We study positive solutions to the steady state reaction diffusion equation of the form: − Δ u=λ f(u); Ω ∂ u∂ η +λ u=0; ∂ Ω $$\begin{array}{} \displaystyle \left\lbrace \begin{matrix} -{\it\Delta} u =\lambda f(u);~ {\it\Omega} \\ \frac{\partial u}{\partial \eta}+ \sqrt{\lambda} u=0;~\partial {\it\Omega}\end{matrix} \right. \end{array}$$ where λ > 0 is a positive parameter, Ω is a bounded domain in ℝ N when N > 1 (with smooth boundary ∂ Ω ) or Ω = (0, 1), and ∂ u∂ η $\begin{array}{} \displaystyle \frac{\partial u}{\partial \eta} \end{array}$ is the outward normal derivative of u . Here f ( s ) = ms + g ( s ) where m ≥ 0 (constant) and g ∈ C 2 [0, r ) ∩ C [0, ∞) for some r > 0. Further, we assume that g is increasing, sublinear at infinity, g (0) = 0, g ′(0) = 1 and g ″(0) > 0. In particular, we discuss the existence of multiple positive solutions for certain ranges of λ leading to the occurrence of Σ -shaped bifurcation diagrams. We establish our multiplicity results via the method of sub-supersolutions.

中文翻译：

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Σ型分叉曲线

我们研究以下形式的稳态反应扩散方程的正解：-Δu =λf（u）; Ω∂uη+λu = 0; ΩΩ$$ \ begin {array} {} \ displaystyle \ left \ lbrace \ begin {matrix}-{\ it \ Delta} u = \ lambda f（u）;〜{\ it \ Omega} \\ \ frac { \ partial u} {\ partial \ eta} + \ sqrt {\ lambda} u = 0;〜\ partial {\ it \ Omega} \ end {matrix} \ right。\ end {array} $$，其中λ> 0是一个正参数，当N> 1（具有平滑边界Ω）或Ω=（0，1）且∂u∂η$时，Ω是ℝN中的有界域。 \ begin {array} {} \ displaystyle \ frac {\ partial u} {\ partial \ eta} \ end {array} $是u的向外正态导数。此处f（s）= ms + g（s）其中，对于某些r> 0，m≥0（常数），且g∈C 2 [0，r）∩C [0，∞）。 ，在无限大处为次线性，g（0）= 0，g'（0）= 1且g''（0）>0。特别是，我们讨论了在某些λ范围内存在多个正解的情况，从而导致了Σ形分叉图的出现。我们通过子超解法建立多重性结果。