Topological Methods in Nonlinear Analysis

Multiple solutions to a Dirichlet eigenvalue problem with $p$-Laplacian

Salvatore A. Marano, Dumitru Motreanu, and Daniele Puglisi

Full-text: Open access

Abstract

The existence of a greatest negative, a smallest positive, and a nodal weak solution to a homogeneous Dirichlet problem with $p$-Laplacian and reaction term depending on a positive parameter is investigated via variational as well as topological methods, besides truncation techniques.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 42, Number 2 (2013), 277-291.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461248980

Mathematical Reviews number (MathSciNet)
MR3203450

Zentralblatt MATH identifier
1290.35178

Citation

Marano, Salvatore A.; Motreanu, Dumitru; Puglisi, Daniele. Multiple solutions to a Dirichlet eigenvalue problem with $p$-Laplacian. Topol. Methods Nonlinear Anal. 42 (2013), no. 2, 277--291. https://projecteuclid.org/euclid.tmna/1461248980


Export citation

References

  • A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems , J. Funct. Anal., 122 (1994), 519–543 \ref\key 2
  • A. Ambrosetti, J. Garcia Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations , J. Funct. Anal., 137(1996), 219–242 \ref\key 3
  • P. Candito, S. Carl and R. Livrea, Multiple solutions for quasilinear elliptic problems via critical points in open sublevels and truncation principles , J. Math. Anal. Appl., 395 (2012), 156–163 \ref\key 4
  • S. Carl, V.K. Le and D. Motreanu, Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer Monogr. Math., Springer, New York (2007) \ref\key 5
  • S. Carl and D. Motreanu, Constant-sign and sign-changing solutions for nonlinear eigenvalue problems , Nonlinear Anal., 68 (2008), 2668–2676 \ref\key 6
  • S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian , Abstr. Appl. Anal., 7 (2002), 613–625 \ref\key 7
  • M. Cuesta, D. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian , J. Differential Equations, 159 (1999), 212–238 \ref\key 8
  • L. Gasiński and N.S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman & Hall/CRC, Boca Raton, FL (2005) \ref\key 9 ––––, Topics in Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, FL (2006) \ref\key 10
  • S. Hu and N.S. Papageorgiou, Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin , Tohoku Math. J., 62 (2010), 137–162 \ref\key 11
  • Y. Jabri, The Mountain Pass Theorem: Variants, Generalizations and some Applications, Encyclopedia Math. Appl., Cambridge Univ. Press, Cambridge (2003) \ref\key 12
  • A. Lê, Eigenvalue problems for the $p$-Laplacian , Nonlinear Anal., 64 (2006), 1057–1099 \ref\key 13
  • S.A. Marano and N.S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter , Comm. Pure Appl. Anal., 12 (2013), 815–829 \ref\key 14
  • B. Ricceri, personal communication \ref\key 15
  • J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations , Appl. Math. Optim., 12 (1984), 191–202