The sub-supersolution method for weak solutions

Montenegro, Marcelo; Ponce, Augusto

doi:10.1090/S0002-9939-08-09231-9

The sub-supersolution method for weak solutions
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by Marcelo Montenegro and Augusto C. Ponce PDF

Proc. Amer. Math. Soc. 136 (2008), 2429-2438 Request permission

Abstract:

We extend the method of sub and supersolutions in order to prove existence of $L^1$-solutions of the equation $-\Delta u = f(x,u)$ in $\Omega$, where $f$ is a Carathéodory function. The proof is based on Schauder’s fixed point theorem.

References

Kiyoshi Akô, On the Dirichlet problem for quasi-linear elliptic differential equations of the second order, J. Math. Soc. Japan 13 (1961), 45–62. MR 147758, DOI 10.2969/jmsj/01310045
Haïm Brezis, Thierry Cazenave, Yvan Martel, and Arthur Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations 1 (1996), no. 1, 73–90. MR 1357955
Haïm Brezis, Moshe Marcus, and Augusto C. Ponce, A new concept of reduced measure for nonlinear elliptic equations, C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 169–174 (English, with English and French summaries). MR 2078069, DOI 10.1016/j.crma.2004.05.012
H. Brezis and L. Nirenberg, Nonlinear functional analysis. In preparation.
Haïm Brezis and Augusto C. Ponce, Reduced measures on the boundary, J. Funct. Anal. 229 (2005), no. 1, 95–120. MR 2180075, DOI 10.1016/j.jfa.2004.12.001
Philippe Clément and Guido Sweers, Getting a solution between sub- and supersolutions without monotone iteration, Rend. Istit. Mat. Univ. Trieste 19 (1987), no. 2, 189–194 (English, with Italian summary). MR 988383
E. N. Dancer and G. Sweers, On the existence of a maximal weak solution for a semilinear elliptic equation, Differential Integral Equations 2 (1989), no. 4, 533–540. MR 996759
Johanna Schoenenberger-Deuel and Peter Hess, A criterion for the existence of solutions of non-linear elliptic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 49–54 (1976). MR 440191, DOI 10.1017/s030821050001653x
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Abdelilah Gmira and Laurent Véron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. J. 64 (1991), no. 2, 271–324. MR 1136377, DOI 10.1215/S0012-7094-91-06414-8
M. Marcus. Nonlinear second order elliptic equations with measure data. To appear in Handbook of Differential Equations 2 (M. Chipot, ed.)
L. Orsina and A. C. Ponce, Semilinear elliptic equations and systems with diffuse measures. In preparation.
Maria Carla Palmeri, Entropy subsolutions and supersolutions for nonlinear elliptic equations in $L^1$, Ricerche Mat. 53 (2004), no. 2, 183–212 (2005). MR 2286838
Guido Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus, Séminaire de Mathématiques Supérieures, No. 16 (Été, vol. 1965, Les Presses de l’Université de Montréal, Montreal, Que., 1966 (French). MR 0251373