Research and Innovation Articles
Published 2019-02-19
Keywords
- A priori estimates,
- subcritical nonlinearity,
- moving planes method,
- Pohozaev identity,
- critical Sobolev hyperbola
- biparameter bifurcation ...More
How to Cite
Pardo, R. (2019). On the existence of a priori bounds for positive solutions of elliptic problems, I. Revista Integración, Temas De matemáticas, 37(1), 77–111. https://doi.org/10.18273/revint.v37n1-2019005
Abstract
This paper gives a survey over the existence of uniform L∞ a priori bounds for positive solutions of subcritical elliptic equations
(P)p -\Delta_p u =f(u), in \Omega, u = 0, on \partial\Omega
widening the known ranges of subcritical nonlinearities for which positive solutions are a-priori bounded. Our arguments rely on the moving planes method, a Pohozaev identity, W1,q regularity for q > N, and Morrey’s Theorem. In this part I, when p = 2, we show that there exists a-priori bounds for classical, positive solutions of (P)2 with f(u) = u2∗−1/[ln(e + u)]α, with 2∗ = 2N/(N − 2), and α > 2/(N − 2). Appealing to the Kelvin transform, we cover non-convex domains.
In a forthcoming paper containing part II, we extend our results for Hamiltonian elliptic systems (see [22]), and for the p-Laplacian (see [10]). We also study the asymptotic behavior of radially symmetric solutions uα = uα(r) of (P)2 as α → 0 (see [24]).
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References
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[7] Castro A. and Pardo R., “Branches of positive solutions for subcritical elliptic equations”, Progr. Nonlinear Differential Equations Appl: Contributions to Nonlinear Elliptic Equations and Systems 86 (2015), 87–98.
[8] Castro A. and Pardo R., “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.
[9] Castro A. and Shivaji R., “Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric”, Comm. Partial Differential Equations 14 (1989), No. 8-9, 1091–1100.
[10] Damascelli L. and Pardo R., “A priori estimates for some elliptic equations involving the p-Laplacian”, Nonlinear Anal. 41 (2018), 475–496.
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[18] Han Z.-C., “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.
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[22] Mavinga N. and Pardo R., “A priori bounds and existence of positive solutions for subcritical semilinear elliptic systems”, J. Math. Anal. Appl. 449 (2017), No. 2, 1172–1188.
[23] Nussbaum R., “Positive solutions of nonlinear elliptic boundary value problems”, J. Math. Anal. Appl. 51 (1975), No.2, 461–482.
[24] Pardo R. and Sanjuán A., “Asymptotics for positive radial solutions of elliptic equations approaching critical growth”, Preprint.
[25] Pohozaev S.I., “On the eigenfunctions of the equation u+f(u) = 0”, Dokl. Akad. Nauk SSSR 165 (1965), 36–39.
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[27] Turner R.E.L., “A priori bounds for positive solutions of nonlinear elliptic equations in two variables”, Duke Math. J. 41 (1974), 759-774.
[2] Bahri A. and Coron J.M., “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.
[3] Brezis H., Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.
[4] Brezis H. and Turner R.E.L., “On a class of superlinear elliptic problems”, Comm. Partial Differential Equations 2 (1977), No. 6, 601–614.
[5] Castro A. and Pardo R., “A priori bounds for positive solutions of subcritical elliptic equations”, Rev. Mat. Complut. 28 (2015), No.3, 715–731.
[6] Castro A. and Pardo R., “Branches of positive solutions of subcritical elliptic equations in convex domains”, Dynamical Systems, Differential Equations and Applications, AIMS Proceedings (2015), 230–238.
[7] Castro A. and Pardo R., “Branches of positive solutions for subcritical elliptic equations”, Progr. Nonlinear Differential Equations Appl: Contributions to Nonlinear Elliptic Equations and Systems 86 (2015), 87–98.
[8] Castro A. and Pardo R., “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.
[9] Castro A. and Shivaji R., “Nonnegative solutions to a semilinear Dirichlet problem in a ball are positive and radially symmetric”, Comm. Partial Differential Equations 14 (1989), No. 8-9, 1091–1100.
[10] Damascelli L. and Pardo R., “A priori estimates for some elliptic equations involving the p-Laplacian”, Nonlinear Anal. 41 (2018), 475–496.
[11] De Figueiredo D.G., Lions P.-L. and Nussbaum R.D., “A priori estimates and existence of positive solutions of semilinear elliptic equations”, J. Math. Pures Appl. (9) 61 (1982), No. 1, 41–63.
[12] Ding W.Y., “Positive solutions of u + u(n+2)/(n−2) = 0 on contractible domains”, J. Partial Differential Equations 2 (1989), No.4, 83–88.
[13] Fenchel W., Convex Cones, Sets and Functions, Lecture Notes at Princeton University, Dept. of Mathematics, Princeton, N.J., 1953.
[14] Gidas B., Ni W.-M. and Nirenberg L., “Symmetry and related properties via the maximum principle”, Comm. Math. Phys. 68 (1979), No. 3, 209-243.
[15] Gidas B., Ni W.-M. and Nirenberg L., “Symmetry of positive solutions of nonlinear elliptic equations in Rn", in Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., 7, Academic Press (1981), 369–402.
[16] Gidas B. and Spruck J., “A priori bounds for positive solutions of nonlinear elliptic equations”, Comm. Partial Differential Equations 6 (1981), No. 8, 883–901.
[17] Gilbarg D. and Trudinger N.S., Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1983.
[18] Han Z.-C., “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.
[19] Joseph D.D. and Lundgren T.S., “Quasilinear Dirichlet problems driven by positive sources”, Arch. Rational Mech. Anal. 49 (1972/73), 241–269.
[20] Ladyzhenskaya O.A. and Ural’tseva N.N., Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968.
[21] Lions P.-L., “On the existence of positive solutions of semilinear elliptic equations”, SIAM Rev. 24 (1982), No. 4, 441–467.
[22] Mavinga N. and Pardo R., “A priori bounds and existence of positive solutions for subcritical semilinear elliptic systems”, J. Math. Anal. Appl. 449 (2017), No. 2, 1172–1188.
[23] Nussbaum R., “Positive solutions of nonlinear elliptic boundary value problems”, J. Math. Anal. Appl. 51 (1975), No.2, 461–482.
[24] Pardo R. and Sanjuán A., “Asymptotics for positive radial solutions of elliptic equations approaching critical growth”, Preprint.
[25] Pohozaev S.I., “On the eigenfunctions of the equation u+f(u) = 0”, Dokl. Akad. Nauk SSSR 165 (1965), 36–39.
[26] Serrin J., “A symmetry problem in potential theory”, Arch. Rational Mech. Anal. 43 (1971), 304–318.
[27] Turner R.E.L., “A priori bounds for positive solutions of nonlinear elliptic equations in two variables”, Duke Math. J. 41 (1974), 759-774.