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, II. Revista Integración, Temas De matemáticas, 37(1), 113–148. https://doi.org/10.18273/revint.v37n1-2019006
Abstract
We continue studying the existence of uniform L∞ a priori bounds for positive solutions of subcritical elliptic equations
(P)p − \Delta_pu = f(u), in \Omega, u = 0, on ∂\Omega,
We provide sufficient conditions for having a-priori L∞ bounds for C1,μ (\overline{\Omega }) positive solutions to a class of subcritical elliptic problems in bounded, convex, C2 domains. In this part II, we extend our results to Hamiltonian elliptic systems −\Delta u = f(v),−\Deltav = g(u), in \Omega, u = v = 0 on ∂\Omega, when f(v) = vp /[ln(e + v)]α, g(u) = uq/[ln(e + u)]β, with α, β > 2/(N − 2), and p, q are lying in the critical Sobolev hyperbolae 1/p+1 + 1/q+1 = N−2/N . For quasilinear elliptic equations involving the p-Laplacian, there exists a-priori bounds for positive solutions of (P)p when f(u) = up⋆−1/[ln(e + u)]α, with p∗ = Np/(N−p), and α > p/(N−p). We also study the asymptotic behavior of radially symmetric solutions uα = uα(r) of (P)2 as α → 0.
Downloads
Download data is not yet available.
References
[1] Agmon S., Douglis A. and Nirenberg L.,“Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I”, Comm. Pure Appl. Math. 12 (1959), 623–727.
[2] Agmon S., Douglis A. and Nirenberg L., “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II”, Comm. Pure Appl. Math. 17 (1964), 35–92.
[3] Alexander J.C. and Antman S.S., “Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems”, Arch. Rational Mech. Anal. 76 (1981), No. 4, 339–354.
[4] Allegretto W. and Huang Y.X., “A Picone’s identity for the p-Laplacian and applications”, Nonlinear Anal. 32 (1998), No. 7, 819–830.
[5] Anane A., “Simplicité et isolation de la première valeur propre du p-laplacien avec poids”, C.R. Acad. Sci. Paris. Sér. I Math. 305 (1987), No. 16, 725–728.
[6] Astarita G. and Marrucci G., Principles of Non-Newtonian Fluid Mechanics, Mc-Graw Hill, New York, 1974.
[7] Atkinson F.V. and Peletier L.A., “Emden-Fowler equations involving critical exponents”, Nonlinear Anal. 10 (1986), No. 8, 755–776.
[8] Atkinson F.V. and Peletier L.A., “Elliptic equations with nearly critical growth”, J. Differential Equations 70 (1987), No. 3, 349–365.
[9] Azizieh C. and Clément P., “A priori estimates and continuation methods for positive solutions of p-Laplace equations”, J. Differential Equations 179 (2002), No. 1, 213–245.
[10] Azizieh C., Clément P. and Mitidieri E., “Existence and a priori estimates for positive solutions of p -Laplace systems”, J. Differential Equations 184 (2002), No. 2, 422–442.
[11] Birindelli I. and Mitidieri E., “Liouville theorems for elliptic inequalities and applications”, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), No. 6, 1217–1247.
[12] Bonheure D., dos Santos E.M. and Tavares H., “Hamiltonian elliptic systems: a guide to variational frameworks”, Port. Math. 71 (2014), No. 3-4, 301–395.
[13] Brezis H., Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.
[14] Busca J. and Manásevich R., “A Liouville-type Theorem for Lane-Emden Systems”, Indiana Univ. Math. J. 51 (2002), No. 1, 37– 51.
[15] Castro A., Mavinga N. and Pardo R., “Equivalence between uniform L^2^⋆(\Omega) a-priori bounds and uniform L^∞(\Omega) a-priori bounds for subcritical elliptic equations", To appear in Topol. Methods Nonlinear Anal. (2019).
[16] Castro A. and Pardo R., “A priori bounds for positive solutions of subcritical elliptic equations”, Rev. Mat. Complut. 28 (2015), No. 3, 715–731.
[17] 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.
[18] Castro A. and Kurepa A., “Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball”, Proc. Amer. Math. Soc. 101 (1987), No. 1, 57–64.
[19] Cianchi A. and Maz’yaW., “Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems”, J. Eur. Math. Soc. 16 (2014), No.3, 571–595.
[20] Cianchi A. and Maz’ya W., “Global gradient estimates in elliptic problems under minimal data and domain regularity”, Commun. Pure Appl. An. 14 (2015), No.1, 285–311.
[21] Clement Ph., de Figueiredo D.G. and Mitidieri E., “Positive Solutions of Semilinear Elliptic Systems”, Comm. Partial Differential Equations 17 (1992), No. 5-6, 923–940.
[22] Clement Ph., de Pagter B., Sweers G. and de Thelin F., “Existence of Solutions to a Semilinear Elliptic System through Orlicz-Sobolev Spaces”, Mediterr. J. Math. 1 (2004), No. 3, 241–267.
[23] Cosner C., “Positive Solutions for Superlinear Elliptic Systems, without variational structure”, Nonlinear Anal. 8 (1984), No. 12, 1427–1436.
[24] Crandall M.G. and Rabinowitz P.H., “Bifurcation from simple eigenvalues”, J. Functional Analysis 8 (1971), 321–340.
[25] Cuesta M. and Takak P., “A strong comparison principle for the Dirichlet plaplacian”, in Lecture notes in Pure and Appl. Math. 194, Dekker, New York (1998), 79–87.
[26] D’Ambrosio L. and Mitidieri E., “A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities”, Adv. Math. 224 (2010), No. 3, 967–1020.
[27] D’Ambrosio L. and Mitidieri E., “Liouville theorems for elliptic systems and applications”, J. Math. Anal. Appl. 413 (2014), No. 1, 121–138.
[28] Damascelli L., “Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results”, Ann. Inst. H. Poincaré. Analyse non linéaire 15 (1998), No. 4, 493–516.
[29] Damascelli L. and Pacella F., “Monotonicity and symmetry of solutions of p-Laplace equations, 1 < p < 2, via the moving plane method”, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 26 (1998), No. 4, 689–707.
[30] Damascelli L. and Pacella F., “Monotonicity and symmetry results for p-Laplace equations and applications”, Adv. Differential Equations 5 (2000), No. 7-9, 1179–1200.
[31] Damascelli L. and Pardo R., “A priori estimates for some elliptic equations involving the p-Laplacian”, Nonlinear Anal. Real World Appl. 41 (2018), 475–496.
[32] Damascelli L. and Sciunzi B., “Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations”, J. Differential Equations 206 (2004), No. 2, 483–515.
[33] Damascelli L. and Sciunzi B., “Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations”, Calc. Var. Partial Differential Equations 25 (2006), No. 2, 139–159.
[34] Dancer E.N., “On the structure of solutions of non-linear eigenvalue problems”, Indiana Univ. Math. J. 23 (1973/74), 1069–1076.
[35] Dancer E.N., “Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one”, Bull. London Math. Soc. 34 (2002), No. 5, 533–538.
[36] De Figueiredo D.G., do Ó J.M. and Ruf B., “On an Inequality by N. Trudinger and J. Moser and Related Elliptic Equations”, Comm. Pure Appl. Math. 55 (2002), No. 2, 135–152.
[37] De Figueiredo D.G., do Ó J.M. and Ruf B., “An Orlicz-space approach to superlinear elliptic systems”, J. Funct. Anal. 224 (2005), No. 2, 471–496.
[38] De Figueiredo D.G., do Ó J.M. and Ruf B., “Semilinear Elliptic Systems With Exponential Nonlinearities in Two Dimensions”, Adv. Nonlinear Stud. 6 (2006), No. 2, 199–213.
[39] De Figueiredo D.G., do Ó J.M. and Ruf B., “Non-variational elliptic systems in dimension two: a priori bounds and existence of positive solutions”, J. Fixed Point Theory Appl. 4 (2008), No. 1, 77–96.
[40] 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.
[41] DiBenedetto E., “C^{1+α} local regularity of weak solutions of degenerate elliptic equations”, Nonlinear Anal. 7 (1983), No. 8, 827 –850.
[42] Farina A. and Serrin J., “Entire solutions of completely coercive quasilinear elliptic equations I”, J. Differential Equations 250 (2011), No. 12, 4367–4408.
[43] Farina A. and Serrin J., “Entire solutions of completely coercive quasilinear elliptic equations II”, J. Differential Equations 250 (2011), No. 12, 4409–4436.
[44] Farina A., Montoro L. and Sciunzi B., “Monotonicity and one-dimensional symmetry for solutions of −\Delta_p u = f(u) in half-spaces”, Calc. Var. Part. Diff. Eq. 43 (2012), No. 1, 123–145.
[45] Farina A., Montoro L. and Sciunzi B., “Monotonicity of solutions of quasilinear degenerate elliptic equations in half-spaces”, Math. Ann. 357 (2013), No. 3, 855–893.
[46] Farina A., Montoro L. and Sciunzi B., “Monotonicity in half-spaces of positive solutions of −pu = f(u) in the case p > 2", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), No. 4, 1207–1229.
[47] Farina A., Montoro L., Riey G. and Sciunzi B., “Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces”, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), No. 1, 1–22.
[48] Fleckinger J. and Pardo R., “Bifurcation for an elliptic system coupled in the linear part”, Nonlinear Anal. RWA 37 (1999), No. 1, 13–30.
[49] Fleckinger J., Pardo R. and de Thélin F., “Four-parameter bifurcation for a p-Laplacian system”, Electron. J. Differential Equations (2001), No. 6, 1–15.
[50] Garcia Azorero J. and Peral I., “Existence and nonuniqueness for the p-laplacian: Nonlinear eigenvalues”, Comm. Partial Differential Equations 12 (1987), No. 12, 1389–1430.
[51] 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.
[52] 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.
[53] Gidas B. and Spruck J., “Global and local behavior of positive solutions of nonlinear elliptic equations”, Comm. Pure Appl. Math. 34 (1981), No. 4, 525–598.
[54] Gilbarg D. and Trudinger N.S., Elliptic partial differential equations of second order, volume 224 of Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1983.
[55] Guedda M. and Veron L., “Quasilinear elliptic equations involving critical Sobolev exponents”,Nonlinear Anal. 13 (1989), No. 8, 879–902.
[56] 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.
[57] Iwaniek T., “Projections onto gradient fields and Lp estimates for degenerated elliptic operators”, Studia Math. 75 (1983), No. 3, 293–312.
[58] KrasnoselskiiM.A., “Fixed point of cone-compressing or cone-extending operators”, Soviet Math. Dokl. 1 (1960), 1285–1288.
[59] Ladyzhenskaya O.A. and Ural’tseva N.N.,Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968.
[60] Lieberman G.M., “Boundary regularity for solutions of degenerate elliptic equations”, Nonlinear Anal. 12 (1988), No. 11, 1203–1219.
[61] Lindqvist P., “On the equation -\Delta_p u+λ|u|^{p−2}u = 0”, Proc. A.M.S. 109 (1990), No. 1, 157–164.
[62] Lions P.-L., “On the existence of positive solutions of semilinear elliptic equations”, SIAM Rev. 24 (1982), No. 4, 441–467.
[63] López-Gómez J., Spectral theory and nonlinear functional analysis, Chapman & Hall/CRC, Boca Raton, FL, 2001.
[64] López-Gómez J. and Pardo R., “Multiparameter nonlinear eigenvalue problems: positive solutions to elliptic Lotka-Volterra systems”, Appl. Anal. 31 (1988), No. 1-2, 103–127.
[65] López-Gómez J. and Pardo R., “Existence and uniqueness for some competition models with diffusion”, C.R. Acad. Sci. Paris Sér. I Math. 313 (1991), No. 13, 933–938.
[66] López-Gómez J. and Pardo R., “Coexistence regions in Lotka-Volterra models with diffusion”, Nonlinear Anal. 19 (1992), No. 1, 11–28.
[67] López-Gómez J. and Pardo R., “Coexistence in a simple food chain with diffusion”, J. Math. Biol. 30 (1992), No. 7, 655–668.
[68] López-Gómez J. and Pardo R., “Invertibility of linear noncooperative elliptic systems”, Nonlinear Anal. 31 (1998), No. 5-6, 687–699.
[69] Marinson L.K. and Pavlov K.B., “The effect of magnetic plasticity in non-Newtonian fluids”, Magnit. Gidrodinamika 3 (1969), 69–75.
[70] Marinson L.K. and Pavlov K.B., “Unsteady shear flows of a conducting fluid with a rheological power law”, Magnit. Gidrodinamika 2 (1970), 50–58.
[71] 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.
[72] Mitidieri E., “A Rellich type identity and applications”, Comm. Partial Differential Equations 18 (1993), No.1-2, 125–151.
[73] Mitidieri E. and Pohozaev S.I., “Absence of global positive solutions of quasilinear elliptic inequalities”, Dokl. Akad. Nauk. 359 (1998), No. 4, 456–460. [In Russian].
[74] Mitidieri E. and Pohozaev S.I., “Nonexistence of positive solutions for quasilinear elliptic problems on RN”, Proc. Steklov Inst. Math. 227 (1999), 186–216.
[75] Pardo R. and Sanjuán A., “Asymptotics for positive radial solutions of elliptic equations approaching critical growth”, Preprint.
[76] Peletier L.A. and Van der Vorst R.C.A.M., “Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation”, Differential Integral Equations 5 (1992), No. 4, 747–767.
[77] Peral I., Multiplicity of solutions for the p-Laplacian, ICTP lectures, Madrid, 1997.
[78] Pucci P. and Serrin J., The maximum principle, Birkhäuser Verlag, Basel, 2007.
[79] Quittner P. and Souplet P.H., “A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces”, Arch. Ration. Mech. Anal. 174 (2004), No. 1, 49–81.
[80] Quittner P., “A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities”, Nonlinear Anal. 102 (2014), 144–158.
[81] Rabinowitz P.H., “Some aspects of nonlinear eigenvalue problems”, Rocky Mountain J. Mat. 3 (1973), 161–202.
[82] Rabinowitz P.H., “Some global results for nonlinear eigenvalue problems”, J. Funct. Anal. 7 (1971), 487–513.
[83] Ruiz D., “ A priori estimates and existence of positive solutions for strongly nonlinear problems”, J. Differential Equations 199 (2004), No. 1, 96–114.
[84] Serrin J. and Zou H., “Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities”, Acta Math. 189 (2002), No. 1, 79–142.
[85] Serrin J. and Zou H., “Existence of positive solutions of Lane-Emden systems”, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 369–380.
[86] Serrin J. and Zou H., “Existence of entire positive solutions of elliptic Hamiltonian systems”, Comm. Partial Differential Equations 23 (1998), No. 3-4, 577–599.
[87] Souplet P.H., “The proof of the Lane-Emden conjecture in four space dimensions”, Adv. Math. 221 (2009), No. 5, 1409–1427.
[88] Tolksdorf P., “Regularity for a more general class of quasilinear elliptic equations”, J. Differential Equations 51 (1984), No. 1, 126–150.
[89] Troy W.C., “Symmetry properties in systems of semilinear elliptic equations”, J. Differential Equations 42 (1981), No. 3, 400–413.
[90] Trudinger N., “Remarks concerning the conformal deformation of Riemannian structures on compact manifolds”, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265–274.
[91] Vazquez J.L., “A strong maximum principle for some quasilinear elliptic equations”, Appl. Math. Optim. 12 (1984), No. 3, 191–202.
[92] Zou H.H., “A priori estimates and existence for quasi-linear elliptic equations”, Calc. Var. Partial Differential Equations 33 (2008), No. 4, 417–437.
[93] Zou H., “A priori estimates for a semilinear elliptic system without variational structure and their applications”, Math. Ann. 323 (2002), No. 4, 713–735.
[2] Agmon S., Douglis A. and Nirenberg L., “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II”, Comm. Pure Appl. Math. 17 (1964), 35–92.
[3] Alexander J.C. and Antman S.S., “Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems”, Arch. Rational Mech. Anal. 76 (1981), No. 4, 339–354.
[4] Allegretto W. and Huang Y.X., “A Picone’s identity for the p-Laplacian and applications”, Nonlinear Anal. 32 (1998), No. 7, 819–830.
[5] Anane A., “Simplicité et isolation de la première valeur propre du p-laplacien avec poids”, C.R. Acad. Sci. Paris. Sér. I Math. 305 (1987), No. 16, 725–728.
[6] Astarita G. and Marrucci G., Principles of Non-Newtonian Fluid Mechanics, Mc-Graw Hill, New York, 1974.
[7] Atkinson F.V. and Peletier L.A., “Emden-Fowler equations involving critical exponents”, Nonlinear Anal. 10 (1986), No. 8, 755–776.
[8] Atkinson F.V. and Peletier L.A., “Elliptic equations with nearly critical growth”, J. Differential Equations 70 (1987), No. 3, 349–365.
[9] Azizieh C. and Clément P., “A priori estimates and continuation methods for positive solutions of p-Laplace equations”, J. Differential Equations 179 (2002), No. 1, 213–245.
[10] Azizieh C., Clément P. and Mitidieri E., “Existence and a priori estimates for positive solutions of p -Laplace systems”, J. Differential Equations 184 (2002), No. 2, 422–442.
[11] Birindelli I. and Mitidieri E., “Liouville theorems for elliptic inequalities and applications”, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), No. 6, 1217–1247.
[12] Bonheure D., dos Santos E.M. and Tavares H., “Hamiltonian elliptic systems: a guide to variational frameworks”, Port. Math. 71 (2014), No. 3-4, 301–395.
[13] Brezis H., Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.
[14] Busca J. and Manásevich R., “A Liouville-type Theorem for Lane-Emden Systems”, Indiana Univ. Math. J. 51 (2002), No. 1, 37– 51.
[15] Castro A., Mavinga N. and Pardo R., “Equivalence between uniform L^2^⋆(\Omega) a-priori bounds and uniform L^∞(\Omega) a-priori bounds for subcritical elliptic equations", To appear in Topol. Methods Nonlinear Anal. (2019).
[16] Castro A. and Pardo R., “A priori bounds for positive solutions of subcritical elliptic equations”, Rev. Mat. Complut. 28 (2015), No. 3, 715–731.
[17] 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.
[18] Castro A. and Kurepa A., “Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball”, Proc. Amer. Math. Soc. 101 (1987), No. 1, 57–64.
[19] Cianchi A. and Maz’yaW., “Gradient regularity via rearrangements for p-Laplacian type elliptic boundary value problems”, J. Eur. Math. Soc. 16 (2014), No.3, 571–595.
[20] Cianchi A. and Maz’ya W., “Global gradient estimates in elliptic problems under minimal data and domain regularity”, Commun. Pure Appl. An. 14 (2015), No.1, 285–311.
[21] Clement Ph., de Figueiredo D.G. and Mitidieri E., “Positive Solutions of Semilinear Elliptic Systems”, Comm. Partial Differential Equations 17 (1992), No. 5-6, 923–940.
[22] Clement Ph., de Pagter B., Sweers G. and de Thelin F., “Existence of Solutions to a Semilinear Elliptic System through Orlicz-Sobolev Spaces”, Mediterr. J. Math. 1 (2004), No. 3, 241–267.
[23] Cosner C., “Positive Solutions for Superlinear Elliptic Systems, without variational structure”, Nonlinear Anal. 8 (1984), No. 12, 1427–1436.
[24] Crandall M.G. and Rabinowitz P.H., “Bifurcation from simple eigenvalues”, J. Functional Analysis 8 (1971), 321–340.
[25] Cuesta M. and Takak P., “A strong comparison principle for the Dirichlet plaplacian”, in Lecture notes in Pure and Appl. Math. 194, Dekker, New York (1998), 79–87.
[26] D’Ambrosio L. and Mitidieri E., “A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities”, Adv. Math. 224 (2010), No. 3, 967–1020.
[27] D’Ambrosio L. and Mitidieri E., “Liouville theorems for elliptic systems and applications”, J. Math. Anal. Appl. 413 (2014), No. 1, 121–138.
[28] Damascelli L., “Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results”, Ann. Inst. H. Poincaré. Analyse non linéaire 15 (1998), No. 4, 493–516.
[29] Damascelli L. and Pacella F., “Monotonicity and symmetry of solutions of p-Laplace equations, 1 < p < 2, via the moving plane method”, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 26 (1998), No. 4, 689–707.
[30] Damascelli L. and Pacella F., “Monotonicity and symmetry results for p-Laplace equations and applications”, Adv. Differential Equations 5 (2000), No. 7-9, 1179–1200.
[31] Damascelli L. and Pardo R., “A priori estimates for some elliptic equations involving the p-Laplacian”, Nonlinear Anal. Real World Appl. 41 (2018), 475–496.
[32] Damascelli L. and Sciunzi B., “Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations”, J. Differential Equations 206 (2004), No. 2, 483–515.
[33] Damascelli L. and Sciunzi B., “Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations”, Calc. Var. Partial Differential Equations 25 (2006), No. 2, 139–159.
[34] Dancer E.N., “On the structure of solutions of non-linear eigenvalue problems”, Indiana Univ. Math. J. 23 (1973/74), 1069–1076.
[35] Dancer E.N., “Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one”, Bull. London Math. Soc. 34 (2002), No. 5, 533–538.
[36] De Figueiredo D.G., do Ó J.M. and Ruf B., “On an Inequality by N. Trudinger and J. Moser and Related Elliptic Equations”, Comm. Pure Appl. Math. 55 (2002), No. 2, 135–152.
[37] De Figueiredo D.G., do Ó J.M. and Ruf B., “An Orlicz-space approach to superlinear elliptic systems”, J. Funct. Anal. 224 (2005), No. 2, 471–496.
[38] De Figueiredo D.G., do Ó J.M. and Ruf B., “Semilinear Elliptic Systems With Exponential Nonlinearities in Two Dimensions”, Adv. Nonlinear Stud. 6 (2006), No. 2, 199–213.
[39] De Figueiredo D.G., do Ó J.M. and Ruf B., “Non-variational elliptic systems in dimension two: a priori bounds and existence of positive solutions”, J. Fixed Point Theory Appl. 4 (2008), No. 1, 77–96.
[40] 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.
[41] DiBenedetto E., “C^{1+α} local regularity of weak solutions of degenerate elliptic equations”, Nonlinear Anal. 7 (1983), No. 8, 827 –850.
[42] Farina A. and Serrin J., “Entire solutions of completely coercive quasilinear elliptic equations I”, J. Differential Equations 250 (2011), No. 12, 4367–4408.
[43] Farina A. and Serrin J., “Entire solutions of completely coercive quasilinear elliptic equations II”, J. Differential Equations 250 (2011), No. 12, 4409–4436.
[44] Farina A., Montoro L. and Sciunzi B., “Monotonicity and one-dimensional symmetry for solutions of −\Delta_p u = f(u) in half-spaces”, Calc. Var. Part. Diff. Eq. 43 (2012), No. 1, 123–145.
[45] Farina A., Montoro L. and Sciunzi B., “Monotonicity of solutions of quasilinear degenerate elliptic equations in half-spaces”, Math. Ann. 357 (2013), No. 3, 855–893.
[46] Farina A., Montoro L. and Sciunzi B., “Monotonicity in half-spaces of positive solutions of −pu = f(u) in the case p > 2", Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), No. 4, 1207–1229.
[47] Farina A., Montoro L., Riey G. and Sciunzi B., “Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces”, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), No. 1, 1–22.
[48] Fleckinger J. and Pardo R., “Bifurcation for an elliptic system coupled in the linear part”, Nonlinear Anal. RWA 37 (1999), No. 1, 13–30.
[49] Fleckinger J., Pardo R. and de Thélin F., “Four-parameter bifurcation for a p-Laplacian system”, Electron. J. Differential Equations (2001), No. 6, 1–15.
[50] Garcia Azorero J. and Peral I., “Existence and nonuniqueness for the p-laplacian: Nonlinear eigenvalues”, Comm. Partial Differential Equations 12 (1987), No. 12, 1389–1430.
[51] 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.
[52] 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.
[53] Gidas B. and Spruck J., “Global and local behavior of positive solutions of nonlinear elliptic equations”, Comm. Pure Appl. Math. 34 (1981), No. 4, 525–598.
[54] Gilbarg D. and Trudinger N.S., Elliptic partial differential equations of second order, volume 224 of Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1983.
[55] Guedda M. and Veron L., “Quasilinear elliptic equations involving critical Sobolev exponents”,Nonlinear Anal. 13 (1989), No. 8, 879–902.
[56] 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.
[57] Iwaniek T., “Projections onto gradient fields and Lp estimates for degenerated elliptic operators”, Studia Math. 75 (1983), No. 3, 293–312.
[58] KrasnoselskiiM.A., “Fixed point of cone-compressing or cone-extending operators”, Soviet Math. Dokl. 1 (1960), 1285–1288.
[59] Ladyzhenskaya O.A. and Ural’tseva N.N.,Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968.
[60] Lieberman G.M., “Boundary regularity for solutions of degenerate elliptic equations”, Nonlinear Anal. 12 (1988), No. 11, 1203–1219.
[61] Lindqvist P., “On the equation -\Delta_p u+λ|u|^{p−2}u = 0”, Proc. A.M.S. 109 (1990), No. 1, 157–164.
[62] Lions P.-L., “On the existence of positive solutions of semilinear elliptic equations”, SIAM Rev. 24 (1982), No. 4, 441–467.
[63] López-Gómez J., Spectral theory and nonlinear functional analysis, Chapman & Hall/CRC, Boca Raton, FL, 2001.
[64] López-Gómez J. and Pardo R., “Multiparameter nonlinear eigenvalue problems: positive solutions to elliptic Lotka-Volterra systems”, Appl. Anal. 31 (1988), No. 1-2, 103–127.
[65] López-Gómez J. and Pardo R., “Existence and uniqueness for some competition models with diffusion”, C.R. Acad. Sci. Paris Sér. I Math. 313 (1991), No. 13, 933–938.
[66] López-Gómez J. and Pardo R., “Coexistence regions in Lotka-Volterra models with diffusion”, Nonlinear Anal. 19 (1992), No. 1, 11–28.
[67] López-Gómez J. and Pardo R., “Coexistence in a simple food chain with diffusion”, J. Math. Biol. 30 (1992), No. 7, 655–668.
[68] López-Gómez J. and Pardo R., “Invertibility of linear noncooperative elliptic systems”, Nonlinear Anal. 31 (1998), No. 5-6, 687–699.
[69] Marinson L.K. and Pavlov K.B., “The effect of magnetic plasticity in non-Newtonian fluids”, Magnit. Gidrodinamika 3 (1969), 69–75.
[70] Marinson L.K. and Pavlov K.B., “Unsteady shear flows of a conducting fluid with a rheological power law”, Magnit. Gidrodinamika 2 (1970), 50–58.
[71] 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.
[72] Mitidieri E., “A Rellich type identity and applications”, Comm. Partial Differential Equations 18 (1993), No.1-2, 125–151.
[73] Mitidieri E. and Pohozaev S.I., “Absence of global positive solutions of quasilinear elliptic inequalities”, Dokl. Akad. Nauk. 359 (1998), No. 4, 456–460. [In Russian].
[74] Mitidieri E. and Pohozaev S.I., “Nonexistence of positive solutions for quasilinear elliptic problems on RN”, Proc. Steklov Inst. Math. 227 (1999), 186–216.
[75] Pardo R. and Sanjuán A., “Asymptotics for positive radial solutions of elliptic equations approaching critical growth”, Preprint.
[76] Peletier L.A. and Van der Vorst R.C.A.M., “Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation”, Differential Integral Equations 5 (1992), No. 4, 747–767.
[77] Peral I., Multiplicity of solutions for the p-Laplacian, ICTP lectures, Madrid, 1997.
[78] Pucci P. and Serrin J., The maximum principle, Birkhäuser Verlag, Basel, 2007.
[79] Quittner P. and Souplet P.H., “A priori estimates and existence for elliptic systems via bootstrap in weighted Lebesgue spaces”, Arch. Ration. Mech. Anal. 174 (2004), No. 1, 49–81.
[80] Quittner P., “A priori estimates, existence and Liouville theorems for semilinear elliptic systems with power nonlinearities”, Nonlinear Anal. 102 (2014), 144–158.
[81] Rabinowitz P.H., “Some aspects of nonlinear eigenvalue problems”, Rocky Mountain J. Mat. 3 (1973), 161–202.
[82] Rabinowitz P.H., “Some global results for nonlinear eigenvalue problems”, J. Funct. Anal. 7 (1971), 487–513.
[83] Ruiz D., “ A priori estimates and existence of positive solutions for strongly nonlinear problems”, J. Differential Equations 199 (2004), No. 1, 96–114.
[84] Serrin J. and Zou H., “Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities”, Acta Math. 189 (2002), No. 1, 79–142.
[85] Serrin J. and Zou H., “Existence of positive solutions of Lane-Emden systems”, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 369–380.
[86] Serrin J. and Zou H., “Existence of entire positive solutions of elliptic Hamiltonian systems”, Comm. Partial Differential Equations 23 (1998), No. 3-4, 577–599.
[87] Souplet P.H., “The proof of the Lane-Emden conjecture in four space dimensions”, Adv. Math. 221 (2009), No. 5, 1409–1427.
[88] Tolksdorf P., “Regularity for a more general class of quasilinear elliptic equations”, J. Differential Equations 51 (1984), No. 1, 126–150.
[89] Troy W.C., “Symmetry properties in systems of semilinear elliptic equations”, J. Differential Equations 42 (1981), No. 3, 400–413.
[90] Trudinger N., “Remarks concerning the conformal deformation of Riemannian structures on compact manifolds”, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265–274.
[91] Vazquez J.L., “A strong maximum principle for some quasilinear elliptic equations”, Appl. Math. Optim. 12 (1984), No. 3, 191–202.
[92] Zou H.H., “A priori estimates and existence for quasi-linear elliptic equations”, Calc. Var. Partial Differential Equations 33 (2008), No. 4, 417–437.
[93] Zou H., “A priori estimates for a semilinear elliptic system without variational structure and their applications”, Math. Ann. 323 (2002), No. 4, 713–735.