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# $\Sigma$-shaped bifurcation curves for classes of elliptic systems

• We study positive solutions to classes of steady state reaction diffusion systems of the form:

$\begin{equation*} \left\lbrace \begin{matrix}-\Delta u = \lambda f(v) ;\; \Omega\\ -\Delta v = \lambda g(u) ;\; \Omega\\ \frac{\partial u}{\partial \eta}+\sqrt{\lambda} u = 0; \; \partial \Omega\\ \frac{\partial v}{\partial \eta}+\sqrt{\lambda}v = 0; \; \partial \Omega\ \end{matrix} \right. \end{equation*}$

where $\lambda>0$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^N$; $N > 1$ with smooth boundary $\partial \Omega$ or $\Omega = (0, 1)$, $\frac{\partial z}{\partial \eta}$ is the outward normal derivative of $z$. Here $f, g \in C^2[0, r) \cap C[0, \infty)$ for some $r>0$. Further, we assume that $f$ and $g$ are increasing functions such that $f(0) = 0 = g(0)$, $f'(0) = g'(0) = 1$, $f''(0)>0, g''(0)>0$, and $\lim\limits_{s \rightarrow \infty} \frac{f(Mg(s))}{s} = 0$ for all $M>0$. Under certain additional assumptions on $f$ and $g$ we prove that the bifurcation diagram for positive solutions of this system is at least $\Sigma-$shaped. We also discuss an example where $f$ is sublinear at $\infty$ and $g$ is superlinear at $\infty$ which satisfy our hypotheses.

Mathematics Subject Classification: Primary: 35J15, 35J25, 35J60.

 Citation: • • Figure 1.  Bifurcation diagram of (1.1) when hypotheses of Theorem 1.1(b) $(H_1-H_3)$ hold

Figure 2.  Bifurcation diagram of (1.1) when hypotheses of Corollary 1 $(H_1-H_4)$ hold

Figure 3.  Shape of $h$ producing multiplicity

Figure 4.  Prototypical shapes of $f$ and $g$ producing a $\Sigma-$shaped bifurcation curve

Figure 5.  Graph of $f$ and corresponding typical bifurcation curves for two sets of parameters $(k, \alpha)$

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