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Existence and multiple solutions for a variable exponent system. (English) Zbl 1203.35094

Summary: We study the following variable exponent system
\[ \begin{cases} -\text{div}(|\nabla u|^{p(x)-2}\nabla u)+ a(x) |u|^{p(x)-2}u= f(x,u,v,\lambda) &\text{in }\Omega,\\ -\text{div}(|\nabla v|^{q(x)-2}\nabla v)+ b(x) |v|^{q(x)-2}v= f(x,u,v,\lambda) &\text{in }\Omega,\\ B(u,v)=0 \end{cases} \]
under some various conditions, we present several results for the existence of multiple nontrivial solutions.

MSC:

35J47 Second-order elliptic systems
35J50 Variational methods for elliptic systems
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