On a class of Hamiltonian strongly degenerate elliptic systems with concave and convex nonlinearities

ABSTRACT

In this paper, we prove the existence of positive and negative energy solutions to the following semilinear strongly degenerate elliptic system $\begin{array}{rl}-{\mathrm{\Delta }}_{\lambda }u& =\beta |v{|}^{q-2}v+\gamma H_v(x,u,v)\text{ in }\mathrm{\Omega },\\ -{\mathrm{\Delta }}_{\lambda }v& =\alpha |u{|}^{p-2}u+\gamma H_u(x,u,v)\text{ in }\mathrm{\Omega },\\ u& =v=0\text{ on }\mathrm{\partial \Omega },\end{array}$ where $\alpha ,\beta ,\gamma \in \mathbb{R},1<p,q<2,$ $H:\overline{\mathrm{\Omega }}\times \mathbb{R}\times \mathbb{R}\to {\mathbb{R}}_{+}$ is a function satisfying the subcritical growth with respect to the second and third variables, Ω is a bounded domain in ${\mathbb{R}}^N$ ($N\ge 3$) with smooth boundary $\mathrm{\partial \Omega }$, and ${\mathrm{\Delta }}_{\lambda }$ is the strongly degenerate operator of the form ${\mathrm{\Delta }}_{\lambda }u=\sum _{j=1}^N\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\left({\lambda }_j^2\left(x\right)\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_j}\right),x=(x_1,\dots ,x_N)\in {\mathbb{R}}^N,$ where ${\lambda }_j,j=1,\dots ,N$, are continuous functions on ${\mathbb{R}}^N$ satisfying some certain conditions.

COMMUNICATED BY:

• X. Xie

KEYWORDS:

• Hamiltonian elliptic system
• existence
• concave-convex nonlinearity
• subcritical growth
• strongly degenerate
• generalized fountain theorem

AMS SUBJECT CLASSIFICATIONS:

• 35J70
• 35D30

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by Vietnam Ministry of Education and Training under grant number B2019-SPH-02.