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Fractional p-Laplacian Kirchhoff-type problem involving a singular term via Nehari manifold

Djamel Abid

TL;DR

The paper investigates a Kirchhoff-type problem with a fractional p-Laplacian, a singular term, and sign-changing nonlinearity on a bounded domain. It develops a variational framework on the Nehari manifold and employs fiber-map analysis together with Ekeland’s variational principle to obtain multiplicity results. In the subcritical regime, it proves the existence of two positive weak solutions for small λ; in the critical regime, it provides conditions ensuring one or two positive solutions, including distinct energy landscapes. These results extend Nehari-manifold techniques to nonlocal Kirchhoff problems with singular terms and critical exponents, contributing to the theory of multiplicity in fractional elliptic problems with singularities.

Abstract

This paper is dedicated to studying the existence of nontrivial positive solutions for a Kirchhoff-type problem with sign change nonlinearities and a singular term, Using the Nehari manifold and EkelandS variational principle we prove that for the appropriate choice of λ our problem has at least two positive solutions for both subcritical and critical cases.

Fractional p-Laplacian Kirchhoff-type problem involving a singular term via Nehari manifold

TL;DR

The paper investigates a Kirchhoff-type problem with a fractional p-Laplacian, a singular term, and sign-changing nonlinearity on a bounded domain. It develops a variational framework on the Nehari manifold and employs fiber-map analysis together with Ekeland’s variational principle to obtain multiplicity results. In the subcritical regime, it proves the existence of two positive weak solutions for small λ; in the critical regime, it provides conditions ensuring one or two positive solutions, including distinct energy landscapes. These results extend Nehari-manifold techniques to nonlocal Kirchhoff problems with singular terms and critical exponents, contributing to the theory of multiplicity in fractional elliptic problems with singularities.

Abstract

This paper is dedicated to studying the existence of nontrivial positive solutions for a Kirchhoff-type problem with sign change nonlinearities and a singular term, Using the Nehari manifold and EkelandS variational principle we prove that for the appropriate choice of λ our problem has at least two positive solutions for both subcritical and critical cases.

Paper Structure

This paper contains 6 sections, 17 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. main results
  4. Compactness properties
  5. Proof of theorem \ref{['th1']}
  6. Proof Of Theorem \ref{['th2']}

Key Result

Theorem 1.1

Suppose that $1 < p < q < p_s^*,$$0 <\alpha<1,$ and $f$ satisfying Euler, e1 Then there exist $\lambda^* > 0$ such that for any $\lambda \in (0, \lambda^*),$ problem p0 has at least two nontrivial positive solutions.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.1
  • Lemma 2.4
  • ...and 19 more