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August  2017, 10(4): 799-813. doi: 10.3934/dcdss.2017040

## On the geometry of the p-Laplacian operator

 1 Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany 2 Fakultät Maschinenbau, TH Ingolstadt, Postfach 21 04 54,85019 Ingolstadt, Germany

* Corresponding author: Bernd Kawohl

Received  April 2016 Revised  August 2016 Published  April 2017

The
 $p$
-Laplacian operator
 $\Delta_pu={\rm div }\left(|\nabla u|^{p-2}\nabla u\right)$
is not uniformly elliptic for any
 $p\in(1,2)\cup(2,\infty)$
and degenerates even more when
 $p\to \infty$
or
 $p\to 1$
. In those two cases the Dirichlet and eigenvalue problems associated with the
 $p$
-Laplacian lead to intriguing geometric questions, because their limits for
 $p\to\infty$
or
 $p\to 1$
can be characterized by the geometry of
 $\Omega$
. In this little survey we recall some well-known results on eigenfunctions of the classical 2-Laplacian and elaborate on their extensions to general
 $p\in[1,\infty]$
. We report also on results concerning the normalized or game-theoretic
 $p$
-Laplacian
 $\Delta_p^Nu:=\tfrac{1}{p}|\nabla u|^{2-p}\Delta_pu=\tfrac{1}{p}\Delta_1^Nu+\tfrac{p-1}{p}\Delta_\infty^Nu$
and its parabolic counterpart
 $u_t-\Delta_p^N u=0$
. These equations are homogeneous of degree 1 and
 $\Delta_p^N$
is uniformly elliptic for any
 $p\in (1,\infty)$
. In this respect it is more benign than the
 $p$
-Laplacian, but it is not of divergence type.
Citation: Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040
##### References:
  T. N. Anoop, P. Drábek and S. Sarath, On the structure of the second eigenfunctions of the p-Laplacian on a ball, Proc. Amer. Math. Soc., 144 (2016), 2503-2512.  doi: 10.1090/proc/12902. Google Scholar  S. N. Armstrong and C. K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games, Trans. Amer. Math. Soc., 364 (2012), 595-636.  doi: 10.1090/S0002-9947-2011-05289-X. Google Scholar  G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561.  doi: 10.1007/BF02591928. Google Scholar  A. Attouch, M. Parviainen and E. Ruosteenoja, C1, α regularity for the normalized p-Poisson problem, preprint, arXiv: 1603.06391, to appear in J. Math. Pures Appl. Google Scholar  A. Banerjee and N. Garofalo, On the Dirichlet boundary value problem for the normalized p-Laplacian evolution, Commun. Pure Appl. Anal., 14 (2015), 1-21.  doi: 10.3934/cpaa.2015.14.1. Google Scholar  T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as p → ∞ of ∆pup = f and related extremal problems, Some topics in nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. , Torino, Special Issue, (1991), 15-68. Google Scholar  I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators, Comm. Pure Applied Anal., 6 (2007), 335-366.   Google Scholar  I. Birindelli and F. Demengel, Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differential Equations, 249 (2010), 1089-1110.  doi: 10.1016/j.jde.2010.03.015. Google Scholar  L. Brasco, C. Nitsch and C. Trombetti, An inequality á la Szegö-Weinberger for the p-Laplacian on convex sets, Communications in Contemporary Mathematics, 18 (2016), 1550086, 23pp.  doi: 10.1142/S0219199715500868. Google Scholar  J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, A Symposium in Honor of Salomon Bochner, (ed. R. C. Gunning), Princeton Univ. Press, (2015), 195-200. doi: 10.1515/9781400869312-013. Google Scholar  M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.   Google Scholar  G. Crasta and I. Fragalá, A C1 regularity result for the inhomogeneous normalized infinity Laplacian, Proc. Amer. Math. Soc., 144 (2016), 2547-2558.  doi: 10.1090/proc/12916. Google Scholar  K. Does, An evolution equation involving the normalized p-Laplacian, Commun. Pure Appl. Anal., 10 (2011), 361-396.  doi: 10.3934/cpaa.2011.10.361. Google Scholar  L. Esposito, B. Kawohl, C. Nitsch and C. Trombetti, The Neumann eigenvalue problem for the ∞-Laplacian, Rend. Lincei Mat.Appl., 26 (2015), 119-134.  doi: 10.4171/RLM/697. Google Scholar  L. Esposito, V. Ferone, B. Kawohl, C. Nitsch and C. Trombetti, The longest shortest fence and sharp Poincaré Sobolev inequalities, Arch. Ration. Mech. Anal., 206 (2012), 821-851.  doi: 10.1007/s00205-012-0545-0. Google Scholar  L. C. Evans and J. Spruck, Motion of level sets by mean curvature Ⅰ, Chapter: Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids, 33 (1991), 328-374.  doi: 10.1007/978-3-642-59938-5_13. Google Scholar  H. Gajewski and K. Gärtner, Domain separation by means of sign changing eigenfunctions of p-Laplacians, Appl. Anal., 79 (2001), 483-501.  doi: 10.1080/00036810108840974. Google Scholar  J. Horák, Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains, Electron. J. Differential Equations, 2011 (2011), 1-30.   Google Scholar  R. Hynd, C. K. Smart and Y. Yu, Nonuniqueness of infinity ground states, Calc. Var. Partial Differential Equations, 48 (2013), 545-554.  doi: 10.1007/s00526-012-0561-9. Google Scholar  R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74.  doi: 10.1007/BF00386368. Google Scholar  T. Jin and L. Silvestre, Hölder gradient estimates for parabolic homogeneous p-Laplacian equations, preprint, arXiv: 1505.05525 doi: 10.1016/j.matpur.2016.10.010. Google Scholar  V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation, Comm. Partial Differential Equations, 37 (2012), 934-946.  doi: 10.1080/03605302.2011.615878. Google Scholar  P. Juutinen, P. Lindqvist and J. Manfredi, The ∞-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105.   Google Scholar  P. Juutinen, p-Harmonic approximation of functions of least gradient, Indiana Univ. Math. J., 54 (2005), 1015-1029.  doi: 10.1512/iumj.2005.54.2658. Google Scholar  P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851.  doi: 10.1007/s00208-006-0766-3. Google Scholar  P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Differential Equations, 236 (2007), 532-550.  doi: 10.1016/j.jde.2007.01.020. Google Scholar  P. Juutinen and P. Lindqvist, On the higher eigenvalues for the ∞-eigenvalue problem, Calc. Var. Partial Differential Equations, 23 (2005), 169-192.  doi: 10.1007/s00526-004-0295-4. Google Scholar  B. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math., 410 (1990), 1-22.   Google Scholar  B. Kawohl and N. Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion, Math. Ann., 311 (1989), 107-123.  doi: 10.1007/s002080050179. Google Scholar  B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolinae, 44 (2003), 659-667.   Google Scholar  B. Kawohl and H. Shahgholian, Gamma limits in some Bernoulli free boundary problems, Archiv D. Math., 84 (2005), 79-87.  doi: 10.1007/s00013-004-1334-2. Google Scholar  B. Kawohl and Th. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane, Pacific J.Math., 225 (2006), 103-118.  doi: 10.2140/pjm.2006.225.103. Google Scholar  B. Kawohl and P. Lindqvist, Positive eigenfunctions for the p-Laplace operator revisited, Analysis, (Munich), 26 (2006), 539-544.  doi: 10.1524/anly.2006.26.4.545. Google Scholar  B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation, Differential Integral Equations, 8 (1995), 1923-1946.   Google Scholar  B. Kawohl, Variational versus PDE-based Approaches in Mathematical Image Processing, CRM Proceedings and Lecture Notes, 44 (2008), 113-126.   Google Scholar  B. Kawohl, Variations on the p-Laplacian in Nonlinear Elliptic Partial Differential Equations, (eds. Bonheure D., P. Takač et al.), Contemporary Mathematics, 540 (2011), 35-46.  doi: 10.1090/conm/540/10657. Google Scholar  B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188.  doi: 10.1016/j.matpur.2011.07.001. Google Scholar  B. Kawohl, S. Krömer and J. Kurtz, Radial eigenfunctions for the game-theoretic p-Laplacian on a ball, Differential and Integral Equations, 27 (2014), 659-670.   Google Scholar  B. Kawohl and F. Schuricht, First eigenfunctions of the 1-Laplacian are viscosity solutions, Commun. Pure Appl. Anal., 14 (2015), 329-339.  doi: 10.3934/cpaa.2015.14.329. Google Scholar  G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868.  doi: 10.1016/j.aim.2007.11.020. Google Scholar  G. Lu and P. Wang, A uniqueness theorem for degenerate elliptic equations, Lecture Notes of Seminario Interdisciplinare di Matematica, Conference on Geometric Methods in PDEs, On the Occasion of 65th Birthday of Ermanno Lanconelli, Bologna, May 27-30,2008, (eds. Giovanna Citti, Annamaria Montanari, Andrea Pascucci, Sergio Polidoro), 207-222. Google Scholar  Th. Lachand-Robert and É. Oudet, Minimizing within convex bodies using a convex hull method, SIAM J. Optim., 16 (2005), 368-379.   Google Scholar  P. J. Martínez-Aparicio, M. Pérez-Llanos and J. D. Rossi, The limit as p → ∞ for the eigenvalue problem of the 1-homogeneous p-Laplacian, Rev. Mat. Complut., 27 (2014), 241-258.   Google Scholar  E. Parini, The second eigenvalue of the p-Laplacian as p goes to 1, Int. J. Diff. Eq. , (2010), Art. ID 984671, 23 pp. Google Scholar  E. Parini, An introduction to the Cheeger problem, Surv. Math. Appl., 6 (2011), 9-22.   Google Scholar  J. D. Rossi and N. Saintier, On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions, Houston J. Math., 42 (2016), 613-635.   Google Scholar  R. P. Sperb, Maximum Principles and Their Applications, Academic Press, New York-London, 1981. Google Scholar  Y. Yu, Some properties of the ground states of the infinity Laplacian, Indiana Univ. Math. J., 56 (2007), 947-964.  doi: 10.1512/iumj.2007.56.2935. Google Scholar

show all references

##### References:
  T. N. Anoop, P. Drábek and S. Sarath, On the structure of the second eigenfunctions of the p-Laplacian on a ball, Proc. Amer. Math. Soc., 144 (2016), 2503-2512.  doi: 10.1090/proc/12902. Google Scholar  S. N. Armstrong and C. K. Smart, A finite difference approach to the infinity Laplace equation and tug-of-war games, Trans. Amer. Math. Soc., 364 (2012), 595-636.  doi: 10.1090/S0002-9947-2011-05289-X. Google Scholar  G. Aronsson, Extension of functions satisfying Lipschitz conditions, Ark. Mat., 6 (1967), 551-561.  doi: 10.1007/BF02591928. Google Scholar  A. Attouch, M. Parviainen and E. Ruosteenoja, C1, α regularity for the normalized p-Poisson problem, preprint, arXiv: 1603.06391, to appear in J. Math. Pures Appl. Google Scholar  A. Banerjee and N. Garofalo, On the Dirichlet boundary value problem for the normalized p-Laplacian evolution, Commun. Pure Appl. Anal., 14 (2015), 1-21.  doi: 10.3934/cpaa.2015.14.1. Google Scholar  T. Bhattacharya, E. DiBenedetto and J. Manfredi, Limits as p → ∞ of ∆pup = f and related extremal problems, Some topics in nonlinear PDEs (Turin, 1989), Rend. Sem. Mat. Univ. Politec. , Torino, Special Issue, (1991), 15-68. Google Scholar  I. Birindelli and F. Demengel, Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators, Comm. Pure Applied Anal., 6 (2007), 335-366.   Google Scholar  I. Birindelli and F. Demengel, Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators, J. Differential Equations, 249 (2010), 1089-1110.  doi: 10.1016/j.jde.2010.03.015. Google Scholar  L. Brasco, C. Nitsch and C. Trombetti, An inequality á la Szegö-Weinberger for the p-Laplacian on convex sets, Communications in Contemporary Mathematics, 18 (2016), 1550086, 23pp.  doi: 10.1142/S0219199715500868. Google Scholar  J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, A Symposium in Honor of Salomon Bochner, (ed. R. C. Gunning), Princeton Univ. Press, (2015), 195-200. doi: 10.1515/9781400869312-013. Google Scholar  M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.   Google Scholar  G. Crasta and I. Fragalá, A C1 regularity result for the inhomogeneous normalized infinity Laplacian, Proc. Amer. Math. Soc., 144 (2016), 2547-2558.  doi: 10.1090/proc/12916. Google Scholar  K. Does, An evolution equation involving the normalized p-Laplacian, Commun. Pure Appl. Anal., 10 (2011), 361-396.  doi: 10.3934/cpaa.2011.10.361. Google Scholar  L. Esposito, B. Kawohl, C. Nitsch and C. Trombetti, The Neumann eigenvalue problem for the ∞-Laplacian, Rend. Lincei Mat.Appl., 26 (2015), 119-134.  doi: 10.4171/RLM/697. Google Scholar  L. Esposito, V. Ferone, B. Kawohl, C. Nitsch and C. Trombetti, The longest shortest fence and sharp Poincaré Sobolev inequalities, Arch. Ration. Mech. Anal., 206 (2012), 821-851.  doi: 10.1007/s00205-012-0545-0. Google Scholar  L. C. Evans and J. Spruck, Motion of level sets by mean curvature Ⅰ, Chapter: Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids, 33 (1991), 328-374.  doi: 10.1007/978-3-642-59938-5_13. Google Scholar  H. Gajewski and K. Gärtner, Domain separation by means of sign changing eigenfunctions of p-Laplacians, Appl. Anal., 79 (2001), 483-501.  doi: 10.1080/00036810108840974. Google Scholar  J. Horák, Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains, Electron. J. Differential Equations, 2011 (2011), 1-30.   Google Scholar  R. Hynd, C. K. Smart and Y. Yu, Nonuniqueness of infinity ground states, Calc. Var. Partial Differential Equations, 48 (2013), 545-554.  doi: 10.1007/s00526-012-0561-9. Google Scholar  R. Jensen, Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient, Arch. Rational Mech. Anal., 123 (1993), 51-74.  doi: 10.1007/BF00386368. Google Scholar  T. Jin and L. Silvestre, Hölder gradient estimates for parabolic homogeneous p-Laplacian equations, preprint, arXiv: 1505.05525 doi: 10.1016/j.matpur.2016.10.010. Google Scholar  V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the p-Laplace equation, Comm. Partial Differential Equations, 37 (2012), 934-946.  doi: 10.1080/03605302.2011.615878. Google Scholar  P. Juutinen, P. Lindqvist and J. Manfredi, The ∞-eigenvalue problem, Arch. Ration. Mech. Anal., 148 (1999), 89-105.   Google Scholar  P. Juutinen, p-Harmonic approximation of functions of least gradient, Indiana Univ. Math. J., 54 (2005), 1015-1029.  doi: 10.1512/iumj.2005.54.2658. Google Scholar  P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851.  doi: 10.1007/s00208-006-0766-3. Google Scholar  P. Juutinen, Principal eigenvalue of a very badly degenerate operator and applications, J. Differential Equations, 236 (2007), 532-550.  doi: 10.1016/j.jde.2007.01.020. Google Scholar  P. Juutinen and P. Lindqvist, On the higher eigenvalues for the ∞-eigenvalue problem, Calc. Var. Partial Differential Equations, 23 (2005), 169-192.  doi: 10.1007/s00526-004-0295-4. Google Scholar  B. Kawohl, On a family of torsional creep problems, J. Reine Angew. Math., 410 (1990), 1-22.   Google Scholar  B. Kawohl and N. Kutev, Maximum and comparison principle for one-dimensional anisotropic diffusion, Math. Ann., 311 (1989), 107-123.  doi: 10.1007/s002080050179. Google Scholar  B. Kawohl and V. Fridman, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolinae, 44 (2003), 659-667.   Google Scholar  B. Kawohl and H. Shahgholian, Gamma limits in some Bernoulli free boundary problems, Archiv D. Math., 84 (2005), 79-87.  doi: 10.1007/s00013-004-1334-2. Google Scholar  B. Kawohl and Th. Lachand-Robert, Characterization of Cheeger sets for convex subsets of the plane, Pacific J.Math., 225 (2006), 103-118.  doi: 10.2140/pjm.2006.225.103. Google Scholar  B. Kawohl and P. Lindqvist, Positive eigenfunctions for the p-Laplace operator revisited, Analysis, (Munich), 26 (2006), 539-544.  doi: 10.1524/anly.2006.26.4.545. Google Scholar  B. Kawohl and N. Kutev, Global behaviour of solutions to a parabolic mean curvature equation, Differential Integral Equations, 8 (1995), 1923-1946.   Google Scholar  B. Kawohl, Variational versus PDE-based Approaches in Mathematical Image Processing, CRM Proceedings and Lecture Notes, 44 (2008), 113-126.   Google Scholar  B. Kawohl, Variations on the p-Laplacian in Nonlinear Elliptic Partial Differential Equations, (eds. Bonheure D., P. Takač et al.), Contemporary Mathematics, 540 (2011), 35-46.  doi: 10.1090/conm/540/10657. Google Scholar  B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188.  doi: 10.1016/j.matpur.2011.07.001. Google Scholar  B. Kawohl, S. Krömer and J. Kurtz, Radial eigenfunctions for the game-theoretic p-Laplacian on a ball, Differential and Integral Equations, 27 (2014), 659-670.   Google Scholar  B. Kawohl and F. Schuricht, First eigenfunctions of the 1-Laplacian are viscosity solutions, Commun. Pure Appl. Anal., 14 (2015), 329-339.  doi: 10.3934/cpaa.2015.14.329. Google Scholar  G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868.  doi: 10.1016/j.aim.2007.11.020. Google Scholar  G. Lu and P. Wang, A uniqueness theorem for degenerate elliptic equations, Lecture Notes of Seminario Interdisciplinare di Matematica, Conference on Geometric Methods in PDEs, On the Occasion of 65th Birthday of Ermanno Lanconelli, Bologna, May 27-30,2008, (eds. Giovanna Citti, Annamaria Montanari, Andrea Pascucci, Sergio Polidoro), 207-222. Google Scholar  Th. Lachand-Robert and É. Oudet, Minimizing within convex bodies using a convex hull method, SIAM J. Optim., 16 (2005), 368-379.   Google Scholar  P. J. Martínez-Aparicio, M. Pérez-Llanos and J. D. Rossi, The limit as p → ∞ for the eigenvalue problem of the 1-homogeneous p-Laplacian, Rev. Mat. Complut., 27 (2014), 241-258.   Google Scholar  E. Parini, The second eigenvalue of the p-Laplacian as p goes to 1, Int. J. Diff. Eq. , (2010), Art. ID 984671, 23 pp. Google Scholar  E. Parini, An introduction to the Cheeger problem, Surv. Math. Appl., 6 (2011), 9-22.   Google Scholar  J. D. Rossi and N. Saintier, On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions, Houston J. Math., 42 (2016), 613-635.   Google Scholar  R. P. Sperb, Maximum Principles and Their Applications, Academic Press, New York-London, 1981. Google Scholar  Y. Yu, Some properties of the ground states of the infinity Laplacian, Indiana Univ. Math. J., 56 (2007), 947-964.  doi: 10.1512/iumj.2007.56.2935. Google Scholar Conceivable nodal lines of the second eigenfunction for $p=\infty$ in the disc Numerical simulation of $u_p$: normalized values along half of the diagonal for $p=2, 3, 4, 6, 8, 10, 15$ (left), and for $p=15$ compared to the line $y=x$ (right)
  Bernd Kawohl, Friedemann Schuricht. First eigenfunctions of the 1-Laplacian are viscosity solutions. Communications on Pure & Applied Analysis, 2015, 14 (1) : 329-339. doi: 10.3934/cpaa.2015.14.329  Nikolaos S. Papageorgiou, Vicenţiu D. Rǎdulescu, Dušan D. Repovš. Nodal solutions for the Robin p-Laplacian plus an indefinite potential and a general reaction term. Communications on Pure & Applied Analysis, 2018, 17 (1) : 231-241. doi: 10.3934/cpaa.2018014  Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure & Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371  Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033  Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922  Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure & Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019  Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021083  Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063  Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069  Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595  Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593  Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2117-2138. doi: 10.3934/cpaa.2021060  Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012  Genni Fragnelli, Dimitri Mugnai, Nikolaos S. Papageorgiou. Robin problems for the p-Laplacian with gradient dependence. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 287-295. doi: 10.3934/dcdss.2019020  Francesca Colasuonno, Benedetta Noris. A p-Laplacian supercritical Neumann problem. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3025-3057. doi: 10.3934/dcds.2017130  Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058  Leszek Gasiński. Positive solutions for resonant boundary value problems with the scalar p-Laplacian and nonsmooth potential. Discrete & Continuous Dynamical Systems, 2007, 17 (1) : 143-158. doi: 10.3934/dcds.2007.17.143  Yinbin Deng, Yi Li, Wei Shuai. Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 683-699. doi: 10.3934/dcds.2016.36.683  Eun Kyoung Lee, R. Shivaji, Inbo Sim, Byungjae Son. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1139-1154. doi: 10.3934/cpaa.2019055  Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171

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