# American Institute of Mathematical Sciences

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

## Nonlinear Neumann equations driven by a nonhomogeneous differential operator

 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).
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, (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. 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. 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. 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. 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, 317 (1993), 465. Google Scholar [7] S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian,, Abstr. Appl. Anal., 7 (2002), 613. 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. doi: doi:10.1016/0022-247X(89)90164-9. Google Scholar [9] K. C. Chang, "Infinite Dimensional Morse theory and Multiple Solution Problems,", Birkhauser, (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. Google Scholar [11] D. Costa and C. Magalhaes, Existence results for perturbations of the $p$-Laplacian,, Nonlinear Anal., 24 (1995), 409. 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. 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. 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. 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. Google Scholar [16] G. Fei, On periodic solutions of superquadratic Hamiltonian systems,, Electronic J. Diff. Equas., 8 (2002), 1. 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. 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. Google Scholar [19] L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Elliptic Boundary Value Problems,", Chapman & Hall / CRC Press, (2005). Google Scholar [20] L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis,", Chapman & Hall/CRC, (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. Google Scholar [22] A. Granas and J. Dugundji, "Fixed Point Theory,", Springer, (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. 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. doi: doi:10.1016/j.na.2006.12.031. Google Scholar [25] O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations,", Acad. Press., (1968). Google Scholar [26] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203. 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. 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. doi: doi:10.1112/S0024609304004023. Google Scholar [29] M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations,, Nonlinear Anal., 37 (1991), 431. 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. 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. doi: doi:10.1512/iumj.2009.58.3565. Google Scholar [32] N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis,", Springer, (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. 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. 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. 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. Google Scholar [37] Z. Q. Wang, On a superlinear elliptic equation,, Ann. Inst. H. Poincare Analyse Non Lineaire, 8 (1991), 43. Google Scholar [38] M. Willem, "Minimax Theorems,", Birkhauser, (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. 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. 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, (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. 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. 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. 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. 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, 317 (1993), 465. Google Scholar [7] S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian,, Abstr. Appl. Anal., 7 (2002), 613. 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. doi: doi:10.1016/0022-247X(89)90164-9. Google Scholar [9] K. C. Chang, "Infinite Dimensional Morse theory and Multiple Solution Problems,", Birkhauser, (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. Google Scholar [11] D. Costa and C. Magalhaes, Existence results for perturbations of the $p$-Laplacian,, Nonlinear Anal., 24 (1995), 409. 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. 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. 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. 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. Google Scholar [16] G. Fei, On periodic solutions of superquadratic Hamiltonian systems,, Electronic J. Diff. Equas., 8 (2002), 1. 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. 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. Google Scholar [19] L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Elliptic Boundary Value Problems,", Chapman & Hall / CRC Press, (2005). Google Scholar [20] L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis,", Chapman & Hall/CRC, (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. Google Scholar [22] A. Granas and J. Dugundji, "Fixed Point Theory,", Springer, (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. 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. doi: doi:10.1016/j.na.2006.12.031. Google Scholar [25] O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations,", Acad. Press., (1968). Google Scholar [26] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations,, Nonlinear Anal., 12 (1988), 1203. 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. 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. doi: doi:10.1112/S0024609304004023. Google Scholar [29] M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations,, Nonlinear Anal., 37 (1991), 431. 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. 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. doi: doi:10.1512/iumj.2009.58.3565. Google Scholar [32] N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis,", Springer, (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. 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. 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. 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. Google Scholar [37] Z. Q. Wang, On a superlinear elliptic equation,, Ann. Inst. H. Poincare Analyse Non Lineaire, 8 (1991), 43. Google Scholar [38] M. Willem, "Minimax Theorems,", Birkhauser, (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. 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. doi: doi:10.1016/j.jde.2004.03.019. Google Scholar
 [1] Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912 [2] Claudianor O. Alves, Giovany M. Figueiredo, Marcelo F. Furtado. Multiplicity of solutions for elliptic systems via local Mountain Pass method. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1745-1758. doi: 10.3934/cpaa.2009.8.1745 [3] Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003 [4] V. Niţicâ. Journé's theorem for $C^{n,\omega}$ regularity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 413-425. doi: 10.3934/dcds.2008.22.413 [5] Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345 [6] Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345 [7] Viktor L. Ginzburg and Basak Z. Gurel. The Generalized Weinstein--Moser Theorem. Electronic Research Announcements, 2007, 14: 20-29. doi: 10.3934/era.2007.14.20 [8] Yuri Berest, Alimjon Eshmatov, Farkhod Eshmatov. On subgroups of the Dixmier group and Calogero-Moser spaces. Electronic Research Announcements, 2011, 18: 12-21. doi: 10.3934/era.2011.18.12 [9] Florian Wagener. A parametrised version of Moser's modifying terms theorem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 719-768. doi: 10.3934/dcdss.2010.3.719 [10] Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637 [11] Björn Sandstede, Arnd Scheel. Relative Morse indices, Fredholm indices, and group velocities. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 139-158. doi: 10.3934/dcds.2008.20.139 [12] Zhenjie Li, Ze Cheng, Dongsheng Li. The Liouville type theorem and local regularity results for nonlinear differential and integral systems. Communications on Pure & Applied Analysis, 2015, 14 (2) : 565-576. doi: 10.3934/cpaa.2015.14.565 [13] Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378 [14] Pedro Teixeira. Dacorogna-Moser theorem on the Jacobian determinant equation with control of support. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4071-4089. doi: 10.3934/dcds.2017173 [15] Yanmin Niu, Xiong Li. An application of Moser's twist theorem to superlinear impulsive differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 431-445. doi: 10.3934/dcds.2019017 [16] Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455 [17] Thierry Champion, Luigi De Pascale. On the twist condition and $c$-monotone transport plans. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1339-1353. doi: 10.3934/dcds.2014.34.1339 [18] Boris Hasselblatt. Critical regularity of invariant foliations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 931-937. doi: 10.3934/dcds.2002.8.931 [19] Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831 [20] Zhong-Zhi Bai. On convergence of the inner-outer iteration method for computing PageRank. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 855-862. doi: 10.3934/naco.2012.2.855

2018 Impact Factor: 0.925