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

1. 

artment of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA

2. 

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040–Madrid, Spain

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.

[2]

A. Castro, M. Hassanpour, 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.

[3]

A. Castro, 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.

[4]

D. G. de Figueiredo, P. L. Lions, R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.

[5]

B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.

[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.

[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.

[8]

S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.

show all references

References:
[1]

H. Brezis, Functional Analysis, {S}obolev Spaces and Partial Differential Equations Universitext. Springer, New York, 2011. ISBN 978-0-387-70913-0.

[2]

A. Castro, M. Hassanpour, 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.

[3]

A. Castro, 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.

[4]

D. G. de Figueiredo, P. L. Lions, R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.

[5]

B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901. doi: 10.1080/03605308108820196.

[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.

[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.

[8]

S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.

Figure 1.  (a) A convex-starlike boundary. (b) A ring-like domain
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