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Singular $p$-biharmonic problem with the Hardy potential

A. Drissi, A. Ghanmi, D. D. Repovš

Abstract

The aim of this paper is to study existence results for a singular problem involving the $p$-biharmonic operator and the Hardy potential. More precisely, by combining monotonicity arguments with the variational method, the existence of solutions is established. By using the Nehari manifold method, the multiplicity of solutions is proved. An example is also given, to illustrate the importance of these results.

Singular $p$-biharmonic problem with the Hardy potential

Abstract

The aim of this paper is to study existence results for a singular problem involving the -biharmonic operator and the Hardy potential. More precisely, by combining monotonicity arguments with the variational method, the existence of solutions is established. By using the Nehari manifold method, the multiplicity of solutions is proved. An example is also given, to illustrate the importance of these results.

Paper Structure

This paper contains 6 sections, 14 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of Theorem \ref{['thm']}
  4. Fibering maps on Nehari manifold sets
  5. The proof of Theorem \ref{['thmm']}
  6. An application

Key Result

Theorem 1

Suppose that hypotheses $(H_1)$-$(H_3)$ hold. Then for all $\delta, \mu>0$, problem p admits at least one nontrivial weak solution $\varphi_\mu$, provided that $\lambda>0$ is small enough.

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 18 more