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

Guowei  Dai

doi:10.3934/dcds.2016034

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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

Guowei Dai ^1,

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

Received April 2015 Revised April 2016 Published July 2016

Abstract
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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.

Keywords: superior limit, topological method., Bifurcation, homogeneous operator, one-sign solution, $p$-Laplacian.

Mathematics Subject Classification: Primary: 47J15, 47J05; Secondary: 35J60, 35B4.

Citation: Guowei Dai. Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5323-5345. doi: 10.3934/dcds.2016034

References:

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[2]	H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rew., 18 (1976), 620-709. doi: 10.1137/1018114. Google Scholar
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[4]	A. Ambrosetti, R. M. Calahorrano and F. R. Dobarro, Global branching for discontinuous problems, Comment. Math. Univ. Carolin., 31 (1990), 213-222. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	C. J. Amick and R. E. L. Turner, A global branch of steady vortex rings, J. Rein. Angew. Math., 384 (1988), 1-23. 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), 215-225. 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), 1879-1911. doi: 10.1081/PDE-100107462. Google Scholar
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[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. Google Scholar
[13]	G. Dai, Global branching for discontinuous problems involving the $p$-Laplacian, Electron. J. Differential Equations, 44 (2013), 1-10. 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), 2448-2468. 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), 119-123. doi: 10.1016/j.jmaa.2012.07.056. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. 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), 452-467. 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), 41-63. 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), 217-242. 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), 27-43. 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), 275-327. 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), 419-431. 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), 1568-1599. 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), 309-327. doi: 10.1016/j.jde.2009.08.008. Google Scholar
[34]	H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer, New York, 2004. doi: 10.1007/b97365. Google Scholar
[35]	M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964. Google Scholar
[36]	Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian, J. Differential Equations, 229 (2006), 229-256. 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), 1203-1219. 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), 441-467. doi: 10.1137/1024101. Google Scholar
[39]	P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. 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), 2269-2299. doi: 10.1016/j.jfa.2013.02.010. Google Scholar
[41]	Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398. doi: 10.1006/jfan.1999.3446. Google Scholar
[42]	J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman and Hall/CRC, Boca Raton, 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, Birkhaüser, Basel, 2007. Google Scholar
[44]	R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems, Appl. Math. Lett., 21 (2008), 754-760. 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), 1443-1459. 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), 877-896. Google Scholar
[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. Google Scholar
[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. Google Scholar
[50]	P. H. Rabinowitz, On bifurcation from infinity, J. Funct. Anal., 14 (1973), 462-475. 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), 161-202. doi: 10.1216/RMJ-1973-3-2-161. Google Scholar
[52]	J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. 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), 2788-2812. 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), 75-94. 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), 126-150. 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), 389-409. doi: 10.1016/j.jde.2004.10.005. Google Scholar
[57]	G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 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, Série I, 305 (1987), 725-728. Google Scholar
[2]	H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rew., 18 (1976), 620-709. 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), 519-543. 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), 213-222. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	C. J. Amick and R. E. L. Turner, A global branch of steady vortex rings, J. Rein. Angew. Math., 384 (1988), 1-23. 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), 215-225. 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), 1879-1911. 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), 4231-4249. 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), 1-10. 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), 2448-2468. 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), 119-123. 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), 1069-1076. Google Scholar
[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. Google Scholar
[19]	K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New-York, 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), 1721-1728. 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), 226-251. 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), 171-188. 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), 397-417. 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), 665-682. 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), 295-318. 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), 452-467. 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), 41-63. 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), 217-242. 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), 27-43. 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), 275-327. 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), 419-431. 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), 1568-1599. 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), 309-327. doi: 10.1016/j.jde.2009.08.008. Google Scholar
[34]	H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer, New York, 2004. doi: 10.1007/b97365. Google Scholar
[35]	M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964. Google Scholar
[36]	Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian, J. Differential Equations, 229 (2006), 229-256. 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), 1203-1219. 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), 441-467. doi: 10.1137/1024101. Google Scholar
[39]	P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600. 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), 2269-2299. doi: 10.1016/j.jfa.2013.02.010. Google Scholar
[41]	Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398. doi: 10.1006/jfan.1999.3446. Google Scholar
[42]	J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman and Hall/CRC, Boca Raton, 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, Birkhaüser, Basel, 2007. Google Scholar
[44]	R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems, Appl. Math. Lett., 21 (2008), 754-760. 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), 1443-1459. 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), 877-896. Google Scholar
[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. Google Scholar
[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. Google Scholar
[50]	P. H. Rabinowitz, On bifurcation from infinity, J. Funct. Anal., 14 (1973), 462-475. 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), 161-202. doi: 10.1216/RMJ-1973-3-2-161. Google Scholar
[52]	J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. 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), 2788-2812. 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), 75-94. 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), 126-150. 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), 389-409. doi: 10.1016/j.jde.2004.10.005. Google Scholar
[57]	G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1964. Google Scholar

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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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