    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 & Continuous Dynamical Systems - A, 2018, 38 (2) : 547-561. doi: 10.3934/dcds.2018024
##### References:
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##### References:
  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  K. Deimling, Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar  P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations Second edition, Birkhäuser, Berlin, 2013. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  R. Kajikiya, Symmetric and asymmetric solutions of p-Laplace elliptic equations in hollow domains, To appear in Adv. Nonlinear Stud. Google Scholar  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.  Google Scholar  R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30.  doi: 10.1007/BF01941322.  Google Scholar  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.  Google Scholar  P. Pucci and J. Serrin, The Maximum Principle Birkhäuser, Berlin, 2007. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  J. L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl Math Optim, 12 (1984), 191-202.  doi: 10.1007/BF01449041.  Google Scholar  E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications Springer, New York, 1995. Google Scholar
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