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S-shaped bifurcation curves of nonlinear elliptic boundary value problems. (English) Zbl 0544.35015
The paper is concerned with the study of the elliptic boundary value problem $$(1)_{\lambda}\quad Lu=\lambda f(u)$$ in $$\Omega$$, $$Bu=0$$ on $$\partial \Omega$$ where $$\lambda$$ is a nonnegative parameter and (L,B) satisfies the strong maximum principle. The nonlinearity $$f:{\mathbb{R}}_+\to {\mathbb{R}}_+$$ is assumed to satisfy $$f(0)>0$$, $$f'(x)>0$$ for all $$x\geq 0$$ and $$f(x)-f'(x)x\geq \theta>0$$ for x sufficiently large. For example $$f(x)=\exp(x/(1+\epsilon x))$$ with $$\epsilon>0$$ which arises in the theory of combustion.
It is shown that the solution set $$\Sigma:=\{(\lambda,u)\in {\mathbb{R}}_+\times C^ 2({\bar \Omega})|$$ ($$\lambda$$,u) satisfies $$(1)_{\lambda}\}$$ of $$(1)_{\lambda}$$ has the same shape as the solution set $$\Sigma_ s:=\{(\nu,x)\in {\mathbb{R}}_+\times {\mathbb{R}}_ x| \nu f(x)=x\}$$ of the scalar equation $$(2)_{\nu} \nu f(x)=x$$, the so-called Semenov approximation. More precisely, if $$(2)_{\nu}$$ is uniquely solvable for large values of $$\nu$$, the same is true for $$(1)_{\lambda}$$. $$(2)_{\nu}$$ and $$(1)_{\lambda}$$ are uniquely solvable for small parameter values. Moreover, either the solution of $$(1)_{\lambda}$$ is unique for each $$\lambda$$ or $$\Sigma$$ is ”S-shaped”, i.e. $$\Sigma$$ contains a subcontinuum which looks like an ”S”. If the ”S- structure” of $$\Sigma_ s$$ is strong enough, it carries over to $$\Sigma$$.

##### MSC:
 35B32 Bifurcations in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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