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

## 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

* 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.

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:
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show all references

##### References:
  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  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  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  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  A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. Google Scholar  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  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  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  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  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  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  J. I. Diaz and J. E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris SÃ©r. I Math., 305 (1987), 521-524. Google Scholar  N. Dunford and J. Schwartz, Linear Operators, Wiles-Interscience, New York, 1958. Google Scholar  G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differential Equations, 8 (2002), 1-12. Google Scholar  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  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  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  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  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  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  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  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  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  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  G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.  Google Scholar  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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