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: |
[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.![]() ![]() ![]() |