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# Branches of positive solutions of subcritical elliptic equations in convex domains

• We provide sufficient conditions for the existence of $L^{\infty}$ a priori estimates for positive solutions to a class of subcritical elliptic equations in bounded $C^2$ convex domains. These sufficient conditions widen the range of nonlinearities for which a priori bounds are known. Using these a priori bounds we prove the existence of positive solutions for a class of problems depending on a parameter
Mathematics Subject Classification: Primary: 35B45; Secondary: 35B09, 35B33, 35J25, 35J60.

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•  [1] H. Brezis., Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. [2] H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems. Comm. Partial Differential Equations, 2 (1977), 601-614. [3] A. Castro and R. Pardo, A priori bounds for positive solutions of subcritical elliptic equations. Revista Matemática Complutense, (2015) 28: 715-731. URL http://dx.doi.org/10.1007/s13163-015-0180-z [4] A. Castro and R. Pardo, Branches of positive solutions for subcritical elliptic equations. To appear in Contributions to Nonlinear Elliptic Equations and Systems, Progress in Nonlinear Differential Equations and Their Applications 86, 87-98. http://dx.doi.org/10.1007/978-3-319-19902-3_7 [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues. J. Functional Analysis, 8 (1971), 321-340. [6] 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 (1982), 41-63. [7] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations, 6 (1981), 883-901. [8] 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. [9] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Academic Press, New York-London, 1968. [10] R. Nussbaum, Positive solutions of nonlinear elliptic boundary value problems. J. Math. Anal. Appl., 51 (1975), 461-482. [11] S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$. Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. [12] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems. J. Functional Analysis, 7 (1971), 487-513. [13] R. E. L. Turner, A priori bounds for positive solutions of nonlinear elliptic equations in two variables. Duke Math. J., 41 (1974), 759-774.
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