American Institute of Mathematical Sciences

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July  2011, 10(4): 1055-1078. doi: 10.3934/cpaa.2011.10.1055

Nonlinear Neumann equations driven by a nonhomogeneous differential operator

Shouchuan Hu 1, and  Nikolaos S. Papageorgiou 2,

1. 

College of Mathematics, Shandong Normal University, Jinan, Shandong
2. 

Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received  January 2010 Revised  November 2010 Published  April 2011

We consider a nonlinear Neumann problem driven by a nonhomogeneous nonlinear differential operator and with a reaction which is $(p-1)$-superlinear without necessarily satisfying the Ambrosetti-Rabinowitz condition. A particular case of our differential operator is the $p$-Laplacian. By combining variational methods based on critical point theory with truncation techniques and Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive and the other negative).
Keywords: nonlinear regularity, Morse relation, Moser iteration method., Mountain Pass theorem, critical group, C-condition.
Mathematics Subject Classification: 35J65, 58E0.
Citation: Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann equations driven by a nonhomogeneous differential operator. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1055-1078. doi: 10.3934/cpaa.2011.10.1055
References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, "Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints," Memoirs of AMS, vol. 196, no. 915, 2008.  Google Scholar

[2]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neumann problems, Annali di Mat Pura Appl., 188 (2009), 679-719. doi: doi:10.1007/s10231-009-0096-7.  Google Scholar

[3]

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. Dynamical Systems, 25 (2009), 431-456.  Google Scholar

[4]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlin. Anal., 7 (1983), 981-1012. doi: doi:10.1016/0362-546X(83)90115-3.  Google Scholar

[5]

T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlin. Anal., 28 (1997), 419-441. doi: doi:10.1016/0362-546X(95)00167-T.  Google Scholar

[6]

H. Brezis and Louis Nirenberg, $H^1$-versus $C^1$ local minimizers, CRAS Paris, Ser. J. Math., 317 (1993), 465-472.  Google Scholar

[7]

S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal., 7 (2002), 613-625. doi: doi:10.1155/S1085337502207010.  Google Scholar

[8]

E. Casas and L. Fernandez, A Green's formula for quasilinear elliptic operators, J. Math. Anal. Appl., 142 (1989), 62-73. doi: doi:10.1016/0022-247X(89)90164-9.  Google Scholar

[9]

K. C. Chang, "Infinite Dimensional Morse theory and Multiple Solution Problems," Birkhauser, Boston, 1993.  Google Scholar

[10]

J. A. Cid and P. J. Torres, Solvability for some boundary value problems with $p$-Laplacian operators, Discrete Contin. Dynamical Systems, 23 (2009), 727-732.  Google Scholar

[11]

D. Costa and C. Magalhaes, Existence results for perturbations of the $p$-Laplacian, Nonlinear Anal., 24 (1995), 409-418. doi: doi:10.1016/0362-546X(94)E0046-J.  Google Scholar

[12]

L. Damascellli, Comparision theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincare Analyse Non linenire, 15 (1998), 493-516.  Google Scholar

[13]

N. Dancer and K. Perera, Some remarks on the Fučik spectrum of the $p$-Laplacian and critical groups, J. Math. Anal. Appl., 254 (2001), 164-177. doi: doi:10.1006/jmaa.2000.7228.  Google Scholar

[14]

P. De Mapoli and C. Mariani, Mountain pass solutions to equations of $p$-Laplacian type, Nonlinear Anal., 54 (2003), 1205-1219. doi: doi:10.1016/S0362-546X(03)00105-6.  Google Scholar

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F. de Paiva and H. R. Quoirin, Resonance and nonresonance for $p$-Laplacian problems with weighted eigenvalues conditions, Discrete Contin. Dynamical Systems, 25 (2009), 1219-1227.  Google Scholar

[16]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electronic J. Diff. Equas., 8 (2002), 1-12.  Google Scholar

[17]

M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nontrivial solutions with precise sign data for a $p$-Laplacian equation, Discrete Contin. Dynamical Systems, 25 (2009), 405-440.  Google Scholar

[18]

J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quarsilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404.  Google Scholar

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L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Elliptic Boundary Value Problems," Chapman & Hall / CRC Press, Baca Raton, 2005.  Google Scholar

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L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar

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L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian type equations, Adv. Nonlin. Studies, 8 (2008), 843-870.  Google Scholar

[22]

A. Granas and J. Dugundji, "Fixed Point Theory," Springer, New York, 2003.  Google Scholar

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Z. Guo and Z. Zhang, $W^{1,p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Anal., 286 (2003), 32-50. doi: doi:10.1016/S0022-247X(03)00282-8.  Google Scholar

[24]

A. Kristaly, H. Lisei and C. Varga, Multiple solutions for $p$-Laplacian type equations, Nonlinear Anal., 68 (2008), 1375-1381. doi: doi:10.1016/j.na.2006.12.031.  Google Scholar

[25]

O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Acad. Press., New York, 1968.  Google Scholar

[26]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: doi:10.1016/0362-546X(88)90053-3.  Google Scholar

[27]

S. Liu, Multiple solutions for coercive $p$-Laplacian equations, J. Math. Anal. Appl., 316 (2006), 229-236. doi: doi:10.1016/j.jmaa.2005.04.034.  Google Scholar

[28]

J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37 (2005), 592-600. doi: doi:10.1112/S0024609304004023.  Google Scholar

[29]

M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations, Nonlinear Anal., 37 (1991), 431-448. doi: doi:10.1016/S0362-546X(98)00057-1.  Google Scholar

[30]

D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems, J. Differential Eqns., 232 (2007), 1-35. doi: doi:10.1016/j.jde.2006.09.008.  Google Scholar

[31]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance, Indiana Univ. Math. J., 58 (2009), 1257-1279. doi: doi:10.1512/iumj.2009.58.3565.  Google Scholar

[32]

N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis," Springer, New York, 2009.  Google Scholar

[33]

E. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian, J. Funct. Anal., 244 (2007), 63-77. doi: doi:10.1016/j.jfa.2006.11.015.  Google Scholar

[34]

E. Papageorgiou and N. S. Papageorgiou, Multiplicity of solutions for a class of resonant $p$-Laplacian Dirichlet problems, Pacific J. Math., 241 (2009), 309-328. doi: doi:10.2140/pjm.2009.241.309.  Google Scholar

[35]

N. S. Papageorgiou, E. M. Rocha and V. Staicu, A multiplicity theorem for hemivariational inequalities with a $p$-Laplacian-like differential operator, Nonlin. Anal., 69 (2008), 1150-1163. doi: doi:10.1016/j.na.2007.06.023.  Google Scholar

[36]

N. S. Trudinger and XuJia Wang, Quasilinear elliptic equations with signed measure, Discrete Contin. Dynamical Systems, 23 (2009), 477-494.  Google Scholar

[37]

Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincare Analyse Non Lineaire, 8 (1991), 43-58.  Google Scholar

[38]

M. Willem, "Minimax Theorems," Birkhauser, Boston, 1996.  Google Scholar

[39]

Q. Zhang, A strong maximum principle for differential equations with nonstandard $p(x)$-growth condition, J. Math. Anal. Appl., 312 (2005), 24-32. doi: doi:10.1016/j.jmaa.2005.03.013.  Google Scholar

[40]

Z. Zhang, J. Q. Chen and S. Li, Construction of pseudo-gradient vector field and sign-changing multiple solutions involving $p$-Laplacian, J. Differential Eqns., 201 (2004), 287-303. doi: doi:10.1016/j.jde.2004.03.019.  Google Scholar

show all references

References:
[1]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, "Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints," Memoirs of AMS, vol. 196, no. 915, 2008.  Google Scholar

[2]

S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neumann problems, Annali di Mat Pura Appl., 188 (2009), 679-719. doi: doi:10.1007/s10231-009-0096-7.  Google Scholar

[3]

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. Dynamical Systems, 25 (2009), 431-456.  Google Scholar

[4]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlin. Anal., 7 (1983), 981-1012. doi: doi:10.1016/0362-546X(83)90115-3.  Google Scholar

[5]

T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlin. Anal., 28 (1997), 419-441. doi: doi:10.1016/0362-546X(95)00167-T.  Google Scholar

[6]

H. Brezis and Louis Nirenberg, $H^1$-versus $C^1$ local minimizers, CRAS Paris, Ser. J. Math., 317 (1993), 465-472.  Google Scholar

[7]

S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal., 7 (2002), 613-625. doi: doi:10.1155/S1085337502207010.  Google Scholar

[8]

E. Casas and L. Fernandez, A Green's formula for quasilinear elliptic operators, J. Math. Anal. Appl., 142 (1989), 62-73. doi: doi:10.1016/0022-247X(89)90164-9.  Google Scholar

[9]

K. C. Chang, "Infinite Dimensional Morse theory and Multiple Solution Problems," Birkhauser, Boston, 1993.  Google Scholar

[10]

J. A. Cid and P. J. Torres, Solvability for some boundary value problems with $p$-Laplacian operators, Discrete Contin. Dynamical Systems, 23 (2009), 727-732.  Google Scholar

[11]

D. Costa and C. Magalhaes, Existence results for perturbations of the $p$-Laplacian, Nonlinear Anal., 24 (1995), 409-418. doi: doi:10.1016/0362-546X(94)E0046-J.  Google Scholar

[12]

L. Damascellli, Comparision theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincare Analyse Non linenire, 15 (1998), 493-516.  Google Scholar

[13]

N. Dancer and K. Perera, Some remarks on the Fučik spectrum of the $p$-Laplacian and critical groups, J. Math. Anal. Appl., 254 (2001), 164-177. doi: doi:10.1006/jmaa.2000.7228.  Google Scholar

[14]

P. De Mapoli and C. Mariani, Mountain pass solutions to equations of $p$-Laplacian type, Nonlinear Anal., 54 (2003), 1205-1219. doi: doi:10.1016/S0362-546X(03)00105-6.  Google Scholar

[15]

F. de Paiva and H. R. Quoirin, Resonance and nonresonance for $p$-Laplacian problems with weighted eigenvalues conditions, Discrete Contin. Dynamical Systems, 25 (2009), 1219-1227.  Google Scholar

[16]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electronic J. Diff. Equas., 8 (2002), 1-12.  Google Scholar

[17]

M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nontrivial solutions with precise sign data for a $p$-Laplacian equation, Discrete Contin. Dynamical Systems, 25 (2009), 405-440.  Google Scholar

[18]

J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quarsilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404.  Google Scholar

[19]

L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Elliptic Boundary Value Problems," Chapman & Hall / CRC Press, Baca Raton, 2005.  Google Scholar

[20]

L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar

[21]

L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian type equations, Adv. Nonlin. Studies, 8 (2008), 843-870.  Google Scholar

[22]

A. Granas and J. Dugundji, "Fixed Point Theory," Springer, New York, 2003.  Google Scholar

[23]

Z. Guo and Z. Zhang, $W^{1,p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Anal., 286 (2003), 32-50. doi: doi:10.1016/S0022-247X(03)00282-8.  Google Scholar

[24]

A. Kristaly, H. Lisei and C. Varga, Multiple solutions for $p$-Laplacian type equations, Nonlinear Anal., 68 (2008), 1375-1381. doi: doi:10.1016/j.na.2006.12.031.  Google Scholar

[25]

O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Acad. Press., New York, 1968.  Google Scholar

[26]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219. doi: doi:10.1016/0362-546X(88)90053-3.  Google Scholar

[27]

S. Liu, Multiple solutions for coercive $p$-Laplacian equations, J. Math. Anal. Appl., 316 (2006), 229-236. doi: doi:10.1016/j.jmaa.2005.04.034.  Google Scholar

[28]

J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37 (2005), 592-600. doi: doi:10.1112/S0024609304004023.  Google Scholar

[29]

M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations, Nonlinear Anal., 37 (1991), 431-448. doi: doi:10.1016/S0362-546X(98)00057-1.  Google Scholar

[30]

D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems, J. Differential Eqns., 232 (2007), 1-35. doi: doi:10.1016/j.jde.2006.09.008.  Google Scholar

[31]

D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance, Indiana Univ. Math. J., 58 (2009), 1257-1279. doi: doi:10.1512/iumj.2009.58.3565.  Google Scholar

[32]

N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis," Springer, New York, 2009.  Google Scholar

[33]

E. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian, J. Funct. Anal., 244 (2007), 63-77. doi: doi:10.1016/j.jfa.2006.11.015.  Google Scholar

[34]

E. Papageorgiou and N. S. Papageorgiou, Multiplicity of solutions for a class of resonant $p$-Laplacian Dirichlet problems, Pacific J. Math., 241 (2009), 309-328. doi: doi:10.2140/pjm.2009.241.309.  Google Scholar

[35]

N. S. Papageorgiou, E. M. Rocha and V. Staicu, A multiplicity theorem for hemivariational inequalities with a $p$-Laplacian-like differential operator, Nonlin. Anal., 69 (2008), 1150-1163. doi: doi:10.1016/j.na.2007.06.023.  Google Scholar

[36]

N. S. Trudinger and XuJia Wang, Quasilinear elliptic equations with signed measure, Discrete Contin. Dynamical Systems, 23 (2009), 477-494.  Google Scholar

[37]

Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincare Analyse Non Lineaire, 8 (1991), 43-58.  Google Scholar

[38]

M. Willem, "Minimax Theorems," Birkhauser, Boston, 1996.  Google Scholar

[39]

Q. Zhang, A strong maximum principle for differential equations with nonstandard $p(x)$-growth condition, J. Math. Anal. Appl., 312 (2005), 24-32. doi: doi:10.1016/j.jmaa.2005.03.013.  Google Scholar

[40]

Z. Zhang, J. Q. Chen and S. Li, Construction of pseudo-gradient vector field and sign-changing multiple solutions involving $p$-Laplacian, J. Differential Eqns., 201 (2004), 287-303. doi: doi:10.1016/j.jde.2004.03.019.  Google Scholar

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