Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $R^3$

Abstract

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: (a+b\ds\int_{\R^3}|D u|^2\right)Δu+V(x)u=|u|^{p-1}u, u\in H^1(\R^3), u>0, $x\in {\mathbb{R}}^3,where$a,b>0$areconstants,$2<p<5$and$V:\R^3\rightarrow\R$.Undercertainassumptionson$V$, we prove that \eqref{0.1} has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. Our main results can be viewed as a partial extension of a recent result of He and Zou in [16] concerning the existence of positive solutions to the nonlinear Kirchhoff problem (\varepsilon^2a+\varepsilon b\ds\int_{\R^3}|D u|^2\right)Δu+V(x)u=f(u), u\in H^1(\R^3), u>0, $x\in \R^3$,where$\varepsilon>0$isaparameter,$V(x)$isapositivecontinuouspotentialand$f(u)\thicksim |u|^{p-1}u$with$3<p<5$ and satisfies the Ambrosetti-Rabinowitz type condition. Our main results extend also the arguments used in [7,36], which deal with Schrödinger-Poisson system with pure power nonlinearities, to the Kirchhoff type problem.

Cited by in corpus (2)

• Uniqueness and Nondegeneracy of positive solutions to Kirchhoff equations and its applications in singular perturbation problems
• Uniqueness and nondegeneracy of positive solutions to a class of Kirchhoff equations in ${\mathbb{R}}^3$