Equivalence between uniform $L^{2^\star}(\Omega)$ a-priori bounds and uniform $L^{\infty}(\Omega)$ a-priori bounds for subcritical elliptic equations
Keywords
A priori estimates, positive solutions, subcritical nonlinearity, radial solutionsAbstract
We provide sufficient conditions for a uniform $L^{2^\star}(\Omega)$ bound to imply a uniform $L^\infty (\Omega)$ bound for positive classical solutions to a class of subcritical elliptic problems in bounded $C^2$ domains in ${\mathbb R}^N$. We also establish an equivalent result for sequences of boundary value problems.References
F.V. Atkinson and L.A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations 70 (1987), no. 3, 349–365.
A. Bahri and J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41(1988), no. 3, 253–294.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential equations, Universitext, Springer, New York, 2011.
H. Brezis and R.E.L. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977), no. 6, 601–614.
A. Castro and A. Kurepa, Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball, Proc. Amer. Math. Soc. 101 (1987), no. 1, 57–64.
A. Castro and R. Pardo, A priori bounds for positive solutions of subcritical elliptic equations, Rev. Mat. Complut. 28 (2015), 715–731.
A. Castro and R. Pardo, Branches of positive solutions of subcritical elliptic equations in convex domains, Dynamical Systems, Differential Equations and Applications, AIMS Proceedings, (2015), 230–238.
A. Castro and R. Pardo, Branches of positive solutions for subcritical elliptic equations, Contributions to Nonlinear Elliptic Equations and Systems, Progress in Nonlinear Differential Equations and Their Applications 86 (2015), 87–98.
A. Castro and R. Pardo, A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 3, 783–790.
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) 61 (1982), no. 1, 41–63.
B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883–901.
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer–Verlag, Berlin, second ed., 1983.
Z.-C .Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 2, 159–174.
D.D. Joseph and T.S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 9 (1972/1973), 241–269.
P.L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), no. 4, 441–467.
N. Mavinga and R. Pardo, A priori bounds and existence of positive solutions for subcritical semilinear elliptic systems, J. Math. Anal. Appl. 449 (2017), no. 2, 1172–1188.
R. Nussbaum, Positive solutions of nonlinear elliptic boundary value problems, J. Math. Anal. Appl. 51 (1975), (1975), no. 2, 461–482.
S.I. Pohozaev, On the eigenfunctions of the equation ∆u + λf (u) = 0, Dokl. Akad. Nauk SSSR 165 (1965), 36–39.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
