# American Institute of Mathematical Sciences

November  2015, 14(6): 2561-2616. doi: 10.3934/cpaa.2015.14.2561

## Nonlinear Neumann problems with indefinite potential and concave terms

 1 College of Mathematics, Shandong Normal University, Jinan, Shandong 2 Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received  December 2014 Revised  July 2015 Published  September 2015

In this paper we conduct a detailed study of Neumann problems driven by a nonhomogeneous differential operator plus an indefinite potential and with concave contribution in the reaction. We deal with both superlinear and sublinear (possibly resonant) problems and we produce constant sign and nodal solutions. We also examine semilinear equations resonant at higher parts of the spectrum and equations with a negative concavity.
Citation: Shouchuan Hu, Nikolaos S. Papageorgiou. Nonlinear Neumann problems with indefinite potential and concave terms. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2561-2616. doi: 10.3934/cpaa.2015.14.2561
##### 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, AMS, vol. 196, no. 905, 2008. doi: 10.1090/memo/0915.  Google Scholar [2] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z., 169 (1979), 127-166. doi: 10.1007/BF01215273.  Google Scholar [3] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Functional Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.  Google Scholar [4] G. Barletta, R. Livrea and N. S. Papageorgiou, A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian, Comm. Pure. Appl. Anal., 13 (2014), 1075-1086. doi: 10.3934/cpaa.2014.13.1075.  Google Scholar [5] A. Castro and A. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl., 120 (1979), 113-137. doi: 10.1007/BF02411940.  Google Scholar [6] A. Castro, J. Cossio and C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems, Discrete Cont Dyn Systems, 33 (2013), 123-140.  Google Scholar [7] L. Cherfils and V. Ilyasov, On the stationary solutions of generalized reaction diffusion equations with p&q Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22.  Google Scholar [8] G. M. Coclite and M. M. Coclite, On a Dirichlet problem in bounded domains with singular nonlinearity, Discrete Contin. Dynam. Systems, 33 (2013), 4923-4944. doi: 10.3934/dcds.2013.33.4923.  Google Scholar [9] 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 [10] M. Filippakis and N. S. Papageorgiou, Nodal solutions for Neumann problems with a nonhomogeneous differential operator, Funkc. Ekv., 56 (2013), 63-79. doi: 10.1619/fesi.56.63.  Google Scholar [11] 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 [12] J. Garcia Azorero, J. Manfredi and J. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404. doi: 10.1142/S0219199700000190.  Google Scholar [13] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar [14] 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 [15] L. Gasinski and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance, Discrete Contin. Dynam. Systems, 34 (2014), 2037-2060.  Google Scholar [16] T. Godoy, J.-P. Gossez and S. Paczka, On the principal eigenvalues of some elliptic problems with large drift, Discrete Contin. Dynam. Systems, 33 (2013), 225-237. doi: 10.3934/dcds.2013.33.225.  Google Scholar [17] 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 [18] Z. Guo and Z. Zhang, $W^{1,p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 286 (2003), 32-50. doi: 10.1016/S0022-247X(03)00282-8.  Google Scholar [19] 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 [20] Shouchuan Hu and N. S. Papageorgiou, Multipcility of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin, Tohoku Math. J., 62 (2010), 137-162. doi: 10.2748/tmj/1270041030.  Google Scholar [21] Shouchuan Hu and N. S. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Comm. Pure Appl. Anal., 10 (2011), 1055-1078. doi: 10.3934/cpaa.2011.10.1055.  Google Scholar [22] 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 [23] Q. Jiu and J. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3.  Google Scholar [24] E. Ko, E. K. Lee and R. Shivaji, Multiplicity results for classes of singular problems on an exterior domain, Discrete Cont Dyn Systems, 33 (2013), 5153-5166. doi: 10.3934/dcds.2013.33.5153.  Google Scholar [25] 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 [26] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.  Google Scholar [27] S. Liu and S. Li, Critical groups at infinity, saddle point reduction and elliptic resonant problems, Comm. Contemp. Math., 5 (2003), 761-773. doi: 10.1142/S0219199703001129.  Google Scholar [28] S. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Comm. Pure Appl. Anal., 12 (2013), 815-829. doi: 10.3934/cpaa.2013.12.815.  Google Scholar [29] V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Topol. Methods Nonl. Anal., 10 (1997), 387-397.  Google Scholar [30] D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.  Google Scholar [31] 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 [32] 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 [33] 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 [34] D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super Pisa Cl. SCI., 11 (2012), 729-788.  Google Scholar [35] D. Mugnai and N. S. Papageorgiou, Wang's multiplicity result for superlinear $(p,q)$-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc., 366 (2014), 4919-4937. doi: 10.1090/S0002-9947-2013-06124-7.  Google Scholar [36] F. D. dePaiva and E. Massa, Multiple solutions for some elliptic equations with nonlinearity concave at the origin, Nonlin. Anal., 66 (2007), 2940-2946. doi: 10.1016/j.na.2006.04.015.  Google Scholar [37] R. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 115-132.  Google Scholar [38] N. S. Papageorgiou and S. Th. Kyritsi, Handbook of Applied Analysis, Springer, New York, 2009. doi: 10.1007/b120946.  Google Scholar [39] N. S. Papageorgiou and V. D. Radulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim., 69 (2014), 393-430. doi: 10.1007/s00245-013-9227-z.  Google Scholar [40] N. S. Papageorgiou and G. Smyrlis, Positive solutions for nonlinear Neumann problems with concave and convex terms, Positivity, 16 (2012), 271-296. doi: 10.1007/s11117-011-0124-x.  Google Scholar [41] N. S. Papageorgiou and G. Smyrlis, On a class of parametric Neumann problems with indefinite and unbounded potential,, \emph{Forum Math.}, ().  doi: 10.1515/forum-2012-0042.  Google Scholar [42] N. S. Papageorgiou and P. Winkert, On a parametric nonlinear Dirichlet problem with subdiffusive and equidiffusive reaction, Adv. Nonlin. Studies, 14 (2014), 565-592.  Google Scholar [43] N. S. Papageorgiou and P. Winkert, Resonant $(p,2)$-equations with concave terms,, \emph{Appl. Anal.}, ().  doi: 10.1080/00036811.2014.895332.  Google Scholar [44] K. Perera, Multiplicity results for some elliptic problems with concave nonlinearities, J. Diff. Equas., 140 (1997), 133-141. doi: 10.1006/jdeq.1997.3310.  Google Scholar [45] P. Poláčik, On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains, Discrete Cont Dyn Systems, 34 (2014), 2657-2667. doi: 10.3934/dcds.2014.34.2657.  Google Scholar [46] P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Basel, 2007.  Google Scholar [47] P. Sacks and M. Warma, Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data, Discrete Cont Dyn Systems, 34 (2014), 761-787.  Google Scholar [48] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Cont Dyn Systems, 33 (2013), 2105-2137.  Google Scholar [49] M. Struwe, Variatioanl Methods, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar [50] A. Szulkin and S. Waliullah, Infinitely many solutions for some singular elliptic problems, Discrete Cont Dyn Systems, 33 (2013), 321-333.  Google Scholar [51] J. Tan, Positive solutions for non local elliptic problems, Discrete Cont Dyn Systems, 33 (2013), 837-859. doi: 10.3934/dcds.2013.33.837.  Google Scholar [52] K. Thews, Nonntrivial solutions of elliptic equations at resonance, Proc. Royal Soc. Edinburgh, 85A (1980), 119-129. doi: 10.1017/S0308210500011732.  Google Scholar [53] X. Wang, Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Diff. Equas., 93 (1991), 283-310. doi: 10.1016/0022-0396(91)90014-Z.  Google Scholar [54] P. Winkert, $L^\infty$-estimates for nonlinear elliptic Neumann boundary value problems, NoDEA Nonlin. Diff. Equ. Appl., 17 (2010), 289-302. doi: 10.1007/s00030-009-0054-5.  Google Scholar [55] X. Yu, Liouville type theorem for nonlinear elliptic equation with general nonlinearity, Discrete Cont Dyn Systems, 34 (2014), 4947-4966. doi: 10.3934/dcds.2014.34.4947.  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, AMS, vol. 196, no. 905, 2008. doi: 10.1090/memo/0915.  Google Scholar [2] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z., 169 (1979), 127-166. doi: 10.1007/BF01215273.  Google Scholar [3] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Functional Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.  Google Scholar [4] G. Barletta, R. Livrea and N. S. Papageorgiou, A nonlinear eigenvalue problem for the periodic scalar $p$-Laplacian, Comm. Pure. Appl. Anal., 13 (2014), 1075-1086. doi: 10.3934/cpaa.2014.13.1075.  Google Scholar [5] A. Castro and A. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Mat. Pura Appl., 120 (1979), 113-137. doi: 10.1007/BF02411940.  Google Scholar [6] A. Castro, J. Cossio and C. Vélez, Existence and qualitative properties of solutions for nonlinear Dirichlet problems, Discrete Cont Dyn Systems, 33 (2013), 123-140.  Google Scholar [7] L. Cherfils and V. Ilyasov, On the stationary solutions of generalized reaction diffusion equations with p&q Laplacian, Commun. Pure Appl. Anal., 4 (2005), 9-22.  Google Scholar [8] G. M. Coclite and M. M. Coclite, On a Dirichlet problem in bounded domains with singular nonlinearity, Discrete Contin. Dynam. Systems, 33 (2013), 4923-4944. doi: 10.3934/dcds.2013.33.4923.  Google Scholar [9] 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 [10] M. Filippakis and N. S. Papageorgiou, Nodal solutions for Neumann problems with a nonhomogeneous differential operator, Funkc. Ekv., 56 (2013), 63-79. doi: 10.1619/fesi.56.63.  Google Scholar [11] 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 [12] J. Garcia Azorero, J. Manfredi and J. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404. doi: 10.1142/S0219199700000190.  Google Scholar [13] L. Gasinski and N. S. Papageorgiou, Nonlinear Analysis, Chapman & Hall/CRC, Boca Raton, 2006.  Google Scholar [14] 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 [15] L. Gasinski and N. S. Papageorgiou, Dirichlet $(p,q)$-equations at resonance, Discrete Contin. Dynam. Systems, 34 (2014), 2037-2060.  Google Scholar [16] T. Godoy, J.-P. Gossez and S. Paczka, On the principal eigenvalues of some elliptic problems with large drift, Discrete Contin. Dynam. Systems, 33 (2013), 225-237. doi: 10.3934/dcds.2013.33.225.  Google Scholar [17] 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 [18] Z. Guo and Z. Zhang, $W^{1,p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 286 (2003), 32-50. doi: 10.1016/S0022-247X(03)00282-8.  Google Scholar [19] 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 [20] Shouchuan Hu and N. S. Papageorgiou, Multipcility of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin, Tohoku Math. J., 62 (2010), 137-162. doi: 10.2748/tmj/1270041030.  Google Scholar [21] Shouchuan Hu and N. S. Papageorgiou, Nonlinear Neumann equations driven by a nonhomogeneous differential operator, Comm. Pure Appl. Anal., 10 (2011), 1055-1078. doi: 10.3934/cpaa.2011.10.1055.  Google Scholar [22] 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 [23] Q. Jiu and J. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601. doi: 10.1016/S0022-247X(03)00165-3.  Google Scholar [24] E. Ko, E. K. Lee and R. Shivaji, Multiplicity results for classes of singular problems on an exterior domain, Discrete Cont Dyn Systems, 33 (2013), 5153-5166. doi: 10.3934/dcds.2013.33.5153.  Google Scholar [25] 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 [26] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlin. Anal., 12 (1988), 1203-1219. doi: 10.1016/0362-546X(88)90053-3.  Google Scholar [27] S. Liu and S. Li, Critical groups at infinity, saddle point reduction and elliptic resonant problems, Comm. Contemp. Math., 5 (2003), 761-773. doi: 10.1142/S0219199703001129.  Google Scholar [28] S. Marano and N. S. Papageorgiou, Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter, Comm. Pure Appl. Anal., 12 (2013), 815-829. doi: 10.3934/cpaa.2013.12.815.  Google Scholar [29] V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Topol. Methods Nonl. Anal., 10 (1997), 387-397.  Google Scholar [30] D. Motreanu, V. Motreanu and N. S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014. doi: 10.1007/978-1-4614-9323-5.  Google Scholar [31] 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 [32] 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 [33] 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 [34] D. Mugnai and N. S. Papageorgiou, Resonant nonlinear Neumann problems with indefinite weight, Ann. Sc. Norm. Super Pisa Cl. SCI., 11 (2012), 729-788.  Google Scholar [35] D. Mugnai and N. S. Papageorgiou, Wang's multiplicity result for superlinear $(p,q)$-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc., 366 (2014), 4919-4937. doi: 10.1090/S0002-9947-2013-06124-7.  Google Scholar [36] F. D. dePaiva and E. Massa, Multiple solutions for some elliptic equations with nonlinearity concave at the origin, Nonlin. Anal., 66 (2007), 2940-2946. doi: 10.1016/j.na.2006.04.015.  Google Scholar [37] R. Palais, Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 115-132.  Google Scholar [38] N. S. Papageorgiou and S. Th. Kyritsi, Handbook of Applied Analysis, Springer, New York, 2009. doi: 10.1007/b120946.  Google Scholar [39] N. S. Papageorgiou and V. D. Radulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim., 69 (2014), 393-430. doi: 10.1007/s00245-013-9227-z.  Google Scholar [40] N. S. Papageorgiou and G. Smyrlis, Positive solutions for nonlinear Neumann problems with concave and convex terms, Positivity, 16 (2012), 271-296. doi: 10.1007/s11117-011-0124-x.  Google Scholar [41] N. S. Papageorgiou and G. Smyrlis, On a class of parametric Neumann problems with indefinite and unbounded potential,, \emph{Forum Math.}, ().  doi: 10.1515/forum-2012-0042.  Google Scholar [42] N. S. Papageorgiou and P. Winkert, On a parametric nonlinear Dirichlet problem with subdiffusive and equidiffusive reaction, Adv. Nonlin. Studies, 14 (2014), 565-592.  Google Scholar [43] N. S. Papageorgiou and P. Winkert, Resonant $(p,2)$-equations with concave terms,, \emph{Appl. Anal.}, ().  doi: 10.1080/00036811.2014.895332.  Google Scholar [44] K. Perera, Multiplicity results for some elliptic problems with concave nonlinearities, J. Diff. Equas., 140 (1997), 133-141. doi: 10.1006/jdeq.1997.3310.  Google Scholar [45] P. Poláčik, On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains, Discrete Cont Dyn Systems, 34 (2014), 2657-2667. doi: 10.3934/dcds.2014.34.2657.  Google Scholar [46] P. Pucci and J. Serrin, The Maximum Principle, Birkhauser, Basel, 2007.  Google Scholar [47] P. Sacks and M. Warma, Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data, Discrete Cont Dyn Systems, 34 (2014), 761-787.  Google Scholar [48] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Cont Dyn Systems, 33 (2013), 2105-2137.  Google Scholar [49] M. Struwe, Variatioanl Methods, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar [50] A. Szulkin and S. 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