S-shaped bifurcation curves of nonlinear elliptic boundary value problems
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 \(\Sigma\).
scientific article; zbMATH DE number 3865789
scientific article