Infinite number of solutions for some elliptic eigenvalue problems of Kirchhoff-type with non-homogeneous material

Baoqiang Yan,Donal O’Regan,Ravi P Agarwal

doi:10.1186/s13661-021-01522-9

Abstract

In this paper, using variational method, we study the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing) for a non-homogeneous elliptic Kirchhoff equation with a nonlinear reaction term.

Highlights

In this paper, we consider the following nonlocal equation: ⎧⎨–M(x, u 2) u = λf (x, u), x ∈, ⎩u|x∈∂ = 0, (1.1)where is a bounded open domain of RN with smooth boundary and⎧ ⎨f ∈ C( × R, R), ⎩M(x, t) = a(x) + b(x)t, u = |∇u|2 dx, with a, b ∈ Cγ ( ), γ ∈ (0, 1), a(x) ≥ a0 > 0, b(x) ≥ 0
To the best of our knowledge, that there are no results in the literature on the existence of a sign-changing solution for Problem (1.1)
In this paper using the steepest descent method for gradient mappings of the isoperimetric variational problem and the method of invariant sets of descending flow in critical point theory, we establish the existence of an infinite number of solutions

Summary

Introduction

To the best of our knowledge, that there are no results in the literature on the existence of a sign-changing solution for Problem (1.1). In this paper (motivated by [21]) using the steepest descent method for gradient mappings of the isoperimetric variational problem (see [6]) and the method of invariant sets of descending flow in critical point theory (see [27]), we establish the existence of an infinite number of solutions (some are positive, some are negative, and others are sign-changing).

Results

Conclusion