Research article

Symmetry of positive solutions of a $ p $-Laplace equation with convex nonlinearites

  • Received: 22 September 2022 Revised: 18 January 2023 Accepted: 25 January 2023 Published: 06 April 2023
  • MSC : 35A21, 35B06

  • In this paper, we consider the symmetry properties of the positive solutions of a $ p $-Laplacian problem of the form

    $ \begin{eqnarray*} \begin{cases} -{{\Delta}}_p u = f(x,u),\ \ \ \ \ \ \ \mathrm{in}\ \ \ \ \ {{\Omega}},\\ \ \ \ \ \ \ \ u = g(x), \ \ \ \ \ \ \ \ \ \ \mathrm{on}\ \ \ \ \partial{{\Omega}}, \end{cases} \end{eqnarray*} $

    where $ {{\Omega}} $ is an open smooth bounded domain in $ R^N, N\geq2 $, and symmetric w.r.t. the hyperplane $ T_0^\nu (\nu $ is a direction vector in $ R^N, |\nu| = 1 $$) $, $ f $: $ {{\Omega}}\times R^+\rightarrow R^+ $ is a continuous function of class $ C^1 $ w.r.t. the second variable, $ g\geq 0 $ is continuous, and both $ f $ and $ g $ are symmetric w.r.t. $ T^\nu_0 $, respectively. Introducing some assumptions on nonlinearities, we get that the positive solutions of the problem above are symmetric w.r.t. the direction $ \nu $ by a new simple idea even if $ {{\Omega}} $ is not convex in the direction $ \nu $.

    Citation: Keqiang Li, Shangjiu Wang. Symmetry of positive solutions of a $ p $-Laplace equation with convex nonlinearites[J]. AIMS Mathematics, 2023, 8(6): 13425-13431. doi: 10.3934/math.2023680

    Related Papers:

  • In this paper, we consider the symmetry properties of the positive solutions of a $ p $-Laplacian problem of the form

    $ \begin{eqnarray*} \begin{cases} -{{\Delta}}_p u = f(x,u),\ \ \ \ \ \ \ \mathrm{in}\ \ \ \ \ {{\Omega}},\\ \ \ \ \ \ \ \ u = g(x), \ \ \ \ \ \ \ \ \ \ \mathrm{on}\ \ \ \ \partial{{\Omega}}, \end{cases} \end{eqnarray*} $

    where $ {{\Omega}} $ is an open smooth bounded domain in $ R^N, N\geq2 $, and symmetric w.r.t. the hyperplane $ T_0^\nu (\nu $ is a direction vector in $ R^N, |\nu| = 1 $$) $, $ f $: $ {{\Omega}}\times R^+\rightarrow R^+ $ is a continuous function of class $ C^1 $ w.r.t. the second variable, $ g\geq 0 $ is continuous, and both $ f $ and $ g $ are symmetric w.r.t. $ T^\nu_0 $, respectively. Introducing some assumptions on nonlinearities, we get that the positive solutions of the problem above are symmetric w.r.t. the direction $ \nu $ by a new simple idea even if $ {{\Omega}} $ is not convex in the direction $ \nu $.



    加载中


    [1] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43 (1971), 304–318. https://doi.org/10.1007/BF00250468 doi: 10.1007/BF00250468
    [2] B. Gidas, Symmetry of positive solutions of nonlinear elliptic equations in $R^{n}$, Adv. Math. Suppl. Stud., 7 (1981), 369–402.
    [3] B. Gidas, W. M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209–243. https://doi.org/10.1007/BF01221125 doi: 10.1007/BF01221125
    [4] H. Berestycki, L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Bras. Mat., 22 (1991), 1–37. https://doi.org/10.1007/BF01244896 doi: 10.1007/BF01244896
    [5] 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 (1998), 689–707.
    [6] L. Damascelli, F. Pacella, Monotonicity and symmetry results for $p$-Laplace equations and applications, Adv. Differ. Equations, 5 (2000), 1179–1200. https://doi.org/10.57262/ade/1356651297 doi: 10.57262/ade/1356651297
    [7] L. Damascelli, F. Pacella, M. Ramaswamy, Symmetry of ground states of $p$-Laplace equations via the moving plane method, Arch. Ration. Mech. Anal., 148 (1999), 291–308. https://doi.org/10.1007/s002050050163 doi: 10.1007/s002050050163
    [8] J. Serrin, H. Zou, Symmetry of ground states of quasilinear elliptic equations, Arch. Ration. Mech. Anal., 148 (1999), 265–290. https://doi.org/10.1007/s002050050162 doi: 10.1007/s002050050162
    [9] J. M. do Ó, R. da Costa, Symmetry properties for nonnegative solutions of non-uniformally elliptic equations in hyperbolic spaces, J. Math. Anal. Appl., 435 (2016), 1753–1711. https://doi.org/10.1016/j.jmaa.2015.11.031 doi: 10.1016/j.jmaa.2015.11.031
    [10] F. Pacella, Symmetry resluts for solutions of semilinear elliptic equations with convex nonlinearities, J. Funct. Anal., 1992 (2002), 271–282. https://doi.org/10.1006/jfan.2001.3901 doi: 10.1006/jfan.2001.3901
    [11] O. A. Ladyzhenskaya, Linear and quasilinear elliptic equations, New York: Academic Press, 1968.
    [12] J. Simon, Régularité de la solution d'un problème aux limites non linéaires, Ann. Fac. Sci. Toulouse, 3 (1981), 247–274. https://doi.org/10.5802/AFST.569 doi: 10.5802/AFST.569
    [13] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), 493–516. https://doi.org/10.1016/S0294-1449(98)80032-2 doi: 10.1016/S0294-1449(98)80032-2
    [14] M. Badiale, E. Nabana, A note on radiality of solutions of $p$-Laplacian, Appl. Anal., 52 (1994), 35–43. https://doi.org/10.1080/00036819408840222 doi: 10.1080/00036819408840222
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(820) PDF downloads(68) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog