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Existence of three solutions for variable exponent elliptic systems. (English) Zbl 1332.35100

The author investigates the existence of solutions for the elliptic system \[ \Delta(|\Delta u|^{p(x)-2}\Delta u)=\lambda F_u(x,u,v)+\mu G_u(x,u,v), \]
\[ \Delta(|\Delta v|^{q(x)-2}\Delta v)=\lambda F_v(x,u,v)+\mu G_v(x,u,v), \] in a smooth open and bounded subset of \({\mathbb R}^N\), \(N\geq 1\). The system is complemented with Navier boundary conditions.
Under some additional conditions the author obtains the existence of at least three weak solutions. The approach relies on a three critical point theorem due to Ricceri.

MSC:

35J35 Variational methods for higher-order elliptic equations
35J62 Quasilinear elliptic equations
47J30 Variational methods involving nonlinear operators
35D30 Weak solutions to PDEs
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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References:

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