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Existence and concentration of ground state solutions for an exponentially critical Choquard equation involving mixed local-nonlocal operators

Shaoxiong Chen, Min Yang, Zhipeng Yang

TL;DR

This work proves the existence and semiclassical concentration of positive ground state solutions for a two-dimensional mixed local-nonlocal Choquard equation with exponential critical growth. By developing a variational framework on $W_{\varepsilon}$, employing the Trudinger--Moser inequality and Hardy–Littlewood–Sobolev convolution estimates, it shows ground states exist for small $\varepsilon$ and concentrate near the minima of the potential $V$. A comparison with an autonomous limit problem yields precise energy-level estimates and a clear concentration description as $\varepsilon\to0^+$. The results extend mixed-operator and nonlocal Choquard analyses to the critical exponential regime in $\mathbb{R}^2$, providing rigorous insight into semiclassical behavior of such systems.

Abstract

We study the Choquard equation involving mixed local and nonlocal operators \[-\varepsilon^{2}Δu+\varepsilon^{2s}(-Δ)^{s}u+V(x)u=\varepsilon^{μ-2}\left(\frac{1}{|x|^μ}*F(u)\right)f(u)\quad \text{in }\R^{2},\] where \(\varepsilon>0\), \(s\in(0,1)\), \(0<μ<2\), \(f\) has Trudinger--Moser critical exponential growth, and \(F(t)=\int_{0}^{t}f(τ)\,dτ\). By variational methods, combined with the Trudinger--Moser inequality and compactness arguments adapted to the critical growth and the nonlocal interaction term, we prove the existence of ground state solutions and describe their concentration behavior as \(\varepsilon\to0^{+}\).

Existence and concentration of ground state solutions for an exponentially critical Choquard equation involving mixed local-nonlocal operators

TL;DR

This work proves the existence and semiclassical concentration of positive ground state solutions for a two-dimensional mixed local-nonlocal Choquard equation with exponential critical growth. By developing a variational framework on , employing the Trudinger--Moser inequality and Hardy–Littlewood–Sobolev convolution estimates, it shows ground states exist for small and concentrate near the minima of the potential . A comparison with an autonomous limit problem yields precise energy-level estimates and a clear concentration description as . The results extend mixed-operator and nonlocal Choquard analyses to the critical exponential regime in , providing rigorous insight into semiclassical behavior of such systems.

Abstract

We study the Choquard equation involving mixed local and nonlocal operators where , \(s\in(0,1)\), , has Trudinger--Moser critical exponential growth, and \(F(t)=\int_{0}^{t}f(τ)\,dτ\). By variational methods, combined with the Trudinger--Moser inequality and compactness arguments adapted to the critical growth and the nonlocal interaction term, we prove the existence of ground state solutions and describe their concentration behavior as .
Paper Structure (6 sections, 21 theorems, 396 equations)

This paper contains 6 sections, 21 theorems, 396 equations.

Table of Contents

  1. Introduction and the main results
  2. Preliminary results
  3. Estimates for the minimax level
  4. Ground state solution of the autonomous problem
  5. Ground state solution of the singularly perturbed problem
  6. Concentration phenomena

Key Result

Theorem 1.1

Assume that $(f_1)$--$(f_6)$ and $(V)$ hold. Then there exists $\varepsilon_{0}>0$ such that, for every $\varepsilon\in(0,\varepsilon_{0})$, problem eq1.1 admits at least one positive ground state solution $u_{\varepsilon}$. Moreover, if $\eta_{\varepsilon}\in\mathbb{R}^{2}$ is a global maximum poin

Theorems & Definitions (37)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 27 more