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July  2017, 16(4): 1169-1188. doi: 10.3934/cpaa.2017057

Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems

Gabriele Bonanno 1, , Pasquale Candito 2, , Roberto Livrea 2,, and  Nikolaos S. Papageorgiou 3,

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

* Corresponding author: P. Candito

Received  May 2016 Revised  February 2017 Published  April 2017

Fund Project: The authors have been supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

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.

Keywords: p-Laplacian, first eigenvalue, generalized Picone's identity, nonlinear regularity, nonlinear maximum principle, variational methods.
Mathematics Subject Classification: Primary: 35J25; Secondary: 35J80.
Citation: Gabriele Bonanno, Pasquale Candito, Roberto Livrea, Nikolaos S. Papageorgiou. Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1169-1188. doi: 10.3934/cpaa.2017057
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

N. Dunford and J. Schwartz, Linear Operators, Wiles-Interscience, New York, 1958. Google Scholar

[14]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differential Equations, 8 (2002), 1-12. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

N. S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, Springer, New York, 2009. doi: 10.1007/b120946. Google Scholar

[25]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013. Google Scholar

[26]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[13]

N. Dunford and J. Schwartz, Linear Operators, Wiles-Interscience, New York, 1958. Google Scholar

[14]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differential Equations, 8 (2002), 1-12. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

N. S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, Springer, New York, 2009. doi: 10.1007/b120946. Google Scholar

[25]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013. Google Scholar

[26]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041. Google Scholar

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