Multiple Solutions for an Elliptic Equation with Hardy-Sobolev Critical Exponent, Hardy-Sobolev-Maz’ya Potential and Sign-Changing Weights

Mohammed El Mokhtar Ould El Mokhtar,Zeid I Almuhiameed

doi:10.4236/jamp.2019.711181

Abstract

In the present paper, an elliptic equation with Hardy-Sobolev critical exponent, Hardy-Sobolev-Maz’ya potential and sign-changing weights, is considered. By using the Nehari manifold and mountain pass theorem, the existence of at least four distinct solutions is obtained.

Highlights

Our motivation of this study is the fact that such equations arise in the search for solitary waves of nonlinear evolution equations of the Schrodinger or Klein-Gordon type
A solitary wave is a nonsingular solution, which travels as a localized packet in such a way that the physical quantities corresponding to the invariances of the equation are finite and conserved in time
Owing to their particle-like behavior, solitary waves can be regarded as a model for extended particles and they arise in many problems of mathematical physics, such as classical and quantum field theory, nonlinear optics, fluid mechanics, and plasma physics

Summary

Preliminaries

Let c ∈ , E a Banach space and I ∈ C1 ( E, ). Let X Banach space, and J ∈ C1 ( X , ) verifying the Palais-Smale condition. 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. Suppose that u0 is a local minimizer for J on. If u0 is a local minimizer for J on , u0 is a solution of the optimization problem min J (u). We refer to [15]

Proof of Theorem 1

Proof of Theorem 2

Proof of Theorem 3

Conclusion