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

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.
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  • 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.

    Mathematics Subject Classification: Primary:35B45;Secondary:35B09, 35B33, 35J25, 35J60.

    Citation:

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

  • [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. CastroM. 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.
    [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.
    [4] D. G. de FigueiredoP. L. Lions and 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 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.
    [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. 
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