Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros

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August 2018, 38(8): 3831-3850. doi: 10.3934/dcds.2018166

Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros

Leonelo Iturriaga ^1, and Eugenio Massa ^2,,

1.	Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Casilla 110-V, Valparaíso, Chile
2.	Departamento de Matemática, Instituto de Ciêencias Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, 13560-970, São Carlos SP, Brazil

* Corresponding author: E. Massa

Received August 2017 Revised April 2018 Published May 2018

Fund Project: The first author gratefully acknowledges financial support from Fondecyt grants 1161635, 1171532 and 1171691
The author E. Massa was supported by: grant #2014/25398-0, São Paulo Research Foundation (FAPESP) and grants #308354/2014-1, #303447/2017-6, CNPq/Brazil.

In this paper we consider the equation $(-Δ)^k\, u = λ f(x, u)+μ g(x, u)$ with Navier boundary conditions, in a bounded and smooth domain. The main interest is when the nonlinearity is nonnegative but admits a zero and $f, g$ are, respectively, identically zero above and below the zero. We prove the existence of multiple positive solutions when the parameters lie in a region of the form $λ>\overline λ$ and $0 < μ< \overlineμ(λ)$, then we provide further conditions under which, respectively, the bound $\overlineμ(λ)$ is either necessary, or can be removed.

Keywords: Poly-Laplacian, multiplicity of positive solutions, existence and nonexistence, variational methods, nonlinearities with zeros.

Mathematics Subject Classification: Primary: 35J40; Secondary: 35J91.

Citation: Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166

References:

[1]	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.
[2]	T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian, Commun. Contemp. Math., 6 (2004), 245-258. doi: 10.1142/S0219199704001306.
[3]	F. Bernis, J. García Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1 (1996), 219-240.
[4]	H. Brezis and L. Nirenberg, H¹ versus C¹ local minimizers, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472.
[5]	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.
[6]	D. G. de Figueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc. (JEMS), 8 (2006), 269-286. doi: 10.4171/JEMS/52.
[7]	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. (9), 61 (1982), 41-63.
[8]	F. O. de Paiva and E. Massa, Multiple solutions for some elliptic equations with a nonlinearity concave at the origin, Nonlinear Anal., 66 (2007), 2940-2946. doi: 10.1016/j.na.2006.04.015.
[9]	J. Díaz and J. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.
[10]	F. Ebobisse and M. Ahmedou, On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52 (2003), 1535-1552. doi: 10.1016/S0362-546X(02)00273-0.
[11]	D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rational Mech. Anal., 112 (1990), 269-289. doi: 10.1007/BF00381236.
[12]	J. García-Melián and L. Iturriaga, Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros, Israel J. Math., 210 (2015), 233-244. doi: 10.1007/s11856-015-1251-z.
[13]	J. García-Melián and J. Sabina de Lis, Stationary profiles of degenerate problems when a parameter is large, Differential Integral Equations, 13 (2000), 1201-1232.
[14]	F. Gazzola, H.-C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143. doi: 10.1007/s00526-002-0182-9.
[15]	F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, vol. 1991 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, Positivity preserving and nonlinear higher order elliptic equations in bounded domains. doi: 10.1007/978-3-642-12245-3.
[16]	M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902. doi: 10.1016/0362-546X(89)90020-5.
[17]	P. Hess, On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. Partial Differential Equations, 6 (1981), 951-961. doi: 10.1080/03605308108820200.
[18]	J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58. doi: 10.1006/jfan.1993.1062.
[19]	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.
[20]	L. Iturriaga, S. Lorca and E. Massa, Multiple positive solutions for the m-Laplacian and a nonlinearity with many zeros, Differential Integral Equations, 30 (2017), 145-159.
[21]	L. Iturriaga, E. Massa, J. Sánchez and P. Ubilla, 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.
[22]	L. Iturriaga, E. Massa, J. Sanchez and P. Ubilla, Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros, Math. Nach., 287 (2014), 1131-1141. doi: 10.1002/mana.201100285.
[23]	A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578. doi: 10.1137/1032120.
[24]	A. C. Lazer and P. J. McKenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl., 181 (1994), 648-655. doi: 10.1006/jmaa.1994.1049.
[25]	P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. doi: 10.1137/1024101.
[26]	Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398. doi: 10.1006/jfan.1999.3446.
[27]	Z. Liu, Positive solutions of a class of nonlinear elliptic eigenvalue problems, Math. Z., 242 (2002), 663-686. doi: 10.1007/s002090100373.
[28]	A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal., 31 (1998), 895-908. doi: 10.1016/S0362-546X(97)00446-X.
[29]	A. M. Micheletti and A. Pistoia, Nontrivial solutions for some fourth order semilinear elliptic problems, Nonlinear Anal., 34 (1998), 509-523. doi: 10.1016/S0362-546X(97)00596-8.
[30]	E. S. Noussair, C. A. Swanson and J. Yang, Critical semilinear biharmonic equations in R^N, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), 139-148. doi: 10.1017/S0308210500014189.
[31]	L. A. Peletier and R. van der Vorst, Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential Integral Equations, 5 (1992), 747-767.
[32]	P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9), 69 (1990), 55-83.
[33]	S. Takeuchi, Coincidence sets in semilinear elliptic problems of logistic type, Differential Integral Equations, 20 (2007), 1075-1080.
[34]	S. Takeuchi, Partial flat core properties associated to the p-Laplace operator, Discrete Contin. Dyn. Syst., (2007), 965-973.
[35]	G. Tarantello, A note on a semilinear elliptic problem, Differential Integral Equations, 5 (1992), 561-565.
[36]	G. Xu and J. Zhang, Existence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity, J. Math. Anal. Appl., 281 (2003), 633-640. doi: 10.1016/S0022-247X(03)00170-7.
[37]	J. Zhang and Z. Wei, Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin, J. Math. Anal. Appl., 383 (2011), 291-306. doi: 10.1016/j.jmaa.2011.05.030.
[38]	Y. Zhang, Positive solutions of semilinear biharmonic equations with critical Sobolev exponents, Nonlinear Anal., 75 (2012), 55-67. doi: 10.1016/j.na.2011.07.065.

show all references

References:

[1]	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.
[2]	T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian, Commun. Contemp. Math., 6 (2004), 245-258. doi: 10.1142/S0219199704001306.
[3]	F. Bernis, J. García Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 1 (1996), 219-240.
[4]	H. Brezis and L. Nirenberg, H¹ versus C¹ local minimizers, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472.
[5]	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.
[6]	D. G. de Figueiredo, J.-P. Gossez and P. Ubilla, Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity, J. Eur. Math. Soc. (JEMS), 8 (2006), 269-286. doi: 10.4171/JEMS/52.
[7]	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. (9), 61 (1982), 41-63.
[8]	F. O. de Paiva and E. Massa, Multiple solutions for some elliptic equations with a nonlinearity concave at the origin, Nonlinear Anal., 66 (2007), 2940-2946. doi: 10.1016/j.na.2006.04.015.
[9]	J. Díaz and J. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.
[10]	F. Ebobisse and M. Ahmedou, On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52 (2003), 1535-1552. doi: 10.1016/S0362-546X(02)00273-0.
[11]	D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Rational Mech. Anal., 112 (1990), 269-289. doi: 10.1007/BF00381236.
[12]	J. García-Melián and L. Iturriaga, Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros, Israel J. Math., 210 (2015), 233-244. doi: 10.1007/s11856-015-1251-z.
[13]	J. García-Melián and J. Sabina de Lis, Stationary profiles of degenerate problems when a parameter is large, Differential Integral Equations, 13 (2000), 1201-1232.
[14]	F. Gazzola, H.-C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143. doi: 10.1007/s00526-002-0182-9.
[15]	F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, vol. 1991 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, Positivity preserving and nonlinear higher order elliptic equations in bounded domains. doi: 10.1007/978-3-642-12245-3.
[16]	M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902. doi: 10.1016/0362-546X(89)90020-5.
[17]	P. Hess, On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. Partial Differential Equations, 6 (1981), 951-961. doi: 10.1080/03605308108820200.
[18]	J. Hulshof and R. van der Vorst, Differential systems with strongly indefinite variational structure, J. Funct. Anal., 114 (1993), 32-58. doi: 10.1006/jfan.1993.1062.
[19]	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.
[20]	L. Iturriaga, S. Lorca and E. Massa, Multiple positive solutions for the m-Laplacian and a nonlinearity with many zeros, Differential Integral Equations, 30 (2017), 145-159.
[21]	L. Iturriaga, E. Massa, J. Sánchez and P. Ubilla, 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.
[22]	L. Iturriaga, E. Massa, J. Sanchez and P. Ubilla, Positive solutions for an elliptic equation in an annulus with a superlinear nonlinearity with zeros, Math. Nach., 287 (2014), 1131-1141. doi: 10.1002/mana.201100285.
[23]	A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578. doi: 10.1137/1032120.
[24]	A. C. Lazer and P. J. McKenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl., 181 (1994), 648-655. doi: 10.1006/jmaa.1994.1049.
[25]	P.-L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467. doi: 10.1137/1024101.
[26]	Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398. doi: 10.1006/jfan.1999.3446.
[27]	Z. Liu, Positive solutions of a class of nonlinear elliptic eigenvalue problems, Math. Z., 242 (2002), 663-686. doi: 10.1007/s002090100373.
[28]	A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal., 31 (1998), 895-908. doi: 10.1016/S0362-546X(97)00446-X.
[29]	A. M. Micheletti and A. Pistoia, Nontrivial solutions for some fourth order semilinear elliptic problems, Nonlinear Anal., 34 (1998), 509-523. doi: 10.1016/S0362-546X(97)00596-8.
[30]	E. S. Noussair, C. A. Swanson and J. Yang, Critical semilinear biharmonic equations in R^N, Proc. Roy. Soc. Edinburgh Sect. A, 121 (1992), 139-148. doi: 10.1017/S0308210500014189.
[31]	L. A. Peletier and R. van der Vorst, Existence and nonexistence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential Integral Equations, 5 (1992), 747-767.
[32]	P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9), 69 (1990), 55-83.
[33]	S. Takeuchi, Coincidence sets in semilinear elliptic problems of logistic type, Differential Integral Equations, 20 (2007), 1075-1080.
[34]	S. Takeuchi, Partial flat core properties associated to the p-Laplace operator, Discrete Contin. Dyn. Syst., (2007), 965-973.
[35]	G. Tarantello, A note on a semilinear elliptic problem, Differential Integral Equations, 5 (1992), 561-565.
[36]	G. Xu and J. Zhang, Existence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity, J. Math. Anal. Appl., 281 (2003), 633-640. doi: 10.1016/S0022-247X(03)00170-7.
[37]	J. Zhang and Z. Wei, Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin, J. Math. Anal. Appl., 383 (2011), 291-306. doi: 10.1016/j.jmaa.2011.05.030.
[38]	Y. Zhang, Positive solutions of semilinear biharmonic equations with critical Sobolev exponents, Nonlinear Anal., 75 (2012), 55-67. doi: 10.1016/j.na.2011.07.065.

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