Weak eigenfunctions for the linearization of extremal elliptic problems. (English) Zbl 0908.35044

Cabré, Xavier; Martel, Yvan

J. Funct. Anal. 156, No. 1, 30-56 (1998).

A semilinear elliptic problem \[ -\Delta u= \lambda g(u) \quad\text{in } \Omega,\qquad u=0 \quad\text{on } \partial \Omega \] is considered. Here \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(\lambda\) is a nonnegative parameter, \(g\in C^1\) is a positive, nondecreasing, convex function, \(\lim_{u\to \infty} g(u)/u= \infty\). It is known that there is an extremal parameter \(\lambda^* \in (0,\infty)\) such that there is a minimal classical solution for any \(\lambda \in [0,\lambda^*)\), there is a weak solution \(u^*\) for \(\lambda= \lambda^*\) and there is no solution for \(\lambda \in (\lambda^*,\infty)\). The linearization of this problem with \(\lambda= \lambda^*\) at the extremal solution \(u^*\) is studied. It is proved that this problem has always a positive weak eigenfunction in \(L_1(\Omega)\) corresponding to the zero eigenvalue, even if it has a discrete and positive \(H_0^1\)-spectrum. The zero eigenvalue is coherent with the nonexistence of solutions of the original semilinear problem for \(\lambda > \lambda^*\) and therefore the \(L_1\)-spectrum seems to be more relevant than the \(H_0^1\)-spectrum. It is shown that also a continuous spectrum of the linearization mentioned can occure. For the case of the unit ball and \(g(u)= e^u\) or \(g(u)= (1+u)^p\), \(p>1\), all weak eigenfunctions and eigenvalues are found.

Reviewer: M.Kučera (Praha)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35P05	General topics in linear spectral theory for PDEs
35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs

Keywords:

positive \(L_1\)-eigenfunction; semilinear elliptic problem; extremal parameter; extremal solution; spectrum of the linearization

References:

[1]	Baras, P.; Cohen, L., Complete blow after \(T_{ max }\) for the solution of a semilinear heat equation, J. Funct. Anal., 71, 142-174 (1987) · Zbl 0653.35037
[2]	Baras, P.; Goldstein, J. A., The heat equation with a singular potential, Trans. Amer. Math. Soc., 284, 121-139 (1984) · Zbl 0556.35063
[3]	Bebernes, J.; Eberly, D., Mathematical Problems from Combustion Theory (1989), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0692.35001
[4]	H. Brezis, X. Cabré, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital.; H. Brezis, X. Cabré, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital. · Zbl 0907.35048
[5]	H. Brezis, T. Cazenave, Nonlinear Evolution Equations; H. Brezis, T. Cazenave, Nonlinear Evolution Equations · Zbl 0868.35058
[6]	Brezis, H.; Cazenave, T.; Martel, Y.; Ramiandrisoa, A., Blow up for\(u_t\)−\(Δugu\), Adv. Differential Equations, 1, 73-90 (1996) · Zbl 0855.35063
[7]	Brezis, H.; Vázquez, J. L., Blow-up solutions of some nonlinear elliptic problems, Revista Mat. Univ. Complutense Madrid, 10, 443-469 (1997) · Zbl 0894.35038
[8]	Crandall, M. G.; Rabinowitz, P. H., Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rational Mech. Anal., 58, 207-218 (1975) · Zbl 0309.35057
[9]	Dautray, R.; Lions, J. L., Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques (1984), Masson: Masson Paris · Zbl 0749.35003
[10]	Folland, G. B., Introduction to Partial Differential Equations (1976), Princeton Univ. Press and Univ. of Tokyo Press: Princeton Univ. Press and Univ. of Tokyo Press Princeton · Zbl 0371.35008
[11]	Gelfand, I. M., Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl., 29, 295-381 (1963) · Zbl 0127.04901
[12]	Joseph, D. D.; Lundgren, T. S., Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49, 241-269 (1973) · Zbl 0266.34021
[13]	Lebedev, N. N., Special Functions and Their Applications (1965), Prentice-Hall: Prentice-Hall New York · Zbl 0131.07002
[14]	Martel, Y., Uniqueness of weak extremal solutions for nonlinear elliptic problems, Houston J. Math., 23, 161-168 (1997) · Zbl 0884.35037
[15]	Y. Martel, Complete blow up and global behavior of solutions of\(u_t}Δugu\); Y. Martel, Complete blow up and global behavior of solutions of\(u_t}Δugu\)
[16]	Y. Martel, Dynamical instability of extremal singular solutions of nonlinear elliptic problems, Adv. Math. Sci. Appl.; Y. Martel, Dynamical instability of extremal singular solutions of nonlinear elliptic problems, Adv. Math. Sci. Appl. · Zbl 0932.35070
[17]	Peral, I.; Vázquez, J. L., On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term, Arch. Rational Mech. Anal., 129, 201-224 (1995) · Zbl 0821.35080
[18]	Watson, G. N., A Treatise on the Theory of Bessel Functions (1922), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · JFM 48.0412.02