Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential

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May 2018, 17(3): 1201-1217. doi: 10.3934/cpaa.2018058

Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential

Song Peng and Aliang Xia ^,

Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

* Corresponding author

Received March 2017 Revised June 2017 Published January 2018

Fund Project: This work was supported by the Foundation of Jiangxi Provincial Education Department (No: GJJ160335), the NSFC (No. 11701239) and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University.

In this article, we prove the existence, multiplicity and concentration of non-trivial solutions for the following indefinite fractional elliptic equation with concave-convex nonlinearities:

$\left\{\begin{array}{*{20}{l}}(-Δ)^α u+V_λ(x)u = a(x)|u|^{q-2}u+b(x)|u|^{p-2}u &{\rm in}\,\,\mathbb{R}^N,\\u≥0\,\,&{\rm in}\,\,\mathbb{R}^N, \end{array} \right.$

where $0<α<1$, $N>2α$, $1<q<2<p<2^*_α$ with $ 2^*_α = 2N/(N-2α)$, the potential $V_λ(x) = λ V^+(x)-V^-(x)$ with $V^± = \max\{± V, 0\}$ and the parameter $λ>0$. Our multiplicity results are based on studying the decomposition of the Nehari manifold.

Keywords: Fractional Laplacian, steep potential, Nehari manifold, concave-convex term.

Mathematics Subject Classification: Primary: 35J60, 47G20.

Citation: Song Peng, Aliang Xia. Multiplicity and concentration of solutions for nonlinear fractional elliptic equations with steep potential. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1201-1217. doi: 10.3934/cpaa.2018058

References:

[1]	K. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function], J. Differential Equations, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.
[2]	B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900. doi: 10.1016/j.anihpc.2014.04.003.
[3]	B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.
[4]	T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149.
[5]	T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494.
[6]	E. Colorado, A. de Pablo and U. Sánchez, Perturbations of a critical fractional equation, Pacific J. Math., 271 (2014), 65-85.
[7]	A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034.
[8]	Y. H. Cheng and T. F. Wu, Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential, Comm. Pure and Applied Ana., 15 (2016), 2457-2473. doi: 10.3934/cpaa.2016044.
[9]	J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892. doi: 10.1016/j.jde.2013.10.006.
[10]	J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235. doi: 10.2140/apde.2015.8.1165.
[11]	S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^N$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. ⅷ+152 pp.
[12]	P. Drábek and S. Pohozaev, Positive solutions for the p-Laplacian: application of the fibrering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.
[13]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[14]	Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419. doi: 10.1007/s00526-006-0071-8.
[15]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[16]	M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961. doi: 10.1088/0951-7715/28/6/1937.
[17]	P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A., 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746.
[18]	N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.
[19]	N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108.
[20]	Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123. doi: 10.2307/1993333.
[21]	S. I. Pohozaev, An approach to nonlinear equations (Russian), Dokl. Akad. Nauk SSSR, 247 (1979), 1327-1331.
[22]	A. Quaas and A. Xia, Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions, Z. Angew. Math. Phys., 67 (2016), 40. doi: 10.1007/s00033-016-0631-5.
[23]	P. Rabinowitz, Variational methods for nonlinear eigenvalue problems of nonlinear problems, Edizioni Cremonese, Rome, 1974,139-195.
[24]	R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.
[25]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
[26]	G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 243-261. doi: 10.1016/S0294-1449(16)30238-4.
[27]	C. Torres, Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well, Comm. Pure and Applied Ana., 15 (2016), 535-547. doi: 10.3934/cpaa.2016.15.535.
[28]	T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in involving sign-changing, J. Funct. Anal., 258 (2010), 99-131. doi: 10.1016/j.jfa.2009.08.005.
[29]	T. F. Wu, Multiplicity results for a semi-linear elliptic equation involving sign-changing weight function, Rocky Mountain J. Math., 39 (2009), 995-1011. doi: 10.1216/RMJ-2009-39-3-995.
[30]	X. Yu, The Nehari manifold for elliptic equation involving the square root of the laplacian, J. Differential Equations, 252 (2012), 1283-1308. doi: 10.1016/j.jde.2011.09.015.

show all references

References:

[1]	K. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function], J. Differential Equations, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.
[2]	B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900. doi: 10.1016/j.anihpc.2014.04.003.
[3]	B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.
[4]	T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. doi: 10.1080/03605309508821149.
[5]	T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494.
[6]	E. Colorado, A. de Pablo and U. Sánchez, Perturbations of a critical fractional equation, Pacific J. Math., 271 (2014), 65-85.
[7]	A. Cotsiolis and N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236. doi: 10.1016/j.jmaa.2004.03.034.
[8]	Y. H. Cheng and T. F. Wu, Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential, Comm. Pure and Applied Ana., 15 (2016), 2457-2473. doi: 10.3934/cpaa.2016044.
[9]	J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892. doi: 10.1016/j.jde.2013.10.006.
[10]	J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235. doi: 10.2140/apde.2015.8.1165.
[11]	S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb{R}^N$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. ⅷ+152 pp.
[12]	P. Drábek and S. Pohozaev, Positive solutions for the p-Laplacian: application of the fibrering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.
[13]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[14]	Y. H. Ding and A. Szulkin, Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29 (2007), 397-419. doi: 10.1007/s00526-006-0071-8.
[15]	I. Ekeland, On the variational principle, J. Math. Anal. Appl., 17 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0.
[16]	M. M. Fall, F. Mahmoudi and E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961. doi: 10.1088/0951-7715/28/6/1937.
[17]	P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A., 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746.
[18]	N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.
[19]	N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. doi: 10.1103/PhysRevE.66.056108.
[20]	Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123. doi: 10.2307/1993333.
[21]	S. I. Pohozaev, An approach to nonlinear equations (Russian), Dokl. Akad. Nauk SSSR, 247 (1979), 1327-1331.
[22]	A. Quaas and A. Xia, Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions, Z. Angew. Math. Phys., 67 (2016), 40. doi: 10.1007/s00033-016-0631-5.
[23]	P. Rabinowitz, Variational methods for nonlinear eigenvalue problems of nonlinear problems, Edizioni Cremonese, Rome, 1974,139-195.
[24]	R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.
[25]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
[26]	G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 243-261. doi: 10.1016/S0294-1449(16)30238-4.
[27]	C. Torres, Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well, Comm. Pure and Applied Ana., 15 (2016), 535-547. doi: 10.3934/cpaa.2016.15.535.
[28]	T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in involving sign-changing, J. Funct. Anal., 258 (2010), 99-131. doi: 10.1016/j.jfa.2009.08.005.
[29]	T. F. Wu, Multiplicity results for a semi-linear elliptic equation involving sign-changing weight function, Rocky Mountain J. Math., 39 (2009), 995-1011. doi: 10.1216/RMJ-2009-39-3-995.
[30]	X. Yu, The Nehari manifold for elliptic equation involving the square root of the laplacian, J. Differential Equations, 252 (2012), 1283-1308. doi: 10.1016/j.jde.2011.09.015.

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