# American Institute of Mathematical Sciences

2015, 2015(special): 230-238. doi: 10.3934/proc.2015.0230

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

 1 Department of Mathematics, Harvey Mudd College, Claremont, CA 91711 2 Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040-Madrid, Spain

Received  September 2014 Revised  March 2015 Published  November 2015

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
Citation: Alfonso Castro, Rosa Pardo. Branches of positive solutions of subcritical elliptic equations in convex domains. Conference Publications, 2015, 2015 (special) : 230-238. doi: 10.3934/proc.2015.0230
##### References:
 [1] H. Brezis., Functional analysis, Sobolev spaces and partial differential equations., Universitext. Springer, (2011). Google Scholar [2] H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems., Comm. Partial Differential Equations, 2 (1977), 601. Google Scholar [3] A. Castro and R. Pardo, A priori bounds for positive solutions of subcritical elliptic equations., Revista Matemática Complutense, (2015), 715. Google Scholar [4] A. Castro and R. Pardo, Branches of positive solutions for subcritical elliptic equations., To appear in Contributions to Nonlinear Elliptic Equations and Systems, (): 87. Google Scholar [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues., J. Functional Analysis, 8 (1971), 321. Google Scholar [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), 9 (1982), 41. Google Scholar [7] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations., Comm. Partial Differential Equations, 6 (1981), 883. Google Scholar [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, (1983). Google Scholar [9] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations., Translated from the Russian by Scripta Technica, (1968). Google Scholar [10] R. Nussbaum, Positive solutions of nonlinear elliptic boundary value problems., J. Math. Anal. Appl., 51 (1975), 461. Google Scholar [11] S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$., Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar [12] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems., J. Functional Analysis, 7 (1971), 487. Google Scholar [13] R. E. L. Turner, A priori bounds for positive solutions of nonlinear elliptic equations in two variables., Duke Math. J., 41 (1974), 759. Google Scholar

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##### References:
 [1] H. Brezis., Functional analysis, Sobolev spaces and partial differential equations., Universitext. Springer, (2011). Google Scholar [2] H. Brézis and R. E. L. Turner, On a class of superlinear elliptic problems., Comm. Partial Differential Equations, 2 (1977), 601. Google Scholar [3] A. Castro and R. Pardo, A priori bounds for positive solutions of subcritical elliptic equations., Revista Matemática Complutense, (2015), 715. Google Scholar [4] A. Castro and R. Pardo, Branches of positive solutions for subcritical elliptic equations., To appear in Contributions to Nonlinear Elliptic Equations and Systems, (): 87. Google Scholar [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues., J. Functional Analysis, 8 (1971), 321. Google Scholar [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), 9 (1982), 41. Google Scholar [7] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations., Comm. Partial Differential Equations, 6 (1981), 883. Google Scholar [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, (1983). Google Scholar [9] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear elliptic equations., Translated from the Russian by Scripta Technica, (1968). Google Scholar [10] R. Nussbaum, Positive solutions of nonlinear elliptic boundary value problems., J. Math. Anal. Appl., 51 (1975), 461. Google Scholar [11] S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$., Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar [12] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems., J. Functional Analysis, 7 (1971), 487. Google Scholar [13] R. E. L. Turner, A priori bounds for positive solutions of nonlinear elliptic equations in two variables., Duke Math. J., 41 (1974), 759. Google Scholar
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