Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy–Sobolev exponent and singular nonlinearity

Abstract

Abstract In this paper, we study a class of critical elliptic problems of Kirchhoff type: [a+b(∫R3|∇u|2−μu2|x|2dx)2−α2](−Δu−μu|x|2)=|u|2∗(α)−2u|x|α+λf(x)|u|q−2u|x|β, $$ \biggl[a+b \biggl( \int_{\mathbb{R}^{3}}\vert \nabla u\vert ^{2}-\mu \frac{u^{2}}{\vert x\vert ^{2}}\,dx \biggr)^{\frac{2-\alpha }{2}} \biggr]\biggl(-\Delta u- \mu \frac{u}{\vert x\vert ^{2}}\biggr) = \frac{\vert u\vert ^{2^{*}(\alpha )-2}u }{\vert x\vert ^{\alpha }}+\lambda \frac{f(x)\vert u\vert ^{q-2}u }{\vert x\vert ^{\beta }}, $$ where a,b>0 $a,b>0$, μ∈[0,1/4) $\mu \in [0,1/4)$, α,β∈[0,2) $\alpha , \beta \in [0,2)$, and q∈(1,2) $q\in (1,2)$ are constants and 2∗(α)=6−2α $2^{*}(\alpha )=6-2\alpha $ is the Hardy–Sobolev exponent in R3 $\mathbb{R}^{3}$. For a suitable function f(x) $f(x)$, we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard b>0 $b>0$ as a parameter to obtain the convergence property of solutions for the given problem as b↘0+ $b\searrow 0^{+}$ by the mountain pass theorem and Ekeland’s variational principle.

Keywords

• Hardy–Sobolev exponent
• Kirchhoff
• Multiplicity
• Nehari manifold
• Fibering map
• Asymptotic behavior