Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T08:45:00.943Z Has data issue: false hasContentIssue false
  • English
  • Français

Monotonicity results in half-spaces for the p-Laplacian

Published online by Cambridge University Press:  03 June 2015

Berardino Sciunzi
Berardino Sciunzi*
Affiliation:
Dipartimento di Matematica, Università della Calabria, Ponte Pietro Bucci 31B, 87036 Arcavacata di Rende, Cosenza, Italy, ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove monotonicity properties of positive weak solutions to – Δpu = f(u) in half-spaces for the case in which (2N + 2)/(N + 2) < p < 2 when zero Dirichlet boundary conditions are imposed. The model nonlinearity is given by f(s) := sq − λsm with q > m ≥ p − 1 and λ > 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 3 , June 2015 , pp. 597 - 610
Copyright
Copyright © Royal Society of Edinburgh 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Alexandrov, A. D.. A characteristic property of the spheres. Annali Mat. Pura Appl. 58 (1962), 303–354.Google Scholar
2 Berestycki, H., Caffarelli, L. A. and Nirenberg, L.. Symmetry for elliptic equations in the halfspace. In Boundary value problems for PDEs and applications (ed. Lions, J. L. et al.), pp. 27–42 (Paris: Masson, 1993).Google Scholar
3 Berestycki, H., Caffarelli, L. and Nirenberg, L.. Inequalities for second order elliptic equations with applications to unbounded domains. Duke Math. J. 81 (1996), 467–494.Google Scholar
4 Berestycki, H., Caffarelli, L. and Nirenberg, L.. Further qualitative properties for elliptic equations in unbounded domains. Annali Scuola Norm. Sup. Pisa IV 25 (1997), 69–94.Google Scholar
5 Berestycki, H., Caffarelli, L. and Nirenberg, L.. Monotonicity for elliptic equations in an unbounded Lipschitz domain. Commun. Pure Appl. Math. 50 (1997), 1089–1111.Google Scholar
6 Berestycki, H. and Nirenberg, L.. On the method of moving planes and the sliding method. Boll. Soc. Brasil. Mat. 22 (1991), 1–37.Google Scholar
7 Charro, F., Montoro, L. and Sciunzi, B.. Monotonicity of solutions of fully nonlinear uniformly elliptic equations in the half-plane. J. Diff. Eqns 251 (2011), 1562–1579.Google Scholar
8 Damascelli, L.. Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Annales Inst. H. Poincaré Analyse Non Linéaire 15 (1998), 493–516.Google Scholar
9 Damascelli, L. and Gladiali, F.. Some nonexistence results for positive solutions of elliptic equations in unbounded domains. Rev. Mat. Ibero. 20 (2004), 67–86.Google Scholar
10 Damascelli, L., Farina, A., Sciunzi, B. and Valdinoci, E.. Liouville results for m-Laplace equations of Lane–Emden–Fowler type. Annales Inst. H. Poincaré Analyse Non Linéaire 26 (2009), 1099–1119.Google Scholar
11 Damascelli, L. and Pacella, F.. Monotonicity and symmetry of solutions of p-Laplace equations, 1 < p < 2, via the moving plane method. Annali Scuola Norm. Sup. Pisa 26 (1998), 689–707.Google Scholar
12 Damascelli, L. and Sciunzi, B.. Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations. J. Diff. Eqns 206 (2004), 483–515.Google Scholar
13 Damascelli, L. an Sciunzi, B.. Harnack inequalities, maximum and comparison principles, and regularity of positive solutions of m-Laplace equations. Calc. Var. PDEs 25 (2006), 139–159.Google Scholar
14 Damascelli, L. and Sciunzi, B.. Monotonicity of the solutions of some quasilinear elliptic equations in the half-plane, and applications. Diff. Integ. Eqns 23 (2010), 419–434.Google Scholar
15 Dancer, E. N.. Some notes on the method of moving planes. Bull. Austral. Math. Soc. 46 (1992), 425–434.Google Scholar
16 Dancer, E. N.. Some remarks on half space problems. Discrete Contin. Dynam. Syst. A 25 (2009), 83–88.Google Scholar
17 Dancer, E. N., Du, Y. and Efendiev, M.. Quasilinear elliptic equations on half- and quarter-spaces. Adv. Nonlin. Studies 13 (2013), 115–136.Google Scholar
18 Benedetto, E. Di. C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlin. Analysis 7 (1983), 827–850.Google Scholar
19 Du, Y. and Guo, Z.. Symmetry for elliptic equations in a half-space without strong maximum principle. Proc. R. Soc. Edinb. A 134 (2004), 259–269.Google Scholar
20 Farina, A.. Rigidity and one-dimensional symmetry for semilinear elliptic equations in the whole of ℝ N and in half spaces. Adv. Math. Sci. Appl. 13 (2003), 65–82.Google Scholar
21 Farina, A., Montoro, L., Riey, G. and Sciunzi, B.. Monotonicity of solutions to quasilin-ear problems with a first-order term in half-spaces. Preprint, 2013. (Available at http://arXiv.org/abs/1306.0381.)Google Scholar
22 Farina, A., Montoro, L. and Sciunzi, B.. Monotonicity and one-dimensional symmetry for solutions of −Δ pu = f(u) in half-spaces. Calc. Var. PDEs 43 (2012), 123–145.Google Scholar
23 Farina, A., Montoro, L. and Sciunzi, B.. Monotonicity of solutions of quasilinear degenerate elliptic equations in half-spaces. Math. Annalen 357 (2013), 855–893.Google Scholar
24 Farina, A., Sciunzi, B. and Valdinoci, E.. Bernstein and De Giorgi type problems: new results via a geometric approach. Annali Scuola Norm. Sup. Pisa V 7 (2008), 741–791.Google Scholar
25 Farina, A., Sciunzi, B. and Valdinoci, E.. On a Poincaré type formula for solutions of singular and degenarate elliptic equations. Manuscr. Math. 132 (2010), 335–342.Google Scholar
26 Galakhov, E.. A comparison principle for quasilinear operators in unbounded domains. Nonlin. Analysis 70 (2009), 4190–4194.Google Scholar
27 Gidas, B., Ni, W. M., and Nirenberg, L.. Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68 (1979), 209–243.Google Scholar
28 Ladyzhenskaya, O. A. and Uraltseva, N. N.. Linear and quasilinear elliptic equations (Academic Press, 1968).Google Scholar
29 Lieberman, G. M.. Boundary regularity for solutions of degenerate elliptic equations. Nonlin. Analysis 12 (1988), 1203–1219.Google Scholar
30 Mercuri, C. and Willem, M.. A global compactness result for the p-Laplacian involving critical nonlinearities. Discrete Contin. Dynam. Syst. 28 (2010), 469–493.Google Scholar
31 Pucci, P. and Serrin, J.. The maximum principle (Birkhäuser, 2007).Google Scholar
32 Quaas, A. and Sirakov, B.. Existence results for nonproper elliptic equations involving the Pucci operator. Commun. PDEs 31 (2006), 987–1003.Google Scholar
33 Sciunzi, B.. Some results on the qualitative properties of positive solutions of quasilinear elliptic equations. Nonlin. Diff. Eqns Applic. 14 (2007), 315–334.Google Scholar
34 Serrin, J.. A symmetry problem in potential theory. Arch. Ration. Mech. Analysis 43 (1971), 304–318.Google Scholar
35 Serrin, J. and Zou, H.. Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189 (2002), 79–142.Google Scholar
36 Tolksdorf, P.. Regularity for a more general class of quasilinear elliptic equations. J. Diff. Eqns 51 (1984), 126–150.Google Scholar
37 Vazquez, J. L.. A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12 (1984), 191–202.Google Scholar
38 Zou, H. H.. A priori estimates and existence for quasi-linear elliptic equations. Calc. Var. PDEs 33 (2008), 417–437.Google Scholar