Nonlinear Neumann equations driven by a nonhomogeneous differential operator

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[6]	H. Brezis and Louis Nirenberg, $H^1$-versus $C^1$ local minimizers, CRAS Paris, Ser. J. Math., 317 (1993), 465-472.
[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.
[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.
[9]	K. C. Chang, "Infinite Dimensional Morse theory and Multiple Solution Problems," Birkhauser, Boston, 1993.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electronic J. Diff. Equas., 8 (2002), 1-12.
[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.
[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.
[19]	L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Elliptic Boundary Value Problems," Chapman & Hall / CRC Press, Baca Raton, 2005.
[20]	L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006.
[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.
[22]	A. Granas and J. Dugundji, "Fixed Point Theory," Springer, New York, 2003.
[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.
[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.
[25]	O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Acad. Press., New York, 1968.
[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.
[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.
[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.
[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.
[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.
[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.
[32]	N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis," Springer, New York, 2009.
[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.
[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.
[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.
[36]	N. S. Trudinger and XuJia Wang, Quasilinear elliptic equations with signed measure, Discrete Contin. Dynamical Systems, 23 (2009), 477-494.
[37]	Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincare Analyse Non Lineaire, 8 (1991), 43-58.
[38]	M. Willem, "Minimax Theorems," Birkhauser, Boston, 1996.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[6]	H. Brezis and Louis Nirenberg, $H^1$-versus $C^1$ local minimizers, CRAS Paris, Ser. J. Math., 317 (1993), 465-472.
[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.
[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.
[9]	K. C. Chang, "Infinite Dimensional Morse theory and Multiple Solution Problems," Birkhauser, Boston, 1993.
[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.
[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.
[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.
[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.
[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.
[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.
[16]	G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electronic J. Diff. Equas., 8 (2002), 1-12.
[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.
[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.
[19]	L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Elliptic Boundary Value Problems," Chapman & Hall / CRC Press, Baca Raton, 2005.
[20]	L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006.
[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.
[22]	A. Granas and J. Dugundji, "Fixed Point Theory," Springer, New York, 2003.
[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.
[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.
[25]	O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Acad. Press., New York, 1968.
[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.
[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.
[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.
[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.
[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.
[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.
[32]	N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis," Springer, New York, 2009.
[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.
[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.
[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.
[36]	N. S. Trudinger and XuJia Wang, Quasilinear elliptic equations with signed measure, Discrete Contin. Dynamical Systems, 23 (2009), 477-494.
[37]	Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincare Analyse Non Lineaire, 8 (1991), 43-58.
[38]	M. Willem, "Minimax Theorems," Birkhauser, Boston, 1996.
[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.
[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.

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