# American Institute of Mathematical Sciences

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

## Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems

 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

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.
Citation: Shouchuan Hu, Nikolaos S. Papageorgiou. Solutions of nonlinear nonhomogeneous Neumann and Dirichlet problems. Communications on Pure & 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.  Google Scholar [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.  Google Scholar [3] A. Allendes and A. Quaas, Multiplicity results for extremal operators through bifurcation, Discrete Cont Dyn Systems, 29 (2011), 51-65.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] H. Brezis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, CRAS Paris t., 317 (1993), 465-472.  Google Scholar [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.  Google Scholar [9] K. C. Chang, "Methods of Nonilnear Analysis," Springer, Berlin (2005).  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] M. Cuesta and P. Takac, A strong comparison principle for positive solutions of degenerate elliptic equations, Diff. Integ. Equas., 13 (2000), 721-746.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] N. Dunford and J.Schwartz, "Linear Operators I,", Wiley-Interscience, ().   Google Scholar [24] G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Elec. J. Diff. Equas., 8 (2002), 1-12.  Google Scholar [25] D. deFigueiredo, Positive solutions of semilinear elliptic problems, in "Lecture Notes Math.," vol. 957, Springer, New York, (1982), 34-85.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] D. Gilberg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] Shouchuan Hu and N. S. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Comm. Pure Appl. Anal., 9 (2010), 1801-1827.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [43] O. A. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York, 1968.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [55] N. S. Papageorgiou and S. Th. 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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.  Google Scholar [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.  Google Scholar [3] A. Allendes and A. Quaas, Multiplicity results for extremal operators through bifurcation, Discrete Cont Dyn Systems, 29 (2011), 51-65.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] H. Brezis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, CRAS Paris t., 317 (1993), 465-472.  Google Scholar [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.  Google Scholar [9] K. C. Chang, "Methods of Nonilnear Analysis," Springer, Berlin (2005).  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] M. Cuesta and P. Takac, A strong comparison principle for positive solutions of degenerate elliptic equations, Diff. Integ. Equas., 13 (2000), 721-746.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] N. Dunford and J.Schwartz, "Linear Operators I,", Wiley-Interscience, ().   Google Scholar [24] G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Elec. J. Diff. Equas., 8 (2002), 1-12.  Google Scholar [25] D. deFigueiredo, Positive solutions of semilinear elliptic problems, in "Lecture Notes Math.," vol. 957, Springer, New York, (1982), 34-85.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] D. Gilberg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] Shouchuan Hu and N. S. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Comm. Pure Appl. Anal., 9 (2010), 1801-1827.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [43] O. A. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York, 1968.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [51] D. Motreanu, V. V. Motreanu and N. S. 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