Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator

Authors:
D. Motreanu and N. S. Papageorgiou

Journal:
Proc. Amer. Math. Soc. **139** (2011), 3527-3535

MSC (2010):
Primary 35J40; Secondary 35J70

Published electronically:
February 18, 2011

MathSciNet review:
2813384

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a nonlinear Neumann problem driven by a nonhomogeneous quasilinear degenerate elliptic differential operator , a special case of which is the -Laplacian. The reaction term is a Carathéodory function which exhibits subcritical growth in . Using variational methods based on the mountain pass and second deformation theorems, together with truncation and minimization techniques, we show that the problem has three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). A crucial tool in our analysis is a result of independent interest which we prove here and which relates and local minimizers of a -functional constructed with the general differential operator .

**1.**S. Aizicovici, N. S. Papageorgiou, and V. Staicu,*The spectrum and an index formula for the Neumann 𝑝-Laplacian and multiple solutions for problems with a crossing nonlinearity*, Discrete Contin. Dyn. Syst.**25**(2009), no. 2, 431–456. MR**2525184**, 10.3934/dcds.2009.25.431**2.**Gabriele Bonanno and Pasquale Candito,*Three solutions to a Neumann problem for elliptic equations involving the 𝑝-Laplacian*, Arch. Math. (Basel)**80**(2003), no. 4, 424–429. MR**1982841****3.**Haïm Brezis and Louis Nirenberg,*𝐻¹ versus 𝐶¹ local minimizers*, C. R. Acad. Sci. Paris Sér. I Math.**317**(1993), no. 5, 465–472 (English, with English and French summaries). MR**1239032****4.**Siegfried Carl and Kanishka Perera,*Sign-changing and multiple solutions for the 𝑝-Laplacian*, Abstr. Appl. Anal.**7**(2002), no. 12, 613–625. MR**1950611**, 10.1155/S1085337502207010**5.**Ph. Clément, M. García-Huidobro, R. Manásevich, and K. Schmitt,*Mountain pass type solutions for quasilinear elliptic equations*, Calc. Var. Partial Differential Equations**11**(2000), no. 1, 33–62. MR**1777463**, 10.1007/s005260050002**6.**Lucio Damascelli,*Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results*, Ann. Inst. H. Poincaré Anal. Non Linéaire**15**(1998), no. 4, 493–516 (English, with English and French summaries). MR**1632933**, 10.1016/S0294-1449(98)80032-2**7.**F. Faraci,*Multiple solutions for two nonlinear problems involving the -Laplacian*, Nonlinear Anal.**63**(2005), e1017-e1029 (electronic).**8.**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**, 10.1142/S0219199700000190**9.**Marta García-Huidobro, Raúl Manásevich, and Pedro Ubilla,*Existence of positive solutions for some Dirichlet problems with an asymptotically homogeneous operator*, Electron. J. Differential Equations (1995), No. 10, approx. 22 pp.}, issn=1072-6691, review=\MR{1345249},.**10.**Leszek Gasiński and Nikolaos S. Papageorgiou,*Nonlinear analysis*, Series in Mathematical Analysis and Applications, vol. 9, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR**2168068****11.**Gary M. Lieberman,*Boundary regularity for solutions of degenerate elliptic equations*, Nonlinear Anal.**12**(1988), no. 11, 1203–1219. MR**969499**, 10.1016/0362-546X(88)90053-3**12.**Shibo Liu,*Multiple solutions for coercive 𝑝-Laplacian equations*, J. Math. Anal. Appl.**316**(2006), no. 1, 229–236. MR**2201759**, 10.1016/j.jmaa.2005.04.034**13.**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**, 10.1112/S0024609304004023**14.**Marcelo Montenegro,*Strong maximum principles for supersolutions of quasilinear elliptic equations*, Nonlinear Anal.**37**(1999), no. 4, Ser. A: Theory Methods, 431–448. MR**1691019**, 10.1016/S0362-546X(98)00057-1**15.**D. Motreanu, V. V. Motreanu, and N. S. Papageorgiou,*A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations*, Manuscripta Math.**124**(2007), no. 4, 507–531. MR**2357796**, 10.1007/s00229-007-0127-x**16.**D. Motreanu, V. V. Motreanu, and N. S. Papageorgiou,*Multiple nontrivial solutions for nonlinear eigenvalue problems*, Proc. Amer. Math. Soc.**135**(2007), no. 11, 3649–3658. MR**2336581**, 10.1090/S0002-9939-07-08927-7**17.**D. Motreanu, V. V. Motreanu, and N. S. Papageorgiou,*A unified approach for multiple constant sign and nodal solutions*, Adv. Differential Equations**12**(2007), no. 12, 1363–1392. MR**2382729****18.**D. Motreanu, V. V. Motreanu, and N. S. Papageorgiou,*Nonlinear Neumann problems near resonance*, Indiana Univ. Math. J.**58**(2009), no. 3, 1257–1279. MR**2541367**, 10.1512/iumj.2009.58.3565**19.**Dumitru Motreanu and Nikolaos S. Papageorgiou,*Existence and multiplicity of solutions for Neumann problems*, J. Differential Equations**232**(2007), no. 1, 1–35. MR**2281188**, 10.1016/j.jde.2006.09.008**20.**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**, 10.1016/j.jfa.2006.11.015**21.**Xian Wu and Kok-Keong Tan,*On existence and multiplicity of solutions of Neumann boundary value problems for quasi-linear elliptic equations*, Nonlinear Anal.**65**(2006), no. 7, 1334–1347. MR**2245508**, 10.1016/j.na.2005.10.010**22.**Qihu Zhang,*A strong maximum principle for differential equations with nonstandard 𝑝(𝑥)-growth conditions*, J. Math. Anal. Appl.**312**(2005), no. 1, 24–32. MR**2175201**, 10.1016/j.jmaa.2005.03.013**23.**Zhitao Zhang, Jianqing Chen, and Shujie Li,*Construction of pseudo-gradient vector field and sign-changing multiple solutions involving 𝑝-Laplacian*, J. Differential Equations**201**(2004), no. 2, 287–303. MR**2059609**, 10.1016/j.jde.2004.03.019

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
35J40,
35J70

Retrieve articles in all journals with MSC (2010): 35J40, 35J70

Additional Information

**D. Motreanu**

Affiliation:
Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France

Email:
motreanu@univ-perp.fr

**N. S. Papageorgiou**

Affiliation:
Department of Mathematics, National Technical University, Athens 15780, Greece

Email:
npapg@math.ntua.gr

DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-10884-0

Keywords:
Nonlinear Neumann problem,
$p$-Laplacian,
local minimizers,
mountain pass theorem,
second deformation theorem,
nonlinear regularity theory

Received by editor(s):
March 17, 2010

Received by editor(s) in revised form:
July 23, 2010, and August 24, 2010

Published electronically:
February 18, 2011

Communicated by:
Walter Craig

Article copyright:
© Copyright 2011
American Mathematical Society