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Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators

Shaoxiong Chen, Hichem Hajaiej, Min Yang, Zhipeng Yang

Abstract

We study the Choquard equation involving mixed local and nonlocal operators $$-Δu+(-Δ)^{s}u+V(x)u=(\frac{1}{|x|^μ}* F(u))f(u)\quad\text{in }\R^{2},$$ where $s\in(0,1)$, $μ\in(0,2)$, $F(t)=\int_{0}^{t} f(τ)\,dτ$, and $f$ has subcritical exponential growth of Trudinger--Moser type. Under suitable assumptions on the potential $V$ and the nonlinearity $f$, we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions.

Existence of positive and sign-changing solutions for a Choquard equation involving mixed local and nonlocal operators

Abstract

We study the Choquard equation involving mixed local and nonlocal operators where , , , and has subcritical exponential growth of Trudinger--Moser type. Under suitable assumptions on the potential and the nonlinearity , we prove the existence of a least energy positive solution by a Nehari manifold approach. We also establish the existence of a sign-changing solution by means of invariant sets of descending flow. If, in addition, the nonlinearity is odd, then the problem admits infinitely many sign-changing solutions.
Paper Structure (9 sections, 25 theorems, 379 equations)

This paper contains 9 sections, 25 theorems, 379 equations.

Table of Contents

  1. Introduction and main results
  2. The Variational Framework and Some Technical Lemmas
  3. Positive solution with least energy
  4. Existence and multiplicity of sign-changing solutions
  5. Abstract critical point principles
  6. The auxiliary operator and basic estimates
  7. Invariant neighborhoods of the positive and negative cones
  8. A locally Lipschitz pseudo-gradient operator
  9. Proof of Theorem \ref{['Thm1.2']}

Key Result

Theorem 1.1

Assume that $V$ satisfies $(V_1)$--$(V_2)$ and $f$ satisfies $(f_1)$--$(f_5)$. Then eq1.1 admits a least energy positive solution.

Theorems & Definitions (51)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Remark 2.1
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Definition 2.1
  • ...and 41 more