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On an elliptic Kirchhoff-type problem depending on two parameters. (English) Zbl 1192.49007
Summary: On a bounded domain $\Omega\subset \Bbb R^n$, we consider a non-local problem of the type $$\cases -K\left(\int_\Omega|\nabla u(x)|^2\,dx\right)\Delta u =\lambda f(x,u)+\mu g(x,u) &\text{in }\Omega, \\ u=0 &\text{on }\partial\Omega.\endcases$$ 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.

49J20Optimal control problems with PDE (existence)
35J20Second order elliptic equations, variational methods
Full Text: DOI arXiv
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