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On Coron problems with Choquard term and mixed operator

Jacques Giacomoni, Tuhina Mukherjee, Lovelesh Sharma

Abstract

In this article, we study a Coron-type problem involving a critical Choquard nonlinearity driven by a mixed operator combining the Laplacian and fractional Laplacian. In annular-type domains, we prove the existence of nontrivial positive solutions when the inner hole is sufficiently small. Using variational methods and concentration compactness arguments, we establish a global compactness result for Palais- Smale sequences and obtain high-energy solutions using topological methods. We also derive regularity results for weak solutions.

On Coron problems with Choquard term and mixed operator

Abstract

In this article, we study a Coron-type problem involving a critical Choquard nonlinearity driven by a mixed operator combining the Laplacian and fractional Laplacian. In annular-type domains, we prove the existence of nontrivial positive solutions when the inner hole is sufficiently small. Using variational methods and concentration compactness arguments, we establish a global compactness result for Palais- Smale sequences and obtain high-energy solutions using topological methods. We also derive regularity results for weak solutions.

Paper Structure

This paper contains 6 sections, 23 theorems, 235 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Regularity of weak solutions
  4. Palais-Smale Sequences and Global Compactness
  5. Positive solutions on annular-shaped domains
  6. Proof of Theorem \ref{['mainthm']}.

Key Result

Proposition 1.1

lieb2001analysis Let $t, r > 1$ and $0 < \mu < n$ with $\frac{1}{t} + \frac{\mu}{n} + \frac{1}{r} = 2$, $g \in L^{t}(\mathbb{R}^n)$ and $h \in L^{r}(\mathbb{R}^n)$. There exists a sharp constant $C(t,n,\mu,r)$, independent of $g,h$such that If $t = r = \frac{2n}{2n-\mu}$, then $C(t,n,\mu,r) = C(n,\mu) = \pi^{\mu/2} \frac{\Gamma\left(\frac{n-\mu}{2}\right)}{\Gamma\left(n-\frac{\mu}{2}\right)} \lef

Figures (1)

  • Figure 1: Geometric condition on the domain $\Omega$. The annulus $\{R_1 < |x| < R_2\}$ (gray region) is contained within $\Omega$ (blue boundary). The ball $\{ |x| < R_1 \}$ (red label, blue boundary) is not entirely contained in $\overline{\Omega}$.

Theorems & Definitions (43)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 33 more