Abstract

In this paper, we establish the existence of at least four distinct solutions to an elliptic problem with singular cylindrical potential, a concave term, and critical Caffarelli-Kohn-Nirenberg exponent, by using the Nehari manifold and mountain pass theorem.

Highlights

  • In the last Section, we prove the Theorem 3

  • ( ) It is well known that J is of class C1 in μ and the solutions of λ,μ are the critical points of J which is not bounded below on μ

  • Taking as a starting point the work of Tarantello [13], we establish the existence of a local minimum for J

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Summary

Introduction

We consider the multiplicity results of nontrivial nonnegative solutions of the following problem ( ) λ,μ. ( ) The starting point for studying λ,μ , is the Hardy-Sobolev-Maz’ya inequality that is particular to the cylindrical case k < N and that was proved by Maz’ya in [4]. It states that there exists positive constant Ca,2∗ such that ( ∫ ) ∫ ( ) Ca,2∗ N y −2∗b v 2∗ dx 2 2∗ ≤ N y −2a ∇v 2 − μ y −2(a+1) v2 dx,. In addition to the assumptions of the Theorem 1, if (H) hold and λ satisfying 0 < λ < (1 2) Λ0 , λ,μ has at least two positive solutions.

Preliminaries
Proof of Theorems 1
Proof of Theorem 2
Proof of Theorem 3

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