American Institute of Mathematical Sciences

Advanced
June  2017, 37(6): 3025-3057. doi: 10.3934/dcds.2017130

A p-Laplacian supercritical Neumann problem

Francesca Colasuonno , and  Benedetta Noris

Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine -CP214, boulevard du Triomphe, 1050 Bruxelles, Belgique

* Corresponding author

Received  November 2016 Revised  January 2017 Published  February 2017

For p > 2, we consider the quasilinear equation $-\Delta_p u+|u|^{p-2}u=g(u)$ in the unit ball B of $\mathbb R^N$, with homogeneous Neumann boundary conditions. The assumptions on g are very mild and allow the nonlinearity to be possibly supercritical in the sense of Sobolev embeddings. We prove the existence of a nonconstant, positive, radially nondecreasing solution via variational methods. In the case $g(u)=|u|^{q-2}u$, we detect the asymptotic behavior of these solutions as $q\to \infty$.

Keywords: Quasilinear elliptic equations, Sobolev-supercritical nonlinearities, Neumann boundary conditions, variational methods.
Mathematics Subject Classification: Primary: 35J92, 35A15, 35B45; Secondary: 35B09, 35B40.
Citation: Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130
References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neumann problems, Ann. Mat. Pura Appl. (4), 188 (2009), 679-719.  doi: 10.1007/s10231-009-0096-7.  Google Scholar

[2]

G. Anello, Existence of infinitely many weak solutions for a Neumann problem, Nonlinear Anal., 57 (2004), 199-209.  doi: 10.1016/j.na.2004.02.009.  Google Scholar

[3]

T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a p-Laplacian equation, Proc. London Math. Soc., 91 (2005), 129-152.  doi: 10.1112/S0024611504015187.  Google Scholar

[4]

T. Bartsch and Z. Liu, On a superlinear elliptic p-Laplacian equation, J. Differential Equations, 198 (2004), 149-175.  doi: 10.1016/j.jde.2003.08.001.  Google Scholar

[5]

G. Bognár and P. Drábek, The p-Laplacian equation with superlinear and supercritical growth, multiplicity of radial solutions, Nonlinear Anal., 60 (2005), 719-728.  doi: 10.1016/j.na.2004.09.047.  Google Scholar

[6]

G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. (Basel), 80 (2003), 424-429.  doi: 10.1007/s00013-003-0479-8.  Google Scholar

[7]

J. F. Bonder and J. D. Rossi, Existence results for the p-Laplacian with nonlinear boundary conditions, J. Math. Anal. Appl., 263 (2001), 195-223.  doi: 10.1006/jmaa.2001.7609.  Google Scholar

[8]

D. Bonheure, B. Noris and T. Weth, Increasing radial solutions for Neumann problems without growth restrictions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 573-588.  doi: 10.1016/j.anihpc.2012.02.002.  Google Scholar

[9]

D. Bonheure and E. Serra, Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 217-235.  doi: 10.1007/s00030-010-0092-z.  Google Scholar

[10]

D. Bonheure, J. -B. Casteras and B. Noris, Multiple positive solutions of the stationary Keller-Segel system, preprint, arXiv: 1603.07374. Google Scholar

[11]

D. Bonheure, M. Grossi, B. Noris and S. Terracini, Multi-layer radial solutions for a supercritical Neumann problem, J. Differential Equations, 261 (2016), 455-504.  doi: 10.1016/j.jde.2016.03.016.  Google Scholar

[12]

D. Bonheure, C. Grumiau and C. Troestler, Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions, Nonlinear Anal., 147 (2016), 236-273.  doi: 10.1016/j.na.2016.09.010.  Google Scholar

[13]

D. Bonheure, E. Serra and P. Tilli, Radial positive solutions of elliptic systems with Neumann boundary conditions, J. Funct. Anal., 265 (2013), 375-398.  doi: 10.1016/j.jfa.2013.05.027.  Google Scholar

[14]

A. Boscaggin and W. Dambrosio, Highly oscillatory solutions of a Neumann problem for a p-Laplacian equation, Nonlinear Anal., 122 (2015), 58-82.  doi: 10.1016/j.na.2015.03.020.  Google Scholar

[15]

M. Clapp and S. Tiwari, Multiple solutions to a pure supercritical problem for the p-Laplacian, Calc. Var. Partial Differential Equations, 55 (2016), 1-23.  doi: 10.1007/s00526-015-0949-4.  Google Scholar

[16]

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.  doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[17]

L. Damascelli, F. Pacella and M. Ramaswamy, Symmetry of ground states of p-Laplace equations via the moving plane method, Arch. Rational Mech. Anal., 148 (1999), 291-308.  doi: 10.1007/s002050050163.  Google Scholar

[18]

F. Faraci, Multiplicity results for a Neumann problem involving the p-Laplacian, J. Math. Anal. Appl., 277 (2003), 180-189.  doi: 10.1016/S0022-247X(02)00530-9.  Google Scholar

[19]

M. Filippakis, L. Gasiński and N. S. Papageorgiou, Multiplicity results for nonlinear Neumann problems, Canad. J. Math., 58 (2006), 64-92.  doi: 10.4153/CJM-2006-004-6.  Google Scholar

[20]

M. Grossi, Asymptotic behaviour of the Kazdan-{W}arner solution in the annulus, J. Differential Equations, 223 (2006), 96-111.  doi: 10.1016/j.jde.2005.08.003.  Google Scholar

[21]

M. Grossi and B. Noris, Positive constrained minimizers for supercritical problems in the ball, Proc. Amer. Math. Soc., 140 (2012), 2141-2154.  doi: 10.1090/S0002-9939-2011-11133-X.  Google Scholar

[22]

L. Iturriaga, S. Lorca and E. Massa, Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 763-771.  doi: 10.1016/j.anihpc.2009.11.003.  Google Scholar

[23]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[24]

P. Lindqvist, Notes on the p-Laplace Equation, Univ., 2006.  Google Scholar

[25]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance, Indiana Univ. Math. J., 58 (2009), 1257-1279.  doi: 10.1512/iumj.2009.58.3565.  Google Scholar

[26]

P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[27]

B. Ricceri, Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian, Bull. London Math. Soc., 33 (2001), 331-340.  doi: 10.1017/S0024609301008001.  Google Scholar

[28]

S. Secchi, Increasing variational solutions for a nonlinear p-Laplace equation without growth conditions, Ann. Mat. Pura Appl., 191 (2012), 469-485.  doi: 10.1007/s10231-011-0191-4.  Google Scholar

[29]

E. Serra and P. Tilli, Monotonicity constraints and supercritical Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 63-74.  doi: 10.1016/j.anihpc.2010.10.003.  Google Scholar

[30]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.  Google Scholar

[31]

X. Wu and K.-K. Tan, On existence and multiplicity of solutions of Neumann boundary value problems for quasi-linear elliptic equations, Nonlinear Anal., 65 (2006), 1334-1347.  doi: 10.1016/j.na.2005.10.010.  Google Scholar

show all references

References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neumann problems, Ann. Mat. Pura Appl. (4), 188 (2009), 679-719.  doi: 10.1007/s10231-009-0096-7.  Google Scholar

[2]

G. Anello, Existence of infinitely many weak solutions for a Neumann problem, Nonlinear Anal., 57 (2004), 199-209.  doi: 10.1016/j.na.2004.02.009.  Google Scholar

[3]

T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a p-Laplacian equation, Proc. London Math. Soc., 91 (2005), 129-152.  doi: 10.1112/S0024611504015187.  Google Scholar

[4]

T. Bartsch and Z. Liu, On a superlinear elliptic p-Laplacian equation, J. Differential Equations, 198 (2004), 149-175.  doi: 10.1016/j.jde.2003.08.001.  Google Scholar

[5]

G. Bognár and P. Drábek, The p-Laplacian equation with superlinear and supercritical growth, multiplicity of radial solutions, Nonlinear Anal., 60 (2005), 719-728.  doi: 10.1016/j.na.2004.09.047.  Google Scholar

[6]

G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. (Basel), 80 (2003), 424-429.  doi: 10.1007/s00013-003-0479-8.  Google Scholar

[7]

J. F. Bonder and J. D. Rossi, Existence results for the p-Laplacian with nonlinear boundary conditions, J. Math. Anal. Appl., 263 (2001), 195-223.  doi: 10.1006/jmaa.2001.7609.  Google Scholar

[8]

D. Bonheure, B. Noris and T. Weth, Increasing radial solutions for Neumann problems without growth restrictions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 573-588.  doi: 10.1016/j.anihpc.2012.02.002.  Google Scholar

[9]

D. Bonheure and E. Serra, Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth, NoDEA Nonlinear Differential Equations Appl., 18 (2011), 217-235.  doi: 10.1007/s00030-010-0092-z.  Google Scholar

[10]

D. Bonheure, J. -B. Casteras and B. Noris, Multiple positive solutions of the stationary Keller-Segel system, preprint, arXiv: 1603.07374. Google Scholar

[11]

D. Bonheure, M. Grossi, B. Noris and S. Terracini, Multi-layer radial solutions for a supercritical Neumann problem, J. Differential Equations, 261 (2016), 455-504.  doi: 10.1016/j.jde.2016.03.016.  Google Scholar

[12]

D. Bonheure, C. Grumiau and C. Troestler, Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions, Nonlinear Anal., 147 (2016), 236-273.  doi: 10.1016/j.na.2016.09.010.  Google Scholar

[13]

D. Bonheure, E. Serra and P. Tilli, Radial positive solutions of elliptic systems with Neumann boundary conditions, J. Funct. Anal., 265 (2013), 375-398.  doi: 10.1016/j.jfa.2013.05.027.  Google Scholar

[14]

A. Boscaggin and W. Dambrosio, Highly oscillatory solutions of a Neumann problem for a p-Laplacian equation, Nonlinear Anal., 122 (2015), 58-82.  doi: 10.1016/j.na.2015.03.020.  Google Scholar

[15]

M. Clapp and S. Tiwari, Multiple solutions to a pure supercritical problem for the p-Laplacian, Calc. Var. Partial Differential Equations, 55 (2016), 1-23.  doi: 10.1007/s00526-015-0949-4.  Google Scholar

[16]

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.  doi: 10.1016/S0294-1449(98)80032-2.  Google Scholar

[17]

L. Damascelli, F. Pacella and M. Ramaswamy, Symmetry of ground states of p-Laplace equations via the moving plane method, Arch. Rational Mech. Anal., 148 (1999), 291-308.  doi: 10.1007/s002050050163.  Google Scholar

[18]

F. Faraci, Multiplicity results for a Neumann problem involving the p-Laplacian, J. Math. Anal. Appl., 277 (2003), 180-189.  doi: 10.1016/S0022-247X(02)00530-9.  Google Scholar

[19]

M. Filippakis, L. Gasiński and N. S. Papageorgiou, Multiplicity results for nonlinear Neumann problems, Canad. J. Math., 58 (2006), 64-92.  doi: 10.4153/CJM-2006-004-6.  Google Scholar

[20]

M. Grossi, Asymptotic behaviour of the Kazdan-{W}arner solution in the annulus, J. Differential Equations, 223 (2006), 96-111.  doi: 10.1016/j.jde.2005.08.003.  Google Scholar

[21]

M. Grossi and B. Noris, Positive constrained minimizers for supercritical problems in the ball, Proc. Amer. Math. Soc., 140 (2012), 2141-2154.  doi: 10.1090/S0002-9939-2011-11133-X.  Google Scholar

[22]

L. Iturriaga, S. Lorca and E. Massa, Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 763-771.  doi: 10.1016/j.anihpc.2009.11.003.  Google Scholar

[23]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.  Google Scholar

[24]

P. Lindqvist, Notes on the p-Laplace Equation, Univ., 2006.  Google Scholar

[25]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance, Indiana Univ. Math. J., 58 (2009), 1257-1279.  doi: 10.1512/iumj.2009.58.3565.  Google Scholar

[26]

P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[27]

B. Ricceri, Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian, Bull. London Math. Soc., 33 (2001), 331-340.  doi: 10.1017/S0024609301008001.  Google Scholar

[28]

S. Secchi, Increasing variational solutions for a nonlinear p-Laplace equation without growth conditions, Ann. Mat. Pura Appl., 191 (2012), 469-485.  doi: 10.1007/s10231-011-0191-4.  Google Scholar

[29]

E. Serra and P. Tilli, Monotonicity constraints and supercritical Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 63-74.  doi: 10.1016/j.anihpc.2010.10.003.  Google Scholar

[30]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.  Google Scholar

[31]

X. Wu and K.-K. Tan, On existence and multiplicity of solutions of Neumann boundary value problems for quasi-linear elliptic equations, Nonlinear Anal., 65 (2006), 1334-1347.  doi: 10.1016/j.na.2005.10.010.  Google Scholar

Figure 1.  The graph of the function $y=\left[\frac{p}{p-1}\left(\frac{x^p}p-\frac{x^q}q\right)\right]^{1/p}$ for $x,\,y\ge0$.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Jason Metcalfe, Jacob Perry. Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions. Communications on Pure & Applied Analysis, 2012, 11 (2) : 547-556. doi: 10.3934/cpaa.2012.11.547
[2]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018
[3]

Shinji Adachi, Masataka Shibata, Tatsuya Watanabe. Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (1) : 97-118. doi: 10.3934/cpaa.2014.13.97
[4]

Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885
[5]

M. Matzeu, Raffaella Servadei. A variational approach to a class of quasilinear elliptic equations not in divergence form. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 819-830. doi: 10.3934/dcdss.2012.5.819
[6]

Zhihua Huang, Xiaochun Liu. Existence theorem for a class of semilinear totally characteristic elliptic equations involving supercritical cone sobolev exponents. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3201-3216. doi: 10.3934/cpaa.2019144
[7]

Zhenhua Zhang. Asymptotic behavior of solutions to the phase-field equations with neumann boundary conditions. Communications on Pure & Applied Analysis, 2005, 4 (3) : 683-693. doi: 10.3934/cpaa.2005.4.683
[8]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061
[9]

Zhong-Qing Wang, Jing-Xia Wu. Generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions in unbounded domains. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 325-346. doi: 10.3934/dcdsb.2012.17.325
[10]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061
[11]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
[12]

Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011
[13]

Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731
[14]

Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1
[15]

Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure & Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191
[16]

Andrea Malchiodi. Topological methods for an elliptic equation with exponential nonlinearities. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 277-294. doi: 10.3934/dcds.2008.21.277
[17]

Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351
[18]

T. J. Christiansen. Resonances and balls in obstacle scattering with Neumann boundary conditions. Inverse Problems & Imaging, 2008, 2 (3) : 335-340. doi: 10.3934/ipi.2008.2.335
[19]

Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175
[20]

Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, , () : -. doi: 10.3934/era.2021043

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (134)
  • HTML views (55)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences