Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems

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November 2013, 12(6): 2889-2922. doi: 10.3934/cpaa.2013.12.2889

Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems

Shouchuan Hu ^1, and Nikolaos S. Papageorgiou ^2,

1.	American Institute of Mathematical Sciences, P.O. Box 2604, Springfield, MO 65801
2.	Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received June 2012 Revised February 2013 Published May 2013

Abstract
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We consider nonlinear Neumann and Dirichlet problems driven by a nonhomogeneous differential operator and a Caratheodory reaction. Our framework incorporates $p$-Laplacian equations and equations with the $(p,q)$-differential operator and with the generalized $p$-mean curvature operator. Using variational methods, together with truncation and comparison techniques and Morse theory, we prove multiplicity theorems, producing three, five or six nontrivial smooth solutions, all with sign information.

Keywords: maximum principle, superlinear reaction., Strong comparison principle, constant sign solutions, Morse theory, nodal solutions.

Mathematics Subject Classification: 35J20, 35J60, 35J92, 58E0.

Citation: Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2889-2922. doi: 10.3934/cpaa.2013.12.2889

References:

[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 Cont Dyn Systems, 25 (2009), 431-456.
[2]	S. Aizicovici, N. S. Papageorgiou and V. Staicu, On a $p$-superlinear Neumann $p$-Laplacian equation, Topol. Methods Nonlin. Anal., 34 (2009), 111-130.
[3]	A. Allendes and A. Quaas, Multiplicity results for extremal operators through bifurcation, Discrete Cont Dyn Systems, 29 (2011), 51-65.
[4]	D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplace operator, Comm. Part. Diff. Equs., 31 (2006), 849-863. doi: 10.1080/03605300500394447.
[5]	V. Benci, P. D'Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Rational Mech. Anal., 154 (2000), 297-324. doi: 10.1007/s002050000101.
[6]	R. Benguria, H. Brezis and E. H. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Physics, 79 (1981), 167-180. doi: 10.1007/BF01942059.
[7]	H. Brezis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, CRAS Paris t., 317 (1993), 465-472.
[8]	S. Cano-Casanova, Coercivity of elliptic mixed boundary value problems in annulus of $\mathbbR^N$, Discrete and Continuous Dynamical Systems, 32 (2012), 3819-3839. doi: 10.3934/dcds.2012.32.3819.
[9]	K. C. Chang, "Methods of Nonilnear Analysis," Springer, Berlin (2005).
[10]	S. Cingolani and M. Degiovanni, Nontrivial solutions for $p$-Laplacian equations with right hand side having $p$-linear growth, Comm. Part. Diff. Equas., 30 (2005), 1191-1203. doi: 10.1080/03605300500257594.
[11]	S. Cingolani and G. Vannella, Critical groups computations on a class of Sobolev Banach spaces via Morse index, ann. Inst. H. Poincare-AN, 20 (2003), 271-292.
[12]	S. Cingolani and G. Vannella, Marino-Prodi perturbation type results and Morse indices of minimaz critical points for a class of functionals in Banach spaces, Annali di Mat. Pura Appl., 186 (2007), 155-183. doi: 10.1007/s10231-005-0176-2.
[13]	P. Clement, M.Garcia Huidobro, R. Manasevich and K. Schmitt, Mountain pass solutions for quasilinear elliptic equations, Calc. Var., 11 (2000), 33-67. doi: 10.1007/s005260050002.
[14]	D. Costa and C. Magalhaes, Existence results for perturbations of the $p$-Laplacian, Nonlinear Anal., 24 (1995), 409-418. doi: 10.1016/0362-546X(94)E0046-J.
[15]	M. Cuesta, D. deFigueiredo and J. P. Gossez, The beginning of the Fu$\brevec$ik spectrum for the $p$-Laplacian, J. Diff. Equas., 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645.
[16]	M. Cuesta and P. Takac, A strong comparison principle for positive solutions of degenerate elliptic equations, Diff. Integ. Equas., 13 (2000), 721-746.
[17]	N. Dancer, On domain perturbation for super-linear Neumann problems and a question of Y. Lou, W-M Ni and L. Su, Discrete Cont Dyn Systems, 32 (2012), 3861-3869.
[18]	P. De Napoli and M. C. Mariani, Mountain pass solutions to equations of $p$-Laplacian type, Nonlinear Anal., (2003), 1205-1219. doi: 10.1016/S0362-546X(03)00105-6.
[19]	M. Degiovanni and M. Scaglia, A variational approach to semilinear elliptic equations with measure data, Discrete and Continuous Dynamical Systems, 31 (2011), 1233-1248. doi: 10.3934/dcds.2011.31.1233.
[20]	Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic equatio involving critical exponent, Discrete Contin. Dynam. Systems, 32 (2012), 795-826.
[21]	C. H. Derrick, Comments on nonlinear wave equations as model elementary particles, J. Math. Phys., 5 (1964), 1252-1254. doi: 10.1063/1.1704233.
[22]	J. I. Diaz and J. E. Saa, Existence et unicité de solutions positives pour certaines equations elliptiques quasilineaires, CRAS, Paris t., 305 (1987), 521-524.
[23]	N. Dunford and J.Schwartz, "Linear Operators I,", Wiley-Interscience, ().
[24]	G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Elec. J. Diff. Equas., 8 (2002), 1-12.
[25]	D. deFigueiredo, Positive solutions of semilinear elliptic problems, in "Lecture Notes Math.," vol. 957, Springer, New York, (1982), 34-85.
[26]	M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation, Discrete Cont Dyn Systems, 24 (2009), 405-440. doi: 10.3934/dcds.2009.24.405.
[27]	J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and globla multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404. doi: 10.1142/S0219199700000190.
[28]	L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006.
[29]	L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian type equations, Adv. Nonlin. Studies, 8 (2008), 843-870.
[30]	L. Gasinski and N. S. Papageorgiou, Multiple solutions for nonlinear coersive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set Valued Var. Anal., 20 (2012), 417-443. doi: 10.1007/s11228-011-0198-4.
[31]	D. Gilberg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.
[32]	M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlin. Anal., 13 (1989), 879-902. doi: 10.1016/0362-546X(89)90020-5.
[33]	Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities, Discrete Cont Dyn Systems, 32 (2012), 3567-3585. doi: 10.3934/dcds.2012.32.3567.
[34]	Z. Guo, Z. Liu, J. Wei and F. Zhou, Bifurcations of some elliptic problems with a singular nonlinearity via Morse index, Comm. Pure. Appl. Anal., 10 (2011), 507-525. doi: 10.3934/cpaa.2011.10.507.
[35]	Shouchuan Hu and N. S. Papageorgiou, Positive solutions for nonlinear hemivariational inequalities, J. Math. Anal. Appl., 310 (2005), 161-176. doi: 10.1016/j.jmaa.2005.01.051.
[36]	Shouchuan Hu and N. S. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Comm. Pure Appl. Anal., 9 (2010), 1801-1827.
[37]	Shouchuan Hu and N. S. Papageorgiou, Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities, Comm. Pure Applied Anal., 11 (2012), 2005-2021. doi: 10.3934/cpaa.2012.11.2005.
[38]	J. Garcia Melian, J. Rossi and J. Sabina de Lis, A convex-concave problem with a parameter on the boundary condition, Discrete Contin. Dynam. Systems, 32 (2012), 1095-1124.
[39]	L. Jeanjean, On the existence of bounded Palais-Smale sequences and applications to Landesmann-Lazer type problems in $\rn$, Proc. Royal Soc. Edinburgh, 129A (1999), 767-809.
[40]	A. Kristaly, M. Mihaileseu and V. Radulescu, Two nontrivial solutions for a nonhomogeneous Neumann problem: an Orlicz-Sobolev space setting, Proc. Royal. Soc. Edinburgh, 139A (2009), 367-379. doi: 10.1017/S030821050700025X.
[41]	S. Kyritsi, D. O'Regan and N. S. Papageorgiou, Multiple solutions for resonant hemivariational inequalities via minimax methods, Adv. Nonlin. Studies, 9 (2009), 453-478.
[42]	S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with asymmetric reaction term, Discrete Cont Dyn Systems, 33 (2013), 2469-2494.
[43]	O. A. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York, 1968.
[44]	G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Part. Diff. Equas., 16 (1991), 311-361. doi: 10.1080/03605309108820761.
[45]	S. Li, S. Wu and H. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities, J. Diff. Equas., 185 (2002), 200-224. doi: 10.1006/jdeq.2001.4167.
[46]	M. Mihailescu, Existence and multiplicity of weak solutions for a class of degenerate nonlinear ellipitc equations, Boundary Value Problems, (2006), article ID 41295, 1-17.
[47]	M. Mihailescu and V. Radulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl., 330 (2007), 416-432. doi: 10.1016/j.jmaa.2006.07.082.
[48]	M. L. Miotto, Multiple solutions for elliptic problems in $R^N$ with critical Spbolev exponent and weight function, Comm. Pure Appl. Anal., 9 (2010), 233-248. doi: 10.3934/cpaa.2010.9.233.
[49]	S. Miyajima, D. Motreanu and M. Tanaka, Multiple existence results of solutions for Neumann problems via super- and sub-solutions, J. Functional Anal., 262 (2012), 1921-1953. doi: 10.1016/j.jfa.2011.11.028.
[50]	D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, On $p$-Laplace equations with concave terms and asymmetric nonlinearities, Proc. Royal Soc. Edinburgh, 141A (2011), 171-192. doi: 10.1017/S0308210509001656.
[51]	D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Annali Scuola Normale Sup. Pisa, X (2011), 729-756.
[52]	D. Motreanu, D. O'Regan and N. S. Papageorgiou, A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems, Comm. Pure Appl. Anal., 10 (2011), 1791-1816. doi: 10.3934/cpaa.2011.10.1791.
[53]	D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems, J. Diff. Equas., 232 (2007), 1-35. doi: 10.1016/j.jde.2006.09.008.
[54]	D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator, Proc. Amer. Math. Soc., 139 (2011), 3527-3535. doi: 10.1090/S0002-9939-2011-10884-0.
[55]	N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis," Springer, New York, 2009.
[56]	N. S. Papageorgiou and E. M. Rocha, On nonlinear parametric problems for $p$-Laplacian like operators, RACSAM, 103 (2009), 177-200. doi: 10.1007/BF03191850.
[57]	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: 10.1016/j.na.2007.06.023.
[58]	P. Pucci and J. Serrin, The strong maximum principle revisited, J. Diff. Equas., 196 (2004), 1-68; Erratum, J. Diff. Equas., 207 (2004), 226-227.
[59]	P. Pucci and J. Serrin, "The Maximum Principle," Birkhauser, Basel, 2007. doi: 10.1016/j.jde.2004.09.002.
[60]	J. M. Rakotoson, Generalized eigenvalue problem for totally discontinuous operator, Discrete Contin. Dynam. Systems, 28 (2010), 343-373. doi: 10.3934/dcds.2010.28.343.
[61]	P. Roselli and B. Sciunzi, A strong comparison principle for the $p$-Laplacian, Proc. Amer. Math. Soc., 135 (2007), 3217-3224. doi: 10.1090/S0002-9939-07-08847-8.
[62]	M. Sun, Multiplicity of solutions for a class of quasilinear elliptic equation at resonance, J. Math. Anal. Appl., 386 (2012), 661-668. doi: 10.1016/j.jmaa.2011.08.030.
[63]	J. Vazquez, A strong maximum principle for some quailinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.
[64]	R. Zhang, J. Chen and F. Zhan, Multiple solutions for superlinear elliptic systems of Hamiltonian type, Discrete Cont Dyn Systems, 30 (2011), 1249-1262. doi: 10.3934/dcds.2011.30.1249.

show all references

References:

[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 Cont Dyn Systems, 25 (2009), 431-456.
[2]	S. Aizicovici, N. S. Papageorgiou and V. Staicu, On a $p$-superlinear Neumann $p$-Laplacian equation, Topol. Methods Nonlin. Anal., 34 (2009), 111-130.
[3]	A. Allendes and A. Quaas, Multiplicity results for extremal operators through bifurcation, Discrete Cont Dyn Systems, 29 (2011), 51-65.
[4]	D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplace operator, Comm. Part. Diff. Equs., 31 (2006), 849-863. doi: 10.1080/03605300500394447.
[5]	V. Benci, P. D'Avenia, D. Fortunato and L. Pisani, Solitons in several space dimensions: Derrick's problem and infinitely many solutions, Arch. Rational Mech. Anal., 154 (2000), 297-324. doi: 10.1007/s002050000101.
[6]	R. Benguria, H. Brezis and E. H. Lieb, The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Physics, 79 (1981), 167-180. doi: 10.1007/BF01942059.
[7]	H. Brezis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, CRAS Paris t., 317 (1993), 465-472.
[8]	S. Cano-Casanova, Coercivity of elliptic mixed boundary value problems in annulus of $\mathbbR^N$, Discrete and Continuous Dynamical Systems, 32 (2012), 3819-3839. doi: 10.3934/dcds.2012.32.3819.
[9]	K. C. Chang, "Methods of Nonilnear Analysis," Springer, Berlin (2005).
[10]	S. Cingolani and M. Degiovanni, Nontrivial solutions for $p$-Laplacian equations with right hand side having $p$-linear growth, Comm. Part. Diff. Equas., 30 (2005), 1191-1203. doi: 10.1080/03605300500257594.
[11]	S. Cingolani and G. Vannella, Critical groups computations on a class of Sobolev Banach spaces via Morse index, ann. Inst. H. Poincare-AN, 20 (2003), 271-292.
[12]	S. Cingolani and G. Vannella, Marino-Prodi perturbation type results and Morse indices of minimaz critical points for a class of functionals in Banach spaces, Annali di Mat. Pura Appl., 186 (2007), 155-183. doi: 10.1007/s10231-005-0176-2.
[13]	P. Clement, M.Garcia Huidobro, R. Manasevich and K. Schmitt, Mountain pass solutions for quasilinear elliptic equations, Calc. Var., 11 (2000), 33-67. doi: 10.1007/s005260050002.
[14]	D. Costa and C. Magalhaes, Existence results for perturbations of the $p$-Laplacian, Nonlinear Anal., 24 (1995), 409-418. doi: 10.1016/0362-546X(94)E0046-J.
[15]	M. Cuesta, D. deFigueiredo and J. P. Gossez, The beginning of the Fu$\brevec$ik spectrum for the $p$-Laplacian, J. Diff. Equas., 159 (1999), 212-238. doi: 10.1006/jdeq.1999.3645.
[16]	M. Cuesta and P. Takac, A strong comparison principle for positive solutions of degenerate elliptic equations, Diff. Integ. Equas., 13 (2000), 721-746.
[17]	N. Dancer, On domain perturbation for super-linear Neumann problems and a question of Y. Lou, W-M Ni and L. Su, Discrete Cont Dyn Systems, 32 (2012), 3861-3869.
[18]	P. De Napoli and M. C. Mariani, Mountain pass solutions to equations of $p$-Laplacian type, Nonlinear Anal., (2003), 1205-1219. doi: 10.1016/S0362-546X(03)00105-6.
[19]	M. Degiovanni and M. Scaglia, A variational approach to semilinear elliptic equations with measure data, Discrete and Continuous Dynamical Systems, 31 (2011), 1233-1248. doi: 10.3934/dcds.2011.31.1233.
[20]	Y. Deng, S. Peng and L. Wang, Existence of multiple solutions for a nonhomogeneous semilinear elliptic equatio involving critical exponent, Discrete Contin. Dynam. Systems, 32 (2012), 795-826.
[21]	C. H. Derrick, Comments on nonlinear wave equations as model elementary particles, J. Math. Phys., 5 (1964), 1252-1254. doi: 10.1063/1.1704233.
[22]	J. I. Diaz and J. E. Saa, Existence et unicité de solutions positives pour certaines equations elliptiques quasilineaires, CRAS, Paris t., 305 (1987), 521-524.
[23]	N. Dunford and J.Schwartz, "Linear Operators I,", Wiley-Interscience, ().
[24]	G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Elec. J. Diff. Equas., 8 (2002), 1-12.
[25]	D. deFigueiredo, Positive solutions of semilinear elliptic problems, in "Lecture Notes Math.," vol. 957, Springer, New York, (1982), 34-85.
[26]	M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation, Discrete Cont Dyn Systems, 24 (2009), 405-440. doi: 10.3934/dcds.2009.24.405.
[27]	J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and globla multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404. doi: 10.1142/S0219199700000190.
[28]	L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006.
[29]	L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian type equations, Adv. Nonlin. Studies, 8 (2008), 843-870.
[30]	L. Gasinski and N. S. Papageorgiou, Multiple solutions for nonlinear coersive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set Valued Var. Anal., 20 (2012), 417-443. doi: 10.1007/s11228-011-0198-4.
[31]	D. Gilberg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.
[32]	M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlin. Anal., 13 (1989), 879-902. doi: 10.1016/0362-546X(89)90020-5.
[33]	Z. Guo and Z. Liu, Perturbed elliptic equations with oscillatory nonlinearities, Discrete Cont Dyn Systems, 32 (2012), 3567-3585. doi: 10.3934/dcds.2012.32.3567.
[34]	Z. Guo, Z. Liu, J. Wei and F. Zhou, Bifurcations of some elliptic problems with a singular nonlinearity via Morse index, Comm. Pure. Appl. Anal., 10 (2011), 507-525. doi: 10.3934/cpaa.2011.10.507.
[35]	Shouchuan Hu and N. S. Papageorgiou, Positive solutions for nonlinear hemivariational inequalities, J. Math. Anal. Appl., 310 (2005), 161-176. doi: 10.1016/j.jmaa.2005.01.051.
[36]	Shouchuan Hu and N. S. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Comm. Pure Appl. Anal., 9 (2010), 1801-1827.
[37]	Shouchuan Hu and N. S. Papageorgiou, Double resonance for Dirichlet problems with unbounded indefinite potential and combined nonlinearities, Comm. Pure Applied Anal., 11 (2012), 2005-2021. doi: 10.3934/cpaa.2012.11.2005.
[38]	J. Garcia Melian, J. Rossi and J. Sabina de Lis, A convex-concave problem with a parameter on the boundary condition, Discrete Contin. Dynam. Systems, 32 (2012), 1095-1124.
[39]	L. Jeanjean, On the existence of bounded Palais-Smale sequences and applications to Landesmann-Lazer type problems in $\rn$, Proc. Royal Soc. Edinburgh, 129A (1999), 767-809.
[40]	A. Kristaly, M. Mihaileseu and V. Radulescu, Two nontrivial solutions for a nonhomogeneous Neumann problem: an Orlicz-Sobolev space setting, Proc. Royal. Soc. Edinburgh, 139A (2009), 367-379. doi: 10.1017/S030821050700025X.
[41]	S. Kyritsi, D. O'Regan and N. S. Papageorgiou, Multiple solutions for resonant hemivariational inequalities via minimax methods, Adv. Nonlin. Studies, 9 (2009), 453-478.
[42]	S. Kyritsi and N. S. Papageorgiou, Multiple solutions for nonlinear elliptic equations with asymmetric reaction term, Discrete Cont Dyn Systems, 33 (2013), 2469-2494.
[43]	O. A. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York, 1968.
[44]	G. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Part. Diff. Equas., 16 (1991), 311-361. doi: 10.1080/03605309108820761.
[45]	S. Li, S. Wu and H. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities, J. Diff. Equas., 185 (2002), 200-224. doi: 10.1006/jdeq.2001.4167.
[46]	M. Mihailescu, Existence and multiplicity of weak solutions for a class of degenerate nonlinear ellipitc equations, Boundary Value Problems, (2006), article ID 41295, 1-17.
[47]	M. Mihailescu and V. Radulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: an Orlicz-Sobolev space setting, J. Math. Anal. Appl., 330 (2007), 416-432. doi: 10.1016/j.jmaa.2006.07.082.
[48]	M. L. Miotto, Multiple solutions for elliptic problems in $R^N$ with critical Spbolev exponent and weight function, Comm. Pure Appl. Anal., 9 (2010), 233-248. doi: 10.3934/cpaa.2010.9.233.
[49]	S. Miyajima, D. Motreanu and M. Tanaka, Multiple existence results of solutions for Neumann problems via super- and sub-solutions, J. Functional Anal., 262 (2012), 1921-1953. doi: 10.1016/j.jfa.2011.11.028.
[50]	D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, On $p$-Laplace equations with concave terms and asymmetric nonlinearities, Proc. Royal Soc. Edinburgh, 141A (2011), 171-192. doi: 10.1017/S0308210509001656.
[51]	D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems, Annali Scuola Normale Sup. Pisa, X (2011), 729-756.
[52]	D. Motreanu, D. O'Regan and N. S. Papageorgiou, A unified treatment using critical point methods of the existence of multiple solutions for superlinear and sublinear Neumann problems, Comm. Pure Appl. Anal., 10 (2011), 1791-1816. doi: 10.3934/cpaa.2011.10.1791.
[53]	D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems, J. Diff. Equas., 232 (2007), 1-35. doi: 10.1016/j.jde.2006.09.008.
[54]	D. Motreanu and N. S. Papageorgiou, Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator, Proc. Amer. Math. Soc., 139 (2011), 3527-3535. doi: 10.1090/S0002-9939-2011-10884-0.
[55]	N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis," Springer, New York, 2009.
[56]	N. S. Papageorgiou and E. M. Rocha, On nonlinear parametric problems for $p$-Laplacian like operators, RACSAM, 103 (2009), 177-200. doi: 10.1007/BF03191850.
[57]	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: 10.1016/j.na.2007.06.023.
[58]	P. Pucci and J. Serrin, The strong maximum principle revisited, J. Diff. Equas., 196 (2004), 1-68; Erratum, J. Diff. Equas., 207 (2004), 226-227.
[59]	P. Pucci and J. Serrin, "The Maximum Principle," Birkhauser, Basel, 2007. doi: 10.1016/j.jde.2004.09.002.
[60]	J. M. Rakotoson, Generalized eigenvalue problem for totally discontinuous operator, Discrete Contin. Dynam. Systems, 28 (2010), 343-373. doi: 10.3934/dcds.2010.28.343.
[61]	P. Roselli and B. Sciunzi, A strong comparison principle for the $p$-Laplacian, Proc. Amer. Math. Soc., 135 (2007), 3217-3224. doi: 10.1090/S0002-9939-07-08847-8.
[62]	M. Sun, Multiplicity of solutions for a class of quasilinear elliptic equation at resonance, J. Math. Anal. Appl., 386 (2012), 661-668. doi: 10.1016/j.jmaa.2011.08.030.
[63]	J. Vazquez, A strong maximum principle for some quailinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202. doi: 10.1007/BF01449041.
[64]	R. Zhang, J. Chen and F. Zhan, Multiple solutions for superlinear elliptic systems of Hamiltonian type, Discrete Cont Dyn Systems, 30 (2011), 1249-1262. doi: 10.3934/dcds.2011.30.1249.

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