Journal of Inequalities and Applications (Aug 2018)
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