Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity

Abstract

In this work, we study a class of Kirchhoff type problems with singularity and nonlinearity, and obtain the uniqueness and existence of positive solutions of those problems by the variational methods. Furthermore, we obtain the multiplicity of positive solutions of those problems by the Nehari method.

Existence and multiplicity of positive solutions for a class of singular Kirchhoff type problems with sign‐changing potential

“…Inspired by Liao et al and Liu and Sun, a natural question is whether there exist solutions for problem . In the present note, we give a positive answer by the Nehari method.…”

“…Later, Lei et al studied problem with s =0, p =5, that is, singular Kirchhoff type equation with critical exponent. They got two positive solutions by the variational method and perturbation method; see Lei et al Liao et al investigated problem with s =0, p =3 and obtained the existence and multiplicity of positive solutions by the Nehari method and variational method. Recently, Liu et al generalized the results of Lei et al to dimension four.…”

“…Later, Lei et al 2 studied problem (2) with s = 0, p = 5, that is, singular Kirchhoff type equation with critical exponent. They got two positive solutions by the variational method and perturbation method; see Lei et al 2 Liao et al 3 When a = 1, b = 0, problem (1) reduces to the classic semilinear singular elliptic problem. It worth pointing out that Sun et al 6 investigated the multiplicity positive solutions for the singular elliptic problems by the variational method at the first time.…”

Existence and Uniqueness of Positive Solution for p-Laplacian Kirchhoff-Schrödinger-Type Equation

“…In recent years, a lot of scholars have studied the singular Kirchhoff problem (for more details, we refer the reader to [1][2][3][4]), the Schrödinger-Poisson system (we refer the reader to [5][6][7][8]), and the Kirchhoff-Schrödinger-Poisson system (we refer the reader to [9][10][11][12]). The authors use various methods to obtain the properties of the solution, which makes such problems very interesting.…”

“…Assume that 1 is also a solution of problem (1). Letting ] = 0 − 1 , according to the definition of the weak solution and (26), we can get…”