Radial solutions for inhomogeneous biharmonic elliptic systems

Abstract

In this article we obtain weak radial solutions for the inhomogeneous elliptic system \displaylines{ \Delta^2u+V_{1}(| x| )| u|^{q-2}u=Q(| x| )F_{u}(u,v)\quad\text{in } \mathbb{R}^N, \cr \Delta^2v+V_2(| x| )| v|^{q-2}v=Q(| x| )F_{v}(u,v)\quad\text{in } \mathbb{R}^N, \cr u,v\in D_0^{2,2}(\mathbb{R}^N),\quad N\geq 5, }$$ where $\Delta^2$ is the biharmonic operator, $V_i$, $ Q\in C^{0 }((0,+\infty ),[0,+\infty ))$, i=1,2, are radially symmetric potentials, $1<q<N$, $q\neq 2$, and F is a s-homogeneous function. Our approach relies on an application of the Symmetric Mountain Pass Theorem and a compact embedding result proved in [17].

Keywords

• Biharmonic operator
• elliptic systems
• existence of solutions
• radial solution
• mountain pass theorem