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Multiple solutions for elliptic equations driven by higher order fractional Laplacian

Fuwei Cheng, Xifeng Su, Jiwen Zhang

TL;DR

This work studies a Dirichlet problem driven by the higher-order fractional Laplacian $(-\Delta)^s$ with $s\in(1,2)$, using a variational framework on $\mathcal{H}_0^s$ to address existence and multiplicity under diverse nonlinearities. It proves a positive-energy Mountain Pass solution for superlinear nonlinearities without the Ambrosetti–Rabinowitz condition and a negative-energy solution via Ekeland's principle for a concave–convex nonlinearity with small parameter. Under a symmetry assumption $f(x,-t)=-f(x,t)$, Fountain and Dual Fountain theorems yield infinitely many weak solutions: an unbounded sequence of positive-energy solutions for the superlinear case and two energy-branch sequences (positive and negative energy) for the symmetric concave–convex problem. The results extend variational multiplicity techniques to higher-order nonlocal operators, including cases without AR-type conditions and with symmetry, broadening the landscape of nonlocal polyharmonic elliptic equations.

Abstract

We consider an elliptic partial differential equation driven by higher order fractional Laplacian $(-Δ)^{s}$, $s \in (1,2)$ with homogeneous Dirichlet boundary condition \begin{equation*} \left\{% \begin{array}{ll} (-Δ)^{s} u=f(x,u) & \text{ in }Ω, u=0 & \text{ in } \mathbb{R}^n \setminus Ω. \end{array}% \right. \end{equation*} The above equation has a variational nature, and we investigate the existence and multiplicity results for its weak solutions under various conditions on the nonlinear term $f$: superlinear growth, concave-convex and symmetric conditions and their combinations. The existence of two different non-trivial weak solutions is established by Mountain Pass Theorem and Ekeland's variational principle, respectively. Furthermore, due to Fountain Theorem and its dual form, both infinitely many weak solutions with positive energy and infinitely many weak solutions with negative energy are obtained.

Multiple solutions for elliptic equations driven by higher order fractional Laplacian

TL;DR

This work studies a Dirichlet problem driven by the higher-order fractional Laplacian with , using a variational framework on to address existence and multiplicity under diverse nonlinearities. It proves a positive-energy Mountain Pass solution for superlinear nonlinearities without the Ambrosetti–Rabinowitz condition and a negative-energy solution via Ekeland's principle for a concave–convex nonlinearity with small parameter. Under a symmetry assumption , Fountain and Dual Fountain theorems yield infinitely many weak solutions: an unbounded sequence of positive-energy solutions for the superlinear case and two energy-branch sequences (positive and negative energy) for the symmetric concave–convex problem. The results extend variational multiplicity techniques to higher-order nonlocal operators, including cases without AR-type conditions and with symmetry, broadening the landscape of nonlocal polyharmonic elliptic equations.

Abstract

We consider an elliptic partial differential equation driven by higher order fractional Laplacian , with homogeneous Dirichlet boundary condition \begin{equation*} \left\{% \begin{array}{ll} (-Δ)^{s} u=f(x,u) & \text{ in }Ω, u=0 & \text{ in } \mathbb{R}^n \setminus Ω. \end{array}% \right. \end{equation*} The above equation has a variational nature, and we investigate the existence and multiplicity results for its weak solutions under various conditions on the nonlinear term : superlinear growth, concave-convex and symmetric conditions and their combinations. The existence of two different non-trivial weak solutions is established by Mountain Pass Theorem and Ekeland's variational principle, respectively. Furthermore, due to Fountain Theorem and its dual form, both infinitely many weak solutions with positive energy and infinitely many weak solutions with negative energy are obtained.
Paper Structure (8 sections, 16 theorems, 107 equations)

This paper contains 8 sections, 16 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of two solutions without symmetry condition
  4. Mountain Pass solution without (AR) condition
  5. Negative energy solution by Ekeland's variational principle
  6. Infinitely many solutions under symmetry condition
  7. Positive energy solutions of superlinear problems
  8. Positive and negative energy solutions of concave-convex problems

Key Result

Theorem 1.1

Assume $s\in (1,2)$. Let $f$ be a continuous function verifying (H1)-(H4). Then, the superlinear problem problem admits a non-trivial Mountain Pass solution $u \in \mathcal{H}_{0}^{s}$ with positive energy $\mathcal{J}(u)$ (see J below).

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 19 more