## Nodal solutions for $\left (p,2\right )$-equations

HTML articles powered by AMS MathViewer

- by Sergiu Aizicovici, Nikolaos S. Papageorgiou and Vasile Staicu PDF
- Trans. Amer. Math. Soc.
**367**(2015), 7343-7372 Request permission

## Abstract:

In this paper, we study a nonlinear elliptic equation driven by the sum of a $p$-Laplacian and a Laplacian ($\left ( p,2\right )$-equation), with a Carathéodory $\left ( p-1\right )$-(sub-)linear reaction. Using variational methods combined with Morse theory, we prove two multiplicity theorems providing precise sign information for all the solutions (constant sign and nodal solutions). In the process, we prove two auxiliary results of independent interest.## References

- Sergiu Aizicovici, Nikolaos S. Papageorgiou, and Vasile Staicu,
*Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints*, Mem. Amer. Math. Soc.**196**(2008), no. 915, vi+70. MR**2459421**, DOI 10.1090/memo/0915 - David Arcoya and David Ruiz,
*The Ambrosetti-Prodi problem for the $p$-Laplacian operator*, Comm. Partial Differential Equations**31**(2006), no. 4-6, 849–865. MR**2233044**, DOI 10.1080/03605300500394447 - Thomas Bartsch,
*Critical point theory on partially ordered Hilbert spaces*, J. Funct. Anal.**186**(2001), no. 1, 117–152. MR**1863294**, DOI 10.1006/jfan.2001.3789 - V. Benci, P. D’Avenia, D. Fortunato, and L. Pisani,
*Solitons in several space dimensions: Derrick’s problem and infinitely many solutions*, Arch. Ration. Mech. Anal.**154**(2000), no. 4, 297–324. MR**1785469**, DOI 10.1007/s002050000101 - Rafael Benguria, Haïm Brézis, and Elliott H. Lieb,
*The Thomas-Fermi-von Weizsäcker theory of atoms and molecules*, Comm. Math. Phys.**79**(1981), no. 2, 167–180. MR**612246**, DOI 10.1007/BF01942059 - Haïm Brezis and Louis Nirenberg,
*$H^1$ versus $C^1$ local minimizers*, C. R. Acad. Sci. Paris Sér. I Math.**317**(1993), no. 5, 465–472 (English, with English and French summaries). MR**1239032** - Kung-Ching Chang,
*Methods in nonlinear analysis*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR**2170995** - L. Cherfils and Y. Il′yasov,
*On the stationary solutions of generalized reaction diffusion equations with $p\&q$-Laplacian*, Commun. Pure Appl. Anal.**4**(2005), no. 1, 9–22. MR**2126276**, DOI 10.3934/cpaa.2005.4.9 - Silvia Cingolani and Marco Degiovanni,
*Nontrivial solutions for $p$-Laplace equations with right-hand side having $p$-linear growth at infinity*, Comm. Partial Differential Equations**30**(2005), no. 7-9, 1191–1203. MR**2180299**, DOI 10.1080/03605300500257594 - Silvia Cingolani and Giuseppina Vannella,
*Critical groups computations on a class of Sobolev Banach spaces via Morse index*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**20**(2003), no. 2, 271–292 (English, with English and French summaries). MR**1961517**, DOI 10.1016/S0294-1449(02)00011-2 - Silvia Cingolani and Giuseppina Vannella,
*Marino-Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces*, Ann. Mat. Pura Appl. (4)**186**(2007), no. 1, 157–185. MR**2263895**, DOI 10.1007/s10231-005-0176-2 - M. Cuesta, D. de Figueiredo, and J.-P. Gossez,
*The beginning of the Fučik spectrum for the $p$-Laplacian*, J. Differential Equations**159**(1999), no. 1, 212–238. MR**1726923**, DOI 10.1006/jdeq.1999.3645 - Djairo Guedes de Figueiredo,
*Positive solutions of semilinear elliptic problems*, Differential equations (S ao Paulo, 1981) Lecture Notes in Math., vol. 957, Springer, Berlin-New York, 1982, pp. 34–87. MR**679140** - Nelson Dunford and Jacob T. Schwartz,
*Linear Operators. I. General Theory*, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR**0117523** - Michael Filippakis, Alexandru Kristály, and Nikolaos S. Papageorgiou,
*Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation*, Discrete Contin. Dyn. Syst.**24**(2009), no. 2, 405–440. MR**2486583**, DOI 10.3934/dcds.2009.24.405 - J. P. García Azorero, I. Peral Alonso, and Juan J. Manfredi,
*Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations*, Commun. Contemp. Math.**2**(2000), no. 3, 385–404. MR**1776988**, DOI 10.1142/S0219199700000190 - Seppo Heikkilä and V. Lakshmikantham,
*Monotone iterative techniques for discontinuous nonlinear differential equations*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 181, Marcel Dekker, Inc., New York, 1994. MR**1280028** - A. D. Ioffe and V. M. Tihomirov,
*Theory of extremal problems*, Studies in Mathematics and its Applications, vol. 6, North-Holland Publishing Co., Amsterdam-New York, 1979. Translated from the Russian by Karol Makowski. MR**528295** - Olga A. Ladyzhenskaya and Nina N. Ural’tseva,
*Linear and quasilinear elliptic equations*, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR**0244627** - Gary M. Lieberman,
*Boundary regularity for solutions of degenerate elliptic equations*, Nonlinear Anal.**12**(1988), no. 11, 1203–1219. MR**969499**, DOI 10.1016/0362-546X(88)90053-3 - Shibo Liu,
*Multiple solutions for coercive $p$-Laplacian equations*, J. Math. Anal. Appl.**316**(2006), no. 1, 229–236. MR**2201759**, DOI 10.1016/j.jmaa.2005.04.034 - Jiaquan Liu and Shibo Liu,
*The existence of multiple solutions to quasilinear elliptic equations*, Bull. London Math. Soc.**37**(2005), no. 4, 592–600. MR**2143739**, DOI 10.1112/S0024609304004023 - Richard S. Palais,
*Homotopy theory of infinite dimensional manifolds*, Topology**5**(1966), 1–16. MR**189028**, DOI 10.1016/0040-9383(66)90002-4 - Evgenia H. Papageorgiou and Nikolaos S. Papageorgiou,
*A multiplicity theorem for problems with the $p$-Laplacian*, J. Funct. Anal.**244**(2007), no. 1, 63–77. MR**2294475**, DOI 10.1016/j.jfa.2006.11.015 - Nikolaos S. Papageorgiou and Sophia Th. Kyritsi-Yiallourou,
*Handbook of applied analysis*, Advances in Mechanics and Mathematics, vol. 19, Springer, New York, 2009. MR**2527754**, DOI 10.1007/b120946 - Patrizia Pucci and James Serrin,
*The maximum principle*, Progress in Nonlinear Differential Equations and their Applications, vol. 73, Birkhäuser Verlag, Basel, 2007. MR**2356201**, DOI 10.1007/978-3-7643-8145-5 - J. L. Vázquez,
*A strong maximum principle for some quasilinear elliptic equations*, Appl. Math. Optim.**12**(1984), no. 3, 191–202. MR**768629**, DOI 10.1007/BF01449041

## Additional Information

**Sergiu Aizicovici**- Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701
- Email: aizicovs@ohio.edu
**Nikolaos S. Papageorgiou**- Affiliation: Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece
- MR Author ID: 135890
- Email: npapg@math.ntua.gr
**Vasile Staicu**- Affiliation: Department of Mathematics, CIDMA, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal
- Email: vasile@ua.pt
- Received by editor(s): February 22, 2013
- Received by editor(s) in revised form: November 7, 2013
- Published electronically: October 21, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**367**(2015), 7343-7372 - MSC (2010): Primary 35J20, 35J60, 35J92, 58E05
- DOI: https://doi.org/10.1090/S0002-9947-2014-06324-1
- MathSciNet review: 3378832