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Positive maximal and minimal solutions for non-homogeneous elliptic equations depending on the gradient. (English) Zbl 1455.35089

Summary: We are concerned with positive maximal and minimal solutions for non-homogeneous elliptic equations of the form \[ - \text{div}(a(| \nabla u |^p) | \nabla u |^{p - 2} \nabla u) = f(x, u, \nabla u) \quad \text{in } \Omega, \] supplied with Dirichlet boundary conditions. First we localize maximal and minimal solutions between not necessarily bounded sub-super solutions. Then using a uniform gradient estimate, which seems of independent interest, we show the existence of positive maximal and minimal solutions in some situations. More precisely, we obtain positive maximal and minimal solution to some classes of non-homogeneous equations depending on the gradient which may be perturbed by unbounded, singular or logistic sources.

MSC:

35J60 Nonlinear elliptic equations
35B09 Positive solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35J75 Singular elliptic equations
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[1] Abdellaoui, B.; Dall’Aglio, A.; Peral, I., Some remarks on elliptic problems with critical growth in the gradient, J. Differ. Equ.. J. Differ. Equ., J. Differ. Equ., 246, 2988-2990 (2009), (Corrigendum) · Zbl 1168.35329
[2] Alama, S.; Tarantello, G., On semilinear elliptic equations with indefinite nonlinearities, Calc. Var., 1, 439-475 (1993) · Zbl 0809.35022
[3] Alves, C. O.; Carrião, P. C.; Faria, L. F.O., Existence of solutions to singular elliptic equations with convection terms via the Galerkin method, Electron. J. Differ. Equ., 12, 1-12 (2010) · Zbl 1188.35081
[4] Alves, C. O.; Moussaoui, A., Existence of solutions for a class of singular elliptic systems with convection term, Asymptot. Anal., 90, 237-248 (2014) · Zbl 1327.35115
[5] Alves, C. O.; Santos, C. A.; Zhou, J., Existence and non-existence of blow-up solutions for a non-autonomous problem with indefinite and gradient terms, Z. Angew. Math. Phys., 66, 891-918 (2015) · Zbl 1327.35121
[6] Amann, H., Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z., 150, 281-295 (1976) · Zbl 0331.35026
[7] Amann, H.; Crandall, M. G., On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J., 27, 779-790 (1978) · Zbl 0391.35030
[8] Arcoya, D.; Boccardo, L.; Leonori, T.; Porretta, A., Existence and nonexistence of solutions for singular quadratic quasilinear equations, J. Differ. Equ., 249, 2771-2795 (2010) · Zbl 1203.35103
[9] Arcoya, D.; Carmona, J.; Martínez-Aparicio, J., Bifurcation for quasilinear elliptic singular BVP, Commun. Partial Differ. Equ., 36, 670-692 (2011) · Zbl 1239.35046
[10] Arcoya, D.; De Coster, C.; Jeanjean, L.; Tanaka, K., Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal., 268, 2298-2335 (2015) · Zbl 1322.35025
[11] Bernstein, S. N., Sur la nature analytique des solutions de certain équations aux derives partielles du second ordre [On the analytic nature of solutions of certain second-order partial differential equations], Math. Ann., 59, 20-76 (1904) · JFM 35.0354.01
[12] Boccardo, L.; Murat, F.; Puel, J.-P., Résultats d’existence pour certains problèmes elliptiques quasilinéaires, Ann. Sc. Norm. Super. Pisa, 11, 213-235 (1984) · Zbl 0557.35051
[13] Brezis, H.; Oswald, L., Remarks on sublinear elliptic equations, Nonlinear Anal., 10, 55-64 (1986) · Zbl 0593.35045
[14] Bueno, H.; Ercole, G.; Zumpano, A.; Ferreira, W. M., Positive solutions for the p-Laplacian with dependence on the gradient, Nonlinearity, 25, 1211-1234 (2012) · Zbl 1241.35105
[15] Brézis, H.; Turner, M. G., On a class of superlinear elliptic problems, Commun. Partial Differ. Equ., 2, 601-614 (1977) · Zbl 0358.35032
[16] Cantrell, R. S.; Cosner, C., Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proc. R. Soc. Edinb., 112, 293-318 (1989) · Zbl 0711.92020
[17] Carl, S.; Le, V. K.; Motreanu, D., Nonsmooth Variational Problems and Their Inequalities (2007), Springer: Springer New York
[18] Cianchi, A.; Maz’ya, V., Global Lipschitz regularity for a class of quasilinear elliptic equations, Commun. Partial Differ. Equ., 36, 100-133 (2011) · Zbl 1220.35065
[19] Cianchi, A.; Maz’ya, V., Global gradient estimates in elliptic problem under minimal data and domain regularity, Commun. Pure Appl. Anal., 14, 285-311 (2015) · Zbl 1325.35027
[20] Corrêa, F. J.S. A.; Corrêa, A. S.S.; Figueiredo, G. M., Positive solution for a class of p&q-singular elliptic equation, Nonlinear Anal., Real World Appl., 16, 163-169 (2014) · Zbl 1297.35102
[21] Corrêa, F. J.S. A.; Leão, A. S.S. C.; Figueiredo, G. M., Existence of positive solution for a singular system involving general quasilinear operators, Differ. Equ. Appl., 6, 481-494 (2014) · Zbl 1310.35099
[22] Cuesta Leon, M., Existence results for quasi-linear problems via ordered sub and supersolutions, Ann. Fac. Sci. Toulouse, Math., 6, 591-608 (1997) · Zbl 0910.35055
[23] Dancer, E. N.; Sweers, G., On the existence of a maximal weak solution for a semilinear elliptic equation, Differ. Integral Equ., 2, 533-540 (1989) · Zbl 0732.35027
[24] de Coster e, C.; Fernández, A. J., Existence and multiplicity for elliptic p-Laplacian problems with critical growth in the gradient, Calc. Var., 57, Article 89 pp. (2018) · Zbl 1401.35131
[25] de Figueiredo, D.; Girardi, M.; Matzeu, M., Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Differ. Integral Equ., 17, 119-126 (2004) · Zbl 1164.35341
[26] de Figueiredo, D. G.; Gossez, J. P.; Quiorin, H. R.; Ubilla, P., Elliptic equations involving the p-Laplacian and a gradient term having natural growth, Rev. Mat. Iberoam., 35, 173-194 (2019) · Zbl 1415.35138
[27] del Pino, M. A., Positive solutions of a semilinear elliptic equation on a compact manifold, Nonlinear Anal., 22, 1423-1430 (1994) · Zbl 0812.58077
[28] Di Benedetto, E., \( C^{1 + \alpha}\) local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7, 827-850 (1983) · Zbl 0539.35027
[29] Deuel, J.; Hess, P., A criterion for the existence of solutions of nonlinear elliptic boundary value problems, Proc. R. Soc. Edinb., 74, 49-54 (1975) · Zbl 0331.35028
[30] Du, Y.; Huang, Q., Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal., 31, 1-18 (1999) · Zbl 0959.35065
[31] Duzaar, F.; Mingione, G., Gradient estimates via non-linear potentials, Am. J. Math., 133, 1093-1149 (2011) · Zbl 1230.35028
[32] Dunford, N.; Schwartz, J. T., Linear Operators. I. General Theory (1958), Interscience Publishers, Inc.: Interscience Publishers, Inc. New York
[33] Evans, L. C., A new proof of local \(C^{1 , \alpha}\) regularity for solutions of certain degenerate elliptic P.D.E., J. Differ. Equ., 45, 356-373 (1982) · Zbl 0508.35036
[34] Faraci, F.; Motreanu, D.; Puglisi, D., Positive solutions of quasi-linear elliptic equations with dependence on the gradient, Calc. Var., 54, 525-538 (2015) · Zbl 1326.35159
[35] Faria, L. F.O.; Miyagaki, O. H.; Motreanu, D., Comparison and positive solutions for problems with (p,q)-Laplacian and convection term, Proc. Edinb. Math. Soc., 57, 687-698 (2014) · Zbl 1315.35114
[36] Faria, L. F.O.; Miyagaki, O. H.; Motreanu, D.; Tanaka, M., Existence results for nonlinear elliptic equations with Leray-Lions operator and dependence on the gradient, Nonlinear Anal., 96, 154-166 (2014) · Zbl 1285.35014
[37] Fazly, M.; Shahgholian, H., Monotonicity formulas for coupled elliptic gradient systems with applications, Adv. Nonlinear Anal., 9, 479-495 (2020) · Zbl 1421.35107
[38] Fraile, J. M.; Koch Medina, P.; López-Gómez, J.; Merino, S., Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differ. Equ., 127, 295-319 (1996) · Zbl 0860.35085
[39] Fukagai, N.; Narukawa, K., On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. Mat. Pura Appl., 186, 539-564 (2007) · Zbl 1223.35132
[40] Gasińsli, L.; Papageorgiou, N. S., Positive solutions for nonlinear elliptic problems with dependence on the gradient, J. Differ. Equ., 263, 1451-1476 (2017) · Zbl 1372.35111
[41] Gueda, M.; Veron, L., Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13, 879-902 (1989) · Zbl 0714.35032
[42] Hess, P., On a second order nonlinear elliptic boundary value problem, (Cesari, L.; Kannan, R.; Weinberger, H. F. (1978), Academic Press: Academic Press New York), 99-107
[43] Kazdan, J. L.; Kramer, R. J., Invariant criteria for existence of solutions to second order quasilinear elliptic equations, Commun. Pure Appl. Math., 31, 619-645 (1978) · Zbl 0368.35031
[44] Ladyzenskaya, O. A.; Ural’ceva, N. N., Linear and Quasilinear Elliptic Equations (1968), Academic Press: Academic Press New York
[45] Leoni, G., A First Course in Sobolev Spaces (2017), AMS: AMS Providence, RI · Zbl 1382.46001
[46] Lieberman, G. M., The Dirichlet problem for quasilinear elliptic equations with continuously differentiable data, Commun. Partial Differ. Equ., 11, 167-229 (1986) · Zbl 0589.35036
[47] Lieberman, G. M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12, 1203-1219 (1988) · Zbl 0675.35042
[48] Lieberman, G. M., The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Commun. Partial Differ. Equ., 16, 311-361 (1991) · Zbl 0742.35028
[49] Liu, Z.; Motreanu, D.; Zeng, S., Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient, Calc. Var., 58, 28 (2019) · Zbl 1409.35091
[50] G.F. Madeira, A.A.F. Nunes, Degenerate or singular elliptic Kirchhoff equations with non-homogeneous operators and concave-convex terms, submitted for publication.
[51] Marcus, M.; Mizel, V., Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Ration. Mech. Anal., 45, 294-320 (1972) · Zbl 0236.46033
[52] Medina, M.; Ochoa, P., On viscosity and weak solutions for non-homogeneous p-Laplace equations, Nonlinear Anal., 8, 468-481 (2019) · Zbl 1419.35075
[53] Nečas, J., Introduction to the Theory of Nonlinear Elliptic Equations (1986), John Willey & Sons: John Willey & Sons United Kingdom · Zbl 0657.76058
[54] Ouyang, T., On the positive solutions of semilinear equations \(\operatorname{\Delta} u + \lambda u - h u^p = 0\) on compact manifolds, Trans. Am. Math. Soc., 331, 503-527 (1992)
[55] Papageorgiou, N. S.; Rădulescu, V. D.; Repovš, D. D., Positive solutions for nonlinear Neumann problems with singular terms and convection, J. Math. Pures Appl., 136, 1-21 (2020) · Zbl 1437.35016
[56] Papageorgiou, N. S.; Rădulescu, V. D.; Repovš, D. D., Positive solutions for nonlinear parametric singular Dirichlet problems, Bull. Math. Sci., 9, Article 1950011 pp. (2019)
[57] Peral, I., On some elliptic and parabolic equations related to growth models, (Figalli, A.; Peral, I.; Valdinoci, E., Partial Differential Equations and Geometric Measure Theory. Partial Differential Equations and Geometric Measure Theory, Lecture Notes in Math., vol. 2211 (2018), Springer: Springer Cham), 43-195 · Zbl 1411.35005
[58] Perera, K.; Silva, E. A.B., Existence and multiplicity of positive solutions for singular quasilinear problems, J. Math. Anal. Appl., 323, 1238-1252 (2006) · Zbl 1168.35358
[59] Pohožaev, S. I., On equations of the form \(\operatorname{\Delta} u = f(x, u, D u)\), Math. USSR Sb., 41, 269-280 (1982) · Zbl 0483.35033
[60] Pucci, P.; Serrin, J., The strong maximum principle revisited, J. Differ. Equ., 196, 1-66 (2004) · Zbl 1109.35022
[61] Puel, J. P., Existence, comportement à l’infini et stabilité dans certains problèmes quasilinéaires elliptiques d’ordre 2, Ann. Sc. Norm. Super. Pisa, 3, 89-119 (1976) · Zbl 0331.35027
[62] Ruiz, D., A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Differ. Equ., 199, 96-114 (2004) · Zbl 1081.35037
[63] Ruiz, D.; Suŕez, A., Existence and uniqueness of positive solution of a logistic equation with nonlinear gradient term, Proc. R. Soc. Edinb., 137, 555-566 (2007) · Zbl 1135.35046
[64] Serrin, J., Local behavior of solutions of quasi-linear equations, Acta Math., 111, 247-302 (1964) · Zbl 0128.09101
[65] Serrin, J., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. R. Soc. London, 264, 413-496 (1969) · Zbl 0181.38003
[66] Serrin, J., The solvability of boundary value problems, (Browder, F. E., Mathematical Developments Arising from Hilbert Problems. Mathematical Developments Arising from Hilbert Problems, Proc. Sympos. Pure Math., vol. 28 (1976), AMS: AMS Providence, RI), 507-524
[67] Tan, Z.; Fang, F., Orlicz-Sobolev versus Hölder local mimizer and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 402, 348-370 (2013) · Zbl 1446.35042
[68] Tanaka, M., Existence of a positive solution for quasilinear elliptic equations with a nonlinearity including the gradient, Bound. Value Probl., 2013, Article 173 pp. (2013) · Zbl 1294.35029
[69] Tolksdorf, P., Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ., 51, 126-150 (1983) · Zbl 0488.35017
[70] Tolksdorf, P., On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Commun. Partial Differ. Equ., 8, 773-817 (1983) · Zbl 0515.35024
[71] Uhlenbeck, K., Regularity for a class of non-linear elliptic systems, Acta Math., 138, 219-240 (1977) · Zbl 0372.35030
[72] Xavier, J. B.M., Some existence theorems for equations of the form \(- \operatorname{\Delta} u = f(x, u, D u)\), Nonlinear Anal., 15, 59-67 (1990) · Zbl 0711.35049
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