Semilinear Neumann boundary value problems on a rectangle
Junping Shi · Transactions of the American Mathematical Society · 2002
We consider a semilinear elliptic equation \begin{equation*} \Delta u+\lambda f(u)=0, \;\; \mathbf {x}\in \Omega ,\;\; \frac {\partial u}{\partial n }=0, \;\; \mathbf {x}\in \partial \Omega , \end{equation*} where $\Omega$ is a rectangle $(0,a)\times (0,b)$ in $\mathbf {R}^2$. For balanced and unbalanced $f$, we obtain partial descriptions of global bifurcation diagrams in $(\lambda ,u)$ space. In particular, we rigorously prove the existence of secondary bifurcation branches from the semi-trivial solutions, which is called dimension-breaking bifurcation. We also study the asymptotic behavior of the monotone solutions when $\lambda \to \infty$. The results can be applied to the Allen-Cahn equation and some equations arising from mathematical biology.