Multiple positive solutions for a class of Kirchhoff type problems involving general critical growth

Abstract

In this paper, we study the following nonlinear Kirchhoff problem involving critical growth: $$ \left\{% \begin{array}{ll} -(a+b\int_Ω|\nabla u|^2dx)Δu=|u|^4u+λ|u|^{q-2}u, u=0\ \ \text{on}\ \ \partialΩ, \end{array}% \right. $$ where $1<q<2$, $\lambda ,a,b>0$ are parameters and $\Omega$ is a bounded domain in ${\mathbb{R}}^3$. We prove that there exists ${\lambda }_1={\lambda }_1(q,\Omega )>0$ such that for any $\lambda \in (0,{\lambda }_1)$ and $a,b>0$, the above Kirchhoff problem possesses at least two positive solutions and one of them is a positive ground state solution. We also establish the convergence property of the ground state solution as the parameter $b\searrow 0$. More generally, we obtain the same results about the following Kirchhoff problem: $$ \left\{% \begin{array}{ll} -(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx)Δu+u=Q(x)|u|^4u+λf(x)|u|^{q-2}u, u\in H^1(\mathbb{R}^3), \end{array}% \right. $$ for any $a,b>0$ and $\lambda \in (0,{\lambda }_0(q,Q,f))$ under certain conditions of $f(x)$ and $Q(x)$. Finally, we investigate the depending relationship between ${\lambda }_0$ and $b$ to show that for any (large) $\lambda >0$, there exists a $b_0(\lambda )>0$ such that the above results hold when $b>b_0(\lambda )$ and $a>0$.

25 Pages

Cited by in corpus (1)

• Multiple positive solutions for Schrodinger-Poisson systems involving critical nonlocal term