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Global bifurcation in a semilinear elliptic free boundary value problem. (Spanish) Zbl 0647.35008
Res. Program 18th Natl. Congr. Mex. Math. Soc., Proc., Merida/Mex. 1984, Aportaciones Mat., Comun. 1, 8-18 (1986).
[For the entire collection see Zbl 0606.00001.]
The free boundary problem $$-\Delta u=\lambda F(u-q)$$ in G, $$u=0$$ on $$\partial G$$, where G is a smooth bounded domain in $${\mathbb{R}}^ 2$$, $$F(t)=t^+$$ or $$F(t)=H(t)$$ (here $$t^+=\max (t,0)$$ and H is the Heaviside function), q is continuous and such that the set $$Q=\{x\in G:$$ $$q(x)<0\}$$ has positive measure, and $$\lambda$$ is a real parameter, is studied in this paper. The main result is that $$\lambda^*_ 1$$, which is the first eigenvalue for $$-\Delta u=\lambda \chi_ Qu$$ in G, $$u=0$$ on G $$(\chi_ Q$$ is the characteristic function for Q) is a bifurcation point and the continuum of positive solutions emanating from $$\lambda^*_ 1$$ either bifurcates at infinity at $$\lambda_ 1$$ (which is the first eigenvalue for $$-\Delta u=\lambda u$$ in G, $$u=0$$ on $$\partial G)$$ either is unbounded in $$\lambda$$.
The main tool in the proof is the degree theory for multivalued operators by K. C. Chang. In particular this improves some previous work by G. Keady and J. Norbury [Proc. R. Soc. Edinb., Sect. A 87, 83-109 (1980; Zbl 0452.35032)]. We point out that K. C. Chang himself, is a paper which is not quoted here [Commun. Partial Differ. Equations 5, 741-751 (1980; Zbl 0444.35039)] gave two very short and elegant proofs of similar results in the special case $$F(t)=t^+$$ and $$q\equiv -C<0$$ constant, by using global bifurcation arguments (Amann, Rabinowitz, Dancer).
Reviewer: J.Hernandez
##### MSC:
 35B32 Bifurcations in context of PDEs 35R35 Free boundary problems for PDEs 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 47J05 Equations involving nonlinear operators (general) 58C06 Set-valued and function-space-valued mappings on manifolds