## Abstract and Applied Analysis

### Nonhomogeneous Nonlinear Dirichlet Problems with a $p$-Superlinear Reaction

#### Abstract

We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reaction $f(z,\zeta )$, whose primitive $f(z,\zeta )$ is $p$-superlinear near $±\infty$, but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like the $p$-Laplacian, the $(p,q)$-Laplacian, and the $p$-generalized mean curvature operator.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 918271, 28 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365125161

Digital Object Identifier
doi:10.1155/2012/918271

Mathematical Reviews number (MathSciNet)
MR2922949

Zentralblatt MATH identifier
1246.35087

#### Citation

Gasiński, Leszek; Papageorgiou, Nikolaos S. Nonhomogeneous Nonlinear Dirichlet Problems with a $p$ -Superlinear Reaction. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 918271, 28 pages. doi:10.1155/2012/918271. https://projecteuclid.org/euclid.aaa/1365125161

#### References

• Z. Q. Wang, “On a superlinear elliptic equation,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 8, no. 1, pp. 43–57, 1991.
• M.-Y. Jiang, “Critical groups and multiple solutions of the $p$-Laplacian equations,” Nonlinear Analysis, vol. 59, no. 8, pp. 1221–1241, 2004.
• L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, vol. 9 of Series in Mathematical Analysis and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2006.
• J. L. Vázquez, “A strong maximum principle for some quasilinear elliptic equations,” Applied Mathematics and Optimization, vol. 12, no. 3, pp. 191–202, 1984.
• V. Benci, D. Fortunato, and L. Pisani, “Soliton like solutions of a Lorentz invariant equation in dimension 3,” Reviews in Mathematical Physics, vol. 10, no. 3, pp. 315–344, 1998.
• S. Cingolani and M. Degiovanni, “Nontrivial solutions for $p$-Laplace equations with right-hand side having $p$-linear growth at infinity,” Communications in Partial Differential Equations, vol. 30, no. 7-9, pp. 1191–1203, 2005.
• G. M. Figueiredo, “Existence of positive solutions for a class of $q$ elliptic problems with critical growth on $n$,” Journal of Mathematical Analysis and Applications, vol. 378, no. 2, pp. 507–518, 2011.
• M. Sun, “Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance,” Journal of Mathematical Analysis and Applications, vol. 386, no. 2, pp. 661–668, 2012.
• Z. Chen and Y. Shen, “Infinitely many solutions of Dirichlet problem for $p$-mean curvature operator,” Applied Mathematics. A Journal of Chinese Universities. Series B, vol. 18, no. 2, pp. 161–172, 2003.
• L. Gasiński and N. S. Papageorgiou, “Existence and multiplicity of solutions for Neumann $p$-Laplacian-type equations,” Advanced Nonlinear Studies, vol. 8, no. 4, pp. 843–870, 2008.
• N. S. Papageorgiou and S. Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, vol. 19 of Advances in Mechanics and Mathematics, Springer, New York, NY, USA, 2009.
• L. Gasiński and N. S. Papageorgiou, “Nodal and multiple constant sign solutions for resonant $p$-Laplacian equations with a nonsmooth potential,” Nonlinear Analysis, vol. 71, no. 11, pp. 5747–5772, 2009.
• O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, vol. 46 of Mathematics in Science and Engineering, Academic Press, New York, Ny, USA, 1968.
• G. M. Lieberman, “Boundary regularity for solutions of degenerate elliptic equations,” Nonlinear Analysis, vol. 12, no. 11, pp. 1203–1219, 1988.
• P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel, Switzerland, 2007.
• A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2003.
• J. Dugundji, Topology, Allyn and Bacon, Boston, Mass, USA, 1966.