Article

Full entry | PDF   (0.3 MB)
Keywords:
elliptic equations; weight function; regularity of solutions
Summary:
We prove the local Hölder continuity of bounded generalized solutions of the Dirichlet problem associated to the equation $\sum_{i =1}^{m} \frac{\partial}{\partial x_i} a_i (x, u, \nabla u) - c_0 |u|^{p-2} u = f(x, u, \nabla u),$ assuming that the principal part of the equation satisfies the following degenerate ellipticity condition $\lambda (|u|) \sum_{i=1}^m a_i (x,u, \eta) \eta_i \geq \nu(x) |\eta|^p,$ and the lower-order term $f$ has a natural growth with respect to $\nabla u$.
References:
[1] Bensoussan A., Boccardo L., Murat F.: On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. Henri Poincaré 5 (1988), no. 4, 347–364. DOI 10.1016/S0294-1449(16)30342-0
[2] Boccardo L., Murat F., Puel J. P.: Existence de solutions faibles pour des équations elliptiques quasi-linéares à croissance quadratique. Nonlinear Partial Differential Equations and Their Applications, College de France Seminar, Vol. IV, Res. Notes in Math., 84, Pitman, London, 1983, 19–73 (French. English summary).
[3] Boccardo L., Murat F., Puel J. P.: Résultat d'existence pour certains problèmes elliptiques quasilinéaires. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 2, 213–235 (French).
[4] Bonafede S.: Quasilinear degenerate elliptic variational inequalities with discontinuous coefficients. Comment. Math. Univ. Carolin. 34 (1993), no. 1, 55–61.
[5] Bonafede S.: Existence and regularity of solutions to a system of degenerate nonlinear elliptic equations. Br. J. Math. Comput. Sci. 18 (2016), no. 5, 1–18. DOI 10.9734/BJMCS/2016/28702
[6] Bonafede S.: Existence of bounded solutions of Neumann problem for a nonlinear degenerate elliptic equation. Electron. J. Differential Equations 2017 (2017), no. 270, 1–21.
[7] Cirmi G. R., D'Asero S., Leonardi S.: Fourth-order nonlinear elliptic equations with lower order term and natural growth conditions. Nonlinear Anal. 108 (2014), 66–86.
[8] Del Vecchio T.: Strongly nonlinear problems with hamiltonian having natural growth. Houston J. Math. 16 (1990), no. 1, 7–24.
[9] Drábek P., Nicolosi F.: Existence of bounded solutions for some degenerated quasilinear elliptic equations. Ann. Mat. Pura Appl. 165 (1993), 217–238. DOI 10.1007/BF01765850
[10] Fabes E. B., Kenig C. E., Serapioni R. P.: The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7 (1982), 77–116. DOI 10.1080/03605308208820218
[11] Gilbarg D., Trudinger N. S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, 1983. Zbl 1042.35002
[12] Guglielmino F., Nicolosi F.: $W$-solutions of boundary value problems for degenerate elliptic operators. Ricerche Mat. 36 (1987), suppl., 59–72.
[13] Guglielmino F., Nicolosi F.: Existence theorems for boundary value problems associated with quasilinear elliptic equations. Ricerche Mat. 37 (1988), 157–176.
[14] John F., Nirenberg L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 (1961), 415–426. DOI 10.1002/cpa.3160140317
[15] Kovalevsky A., Nicolosi F.: Boundedness of solutions of variational inequalities with nonlinear degenerated elliptic operators of high order. Appl. Anal. 65 (1997), 225–249. DOI 10.1080/00036819708840560
[16] Kovalevsky A., Nicolosi F.: On Hölder continuity of solutions of equations and variational inequalities with degenerate nonlinear elliptic high order operators. Current Problems of Analysis and Mathematical Physics, Taormina 1998, Aracne, Rome, 2000, 205–220.
[17] Kovalevsky A., Nicolosi F.: Boundedness of solutions of degenerate nonlinear elliptic variational inequalities. Nonlinear Anal. 35 (1999), 987–999. DOI 10.1016/S0362-546X(98)00110-2
[18] Kovalevsky A., Nicolosi F.: On regularity up to the boundary of solutions to degenerate nonlinear elliptic high order equations. Nonlinear Anal. 40 (2000), 365–379.
[19] Ladyzhenskaya O., Ural'tseva N.: Linear and Quasilinear Elliptic Equations. translated from the Russian, Academic Press, New York-London, 1968. Zbl 0177.37404
[20] Landes R.: Solvability of perturbed elliptic eqautions with critical growth exponent for the gradient. J. Math. Anal. Appl. 139 (1989), 63–77. DOI 10.1016/0022-247X(89)90230-8
[21] Moser J.: A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math. 13 (1960), pp. 457–468. DOI 10.1002/cpa.3160130308
[22] Murthy M. K. V., Stampacchia G.: Boundary value problems for some degenarate elliptic operators. Ann. Mat. Pura Appl. (4) 80 (1968), 1–122. DOI 10.1007/BF02413623
[23] Serrin J. B.: Local behavior of solutions of quasi-linear equations. Acta Math. 111 (1964), 247–302. DOI 10.1007/BF02391014
[24] Skrypnik I. V.: Nonlinear Higher Order Elliptic Equations. Naukova dumka, Kiev, 1973 (Russian).
[25] Skrypnik I. V.: Higher order quasilinear elliptic equations with continuous generalized solutions. Differ. Equ. 14 (1978), no. 6, 786–795.
[26] Trudinger N. S.: On Harnack type inequalities and their application to quasilinear elliptic equations. Comm. Pure Appl. Math. 20 (1967), 721–747. DOI 10.1002/cpa.3160200406
[27] Trudinger N. S.: On the regularity of generalized solutions of linear non-uniformly elliptic equations. Arch. Ration. Mech. Anal. 42 (1971), 51–62. DOI 10.1007/BF00282317
[28] Trudinger N. S.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa 27 (1973), 265–308.
[29] Voitovich M. V.: Existence of bounded solutions for a class of nonlinear fourth-order equations. Differ. Equ. Appl. 3 (2011), no. 2, 247–266.
[30] Voitovich M. V.: Existence of bounded solutions for nonlinear fourth-order elliptic equations with strengthened coercivity and lower-terms with natural growth. Electron. J. Differential Equations 2013 (2013), no. 102, 25 pages.
[31] Voitovich M. V.: On the existence of bounded generalized solutions of the Dirichlet problem for a class of nonlinear high-order elliptic equations. J. Math. Sci. (N.Y.) 210 (2015), no. 1, 86–113. DOI 10.1007/s10958-015-2550-y
[32] Voitovych M. V.: Hölder continuity of bounded generalized solutions for nonlinear fourth-order elliptic equations with strengthened coercivity and natural growth terms. Electron. J. Differential Equations 2017 (2017), no. 63, 18 pages.
[33] Zamboni P.: Hölder continuity for solutions of linear degenerate elliptic equations under minimal assumptions. J. Differential Equations 182 (2002), 121–140. DOI 10.1006/jdeq.2001.4094

Partner of