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A convex-concave elliptic problem with a parameter on the boundary condition

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  • In this paper we study existence and multiplicity of nonnegative solutions to $$ \begin{equation} \left\{\begin{array}{ll} \Delta u = u^p + u^q \qquad & \mbox{in }\Omega, \\ \frac{\partial u }{\partial \nu} =\lambda u \qquad & \mbox{on }\partial \Omega. \end{array}\right. \end{equation} $$ Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\nu$ stands for the outward unit normal and $p$, $q$ are in the convex-concave case, that is $0 < q < 1 < p$. We prove that there exists $\Lambda^* >0$ such that there are no nonnegative solutions for $\lambda < \Lambda^*$, and there is a maximal nonnegative solution for $\lambda \ge \Lambda^{*}$. If $\lambda$ is large enough, then there exist at least two nonnegative solutions. We also study the asymptotic behavior of solutions when $\lambda\to \infty$ and the occurrence of dead cores. In the particular case where $\Omega$ is the unit ball of $\mathbb{R}^N$ we show exact multiplicity of radial nonnegative solutions when $\lambda$ is large enough, and also the existence of nonradial nonnegative solutions.
    Mathematics Subject Classification: Primary: 35J25, 35J61; Secondary: 35B40.


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