Electron. J. Differential Equations, Vol. 2018 (2018), No. 67, pp. 1-14.

Radial solutions for inhomogeneous biharmonic elliptic systems

Reginaldo Demarque, Narciso da Hora Lisboa

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].

Submitted June 16, 2017. Published March 14, 2018.
Math Subject Classifications: 35J50, 31A30.
Key Words: Biharmonic operator; elliptic systems; existence of solutions; radial solution; mountain pass theorem.

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Reginaldo Demarque
Departamento de Ciências da Natureza
Universidade Federal Fluminense
Rio das Ostras, RJ, 28895-532, Brazil
email: r.demarque@gmail.com
Narciso da Hora Lisboa
Departamento de Ciências Exatas
Universidade Estadual de Montes Claros
Montes Claros, MG, 39401-089, Brazil
email: narciso.lisboa@unimontes.br

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