Opuscula Mathematica
Opuscula Math. 39, no. 2 (2019), 159-174

Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions

Gabriele Bonanno
Giuseppina D'Aguì
Angela Sciammetta
Abstract. In this paper, a nonlinear differential problem involving the \(p\)-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results.
Keywords: mixed problem, critical points.
Mathematics Subject Classification: 35J25, 35J20.
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References ▼
  1. [1] G. Barletta, R. Livrea, 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.
  2. [2] G. Bonanno, P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. 80 (2003), 424-429.
  3. [3] G. Bonanno, P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities, J. Differential Equations 244 (2008), 3031-3059.
  4. [4] G. Bonanno, G. D'Aguì, Mixed elliptic problems involving the \(p\)-Laplacian with nonhomogeneous boundary conditions, Discrete Contin. Dyn. Syst. 37 (2017) 11, 5797-5817.
  5. [5] G. Bonanno, G. D'Aguì, N.S. Papageorgiou Infinitely many solutions for mixed elliptic problems involving the \(p\)-Laplacian, Adv. Nonlinear Stud. 15 (2015), 939-950.
  6. [6] G. Bonanno, S.A. Marano, On the structure of the critical set of non-differentiable functions with a weak compactness condition, Appl. Anal. 89 (2010), 1-10.
  7. [7] V. Bonfim, A.F. Neves, A one-dimensional heat equation with mixed boundary conditions, J. Differential Equations 139 (1997), 319-338.
  8. [8] E. Colorado, I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal. 199 (2003), 468-507.
  9. [9] J. Dávila, A strong maximum principle for the Laplace equation with mixed boundary condition, J. Funct. Anal. 183 (2001), 231-244.
  10. [10] G. D'Aguì, S.A. Marano, N.S. Papageorgiou, Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction, J. Math. Anal. Appl. 433 (2016) 2, 1821-1845.
  11. [11] J. Garcia Azorero, A. Malchiodi, L. Montoro, 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.
  12. [12] R. Haller-Dintelmann, H.C. Kaiser, J. Rehberg, Elliptic model problems including mixed boundary conditions and material heterogeneities, J. Math. Pures. Appl. 89 (2008), 25-48.
  13. [13] I. Mitrea, 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.
  14. [14] N.S. Papageorgiou, V.D. Radulescu, Nonlinear nonhomogeneous Robin problems with superlinear reaction term, Adv. Nonlinear Stud. 16 (2016), 737-764.
  15. [15] N.S. Papageorgiou, V.D. Radulescu, Multiplicity of solutions for nonlinear nonhomogeneous Robin problems, Proc. Amer. Math. Soc. 146 (2018), 601-611.
About the authors ▼
Gabriele Bonanno
https://orcid.org/0000-0003-4115-7963
University of Messina, Department of Engineering, C. da Di Dio (S. Agata), 98166 Messina, Italy
Giuseppina D'Aguì
https://orcid.org/0000-0003-2080-8181
University of Messina, Department of Engineering, C. da Di Dio (S. Agata), 98166 Messina, Italy
Angela Sciammetta
University of Palermo, Department of Mathematics and Computer Science, Via Archirafi 34, 90123 Palermo, Italy
About the article ▼
Communicated by Dušan Repovš.
Received: 2018-10-23.
Accepted: 2018-10-30.
Published online: 2018-12-07.
Cite this article as ▼
Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta, Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions, Opuscula Math. 39, no. 2 (2019), 159-174, https://doi.org/10.7494/OpMath.2019.39.2.159.
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