Three positive solutions for Kirchhoff equation with singular and sub-cubic nonlinearities

Abstract

In this article, we consider the following Kirchhoff equation arising in the string vibrations model

$$\begin{aligned} \left\{ \begin{array}{ll} -M\left( \int \limits _\Omega |\nabla u|^{2}\textrm{d}x\right) \Delta u=\mu f(x)u^{-\gamma }+\lambda g(x)u^p &{}\text {in} \quad \Omega ,\\ u=0 &{}\text {on} \quad \partial \Omega , \end{array} \right. \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^{3}$ is a bounded smooth domain, $M(s)=as+1$, $\gamma \in (0,1)$, $p\in (1,3)$, the parameters $a,\mu ,\lambda >0$ and the weight functions $f, g\in C({\overline{\Omega }},{\mathbb {R}}^{+})$. In order to obtain multiple positive solutions, we have to face two main difficulties: (i) the singular term leads to the non-differentiability of the corresponding energy functional, the critical point theory, such as mountain pass theorem, cannot be applied directly; (ii) due to the complicated competition between nonlocal Kirchhoff term and sub-cubic nonlinearity, the standard Nehari manifold method does not work. By constructing two different truncation functions and using the decomposition of Nehari manifold, we obtain the existence and multiplicity of positive solutions for the above problem. In particular, there exist at least three positive solutions when $a,\lambda$ and $\mu$ given in specific intervals. Moreover, we also investigate the asymptotic behavior of positive solutions as $a\rightarrow 0^{+}$. Our results extend the existence results from the super-cubic (Lei et al. in J Math Anal Appl 421:521–538, 2015; Liu and Sun in Commun Pure Appl Anal 12:721–733, 2013) and cubic (Liao et al. J Math Anal Appl 430:1124–1148, 2015) cases to the sub-cubic case.