Singular reaction diffusion equations where a parameter infuluences the reaction term and the boundary conditions

Abstract

We analyse positive solutions to the steady state reaction diffusion equation: \begin{equation*} \begin{cases} -u''=\lambda h(t) f(u) \quad \text{in } (0,1), \\ -du'(0)+\mu(\lambda) u(0)=0,\\ u'(1)+\mu(\lambda) u(1)=0, \end{cases} \end{equation*} where $\lambda>0$ is a parameter, $d\geq 0$ is a constant, $f \in C^2([0,\infty),\mathbb{R}) $ is an increasing function which is sublinear at infinity $\Big (\lim\limits_{s \rightarrow \infty}{f(s)}/{s}=0\Big)$, $h \in C^1((0,1],(0,\infty))$ is a nonincreasing function with $h_1:=h(1)>0$ and there exist constants $d_0>0$, $\alpha \in [0,1)$ such that $h(t)\leq {d_0}/{t^\alpha}$ for all $t \in (0,1]$, and $\mu \in C([0,\infty),[0,\infty))$ is an increasing function such that $\mu(0)\geq 0$. We consider three cases of $f$, namely, $f(0)=0$, $f(0)> 0$ and $f(0) <0$. We will discuss existence and multiplicity results via the method of sub-supersolutions. Further, we will establish uniqueness results for $\lambda\approx 0$ and $\lambda\gg 1$.