {R}$ be two positive continuous radial functions such that

$$\begin{aligned} u(x) \sim \left| x\right| ^a e^{-b\left| x\right| }, \quad v(x) \sim \left| x\right| ^{a'} e^{-b'\left| x\right| } \end{aligned}$$

as $\left| x\right| \rightarrow \infty$, where $a, a' \in \mathbb {R}$, and $b, b' >0$. Let $\xi \in \mathbb {R}^N$ such that $\left| \xi \right| \rightarrow \infty$. We denote $u_{\xi }(x)=u(x-\xi )$. Then the following asymptotic estimates hold:

• (i) If $b < b'$,

  $$\begin{aligned} \int _{\mathbb {R}^N} u_{\xi } v\sim e^{-b\left| \xi \right| } \left| \xi \right| ^{a}. \end{aligned}$$

  A similar expression holds if $b > b'$, by replacing a and b with $a'$ and $b'$.

• (ii) If $b=b'$, suppose that $a \ge a'$. Then:

  $$\begin{aligned} \int _{\mathbb {R}^N} u_{\xi } v\sim {\left\{ \begin{array}{ll} e^{-b\left| \xi \right| } \left| \xi \right| ^{a+a'+\frac{N+1}{2}} &{} \text { if } a' > -\frac{N+1}{2},\\ e^{-b\left| \xi \right| } \left| \xi \right| ^{a} \log |\xi | &{} \text { if } a' = -\frac{N+1}{2},\\ e^{-b\left| \xi \right| } \left| \xi \right| ^{a}&{} \text { if } a' < -\frac{N+1}{2}. \end{array}\right. } \end{aligned}$$