# 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 & Continuous Dynamical Systems - A, 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] K. Deimling, Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7. Google Scholar [9] P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations Second edition, Birkhäuser, Berlin, 2013.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] R. Kajikiya, Symmetric and asymmetric solutions of p-Laplace elliptic equations in hollow domains, To appear in Adv. Nonlinear Stud.Google Scholar [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. Google Scholar [17] R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322. Google Scholar [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. Google Scholar [19] P. Pucci and J. Serrin, The Maximum Principle Birkhäuser, Berlin, 2007. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] 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 [24] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications Springer, New York, 1995. Google Scholar

show all references

##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] K. Deimling, Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7. Google Scholar [9] P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations Second edition, Birkhäuser, Berlin, 2013.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] R. Kajikiya, Symmetric and asymmetric solutions of p-Laplace elliptic equations in hollow domains, To appear in Adv. Nonlinear Stud.Google Scholar [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. Google Scholar [17] R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. doi: 10.1007/BF01941322. Google Scholar [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. Google Scholar [19] P. Pucci and J. Serrin, The Maximum Principle Birkhäuser, Berlin, 2007. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [23] 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 [24] E. Zeidler, Applied Functional Analysis: Main Principles and Their Applications Springer, New York, 1995. Google Scholar
 [1] Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055 [2] Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237 [3] Arrigo Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4071-4085. doi: 10.3934/dcds.2018177 [4] Sihem Guerarra. Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 15-22. doi: 10.3934/naco.2019002 [5] Antonio Greco, Giovanni Porru. Optimization problems for the energy integral of p-Laplace equations. Conference Publications, 2013, 2013 (special) : 301-310. doi: 10.3934/proc.2013.2013.301 [6] Shun Kodama. A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential. Communications on Pure & Applied Analysis, 2017, 16 (2) : 671-698. doi: 10.3934/cpaa.2017033 [7] Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857 [8] Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 [9] Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335 [10] Xiang-Dong Fang. A positive solution for an asymptotically cubic quasilinear Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (1) : 51-64. doi: 10.3934/cpaa.2019004 [11] Roman Chapko, B. Tomas Johansson. On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems & Imaging, 2012, 6 (1) : 25-38. doi: 10.3934/ipi.2012.6.25 [12] Jagadeesh R. Sonnad, Chetan T. Goudar. Solution of the Michaelis-Menten equation using the decomposition method. Mathematical Biosciences & Engineering, 2009, 6 (1) : 173-188. doi: 10.3934/mbe.2009.6.173 [13] Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 [14] Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099 [15] Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solution of the Novikov equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2865-2899. doi: 10.3934/dcdsb.2018290 [16] Diane Denny. A unique positive solution to a system of semilinear elliptic equations. Conference Publications, 2013, 2013 (special) : 193-195. doi: 10.3934/proc.2013.2013.193 [17] GUANGBING LI. Positive solution for quasilinear Schrödinger equations with a parameter. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1803-1816. doi: 10.3934/cpaa.2015.14.1803 [18] Yanqin Fang, Jihui Zhang. Nonexistence of positive solution for an integral equation on a Half-Space $R_+^n$. Communications on Pure & Applied Analysis, 2013, 12 (2) : 663-678. doi: 10.3934/cpaa.2013.12.663 [19] Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185 [20] Brian D. O. Anderson, Shaoshuai Mou, A. Stephen Morse, Uwe Helmke. Decentralized gradient algorithm for solution of a linear equation. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 319-328. doi: 10.3934/naco.2016014

2018 Impact Factor: 1.143