## 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 $\begin{gathered} -\sum_{i,j=1}^n a_{ij}(\cdot) u_{x_ix_j}(x) + \sum_{i=1}^n b_i(\cdot) u_{x_i}(x) + c(\cdot)u(x) = \lambda d(x) u(x) + f(\lambda,\cdot) \quad \text{in } \Omega \\ u(x) = 0 \quad\text{on } \partial \Omega \end{gathered}$ are studied. Here $$(\cdot)$$ denotes $$(x,u(x),Du(x))$$, $$\Omega$$ is a bounded $$C^{3,\alpha}$$ domain in $$\mathbb{R}^n$$, $$\lambda$$ 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 $$\mathbb{R}\times E$$, $$E=\{u \in C^{2,\alpha}(\overline \Omega);u=0 \text{ on } \partial \Omega\}$$. 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).
Reviewer: M.Kučera (Praha)

### MSC:

 35B32 Bifurcations in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

genericity; set of nontrivial solutions
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### References:

  Rabinowitz, P. H., Some global results for nonlinear eigenvalue problems, J. Funct. Analysis, 7, 487-513 (1971) · Zbl 0212.16504  Healey, T. J.; Kielhöfer, H., Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations, Arch. Rat. Mech. Anal., 113, 299-311 (1991) · Zbl 0726.35013  Healey, T. J.; Kielhöfer, H., Preservation of nodal structure on global bifurcating solution branches of elliptic equations with symmetry, J. Diff. Eqns, 106, 70-89 (1993) · Zbl 0788.35006  Cosner, C., Bifurcation from higher eigenvalues in nonlinear elliptic equations: continua that meet infinity, Nonlinear Analysis, 12, 271-277 (1988) · Zbl 0695.35011  Saut, J. C.; Temam, R., Generic properties of nonlinear boundary value problems, Comm. in PDE, 4, 293-319 (1979) · Zbl 0462.35016  Henry, D. B., Generic properties of equilibrium solutions by perturbation of the boundary, (Chow, S-N.; Hale, J. K., Dynamics of Infinite Dimensional Systems (1987), Springer) · Zbl 0656.35069  Uhlenbeck, K., Generic properties of eigenfunctions, Am. J. Math., 98, 1059-1078 (1976) · Zbl 0355.58017  Kielhöfer, H., Smoothness and asymptotics of global positive branches of $$Δu + λf (u) = 0$$, Z. Angew Math. Phys., 43, 139-153 (1992) · Zbl 0766.35002  Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer · Zbl 0691.35001  Crandall, M. G.; Rabinowitz, P. H., Bifurcation from simple eigenvalues, J. Funct. Anal., 8, 321-340 (1971) · Zbl 0219.46015  Magnus, R. J., A generalization of multiplicity and the problem of bifurcation, (Proc. Lond. Math. Soc., 32 (1976)), 251-278 · Zbl 0316.47042  Dancer, E. N., Counterexamples to some conjectures on the number of solutions of nonlinear equations, Math. Ann., 272, 421-440 (1985) · Zbl 0556.35001
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