-
Previous Article
A critical exponent of Joseph-Lundgren type for an Hénon equation on the hyperbolic space
- CPAA Home
- This Issue
-
Next Article
Nonlinear Dirichlet problems with double resonance
Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems
1. | Department of Engineering, University of Messina, Messina, 98166, Italy |
2. | Department DICEAM, University of Reggio Calabria, Reggio Calabria, 89122, Italy |
3. | Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece |
We study the existence of positive solutions for perturbations of the classical eigenvalue problem for the Dirichlet $p-$Laplacian. We consider three cases. In the first the perturbation is $(p-1)-$sublinear near $+\infty$, while in the second the perturbation is $(p-1)-$superlinear near $+\infty$ and in the third we do not require asymptotic condition at $+\infty$. Using variational methods together with truncation and comparison techniques, we show that for $\lambda\in (0, \widehat{\lambda}_1)$ -$\lambda>0$ is the parameter and $\widehat{\lambda}_1$ being the principal eigenvalue of $\left(-\Delta_p, W^{1, p}_0(\Omega)\right)$ -we have positive solutions, while for $\lambda\geq \widehat{\lambda}_1$, no positive solutions exist. In the "sublinear case" the positive solution is unique under a suitable monotonicity condition, while in the "superlinear case" we produce the existence of a smallest positive solution. Finally, we point out an existence result of a positive solution without requiring asymptotic condition at $+\infty$, provided that the perturbation is damped by a parameter.
References:
[1] |
S. Aizicovici, N. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc. , 196 (2008).
doi: 10.1090/memo/0915. |
[2] |
S. Aizicovici, N. Papageorgiou and V. Staicu,
On p-superlinear equations with a nonhomogeneous differential operator, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 151-175.
doi: 10.1007/s00030-012-0187-9. |
[3] |
W. Allegretto and Y. X. Huang,
A Picone's identity for the p-Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.
doi: 10.1016/S0362-546X(97)00530-0. |
[4] |
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. |
[5] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
|
[6] |
D. Arcoya and D. Ruiz,
The Ambrosetti-Prodi problem for the p-Laplacian operator, Comm. Partial Differential Equations, 31 (2006), 849-865.
doi: 10.1080/03605300500394447. |
[7] |
G. Bonanno,
A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75 (2012), 2992-3007.
doi: 10.1016/j.na.2011.12.003. |
[8] |
G. Bonanno,
Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal., 1 (2012), 205-220.
doi: 10.1515/anona-2012-0003. |
[9] |
G. Bonanno and R. Livrea,
Existence and multiplicity of periodic solutions for second order Hamiltonian systems depending on a parameter, J. Convex Anal., 20 (2013), 1075-1094.
|
[10] |
P. Candito, G. D'Aguí and N. S. Papageorgiou,
Nonlinear noncoercive Neumann problems with a reaction concave near the origin, Topol. Methods Nonlinear Anal., 46 (2016), 289-317.
|
[11] |
D. Costa and C. Magalhaes,
Existence results for perturbations of the p-Laplacian, Nonlinear Anal., 24 (1995), 409-418.
doi: 10.1016/0362-546X(94)E0046-J. |
[12] |
J. I. Diaz and J. E. Saá,
Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.
|
[13] |
N. Dunford and J. Schwartz, Linear Operators, Wiles-Interscience, New York, 1958. |
[14] |
G. Fei,
On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differential Equations, 8 (2002), 1-12.
|
[15] |
M. Filippakis, A. Kristály and N. S. Papageorgiou,
Existence of five nonzero solutions with exact sign for a p-Laplacian equation, Discrete Contin. Dyn. Syst., 24 (2009), 405-440.
doi: 10.3934/dcds.2009.24.405. |
[16] |
J. Garc′ıa Azorero, I. Peral Alonso and J. Manfredi,
Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2 (2000), 385-404.
doi: 10.1142/S0219199700000190. |
[17] |
Z. M. Guo and Z. T. Zhang,
W1,p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 286 (2003), 32-50.
doi: 10.1016/S0022-247X(03)00282-8. |
[18] |
Hu Shouchuan and N. S. Papageorgiou,
Multiplicity of solutions for parametric p-Laplacian equations with nonlinearity concave near the origin, Tohoku Math. J. (2), 62 (2010), 137-162.
doi: 10.2748/tmj/1270041030. |
[19] |
Hu Shouchuan and N. S. Papageorgiou,
Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Commun. Pure Appl. Anal., 10 (2011), 1055-1078.
doi: 10.3934/cpaa.2011.10.1055. |
[20] |
S. Li, S. Wu and H. S. Zhou,
Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations, 185 (2002), 200-224.
doi: 10.1006/jdeq.2001.4167. |
[21] |
S. A. Marano and N. S. Papageorgiou,
Positive solutions to a Dirichlet problem with pLaplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., 12 (2013), 815-829.
doi: 10.3934/cpaa.2013.12.815. |
[22] |
S. A. Marano and N. S. Papageorgiou,
Multiple solutions to a Dirichlet problem with pLaplacian and nonlinearity depending on a parameter, Adv. Nonlinear Anal., 1 (2012), 257-275.
doi: 10.1515/anona-2012-0005. |
[23] |
S. A. Marano and N. S. Papageorgiou,
Constant-sign and nodal solutions of coercive (p. q)-Laplacian problems, Nonlinear Anal., 77 (2013), 118-129.
doi: 10.1016/j.na.2012.09.007. |
[24] |
N. S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, Springer, New York, 2009.
doi: 10.1007/b120946. |
[25] |
G. Talenti,
Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[26] |
J. L. Vázquez,
A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
show all references
References:
[1] |
S. Aizicovici, N. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc. , 196 (2008).
doi: 10.1090/memo/0915. |
[2] |
S. Aizicovici, N. Papageorgiou and V. Staicu,
On p-superlinear equations with a nonhomogeneous differential operator, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 151-175.
doi: 10.1007/s00030-012-0187-9. |
[3] |
W. Allegretto and Y. X. Huang,
A Picone's identity for the p-Laplacian and applications, Nonlinear Anal., 32 (1998), 819-830.
doi: 10.1016/S0362-546X(97)00530-0. |
[4] |
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. |
[5] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
|
[6] |
D. Arcoya and D. Ruiz,
The Ambrosetti-Prodi problem for the p-Laplacian operator, Comm. Partial Differential Equations, 31 (2006), 849-865.
doi: 10.1080/03605300500394447. |
[7] |
G. Bonanno,
A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75 (2012), 2992-3007.
doi: 10.1016/j.na.2011.12.003. |
[8] |
G. Bonanno,
Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal., 1 (2012), 205-220.
doi: 10.1515/anona-2012-0003. |
[9] |
G. Bonanno and R. Livrea,
Existence and multiplicity of periodic solutions for second order Hamiltonian systems depending on a parameter, J. Convex Anal., 20 (2013), 1075-1094.
|
[10] |
P. Candito, G. D'Aguí and N. S. Papageorgiou,
Nonlinear noncoercive Neumann problems with a reaction concave near the origin, Topol. Methods Nonlinear Anal., 46 (2016), 289-317.
|
[11] |
D. Costa and C. Magalhaes,
Existence results for perturbations of the p-Laplacian, Nonlinear Anal., 24 (1995), 409-418.
doi: 10.1016/0362-546X(94)E0046-J. |
[12] |
J. I. Diaz and J. E. Saá,
Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math., 305 (1987), 521-524.
|
[13] |
N. Dunford and J. Schwartz, Linear Operators, Wiles-Interscience, New York, 1958. |
[14] |
G. Fei,
On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differential Equations, 8 (2002), 1-12.
|
[15] |
M. Filippakis, A. Kristály and N. S. Papageorgiou,
Existence of five nonzero solutions with exact sign for a p-Laplacian equation, Discrete Contin. Dyn. Syst., 24 (2009), 405-440.
doi: 10.3934/dcds.2009.24.405. |
[16] |
J. Garc′ıa Azorero, I. Peral Alonso and J. Manfredi,
Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2 (2000), 385-404.
doi: 10.1142/S0219199700000190. |
[17] |
Z. M. Guo and Z. T. Zhang,
W1,p versus C1 local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 286 (2003), 32-50.
doi: 10.1016/S0022-247X(03)00282-8. |
[18] |
Hu Shouchuan and N. S. Papageorgiou,
Multiplicity of solutions for parametric p-Laplacian equations with nonlinearity concave near the origin, Tohoku Math. J. (2), 62 (2010), 137-162.
doi: 10.2748/tmj/1270041030. |
[19] |
Hu Shouchuan and N. S. Papageorgiou,
Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Commun. Pure Appl. Anal., 10 (2011), 1055-1078.
doi: 10.3934/cpaa.2011.10.1055. |
[20] |
S. Li, S. Wu and H. S. Zhou,
Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations, 185 (2002), 200-224.
doi: 10.1006/jdeq.2001.4167. |
[21] |
S. A. Marano and N. S. Papageorgiou,
Positive solutions to a Dirichlet problem with pLaplacian and concave-convex nonlinearity depending on a parameter, Commun. Pure Appl. Anal., 12 (2013), 815-829.
doi: 10.3934/cpaa.2013.12.815. |
[22] |
S. A. Marano and N. S. Papageorgiou,
Multiple solutions to a Dirichlet problem with pLaplacian and nonlinearity depending on a parameter, Adv. Nonlinear Anal., 1 (2012), 257-275.
doi: 10.1515/anona-2012-0005. |
[23] |
S. A. Marano and N. S. Papageorgiou,
Constant-sign and nodal solutions of coercive (p. q)-Laplacian problems, Nonlinear Anal., 77 (2013), 118-129.
doi: 10.1016/j.na.2012.09.007. |
[24] |
N. S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, Springer, New York, 2009.
doi: 10.1007/b120946. |
[25] |
G. Talenti,
Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.
doi: 10.1007/BF02418013. |
[26] |
J. L. Vázquez,
A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.
doi: 10.1007/BF01449041. |
[1] |
Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063 |
[2] |
E. N. Dancer, Zhitao Zhang. Critical point, anti-maximum principle and semipositone p-laplacian problems. Conference Publications, 2005, 2005 (Special) : 209-215. doi: 10.3934/proc.2005.2005.209 |
[3] |
Giuseppina Barletta, Roberto Livrea, Nikolaos S. Papageorgiou. A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1075-1086. doi: 10.3934/cpaa.2014.13.1075 |
[4] |
Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure and Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012 |
[5] |
Shun Uchida. Solvability of doubly nonlinear parabolic equation with p-laplacian. Evolution Equations and Control Theory, 2022, 11 (3) : 975-1000. doi: 10.3934/eect.2021033 |
[6] |
Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173 |
[7] |
Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure and Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335 |
[8] |
Kanishka Perera, Andrzej Szulkin. p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 743-753. doi: 10.3934/dcds.2005.13.743 |
[9] |
Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254 |
[10] |
Eun Kyoung Lee, R. Shivaji, Inbo Sim, Byungjae Son. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1139-1154. doi: 10.3934/cpaa.2019055 |
[11] |
Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure and Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475 |
[12] |
Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897 |
[13] |
Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure and Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735 |
[14] |
Julián Fernández Bonder, Leandro M. Del Pezzo. An optimization problem for the first eigenvalue of the $p-$Laplacian plus a potential. Communications on Pure and Applied Analysis, 2006, 5 (4) : 675-690. doi: 10.3934/cpaa.2006.5.675 |
[15] |
Adam Lipowski, Bogdan Przeradzki, Katarzyna Szymańska-Dębowska. Periodic solutions to differential equations with a generalized p-Laplacian. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2593-2601. doi: 10.3934/dcdsb.2014.19.2593 |
[16] |
Dimitri Mugnai. Bounce on a p-Laplacian. Communications on Pure and Applied Analysis, 2003, 2 (3) : 371-379. doi: 10.3934/cpaa.2003.2.371 |
[17] |
VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure and Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003 |
[18] |
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 |
[19] |
Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 |
[20] |
Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]