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The structure of Rabinowitz’ global bifurcating continua for generic quasilinear elliptic equations. (English) Zbl 0901.35007

Global bifurcation branches for boundary value problems of the type

$\begin{array}{c}-\sum _{i,j=1}^{n}{a}_{ij}\left(·\right){u}_{{x}_{i}{x}_{j}}\left(x\right)+\sum _{i=1}^{n}{b}_{i}\left(·\right){u}_{{x}_{i}}\left(x\right)+c\left(·\right)u\left(x\right)=\lambda d\left(x\right)u\left(x\right)+f\left(\lambda ,·\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }\\ u\left(x\right)=0\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }\end{array}$

are studied. Here $\left(·\right)$ denotes $\left(x,u\left(x\right),Du\left(x\right)\right)$, ${\Omega }$ is a bounded ${C}^{3,\alpha }$ domain in ${ℝ}^{n}$, $\lambda$ is a real parameter, the coefficients satisfy certain differentiability assumptions. It is shown that the set of nontrivial solutions generically consists of a finite or countable system of smooth, one-dimensional curves in $ℝ×E$, $E=\left\{u\in {C}^{2,\alpha }\left(\overline{{\Omega }}\right);u=0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }\right\}$. Further, also generically, all eigenvalues ${\lambda }_{i}$ of the linearized problem are simple, and the branches of nontrivial solutions bifurcating from these eigenvalues are smooth curves which either meet infinity or are closed loops which meet an even number of distinct eigenvalues. Some examples are given showing that both alternatives really can occur. (It is known from the Rabinowitz result that the second one is excluded in the case of ordinary differential equations).

##### MSC:
 35B32 Bifurcation (PDE) 35J65 Nonlinear boundary value problems for linear elliptic equations
##### Keywords:
genericity; set of nontrivial solutions