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The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form
Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros
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 |
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.
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 RN, 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 RN, 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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