# American Institute of Mathematical Sciences

February  2018, 38(2): 547-561. doi: 10.3934/dcds.2018024

## Nonradial least energy solutions of the p-Laplace elliptic equations

 Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840-8502, Japan

Received  April 2017 Revised  August 2017 Published  February 2018

Fund Project: This work was supported by JSPS KAKENHI Grant Number 16K05236.

We study the p-Laplace elliptic equations in the unit ball under the Dirichlet boundary condition. We call u a least energy solution if it is a minimizer of the Lagrangian functional on the Nehari manifold. A least energy solution becomes a positive solution. Assume that the nonlinear term is radial and it vanishes in $|x| <a$ and it is positive in $a<|x|<1$. We prove that if a is close enough to 1, then no least energy solution is radial. Therefore there exist both a positive radial solution and a positive nonradial solution.

Citation: Ryuji Kajikiya. Nonradial least energy solutions of the p-Laplace elliptic equations. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 547-561. doi: 10.3934/dcds.2018024
##### References:
 [1] M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453-467.  doi: 10.1515/ans-2004-0406. [2] V. Barutello, S. Secchi and E. Serra, A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl., 341 (2008), 720-728.  doi: 10.1016/j.jmaa.2007.10.052. [3] J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states, Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 803-828.  doi: 10.1016/j.anihpc.2006.04.001. [4] J. Byeon and Z.-Q. Wang, On the Hénon equation: Asymptotic profile of ground states, Ⅱ, J. Differential Equations, 216 (2005), 78-108.  doi: 10.1016/j.jde.2005.02.018. [5] M. Calanchi, S. Secchi and E. Terraneo, Multiple solutions for a Hénon-like equation on the annulus, J. Differential Equations, 245 (2008), 1507-1525.  doi: 10.1016/j.jde.2008.06.018. [6] D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17.  doi: 10.1016/S0022-247X(02)00292-5. [7] J.-L. Chern and C.-S. Lin, The symmetry of least-energy solutions for semilinear elliptic equations, J. Differential Equations, 187 (2003), 240-268.  doi: 10.1016/S0022-0396(02)00080-3. [8] K. Deimling, Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7. [9] P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations Second edition, Birkhäuser, Berlin, 2013. [10] P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbb{R}^2$, J. Anal. Math., 100 (2006), 249-280.  doi: 10.1007/BF02916763. [11] N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differential Equations, 247 (2009), 1311-1333.  doi: 10.1016/j.jde.2009.06.008. [12] R. Kajikiya, Non-even least energy solutions of the Emden-Fowler equation, Proc. Amer. Math. Soc., 140 (2012), 1353-1362.  doi: 10.1090/S0002-9939-2011-11172-9. [13] R. Kajikiya, Non-radial least energy solutions of the generalized Hénon equation, J. Differential Equations, 252 (2012), 1987-2003.  doi: 10.1016/j.jde.2011.08.032. [14] R. Kajikiya, Nonradial positive solutions of the p-Laplace Emden-Fowler equation with sign-changing weight, Mathematische Nachrichten, 289 (2016), 290-299.  doi: 10.1002/mana.201500103. [15] R. Kajikiya, Symmetric and asymmetric solutions of p-Laplace elliptic equations in hollow domains, To appear in Adv. Nonlinear Stud. [16] R. A. Moore and Z. Nehari, Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93 (1959), 30-52.  doi: 10.1090/S0002-9947-1959-0111897-8. [17] R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322. [18] A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), 75-97.  doi: 10.1007/s00209-006-0060-9. [19] P. Pucci and J. Serrin, The Maximum Principle Birkhäuser, Berlin, 2007. [20] E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301-326.  doi: 10.1007/s00526-004-0302-9. [21] D. Smets, M. Willem and J. Su, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480.  doi: 10.1142/S0219199702000725. [22] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations, 8 (1983), 773-817.  doi: 10.1080/03605308308820285. [23] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl Math Optim, 12 (1984), 191-202.  doi: 10.1007/BF01449041. [24] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications Springer, New York, 1995.

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##### References:
 [1] M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453-467.  doi: 10.1515/ans-2004-0406. [2] V. Barutello, S. Secchi and E. Serra, A note on the radial solutions for the supercritical Hénon equation, J. Math. Anal. Appl., 341 (2008), 720-728.  doi: 10.1016/j.jmaa.2007.10.052. [3] J. Byeon and Z.-Q. Wang, On the Hénon equation: asymptotic profile of ground states, Ⅰ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 803-828.  doi: 10.1016/j.anihpc.2006.04.001. [4] J. Byeon and Z.-Q. Wang, On the Hénon equation: Asymptotic profile of ground states, Ⅱ, J. Differential Equations, 216 (2005), 78-108.  doi: 10.1016/j.jde.2005.02.018. [5] M. Calanchi, S. Secchi and E. Terraneo, Multiple solutions for a Hénon-like equation on the annulus, J. Differential Equations, 245 (2008), 1507-1525.  doi: 10.1016/j.jde.2008.06.018. [6] D. Cao and S. Peng, The asymptotic behaviour of the ground state solutions for Hénon equation, J. Math. Anal. Appl., 278 (2003), 1-17.  doi: 10.1016/S0022-247X(02)00292-5. [7] J.-L. Chern and C.-S. Lin, The symmetry of least-energy solutions for semilinear elliptic equations, J. Differential Equations, 187 (2003), 240-268.  doi: 10.1016/S0022-0396(02)00080-3. [8] K. Deimling, Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7. [9] P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations Second edition, Birkhäuser, Berlin, 2013. [10] P. Esposito, A. Pistoia and J. Wei, Concentrating solutions for the Hénon equation in $\mathbb{R}^2$, J. Anal. Math., 100 (2006), 249-280.  doi: 10.1007/BF02916763. [11] N. Hirano, Existence of positive solutions for the Hénon equation involving critical Sobolev terms, J. Differential Equations, 247 (2009), 1311-1333.  doi: 10.1016/j.jde.2009.06.008. [12] R. Kajikiya, Non-even least energy solutions of the Emden-Fowler equation, Proc. Amer. Math. Soc., 140 (2012), 1353-1362.  doi: 10.1090/S0002-9939-2011-11172-9. [13] R. Kajikiya, Non-radial least energy solutions of the generalized Hénon equation, J. Differential Equations, 252 (2012), 1987-2003.  doi: 10.1016/j.jde.2011.08.032. [14] R. Kajikiya, Nonradial positive solutions of the p-Laplace Emden-Fowler equation with sign-changing weight, Mathematische Nachrichten, 289 (2016), 290-299.  doi: 10.1002/mana.201500103. [15] R. Kajikiya, Symmetric and asymmetric solutions of p-Laplace elliptic equations in hollow domains, To appear in Adv. Nonlinear Stud. [16] R. A. Moore and Z. Nehari, Nonoscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc., 93 (1959), 30-52.  doi: 10.1090/S0002-9947-1959-0111897-8. [17] R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322. [18] A. Pistoia and E. Serra, Multi-peak solutions for the Hénon equation with slightly subcritical growth, Math. Z., 256 (2007), 75-97.  doi: 10.1007/s00209-006-0060-9. [19] P. Pucci and J. Serrin, The Maximum Principle Birkhäuser, Berlin, 2007. [20] E. Serra, Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005), 301-326.  doi: 10.1007/s00526-004-0302-9. [21] D. Smets, M. Willem and J. Su, Non-radial ground states for the Hénon equation, Commun. Contemp. Math., 4 (2002), 467-480.  doi: 10.1142/S0219199702000725. [22] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations, 8 (1983), 773-817.  doi: 10.1080/03605308308820285. [23] J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl Math Optim, 12 (1984), 191-202.  doi: 10.1007/BF01449041. [24] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications Springer, New York, 1995.
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