# American Institute of Mathematical Sciences

November  2017, 37(11): 5797-5817. doi: 10.3934/dcds.2017252

## Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions

 Department of Engineering, University of Messina, C.da Di Dio, Sant'Agata, 98166 Messina, Italy

Received  September 2016 Revised  June 2017 Published  July 2017

In this paper, mixed elliptic problems involving the p-Laplacian and with nonhomogeneous boundary conditions are investigated. At first, the existence of one non-trivial solution, under a suitable behaviour on the nonlinearity and without requiring neither conditions at zero nor conditions at infinity, is established. Then, by adding a condition at infinity on the nonlinearity, also a second non-trivial solution is guaranteed. Some special cases are pointed out as, in particular, the existence of one non-trivial solution when the datum is $(p-1)-$sublinear at zero and the existence of two non-trivial solutions when the nonlinear term is again $(p-1)-$sublinear at zero and, in addition, more than $(p-1)-$superlinear at infinity. As a consequence, the existence of two non-trivial solutions for concave-convex nonlinearities is emphasized. Finally, the case of a simple $(p-1)-$superlinearity at infinity is considered and it is also observed that the same results hold when the nonlinear behaviour, described before for the datum, is instead assumed by the nonhomogeneous Neumann boundary conditions. Concrete examples of applications are also given. The approach is based on variational methods and critical point theory. Precisely, a non-zero local minimum theorem and a two non-zero critical points theorem are applied.

Citation: Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252
##### References:
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##### References:
 [1] A. Ambrosetti, H. Brézis 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] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [3] G. Barletta, R. Livrea and N. S. Papageorgiou, Bifurcation phenomena for the positive solutions on semilinear elliptic problems with mixed boundary conditions, J. Nonlinear Convex Anal., 17 (2016), 1497-1516. [4] 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. [5] G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal., 1 (2012), 205-220.  doi: 10.1515/anona-2012-0003. [6] G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math., 80 (2003), 424-429.  doi: 10.1007/s00013-003-0479-8. [7] G. Bonanno, P. Candito and G. D'Aguí, Variational methods on finite dimensional Banach spaces and discrete problems, Adv. Nonlinear Stud., 14 (2014), 915-939.  doi: 10.1515/ans-2014-0406. [8] G. Bonanno and G. D'Aguí, Two non-zero solutions for elliptic Dirichlet problems, Z. Anal. Anwend., 35 (2016), 449-464.  doi: 10.4171/ZAA/1573. [9] G. Bonanno, G. D'Aguì and N. S. Papageorgiou, Infinitely many solutions for mixed elliptic problems involving the $p-$Laplacian, Adv. Nonlinear Stud., 15 (2015), 939-950.  doi: 10.1515/ans-2015-0410. [10] G. Bonanno, G. D'Aguì and P. Winkert, Sturm-Liouville equations involving discontinuous nonlinearities, Minimax Theory Appl., 1 (2016), 125-143. [11] G. Bonanno, R. Livrea and M. Schechter, Some notes on a superlinear second order Hamiltonian system, Manuscripta Math., (2016), 1-19.  doi: 10.1007/s00229-016-0903-6. [12] V. Bonfim and A. F. Neves, A one-dimensional heat equation with mixed boundary conditions, J. Differential Equations, 139 (1997), 319-338.  doi: 10.1006/jdeq.1997.3299. [13] E. Colorado and I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal., 199 (2003), 468-507.  doi: 10.1016/S0022-1236(02)00101-5. [14] G. D'Aguì, Existence results for a mixed boundary value problem with Sturm-Liouville equation, Adv. Pure Appl. Math., 2 (2011), 237-248.  doi: 10.1515/APAM.2010.043. [15] G. D'Aguì, B. Di Bella and S. Tersian, Multiplicity results for superlinear boundary value problems with impulsive effects, Math. Methods Appl. Sci., 39 (2016), 1060-1068.  doi: 10.1002/mma.3545. [16] J. Dávila, A strong maximum principle for the Laplace equation with mixed boundary condition, J. Funct. Anal., 183 (2001), 231-244.  doi: 10.1006/jfan.2000.3729. [17] J. Garcia Azorero, A. Malchiodi, L. Montoro and I. Peral, Concentration of solutions for some singularly perturbed mixed problems: Asymptotics of minimal energy solutions, Ann. I. H. Poincaré AN, 27 (2010), 37-56.  doi: 10.1016/j.anihpc.2009.06.005. [18] R. Haller-Dintelmann, H. C. Kaiser and J. Rehberg, Elliptic model problems including mixed boundary conditions and material heterogeneities, J. Math. Pures. Appl., 89 (2008), 25-48.  doi: 10.1016/j.matpur.2007.09.001. [19] S. A. Marano and S. Mosconi, Non-Smooth critical point theory on closed convex sets, Commun. Pure Appl. Anal., 13 (2014), 1187-1202.  doi: 10.3934/cpaa.2014.13.1187. [20] S. A. Marano and N. S. Papageorgiou, Multiple solutions to a Dirichlet problem with p-Laplacian and nonlinearity depending on a parameter, Adv. Nonlinear Anal., 1 (2012), 257-275.  doi: 10.1515/anona-2012-0005. [21] I. Mitrea and M. Mitrea, The Poisson problem with mixed boundary conditions in Sobolev and Besov spaces in non-smooth domains, Trans. Amer. Math. Soc., 359 (2007), 4143-4182.  doi: 10.1090/S0002-9947-07-04146-3. [22] D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5. [23] G. Savaré, Regularity and perturbation results for mixed second order elliptic problems, Comm. Partial Differential Equations, 22 (1997), 869-899.  doi: 10.1080/03605309708821287.
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