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Existence of infinitely many solutions for a critical Hartree type equation with potential: local Pohožaev identities methods

Daniele Cassani, Minbo Yang, Xinyun Zhang

Abstract

This paper deals with the following equation $$-Δu =K(|x'|, x'')\Big(|x|^{-α}\ast (K(|x'|, x'')|u|^{2^{\ast}_α})\Big) |u|^{2^{\ast}_α-2}u\quad\mbox{in}\ \mathbb{R}^N,$$ where $N\geq5$, $α>5-\frac{6}{N-2}$, $2^{\ast}_α=\frac{2N-α}{N-2}$ is the so-called upper critical exponent in the Hardy-Littlewood-Sobolev inequality and $K(|x'|, x'')$, where $(x',x'')\in \mathbb{R}^2\times\mathbb{R}^{N-2}$, is bounded and nonnegative. Under proper assumptions on the potential function $K$, we obtain the existence of infinitely many solutions for the nonlocal critical equation by using a finite dimensional reduction argument and local Pohožaev identities. It is a remarkable fact that the order of the Riesz potential influences the existence/non-existence of solutions.

Existence of infinitely many solutions for a critical Hartree type equation with potential: local Pohožaev identities methods

Abstract

This paper deals with the following equation where , , is the so-called upper critical exponent in the Hardy-Littlewood-Sobolev inequality and , where , is bounded and nonnegative. Under proper assumptions on the potential function , we obtain the existence of infinitely many solutions for the nonlocal critical equation by using a finite dimensional reduction argument and local Pohožaev identities. It is a remarkable fact that the order of the Riesz potential influences the existence/non-existence of solutions.
Paper Structure (5 sections, 19 theorems, 224 equations)

This paper contains 5 sections, 19 theorems, 224 equations.

Table of Contents

  1. Introduction and main results
  2. Finite-dimensional reduction
  3. The energy expansion
  4. Proof of the main result
  5. Some basic estimates

Key Result

Proposition 1.1

Let $t,\,r>1$ and $0<\alpha<N$ be such that $\frac{1}{t}+\frac{\alpha}{N}+\frac{1}{r}=2$. Then, there exists a constant $C(N,\alpha,t)$ such that, for $f\in L^{t}(\mathbb{R}^N)$ and $h\in L^{r}(\mathbb{R}^N)$, If $t=r=2N/(2N-\alpha)$, then and the equality holds if and only if $f\equiv Ch$ and for some $A\in \mathbb{R}$, $0\neq\gamma\in\mathbb{R}$ and $a\in \mathbb{R}^{N}$.

Theorems & Definitions (32)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 22 more