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 & Continuous Dynamical Systems - A, 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, 305 (1987), 725. Google Scholar

[2]

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

[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. doi: 10.1006/jfan.1994.1078. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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. doi: 10.1006/jdeq.1998.3502. Google Scholar

[10]

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

[11]

M. S. Berger, Nonlinearity and Functional Analysis,, Academic Press, (1977). Google Scholar

[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. doi: 10.1090/S0002-9947-97-01947-8. Google Scholar

[13]

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

[14]

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

[15]

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

[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. doi: 10.1016/j.jmaa.2012.07.056. Google Scholar

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

[18]

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

[19]

K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (1985). doi: 10.1007/978-3-662-00547-7. Google Scholar

[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. doi: 10.1090/S0002-9939-04-07233-8. Google Scholar

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

[22]

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

[23]

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

[24]

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

[25]

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

[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. doi: 10.1016/S0022-1236(02)00060-5. Google Scholar

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. doi: 10.1007/s00023-008-0356-x. Google Scholar

[31]

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

[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. doi: 10.1016/j.jde.2008.10.035. Google Scholar

[33]

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

[34]

H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs,, Springer, (2004). doi: 10.1007/b97365. Google Scholar

[35]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations,, Macmillan, (1964). Google Scholar

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

[42]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, Chapman and Hall/CRC, (2001). doi: 10.1201/9781420035506. Google Scholar

[43]

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

[44]

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

[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. doi: 10.1016/j.jfa.2013.06.017. Google Scholar

[46]

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

[47]

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

[48]

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

[49]

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

[50]

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

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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

[56]

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

[57]

G. T. Whyburn, Topological Analysis,, Princeton University Press, (1964). Google Scholar

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, 305 (1987), 725. Google Scholar

[2]

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

[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. doi: 10.1006/jfan.1994.1078. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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. doi: 10.1006/jdeq.1998.3502. Google Scholar

[10]

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

[11]

M. S. Berger, Nonlinearity and Functional Analysis,, Academic Press, (1977). Google Scholar

[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. doi: 10.1090/S0002-9947-97-01947-8. Google Scholar

[13]

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

[14]

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

[15]

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

[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. doi: 10.1016/j.jmaa.2012.07.056. Google Scholar

[17]

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

[18]

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

[19]

K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (1985). doi: 10.1007/978-3-662-00547-7. Google Scholar

[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. doi: 10.1090/S0002-9939-04-07233-8. Google Scholar

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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. doi: 10.1016/S0022-1236(02)00060-5. Google Scholar

[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. Google Scholar

[28]

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

[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. Google Scholar

[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. doi: 10.1007/s00023-008-0356-x. Google Scholar

[31]

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

[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. doi: 10.1016/j.jde.2008.10.035. Google Scholar

[33]

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

[34]

H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs,, Springer, (2004). doi: 10.1007/b97365. Google Scholar

[35]

M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations,, Macmillan, (1964). Google Scholar

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

[42]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, Chapman and Hall/CRC, (2001). doi: 10.1201/9781420035506. Google Scholar

[43]

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

[44]

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

[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. doi: 10.1016/j.jfa.2013.06.017. Google Scholar

[46]

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

[47]

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

[48]

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

[49]

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

[50]

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

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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

[56]

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

[57]

G. T. Whyburn, Topological Analysis,, Princeton University Press, (1964). Google Scholar

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