On a class of Kirchhoff equations involving an anisotropic operator and potential

Mohammed Massar

doi:10.1007/s40065-020-00304-y

Abstract

In this work, we are concerned with a class of fractional equations of Kirchhoff type with potential. Using variational methods and a variant of quantitative deformation lemma, we prove the existence of a least energy sign-changing solution. Moreover, the existence of infinitely many solution is established.

Highlights

Introduction and main resultConsider the following fractional Kirchhoff equation1 + b[u]2α (− x )αu − yu + V (x, y)u = f (u), (x, y) ∈ RN = Rn × Rm, (1.1) where [u]α = RN |(− x ) α 2 u|2 +|∇ y u |2 dxdy 2, α ∈ (0, 1), n, m ≥ 1
Under suitable assumptions on f ∈ C1(R, R), by adapting some arguments developed in [7,8,9] and using a variant of deformation lemma, the author shows that the equation admits a least energy sign-changing solution
To the best of our knowledge, there are no result about existence of solutions for a Kirchhoff-type equation with an operator containing local and nonlocal diffusions except [24]

Summary

Introduction

Under suitable assumptions on f ∈ C1(R, R), by adapting some arguments developed in [7,8,9] and using a variant of deformation lemma, the author shows that the equation admits a least energy sign-changing solution. To the best of our knowledge, there are no result about existence of solutions for a Kirchhoff-type equation with an operator containing local and nonlocal diffusions except [24]. Lemma 2.1 (Cf.[16]) The fractional Sobolev–Liouville space Hα(RN ) is continuously embedded into Lq (RN ) for q ∈ [2, p∗]; and is compactly embedded into Lq (RN ) for q ∈ (2, p∗).

Results

Conclusion