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).

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 |