×

Bounded solutions of fourth-order nonlinear elliptic equations with convection terms. (English) Zbl 1479.35316

Summary: We study a Dirichlet problem associated to a nonlinear fourth-order equation with a convective lower order term and with coefficients of principal part satisfying a strengthened coercivity condition. We obtain the existence of a weak solution and its Hölder’s continuity.

MSC:

35J30 Higher-order elliptic equations
35J60 Nonlinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B65 Smoothness and regularity of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Skripnik, IV., High-order quasilinear elliptic equations with continuous generalized solutions, Differ Uravn, 14, 1104-1118 (1978) · Zbl 0408.35032
[2] De Giorgi, E., Un esempio di estremali discontinue per un problema variazionale di tipo ellittico, Boll Unione Mat Italiana, 1, 135-138 (1968) · Zbl 0155.17603
[3] Maz’ya, VG., Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients, Funkts Anal, 2, 3, 230-234 (1968) · Zbl 0179.43601 · doi:10.1007/BF01076124
[4] Giusti, E.; Miranda, M., Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni, Boll Unione Mat Ital, 2, 219-226 (1968) · Zbl 0155.44501
[5] Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann Ist Fourier, Grenoble, 15, 1, 189-257 (1965) · Zbl 0151.15401 · doi:10.5802/aif.204
[6] Stampacchia, G., Équations elliptiques du second ordre à coefficients discontinus (1966), Montréal: Université de Montréal, Montréal · Zbl 0151.15501
[7] Del Vecchio, T.; Posteraro, MR., An existence result for nonlinear and noncoercive problems, Nonlinear Anal, 31, 1-2, 191-206 (1998) · Zbl 0911.35045 · doi:10.1016/S0362-546X(96)00304-5
[8] Boccardo, L., Some developments on Dirichlet problems with discontinuous coefficients, Boll Un Mat Italiana (9), 2, 1, 285-297 (2009) · Zbl 1178.35367
[9] Boccardo, L.; Buccheri, S.; Cirmi, GR., Two linear noncoercive Dirichlet problems in duality, Milan J Math, 86, 1, 97-104 (2018) · Zbl 1398.35036 · doi:10.1007/s00032-018-0281-5
[10] Cirmi, GR; D’Asero, S.; Leonardi, S., Morrey estimates for a class of elliptic equations with drift term, Adv Nonlinear Anal · Zbl 1436.35120
[11] Cirmi, GR; D’Asero, S.; Leonardi, S.; Porzio, MM., Local regularity results for solutions of linear elliptic equations with drift term, Adv Calc Var · Zbl 1486.35202
[12] Dal Maso, G.; Skrypnik, IV., Asymptotic behaviour of nonlinear elliptic higher order equations in perforated domains, J Anal Math, 79, 63-112 (1999) · Zbl 0989.35047 · doi:10.1007/BF02788237
[13] D’Asero, S.; Larin, DV., Degenerate nonlinear higher-order elliptic problems in domains with fine-grained boundary, Nonlinear Anal, 64, 788-825 (2006) · Zbl 1208.35040 · doi:10.1016/j.na.2005.04.051
[14] D’Asero, S., On Harnack inequality for degenerate nonlinear higher-order elliptic equations, Appl Anal, 85, 8, 971-985 (2006) · Zbl 1207.35162 · doi:10.1080/00036810600793327
[15] D’Asero, S., On removability of the isolated singularity for solutions of high-order elliptic equations, Complex Var Elliptic Equ, 55, 5-6, 525-536 (2010) · Zbl 1187.35107 · doi:10.1080/17476931003728362
[16] Kufner, A.; Leonardi, S., Solvability of degenerate elliptic boundary value problems: another approach, Math Bohem, 119, 3, 255-274 (1994) · Zbl 0816.35039
[17] Cirmi, GR; D’Asero, S.; Leonardi, S., Fourth-order nonlinear elliptic equations with lower order term and natural growth conditions, Nonlinear Anal, 108, 66-86 (2014) · Zbl 1295.35213 · doi:10.1016/j.na.2014.05.014
[18] Voitovich, MV., Existence of bounded solutions for a class of nonlinear fourth-order equations, DEA, 3, 2, 247-266 (2011) · Zbl 1220.35056
[19] Voitovich, MV., Hölder continuity of bounded generalized solutions for nonlinear fourth-order elliptic equations with strengthened coercivity and natural growth terms, Electron J Differ Equ, 63, 1-18 (2017) · Zbl 1370.35065
[20] Boccardo, L.; Murat, F.; Puel, JP., Existence of bounded solutions for non linear elliptic unilateral problems, Ann Mat Pura Appl, 152, 183-196 (1988) · Zbl 0687.35042 · doi:10.1007/BF01766148
[21] Boccardo, L.; Murat, F.; Puel, JP., \(####\)-estimate for some nonlinear elliptic partial differential equations and application to an existence result, SIAM J Math Anal, 23, 326-333 (1992) · Zbl 0785.35033 · doi:10.1137/0523016
[22] Arcoya, D.; Carmona, J.; Leonori, T.; Martínez-Aparicio, PJ; Orsina, L.; Petitta, F., Existence and nonexistence of solutions for singular quadratic quasilinear equations, J Differ Equ, 246, 4006-4042 (2009) · Zbl 1173.35051 · doi:10.1016/j.jde.2009.01.016
[23] Cirmi, GR., Nonlinear elliptic equations with lower order term and \(####\)-data, Nonlinear Anal, 68, 2741-2749 (2008) · Zbl 1138.35021 · doi:10.1016/j.na.2007.02.020
[24] Cianci, P.; Cirmi, GR; D’Asero, S.; Leonardi, S., Morrey estimates for solutions of singular quadratic nonlinear equations, Ann Mat Pura Appl, 196, 5, 1739-1758 (2017) · Zbl 1378.35133 · doi:10.1007/s10231-017-0636-5
[25] Cirmi, GR, D’Asero, S, Leonardi, S.Gradient estimate for solutions of a class of nonlinear elliptic equations below the duality exponent. AIP Conference Proceedings; 2017. p. 1863.
[26] Cirmi, GR; D’Asero, S.; Leonardi, S., Gradient estimate for solutions of nonlinear singular elliptic equations below the duality exponent, Math Methods Appl Sci, 41, 261-269 (2018) · Zbl 1387.35268 · doi:10.1002/mma.4609
[27] Cirmi, GR; Leonardi, S., Regularity results for the gradient of solutions of linear elliptic equations with \(####\) data, Ann Mat Pura Appl, 185, 537-553 (2006) · Zbl 1232.35042 · doi:10.1007/s10231-005-0167-3
[28] Cirmi, GR; Leonardi, S., Regularity results for solutions of nonlinear elliptic equations with \(####\) data, Nonlinear Anal, 69, 1, 230-244 (2008) · Zbl 1187.35068 · doi:10.1016/j.na.2007.05.014
[29] Cirmi, GR; Leonardi, S., Higher differentiability for the solutions of nonlinear elliptic systems with lower order terms and \(####\)-data, Ann Mat Pura Appl, 193, 115-131 (2014) · Zbl 1305.35007 · doi:10.1007/s10231-012-0269-7
[30] Cirmi, GR; Leonardi, S.; Stará, J., Regularity results for the gradient of solutions of a class of linear elliptic systems with \(####\) data, Nonlinear Analysis: Theory, Methods & Applications, 68, 12, 3609-3624 (2008) · Zbl 1187.35050 · doi:10.1016/j.na.2007.04.004
[31] Leonardi, S.; Kottas, J.; Starà, J., Hölder regularity of the solutions of some classes of elliptic systems in convex non smooth domains, Nonlinear Anal, 60, 5, 925-944 (2005) · Zbl 1161.35373 · doi:10.1016/j.na.2004.09.056
[32] Hao, W.; Leonardi, S.; Steinhauer, M., Examples of discontinuous divergence-free solutions to elliptic variational problems, Comm Math Univ Carolinae, 36, 3, 511-517 (1995) · Zbl 0837.35041
[33] Hao, W.; Leonardi, S.; Nečas, J., An example of irregular solution to a nonlinear Euler-Lagrange elliptic system with real analytic coefficients, Annali Scuola Normale Superiore Pisa (IV), 23, 57-67 (1996) · Zbl 0864.35031
[34] Leonardi, S., On constants of some regularity theorems. De Giorgi’s type counterexample, Mathematische Nachrichten, 192, 1, 191-204 (1998) · Zbl 0909.35030 · doi:10.1002/mana.19981920111
[35] Leonardi, S., Remarks on the regularity of solutions of elliptic systems (1999), New York: Kluwer/Plenum, New York · Zbl 0952.35034
[36] Leonardi, S., Gradient estimates below duality exponent for a class of linear elliptic systems, Nonlinear Differ Equ Appl, 18, 3, 237-254 (2011) · Zbl 1219.35344 · doi:10.1007/s00030-010-0093-y
[37] Leonardi, S., Fractional differentiability for solutions of a class of parabolic systems with \(####\)-data, Nonlinear Anal, 95, 530-542 (2014) · Zbl 1286.35131 · doi:10.1016/j.na.2013.10.003
[38] Leonardi, S., Morrey estimates for some classes of elliptic equations with a lower order term, Nonlinear Anal, 177, part. B, 611-627 (2018) · Zbl 1403.35111 · doi:10.1016/j.na.2018.05.010
[39] Leonardi, S.; Nicolosi, F.; Shishkov, AE., Cauchy-Dirichlet problem for quasilinear degenerate parabolic equations of higher order with initial data increasing at infinity, Nonlinear World, 4, 4 (1997) · Zbl 0912.35074
[40] Leonardi, S., Weighted Miranda-Talenti inequality and applications to linear elliptic equations with discontinuous coefficients, Comm Math Univ Carolinae, 43, 1, 43-59 (2002) · Zbl 1090.35045
[41] Leonardi, S.; Leonetti, F.; Pignotti, C.; Rocha, E.; Staicu, V., Maximum principles for some quasilinear elliptic systems, Nonlinear Anal · Zbl 1435.35096
[42] Leonardi, S.; Stará, J., Regularity results for the gradient of solutions of linear elliptic systems with VMO-coefficients and \(####\) data, Forum Math, 22, 5, 913-940 (2010) · Zbl 1200.35084 · doi:10.1515/forum.2010.048
[43] Leonardi, S.; Stará, J., Regularity up to the boundary for the gradient of solutions of linear elliptic systems with VMO coefficients and \(####\) data, Complex Var Elliptic Equ, 56, 12, 1085-1098 (2011) · Zbl 1234.35088 · doi:10.1080/17476933.2010.534152
[44] Leonardi, S.; Stará, J., Regularity results for solutions of a class of parabolic systems with measure data, Nonlinear Anal, 75, 4, 2069-2089 (2012) · Zbl 1242.35073 · doi:10.1016/j.na.2011.10.008
[45] Leonardi, S.; Stará, J., Higher differentiability for solutions of a class of parabolic systems with \(####\)-data, Quart J Math, 66, 2, 659-676 (2015) · Zbl 1515.35084 · doi:10.1093/qmath/hau031
[46] Leray, J.; Lions, JL., Quelques résultats de Visik sur les problèmes aux limites non linéaires (1969), Paris: Dunod et Gauthier Villars, Paris · Zbl 0189.40603
[47] Kovalevsky, AA; Voitovich, MV., On the improvement of summability of generalized solutions of the Dirichlet problem for nonlinear equations of the fourth-order with strengthened ellipticity, Ukrainian Math J, 58, 11, 1717-1733 (2006) · Zbl 1150.35034 · doi:10.1007/s11253-006-0164-8
[48] Kovalevsky, AA., Entropy solutions of the Dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in \(####\), Izvestiya Math, 65, 2, 7-80 (2001) · Zbl 1052.35063
[49] Skripnik, IV., Nonlinear higher order elliptic equations (1973), Kiev: Naukova Dumka, Kiev · Zbl 0276.35043
[50] Serrin, J., Local behavior of solutions of quasi-linear equations, Acta Math, 111, 247-302 (1964) · Zbl 0128.09101 · doi:10.1007/BF02391014
[51] Ladyzhenskaya, OA; Uralt’seva, NN., Linear and quasilinear elliptic equations (1968), New York: Academic Press INC, New York · Zbl 0164.13002
[52] Kovalevsky, A.; Nicolosi, F., On regularity up to the boundary of solutions to degenerate nonlinear elliptic high-order equations., Nonlinear Anal., 40, 1-8, 365-379 (2000) · Zbl 0951.35053 · doi:10.1016/S0362-546X(00)85022-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.