Monotonicity of solutions to quasilinear problems with a first-order term in half-spaces

Farina, Alberto; Montoro, Luigi; Riey, Giuseppe; Sciunzi, Berardino

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 1-22.

We consider a quasilinear elliptic equation involving a first-order term, under zero Dirichlet boundary condition in half-spaces. We prove that any positive solution is monotone increasing with respect to the direction orthogonal to the boundary. The main ingredient in the proof is a new comparison principle in unbounded domains. As a consequence of our analysis, we also obtain some new Liouville type theorems.

[1] A.D. Alexandrov, A characteristic property of the spheres, Ann. Mat. Pura Appl. 58 (1962), 303 -354 | MR

[2] H. Berestycki, L.A. Caffarelli, L. Nirenberg, Symmetry for elliptic equations in a half space, J.L. Lions, et al. (ed.), Boundary Value Problems for PDEs and Applications, Masson, Paris (1993), 27 -42 | MR | Zbl

[3] H. Berestycki, L. Caffarelli, L. Nirenberg, Inequalities for second-order elliptic equations with applications to unbounded domains, Duke Math. J. 81 no. 2 (1996), 467 -494 | MR | Zbl

[4] H. Berestycki, L. Caffarelli, L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 25 no. 1–2 (1997), 69 -94 | EuDML | Numdam | MR | Zbl

[5] H. Berestycki, L. Caffarelli, L. Nirenberg, Monotonicity for elliptic equations in an unbounded Lipschitz domain, Commun. Pure Appl. Math. 50 (1997), 1089 -1111 | MR | Zbl

[6] H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat. Nova Ser. 22 no. 1 (1991), 1 -37 | MR | Zbl

[7] F. Charro, L. Montoro, B. Sciunzi, Monotonicity of solutions of Fully nonlinear uniformly elliptic equations in the half-plane, J. Differ. Equ. 251 no. 6 (2011), 1562 -1579 | MR | Zbl

[8] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15 no. 4 (1998), 493 -516 | EuDML | Numdam | MR | Zbl

[9] L. Damascelli, F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoam. 20 no. 1 (2004), 67 -86 | EuDML | MR | Zbl

[10] L. Damascelli, A. Farina, B. Sciunzi, E. Valdinoci, Liouville results for m-Laplace equations of Lane–Emden–Fowler type, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 no. 4 (2009), 1099 -1119 | EuDML | Numdam | MR | Zbl

[11] L. Damascelli, F. Pacella, Monotonicity and symmetry of solutions of p-Laplace equations, $1<p<2$, via the moving plane method, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 26 no. 4 (1998), 689 -707 | EuDML | Numdam | MR | Zbl

[12] L. Damascelli, B. Sciunzi, Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations, J. Differ. Equ. 206 no. 2 (2004), 483 -515 | MR | Zbl

[13] L. Damascelli, B. Sciunzi, Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations, Calc. Var. Partial Differ. Equ. 25 no. 2 (2006), 139 -159 | MR | Zbl

[14] L. Damascelli, B. Sciunzi, Monotonicity of the solutions of some quasilinear elliptic equations in the half-plane, and applications, Differ. Integral Equ. 23 no. 5–6 (2010), 419 -434 | MR | Zbl

[15] E.N. Dancer, Some notes on the method of moving planes, Bull. Aust. Math. Soc. 46 no. 3 (1992), 425 -434 | MR | Zbl

[16] E.N. Dancer, Some remarks on half space problems, Discrete Contin. Dyn. Syst., Ser. A 25 no. 1 (2009), 83 -88 | MR | Zbl

[17] E.N. Dancer, Y. Du, M. Efendiev, Quasilinear elliptic equations on half- and quarter-spaces, Adv. Nonlinear Stud. 13 (2013), 115 -136 | MR | Zbl

[18] E. Di Benedetto, $C^{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 no. 8 (1983), 827 -850 | MR | Zbl

[19] Y. Du, Z. Guo, Symmetry for elliptic equations in a half-space without strong maximum principle, Proc. R. Soc. Edinb., Sect. A 134 no. 2 (2004), 259 -269 | MR | Zbl

[20] A. Farina, Rigidity and one-dimensional symmetry for semilinear elliptic equations in the whole of ${ℝ}^N$ and in half spaces, Adv. Math. Sci. Appl. 13 no. 1 (2003), 65 -82 | MR | Zbl

[21] A. Farina, L. Montoro, B. Sciunzi, Monotonicity and one-dimensional symmetry for solutions of $-{\Delta }_{p}u=f\left(u\right)$ in half-spaces, Calc. Var. Partial Differ. Equ. 43 (2012), 123 -145 | MR | Zbl

[22] A. Farina, L. Montoro, B. Sciunzi, Monotonicity of solutions of quasilinear degenerate elliptic equations in half-spaces, Math. Ann. (2013), http://dx.doi.org/10.1007/s00208-013-0919-0 | MR | Zbl

[23] A. Farina, B. Sciunzi, E. Valdinoci, Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 7 no. 4 (2008), 741 -791 | EuDML | Numdam | MR | Zbl

[24] A. Farina, B. Sciunzi, E. Valdinoci, On a Poincaré type formula for solutions of singular and degenerate elliptic equations, Manuscr. Math. 132 no. 3–4 (2010), 335 -342 | MR | Zbl

[25] A. Farina, E. Valdinoci, Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems, Arch. Ration. Mech. Anal. 195 no. 3 (2010) | MR | Zbl

[26] E. Galakhov, A comparison principle for quasilinear operators in unbounded domains, Nonlinear Anal. 70 no. 12 (2009), 4190 -4194 | MR | Zbl

[27] B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys. 68 no. 3 (1979), 209 -243 | MR | Zbl

[28] B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Commun. Partial Differ. Equ. 6 (1981), 883 -901 | MR | Zbl

[29] T. Kilpeläinen, H. Shahgholian, X. Zhong, Growth estimates through scaling for quasilinear partial differential equations, Ann. Acad. Sci. Fenn., Math. 32 no. 2 (2007), 595 -599 | MR | Zbl

[30] O.A. Ladyzhenskaya, N.N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968) | MR | Zbl

[31] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 no. 11 (1988), 1203 -1219 | MR | Zbl

[32] S. Merchán, L. Montoro, I. Peral, B. Sciunzi, Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy–Leray potential, Ann. Inst. Henri Poincaré, Anal. Non Linéaire (2013), http://dx.doi.org/10.1016/j.anihpc.2013.01.003 | Numdam | MR | Zbl

[33] E. Mitidieri, S.I. Pokhozhaev, The absence of positive solutions for quasilinear elliptic inequalities, Dokl. Akad. Nauk 359 (1998), 456 -460 , Dokl. Math. 57 (1998), 250 -253 | MR | Zbl

[34] L. Montoro, B. Sciunzi, M. Squassina, Asymptotic symmetry for a class of quasi-linear parabolic problems, Adv. Nonlinear Stud. 10 no. 4 (2010), 789 -818 | MR | Zbl

[35] P. Pucci, J. Serrin, The Maximum Principle, Birkhäuser, Boston (2007) | MR | Zbl

[36] A. Quaas, B. Sirakov, Existence results for nonproper elliptic equations involving the Pucci operator, Commun. Partial Differ. Equ. 31 no. 7–9 (2006), 987 -1003 | MR | Zbl

[37] O. Savin, E. Valdinoci, Some monotonicity results for minimizers in the calculus of variations, J. Funct. Anal. 264 no. 10 (2013), 2469 -2496 | MR | Zbl

[38] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal. 43 no. 4 (1971), 304 -318 | MR | Zbl

[39] J. Serrin, H. Zou, Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 no. 1 (2002), 79 -142 | MR | Zbl

[40] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ. 51 no. 1 (1984), 126 -150 | MR | Zbl

[41] J.L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 no. 3 (1984), 191 -202 | MR | Zbl

[42] H.H. Zou, A priori estimates and existence for quasi-linear elliptic equations, Calc. Var. Partial Differ. Equ. 33 no. 4 (2008), 417 -437 | MR | Zbl