Nodal solution for Kirchhoff-type problems with concave-convex nonlinearities

Abstract

In this paper, we study the existence of nodal solutions for the Kirchhoff-type problem with concave-convex nonlinearities $\begin{array}{cc}-\left(a+b{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{\Delta }u=\lambda |u{|}^{q-1}u+|u{|}^{p-1}u& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=0& \mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$ where Ω is a bounded domain with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$ in $R^N(N=1,2,3)$, a $0<q<1,3<p<5$. By constraining the energy functional of the above problem on a subset of the nodal Nehari set ${\mathcal{M}}_{\lambda }$, we show that there exists a constant ${\lambda }^{\ast }>0$ such that for any $\lambda <{\lambda }^{\ast }$, the above problem has a nodal solution $u_{\lambda }$ with positive energy.

COMMUNICATED BY:

• A. Pankov

Keywords:

• Kirchhoff-type equation
• nodal solution
• Nehari manifold
• concave-convex nonlinearities

AMS Subject classifications:

• Primary: 35J20
• Secondary: 35J25
• 35J60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Supported by National Natural Science Foundation of China [No. 11801465, 11971393].