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On an elliptic Kirchhoff-type problem depending on two parameters. (English) Zbl 1192.49007

Summary: On a bounded domain \(\Omega\subset \mathbb R^n\), we consider a non-local problem of the type
\[ \begin{cases} -K\left(\int_\Omega|\nabla u(x)|^2\,dx\right)\Delta u =\lambda f(x,u)+\mu g(x,u) &\text{in }\Omega, \\ u=0 &\text{on }\partial\Omega.\end{cases} \]
Under rather general assumptions on \(K\) and \(f\), we prove, in particular, that there exists \(\lambda^*> 0\) such that, for each \(\lambda > \lambda ^*\) and each Carathéodory function \(g\) with a sub-critical growth, the above problem has at least three weak solutions for every \(\mu \geq 0\) small enough.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
35J20 Variational methods for second-order elliptic equations
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References:

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