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

Summary: On a bounded domain ${\Omega }\subset {ℝ}^{n}$, we consider a non-local problem of the type

$\left\{\begin{array}{cc}-K\left({\int }_{{\Omega }}{|\nabla u\left(x\right)|}^{2}\phantom{\rule{0.166667em}{0ex}}dx\right){\Delta }u=\lambda f\left(x,u\right)+\mu g\left(x,u\right)\hfill & \text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\hfill \\ u=0\hfill & \text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }·\hfill \end{array}\right\$

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 $\mu \ge 0$ small enough.

##### MSC:
 49J20 Optimal control problems with PDE (existence) 35J20 Second order elliptic equations, variational methods
##### References:
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