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

show all references

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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