American Institute of Mathematical Sciences

May  2017, 22(3): 783-790. doi: 10.3934/dcdsb.2017038

A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions

Received  September 2015 Revised  October 2016 Published  January 2017

Fund Project: This work was partially supported by a grant from the Simons Foundations (# 245966 to Alfonso Castro). The second author is partially supported by grants MTM2012-31298, MTM2016-75465, MINECO, Spain and Grupo de Investigaci´on CADEDIF, UCM.

Building on the a priori estimates established in [3], we obtain a priori estimates for classical solutions to ellipticproblems with Dirichlet boundary conditions on regions with convex-starlike boundary. This includes ring-like regions. Arguments that go back to [4] are used to prove a priori bounds near the convex part of the boundary.Using that the boundary term in the Pohozaev identity on the boundary of a star-like region does not change sign, the proof isconcluded.

Citation: Alfonso Castro, Rosa Pardo. A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 783-790. doi: 10.3934/dcdsb.2017038
References:
 [1] H. Brezis, Functional Analysis, {S}obolev Spaces and Partial Differential Equations Universitext. Springer, New York, 2011. ISBN 978-0-387-70913-0. Google Scholar [2] A. Castro, M. Hassanpour and R. Shivaji, Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity, Comm. Partial Differential Equations, 20 (1995), 1927-1936.  doi: 10.1080/03605309508821157.  Google Scholar [3] A. Castro and R. Pardo, A priori bounds for positive solutions of subcritical elliptic equations, Revista Matemática Complutense, 28 (2015), 715-731.  doi: 10.1007/s13163-015-0180-z.  Google Scholar [4] D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl.(9), 61 (1982), 41-63.   Google Scholar [5] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.  Google Scholar [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 1983. ISBN 3-540-13025-X. Google Scholar [7] O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London, 1968. Google Scholar [8] S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.   Google Scholar

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References:
 [1] H. Brezis, Functional Analysis, {S}obolev Spaces and Partial Differential Equations Universitext. Springer, New York, 2011. ISBN 978-0-387-70913-0. Google Scholar [2] A. Castro, M. Hassanpour and R. Shivaji, Uniqueness of non-negative solutions for a semipositone problem with concave nonlinearity, Comm. Partial Differential Equations, 20 (1995), 1927-1936.  doi: 10.1080/03605309508821157.  Google Scholar [3] A. Castro and R. Pardo, A priori bounds for positive solutions of subcritical elliptic equations, Revista Matemática Complutense, 28 (2015), 715-731.  doi: 10.1007/s13163-015-0180-z.  Google Scholar [4] D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl.(9), 61 (1982), 41-63.   Google Scholar [5] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.  Google Scholar [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second edition, 1983. ISBN 3-540-13025-X. Google Scholar [7] O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London, 1968. Google Scholar [8] S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.   Google Scholar
(a) A convex-starlike boundary. (b) A ring-like domain
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