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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 - A, 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

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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

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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

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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 $x,\,y\ge0$.
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