# American Institute of Mathematical Sciences

August  2019, 39(8): 4863-4873. doi: 10.3934/dcds.2019198

## Regularity and weak comparison principles for double phase quasilinear elliptic equations

 Dipartimento di Matematica e Informatica, Università della Calabria, Via P. Bucci, 87036 Rende (CS), Italy

Received  November 2018 Revised  February 2019 Published  May 2019

We consider the Euler equation of functionals involving a term of the form
 $\int_{\Omega}(| \nabla u|^p+a(x)| \nabla u|^q) \,{ {\rm{d}}} x,$
with
 $1 and $ a(x)\geq 0 $. We prove weak comparison principle and summability results for the second derivatives of solutions. Citation: Giuseppe Riey. Regularity and weak comparison principles for double phase quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4863-4873. doi: 10.3934/dcds.2019198 ##### References:  [1] P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), Art. 62, 48 pp. doi: 10.1007/s00526-018-1332-z. [2] D. Castorina, G. Riey and B. Sciunzi, Hopf Lemma and Regularity Results for Quasilinear Anisotropic Elliptic Equations, Calc. Var. Partial Differential Equations, To appear. [3] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 215 (2015), 443-496. [4] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273. doi: 10.1007/s00205-015-0859-9. [5] M. Colombo and G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270 (2016), 1416-1478. doi: 10.1016/j.jfa.2015.06.022. [6] G. Cupini, F. Leonetti and E. Mascolo, Existence of weak solutions for elliptic systems with$p, q$-growth conditions, Ann. Acad. Sci. Fenn. Ser A I Math., 40 (2015), 645-658. doi: 10.5186/aasfm.2015.4035. [7] G. Cupini, P. Marcellini and E. Mascolo, Existence for elliptic equations under$p, q$-growth, Adv. Differential Equations, 19 (2014), 693-724. [8] L. Damascelli, Comparison theorems for some quesilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré. Anal. non linéaire, 15 (1998), 493-516. doi: 10.1016/S0294-1449(98)80032-2. [9] L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of$m$-Laplace equations, J. Differential Equations, 206 (2004), 483-515. doi: 10.1016/j.jde.2004.05.012. [10] L. Damascelli and B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differential Equations, 25 (2006), 139-159. doi: 10.1007/s00526-005-0337-6. [11] E. Di Benedetto,$C^{1+\alpha}$local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5. [12] L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with (p, q) growth, J. Differential Equations, 204 (2004), 5-55. doi: 10.1016/j.jde.2003.11.007. [13] A. Farina, L. Montoro, G. Riey and B. Sciunzi, Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 32 (2015), 1-22. doi: 10.1016/j.anihpc.2013.09.005. [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. doi: 10.1007/978-3-642-61798-0. [15] T. Leonori, A. Porretta and G. Riey, Comparison principles for p-Laplace equations with lower order terms, Ann. Mat. Pura Appl., 196 (2017), 877-903. doi: 10.1007/s10231-016-0600-9. [16] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth cconditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284. doi: 10.1007/BF00251503. [17] P. Marcellini, Regularity and existence of solutions of elliptic equations with$p, q$-growth conditions, J. Differential Equations, 90 (1991), 1-30. doi: 10.1016/0022-0396(91)90158-6. [18] C. Mercuri, G. Riey and B. Sciunzi, A regularity result for the p-Laplacian near uniform ellipticity, SIAM J. Math. Anal., 48 (2016), 2059-2075. doi: 10.1137/16M1058546. [19] N. G. Meyers and J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055-1056. doi: 10.1073/pnas.51.6.1055. [20] G. Montoro, G. Riey and B. Sciunzi, Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differential Equations, 20 (2015), 717-740. [21] M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1-122. doi: 10.1007/BF02413623. [22] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007. [23] G. Riey, Boundary regularity for quasi-linear elliptic equations with lower order term, Electron. J. Differential Equations, 283 (2017), 1-9. [24] G. Riey and B. Sciunzi, A note on the boundary regularity of solutions to quasilinear elliptic equations, ESAIM Control Optim. Calc. Var., 24 (2018), 849-858. doi: 10.1051/cocv/2017040. [25] B. Sciunzi, Some results on the qualitative properties of positive solutions of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 315-334. doi: 10.1007/s00030-007-5047-7. [26] B. Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term, Commum. Contemp. Math., 16 (2014), 1450013, 20pp. doi: 10.1142/S0219199714500138. [27] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0. [28] N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa, 27 (1973), 265-308. [29] V. V. Zhykov, Averaging of functional of the calculus of variations and elasticity theory, Izk. Akad. Nauk. SSSR Ser. Mat., 50 (1986), 675-710. [30] V. V. Zhykov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5. show all references ##### References:  [1] P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations, 57 (2018), Art. 62, 48 pp. doi: 10.1007/s00526-018-1332-z. [2] D. Castorina, G. Riey and B. Sciunzi, Hopf Lemma and Regularity Results for Quasilinear Anisotropic Elliptic Equations, Calc. Var. Partial Differential Equations, To appear. [3] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 215 (2015), 443-496. [4] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal., 218 (2015), 219-273. doi: 10.1007/s00205-015-0859-9. [5] M. Colombo and G. Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal., 270 (2016), 1416-1478. doi: 10.1016/j.jfa.2015.06.022. [6] G. Cupini, F. Leonetti and E. Mascolo, Existence of weak solutions for elliptic systems with$p, q$-growth conditions, Ann. Acad. Sci. Fenn. Ser A I Math., 40 (2015), 645-658. doi: 10.5186/aasfm.2015.4035. [7] G. Cupini, P. Marcellini and E. Mascolo, Existence for elliptic equations under$p, q$-growth, Adv. Differential Equations, 19 (2014), 693-724. [8] L. Damascelli, Comparison theorems for some quesilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré. Anal. non linéaire, 15 (1998), 493-516. doi: 10.1016/S0294-1449(98)80032-2. [9] L. Damascelli and B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of$m$-Laplace equations, J. Differential Equations, 206 (2004), 483-515. doi: 10.1016/j.jde.2004.05.012. [10] L. Damascelli and B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differential Equations, 25 (2006), 139-159. doi: 10.1007/s00526-005-0337-6. [11] E. Di Benedetto,$C^{1+\alpha}$local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1983), 827-850. doi: 10.1016/0362-546X(83)90061-5. [12] L. Esposito, F. Leonetti and G. Mingione, Sharp regularity for functionals with (p, q) growth, J. Differential Equations, 204 (2004), 5-55. doi: 10.1016/j.jde.2003.11.007. [13] A. Farina, L. Montoro, G. Riey and B. Sciunzi, Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 32 (2015), 1-22. doi: 10.1016/j.anihpc.2013.09.005. [14] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1983. doi: 10.1007/978-3-642-61798-0. [15] T. Leonori, A. Porretta and G. Riey, Comparison principles for p-Laplace equations with lower order terms, Ann. Mat. Pura Appl., 196 (2017), 877-903. doi: 10.1007/s10231-016-0600-9. [16] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth cconditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284. doi: 10.1007/BF00251503. [17] P. Marcellini, Regularity and existence of solutions of elliptic equations with$p, q$-growth conditions, J. Differential Equations, 90 (1991), 1-30. doi: 10.1016/0022-0396(91)90158-6. [18] C. Mercuri, G. Riey and B. Sciunzi, A regularity result for the p-Laplacian near uniform ellipticity, SIAM J. Math. Anal., 48 (2016), 2059-2075. doi: 10.1137/16M1058546. [19] N. G. Meyers and J. Serrin, H = W, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 1055-1056. doi: 10.1073/pnas.51.6.1055. [20] G. Montoro, G. Riey and B. Sciunzi, Qualitative properties of positive solutions to systems of quasilinear elliptic equations, Adv. Differential Equations, 20 (2015), 717-740. [21] M. K. V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1-122. doi: 10.1007/BF02413623. [22] P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, 2007. [23] G. Riey, Boundary regularity for quasi-linear elliptic equations with lower order term, Electron. J. Differential Equations, 283 (2017), 1-9. [24] G. Riey and B. Sciunzi, A note on the boundary regularity of solutions to quasilinear elliptic equations, ESAIM Control Optim. Calc. Var., 24 (2018), 849-858. doi: 10.1051/cocv/2017040. [25] B. Sciunzi, Some results on the qualitative properties of positive solutions of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 315-334. doi: 10.1007/s00030-007-5047-7. [26] B. Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term, Commum. Contemp. Math., 16 (2014), 1450013, 20pp. doi: 10.1142/S0219199714500138. [27] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0. [28] N. S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. 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