| Abstract: |
| In this talk, we consider positive solutions of the sublinear elliptic equation $-\Delta u = a(x)u^{q}$ in a smooth bounded domain $\Omega \subset \mathbb{R}^{N}$, $N\geq1$, under the Robin type boundary condition $\frac{\partial u}{\partial \nu}=\alpha u$ on $\partial \Omega$, where $q\in (0,1)$, $\alpha\geq 0$, and $a\in C^{\theta}(\overline{\Omega})$, $\theta \in (0,1)$, changes sign in $\overline{\Omega}$. We are interested in positive solutions belonging to $\mathcal{P}^\circ = \{ u\in C(\overline{\Omega}) : u>0 \ \mbox{ in } \overline{\Omega}\}$. First, we observe that if this problem, $(P_{\alpha})$, has a positive solution, then $\int_{\Omega} a < 0$. Next, in some case of $a$ and $q$, we show the existence of $\alpha_s>0$ such that $(P_\alpha)$ has a unique positive solution in $\mathcal{P}^\circ$ for $\alpha = 0, \alpha_{s}$, exactly two (ordered) positive solutions in $\mathcal{P}^\circ$ for every $\alpha\in (0,\alpha_{s})$, and no positive solutions in $\mathcal{P}^\circ$ for any \alpha>\alpha_{s}$. We shall discuss it in the case when $q \simeq 1$ and $\int_{\Omega} a \simeq 0$. |
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