October  2016, 36(10): 5323-5345. doi: 10.3934/dcds.2016034

Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros

1. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received  April 2015 Revised  April 2016 Published  July 2016

In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.
Citation: Guowei Dai. Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5323-5345. doi: 10.3934/dcds.2016034
References:
[1]

A. Anane, Simplicité et isolation de la première valeur propre du $p$-Laplacien avec poids, Comptes Rendus Acad. Sc. Paris, Série I, 305 (1987), 725-728.

[2]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rew., 18 (1976), 620-709. doi: 10.1137/1018114.

[3]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[4]

A. Ambrosetti, R. M. Calahorrano and F. R. Dobarro, Global branching for discontinuous problems, Comment. Math. Univ. Carolin., 31 (1990), 213-222.

[5]

A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl., 73 (1980), 411-422. doi: 10.1016/0022-247X(80)90287-5.

[6]

A. Ambrosetti, J. G. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242. doi: 10.1006/jfan.1996.0045.

[7]

A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics No. 104, Cambridge Univ. Press, 2007. doi: 10.1017/CBO9780511618260.

[8]

C. J. Amick and R. E. L. Turner, A global branch of steady vortex rings, J. Rein. Angew. Math., 384 (1988), 1-23.

[9]

D. Arcoya, J. I. Diaz and L. Tello, $S$-shaped bifurcation branch in a quasilinear multivalued model arising in climatoloty, J. Differential Equations, 150 (1998), 215-225. doi: 10.1006/jdeq.1998.3502.

[10]

D. Arcoya and J. L. Gámez, Bifurcation theory and related problems: anti-maximum principle and resonance, Comm. Partial Differential Equations, 26 (2001), 1879-1911. doi: 10.1081/PDE-100107462.

[11]

M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.

[12]

A. Cañada, P. Drábek and J. L. Gámez, Existence of positive solutions for some problems with nonlinear diffusion, Trans. Amer. Math. Soc., 349 (1997), 4231-4249. doi: 10.1090/S0002-9947-97-01947-8.

[13]

G. Dai, Global branching for discontinuous problems involving the $p$-Laplacian, Electron. J. Differential Equations, 44 (2013), 1-10.

[14]

G. Dai, Eigenvalue, global bifurcation and positive solutions for a class of nonlocal elliptic equations,, Topol. Methods Nonlinear Anal., (). 

[15]

G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian, J. Differential Equations, 252 (2012), 2448-2468. doi: 10.1016/j.jde.2011.09.026.

[16]

G. Dai, R. Ma and Y. Lu, Bifurcation from infinity and nodal solutions of quasilinear problems without the signum condition, J. Math. Anal. Appl., 397 (2013), 119-123. doi: 10.1016/j.jmaa.2012.07.056.

[17]

E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1974), 1069-1076.

[18]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc., 34 (2002), 533-538. doi: 10.1112/S002460930200108X.

[19]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New-York, 1985. doi: 10.1007/978-3-662-00547-7.

[20]

M. Delgado and A. Suárez, On the existence and multiplicity of positive solutions for some indefinite nonlinear eigenvalue problem, Proc. Amer. Math. Soc., 132 (2004), 1721-1728. doi: 10.1090/S0002-9939-04-07233-8.

[21]

M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Laplacian, J. Differential Equations, 92 (1991), 226-251. doi: 10.1016/0022-0396(91)90048-E.

[22]

P. Drábek and Y. X. Huang, Bifurcation problems for the $p$-Laplacian in $\mathbbR^N$, Trans. Amer. Math. Soc., 349 (1997), 171-188. doi: 10.1090/S0002-9947-97-01788-1.

[23]

X. L. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417. doi: 10.1016/j.jde.2007.01.008.

[24]

X. L. Fan, On the sub-supersolution method for $p(x)$-Laplacian equations, J. Math. Anal. Appl., 330 (2007), 665-682. doi: 10.1016/j.jmaa.2006.07.093.

[25]

X. L. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36 (1999), 295-318. doi: 10.1016/S0362-546X(97)00628-7.

[26]

D. G. De Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467. doi: 10.1016/S0022-1236(02)00060-5.

[27]

D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.

[28]

J. Fleckinger, R. Manásevich and Thélin, Global bifurcation from the first eigenvalue for a system of $p$-Laplacians, Math. Nachr., 182 (1996), 217-242. doi: 10.1002/mana.19961820110.

[29]

J. García-Melián and J. Sabina de Lis, A local bifurcation theorem for degenerate elliptic equations with radial symmetry, J. Differential Equations, 179 (2002), 27-43.

[30]

P. Girg and P. Takáč, Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (2008), 275-327. doi: 10.1007/s00023-008-0356-x.

[31]

M. Guedda and L. Véron, Bifurcation phenomena associated to the $p$-Laplace operator, Trans. Amer. Math. Soc., 310 (1988), 419-431. doi: 10.2307/2001132.

[32]

K. C. Hung and S. H. Wang, A complete classification of bifurcation diagrams of classes of multiparameter $p$-Laplacian boundary value problems, J. Differential Equations, 246 (2009), 1568-1599. doi: 10.1016/j.jde.2008.10.035.

[33]

L. Iturriaga et al., Positive solutions of the $p$-Laplacian involving a superlinear nonlinearity with zeros, J. Differential Equations, 248 (2010), 309-327. doi: 10.1016/j.jde.2009.08.008.

[34]

H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer, New York, 2004. doi: 10.1007/b97365.

[35]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964.

[36]

Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian, J. Differential Equations, 229 (2006), 229-256. doi: 10.1016/j.jde.2006.03.021.

[37]

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.

[38]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. doi: 10.1137/1024101.

[39]

P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. doi: 10.1016/j.jfa.2007.06.015.

[40]

P. Liu, J. Shi and Y. Wang, Bifurcation from a degenerate simple eigenvalue, J. Funct. Anal., 264 (2013), 2269-2299. doi: 10.1016/j.jfa.2013.02.010.

[41]

Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398. doi: 10.1006/jfan.1999.3446.

[42]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman and Hall/CRC, Boca Raton, 2001. doi: 10.1201/9781420035506.

[43]

J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators, Advances in Operator Theory and Applications Vol. 177, Birkhaüser, Basel, 2007.

[44]

R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems, Appl. Math. Lett., 21 (2008), 754-760. doi: 10.1016/j.aml.2007.07.029.

[45]

R. Ma and G. Dai, Global bifurcation and nodal solutions for a Sturm-Liouville problem with a nonsmooth nonlineariy, J. Funct. Anal., 265 (2013), 1443-1459. doi: 10.1016/j.jfa.2013.06.017.

[46]

I. Peral, Multiplicity of solutions for the $p$-Laplacian, ICTP SMR 990/1, 1997.

[47]

S. Prashanth and K. Sreenadh, Multiplicity results in a ball for $p$-Laplace equation with positive nonlinearity, Adv. Differential Equations, 7 (2002), 877-896.

[48]

P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1-66. doi: 10.1016/j.jde.2003.05.001.

[49]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[50]

P. H. Rabinowitz, On bifurcation from infinity, J. Funct. Anal., 14 (1973), 462-475. doi: 10.1016/0022-0396(73)90061-2.

[51]

P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202. doi: 10.1216/RMJ-1973-3-2-161.

[52]

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.

[53]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009.

[54]

P. Takáč, L. Tello and M. Ulm, Variational problems with a $p$-homogeneous energy, Positivity, 6 (2002), 75-94. doi: 10.1023/A:1012088127719.

[55]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.

[56]

M. Väth, Global bifurcation of the $p$-Laplacian and related operators, J. Differential Equations, 213 (2005), 389-409. doi: 10.1016/j.jde.2004.10.005.

[57]

G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1964.

show all references

References:
[1]

A. Anane, Simplicité et isolation de la première valeur propre du $p$-Laplacien avec poids, Comptes Rendus Acad. Sc. Paris, Série I, 305 (1987), 725-728.

[2]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rew., 18 (1976), 620-709. doi: 10.1137/1018114.

[3]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[4]

A. Ambrosetti, R. M. Calahorrano and F. R. Dobarro, Global branching for discontinuous problems, Comment. Math. Univ. Carolin., 31 (1990), 213-222.

[5]

A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl., 73 (1980), 411-422. doi: 10.1016/0022-247X(80)90287-5.

[6]

A. Ambrosetti, J. G. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242. doi: 10.1006/jfan.1996.0045.

[7]

A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics No. 104, Cambridge Univ. Press, 2007. doi: 10.1017/CBO9780511618260.

[8]

C. J. Amick and R. E. L. Turner, A global branch of steady vortex rings, J. Rein. Angew. Math., 384 (1988), 1-23.

[9]

D. Arcoya, J. I. Diaz and L. Tello, $S$-shaped bifurcation branch in a quasilinear multivalued model arising in climatoloty, J. Differential Equations, 150 (1998), 215-225. doi: 10.1006/jdeq.1998.3502.

[10]

D. Arcoya and J. L. Gámez, Bifurcation theory and related problems: anti-maximum principle and resonance, Comm. Partial Differential Equations, 26 (2001), 1879-1911. doi: 10.1081/PDE-100107462.

[11]

M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.

[12]

A. Cañada, P. Drábek and J. L. Gámez, Existence of positive solutions for some problems with nonlinear diffusion, Trans. Amer. Math. Soc., 349 (1997), 4231-4249. doi: 10.1090/S0002-9947-97-01947-8.

[13]

G. Dai, Global branching for discontinuous problems involving the $p$-Laplacian, Electron. J. Differential Equations, 44 (2013), 1-10.

[14]

G. Dai, Eigenvalue, global bifurcation and positive solutions for a class of nonlocal elliptic equations,, Topol. Methods Nonlinear Anal., (). 

[15]

G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian, J. Differential Equations, 252 (2012), 2448-2468. doi: 10.1016/j.jde.2011.09.026.

[16]

G. Dai, R. Ma and Y. Lu, Bifurcation from infinity and nodal solutions of quasilinear problems without the signum condition, J. Math. Anal. Appl., 397 (2013), 119-123. doi: 10.1016/j.jmaa.2012.07.056.

[17]

E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana Univ. Math. J., 23 (1974), 1069-1076.

[18]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc., 34 (2002), 533-538. doi: 10.1112/S002460930200108X.

[19]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New-York, 1985. doi: 10.1007/978-3-662-00547-7.

[20]

M. Delgado and A. Suárez, On the existence and multiplicity of positive solutions for some indefinite nonlinear eigenvalue problem, Proc. Amer. Math. Soc., 132 (2004), 1721-1728. doi: 10.1090/S0002-9939-04-07233-8.

[21]

M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Laplacian, J. Differential Equations, 92 (1991), 226-251. doi: 10.1016/0022-0396(91)90048-E.

[22]

P. Drábek and Y. X. Huang, Bifurcation problems for the $p$-Laplacian in $\mathbbR^N$, Trans. Amer. Math. Soc., 349 (1997), 171-188. doi: 10.1090/S0002-9947-97-01788-1.

[23]

X. L. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417. doi: 10.1016/j.jde.2007.01.008.

[24]

X. L. Fan, On the sub-supersolution method for $p(x)$-Laplacian equations, J. Math. Anal. Appl., 330 (2007), 665-682. doi: 10.1016/j.jmaa.2006.07.093.

[25]

X. L. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36 (1999), 295-318. doi: 10.1016/S0362-546X(97)00628-7.

[26]

D. G. De Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467. doi: 10.1016/S0022-1236(02)00060-5.

[27]

D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.

[28]

J. Fleckinger, R. Manásevich and Thélin, Global bifurcation from the first eigenvalue for a system of $p$-Laplacians, Math. Nachr., 182 (1996), 217-242. doi: 10.1002/mana.19961820110.

[29]

J. García-Melián and J. Sabina de Lis, A local bifurcation theorem for degenerate elliptic equations with radial symmetry, J. Differential Equations, 179 (2002), 27-43.

[30]

P. Girg and P. Takáč, Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (2008), 275-327. doi: 10.1007/s00023-008-0356-x.

[31]

M. Guedda and L. Véron, Bifurcation phenomena associated to the $p$-Laplace operator, Trans. Amer. Math. Soc., 310 (1988), 419-431. doi: 10.2307/2001132.

[32]

K. C. Hung and S. H. Wang, A complete classification of bifurcation diagrams of classes of multiparameter $p$-Laplacian boundary value problems, J. Differential Equations, 246 (2009), 1568-1599. doi: 10.1016/j.jde.2008.10.035.

[33]

L. Iturriaga et al., Positive solutions of the $p$-Laplacian involving a superlinear nonlinearity with zeros, J. Differential Equations, 248 (2010), 309-327. doi: 10.1016/j.jde.2009.08.008.

[34]

H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer, New York, 2004. doi: 10.1007/b97365.

[35]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964.

[36]

Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian, J. Differential Equations, 229 (2006), 229-256. doi: 10.1016/j.jde.2006.03.021.

[37]

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.

[38]

P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. doi: 10.1137/1024101.

[39]

P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. doi: 10.1016/j.jfa.2007.06.015.

[40]

P. Liu, J. Shi and Y. Wang, Bifurcation from a degenerate simple eigenvalue, J. Funct. Anal., 264 (2013), 2269-2299. doi: 10.1016/j.jfa.2013.02.010.

[41]

Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398. doi: 10.1006/jfan.1999.3446.

[42]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman and Hall/CRC, Boca Raton, 2001. doi: 10.1201/9781420035506.

[43]

J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators, Advances in Operator Theory and Applications Vol. 177, Birkhaüser, Basel, 2007.

[44]

R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems, Appl. Math. Lett., 21 (2008), 754-760. doi: 10.1016/j.aml.2007.07.029.

[45]

R. Ma and G. Dai, Global bifurcation and nodal solutions for a Sturm-Liouville problem with a nonsmooth nonlineariy, J. Funct. Anal., 265 (2013), 1443-1459. doi: 10.1016/j.jfa.2013.06.017.

[46]

I. Peral, Multiplicity of solutions for the $p$-Laplacian, ICTP SMR 990/1, 1997.

[47]

S. Prashanth and K. Sreenadh, Multiplicity results in a ball for $p$-Laplace equation with positive nonlinearity, Adv. Differential Equations, 7 (2002), 877-896.

[48]

P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1-66. doi: 10.1016/j.jde.2003.05.001.

[49]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9.

[50]

P. H. Rabinowitz, On bifurcation from infinity, J. Funct. Anal., 14 (1973), 462-475. doi: 10.1016/0022-0396(73)90061-2.

[51]

P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202. doi: 10.1216/RMJ-1973-3-2-161.

[52]

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483.

[53]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009.

[54]

P. Takáč, L. Tello and M. Ulm, Variational problems with a $p$-homogeneous energy, Positivity, 6 (2002), 75-94. doi: 10.1023/A:1012088127719.

[55]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150. doi: 10.1016/0022-0396(84)90105-0.

[56]

M. Väth, Global bifurcation of the $p$-Laplacian and related operators, J. Differential Equations, 213 (2005), 389-409. doi: 10.1016/j.jde.2004.10.005.

[57]

G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1964.

[1]

Guowei Dai, Ruyun Ma, Haiyan Wang. Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2839-2872. doi: 10.3934/cpaa.2013.12.2839

[2]

Bernd Kawohl, Jiří Horák. On the geometry of the p-Laplacian operator. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 799-813. doi: 10.3934/dcdss.2017040

[3]

Jinguo Zhang, Dengyun Yang. Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, 2021, 29 (5) : 3243-3260. doi: 10.3934/era.2021036

[4]

Po-Chun Huang, Shin-Hwa Wang, Tzung-Shin Yeh. Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2297-2318. doi: 10.3934/cpaa.2013.12.2297

[5]

Michael Filippakis, Alexandru Kristály, Nikolaos S. Papageorgiou. Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 405-440. doi: 10.3934/dcds.2009.24.405

[6]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem. Conference Publications, 2013, 2013 (special) : 51-59. doi: 10.3934/proc.2013.2013.51

[7]

Marta García-Huidobro, Raul Manásevich, J. R. Ward. Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 299-321. doi: 10.3934/dcds.2007.19.299

[8]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107

[9]

Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971

[10]

Guowei Dai, Ruyun Ma. Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 99-116. doi: 10.3934/dcds.2015.35.99

[11]

Mihai Mihăilescu, Julio D. Rossi. Monotonicity with respect to $ p $ of the First Nontrivial Eigenvalue of the $ p $-Laplacian with Homogeneous Neumann Boundary Conditions. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4363-4371. doi: 10.3934/cpaa.2020198

[12]

Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147

[13]

María del Mar González, Regis Monneau. Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1255-1286. doi: 10.3934/dcds.2012.32.1255

[14]

K. D. Chu, D. D. Hai. Positive solutions for the one-dimensional singular superlinear $ p $-Laplacian problem. Communications on Pure and Applied Analysis, 2020, 19 (1) : 241-252. doi: 10.3934/cpaa.2020013

[15]

Alexander Krasnosel'skii, Alexei Pokrovskii. On subharmonics bifurcation in equations with homogeneous nonlinearities. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 747-762. doi: 10.3934/dcds.2001.7.747

[16]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[17]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[18]

Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062

[19]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

[20]

Pavel Jirásek. On Compactness Conditions for the $p$-Laplacian. Communications on Pure and Applied Analysis, 2016, 15 (3) : 715-726. doi: 10.3934/cpaa.2016.15.715

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (158)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]