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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June
2017, 37(6): 3025-3057.
doi: 10.3934/dcds.2017130
A p-Laplacian supercritical Neumann problem
Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine -CP214, boulevard du Triomphe, 1050 Bruxelles, Belgique |
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. |
| [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. |
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [24] |
P. Lindqvist,
Notes on the p-Laplace Equation,
Univ., 2006. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [30] |
J. L. Vázquez,
A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
D. Bonheure, J. -B. Casteras and B. Noris, Multiple positive solutions of the stationary Keller-Segel system, preprint, arXiv: 1603.07374. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [24] |
P. Lindqvist,
Notes on the p-Laplace Equation,
Univ., 2006. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [30] |
J. L. Vázquez,
A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
| [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. |
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