January  2017, 16(1): 99-114. doi: 10.3934/cpaa.2017004

Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$

Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo Ⅱ s/n, Trujillo, Perú

Received  November 2015 Revised  July 2016 Published  November 2016

In this article we are interested in the following fractional $p$-Laplacian equation in $\mathbb{R}^n$
$ (-\Delta)_{p}^{s}u + V(x)|u|^{p-2}u = f(x,u) \mbox{ in } \mathbb{R}^{n}, $
where $p\geq 2$, $0 < s < 1$, $n\geq 2$ and $f$ is $p$-superlinear. By using mountain pass theorem with Cerami condition we prove the existence of nontrivial solution. Furthermore, we show that this solution is radially simmetry.
Citation: CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure & Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004
References:
[1]

F. Almgren and E. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773.  doi: 10.2307/1990893.  Google Scholar

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C. Alves and S. Liu, On superlinear p(x)-Laplacian equations in $\mathbb{R}^N$, Nonlinear Analysis, 73 (2010), 2566-2579.  doi: 10.1016/j.na.2010.06.033.  Google Scholar

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A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Func. Anal., 14 (1973), 349-381.   Google Scholar

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L. AmbrosioG. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.  doi: 10.1007/s00229-010-0399-4.  Google Scholar

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D. Appleabeaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

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B. BarriosE. Colorado and A. De Pablo, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar

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H. Berestycki and P. -L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

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L. Caffarelli, Nonlocal equations, drifts and games, in Nonlinear Partial Differential Equations, The Abel Symposium 2010 (eds. H. Holden and K. Karlsen) Springer-Verlag Berlin Heidelberg, (2012), 37-52. doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

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L. CaffarelliJ. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.  Google Scholar

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A. Di CastroT. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.  doi: 10.1016/j.jfa.2014.05.023.  Google Scholar

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A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré, (2015). doi: 10.1016/j.anihpc.2015.04.003.  Google Scholar

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E. Di NezzaG. 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.  Google Scholar

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S. DipierroG. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216.   Google Scholar

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P. FelmerA. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

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P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var., 54 (2015), 75-98.  doi: 10.1007/s00526-014-0778-x.  Google Scholar

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P. Felmer and C. Torres, Radial symmetry of ground states for a regional fractional nonlinear Schrödinger equation, Comm. Pure Appl. Math., 13 (2014), 2395-2406.  doi: 10.3934/cpaa.2014.13.2395.  Google Scholar

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G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 315-328.   Google Scholar

[19]

A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385.  doi: 10.1016/j.jmaa.2013.12.059.  Google Scholar

[20]

A. Iannizzotto and M. Squassina, Weyl-type laws for fractional p-eigenvalue problems, Asymptotic Anal., 88 (2014), 233-245.   Google Scholar

[21]

A. IannizzottoS. LiuK. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.  doi: 10.1515/acv-2014-0024.  Google Scholar

[22]

T. KuusiG. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368.  doi: 10.1007/s00220-015-2356-2.  Google Scholar

[23]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[24]

E. Lieb and M. Loss, Analysis, Grad. Stud. Math. , vol. 14, Amer. Math. Soc. , Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

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J. Mawhin and M. Willen, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74, Springer, Berlin, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[26]

R. Metzler and J. Klafter, The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), 161-208.  doi: 10.1088/0305-4470/37/31/R01.  Google Scholar

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P. Rabinowitz, Minimax Method in Critical Point Theory with Applications to Differential Equations, CBMS Amer. Math. Soc. , No 65,1986. doi: 10.1090/cbms/065.  Google Scholar

[28]

K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. RWA, 14 (2013), 867-874.  doi: 10.1016/j.nonrwa.2012.08.008.  Google Scholar

[29]

C. Torres, Multiplicity and symmetry results for a nonlinear Schrödinger equation with nonlocal regional diffusion, Math. Meth. Appl. Sci., 39 (2016), 2808-2820.  doi: 10.1002/mma.3731.  Google Scholar

[30]

C. Torres, Symmetric ground state solution for a non-linear Schrödinger equation with nonlocal regional diffusion, Complex Variables and Elliptic equations, (2016). doi: 10.1080/17476933.2016.1178730.  Google Scholar

[31]

C. Torres, Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well, Comm. Pure Appl. Math., 15 (2016), 353-547.  doi: 10.3934/cpaa.2016.15.535.  Google Scholar

[32]

M. Willen, Minimax Theorems, Birkhäuser, Boston, Basel, Berlin, 1996 doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[33]

J. Wloka, Partial Differential Equations, Cambridge University Press, 1987. doi: 10.1017/CBO9781139171755.  Google Scholar

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W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006.  Google Scholar

show all references

References:
[1]

F. Almgren and E. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773.  doi: 10.2307/1990893.  Google Scholar

[2]

C. Alves and S. Liu, On superlinear p(x)-Laplacian equations in $\mathbb{R}^N$, Nonlinear Analysis, 73 (2010), 2566-2579.  doi: 10.1016/j.na.2010.06.033.  Google Scholar

[3]

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Func. Anal., 14 (1973), 349-381.   Google Scholar

[4]

L. AmbrosioG. De Philippis and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.  doi: 10.1007/s00229-010-0399-4.  Google Scholar

[5]

D. Appleabeaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

[6]

B. BarriosE. Colorado and A. De Pablo, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar

[7]

H. Berestycki and P. -L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[8]

L. Caffarelli, Nonlocal equations, drifts and games, in Nonlinear Partial Differential Equations, The Abel Symposium 2010 (eds. H. Holden and K. Karlsen) Springer-Verlag Berlin Heidelberg, (2012), 37-52. doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

[9]

L. CaffarelliJ. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.  doi: 10.1002/cpa.20331.  Google Scholar

[10]

X. Chang, Ground state solutions of asymptotically linear fractional Schrödinger equations, J. Math. Physics, 54 (2013), 04 (0615).  doi: 10.1063/1.4809933.  Google Scholar

[11]

A. Di CastroT. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807-1836.  doi: 10.1016/j.jfa.2014.05.023.  Google Scholar

[12]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré, (2015). doi: 10.1016/j.anihpc.2015.04.003.  Google Scholar

[13]

E. Di NezzaG. 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.  Google Scholar

[14]

S. DipierroG. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216.   Google Scholar

[15]

P. FelmerA. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

[16]

P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var., 54 (2015), 75-98.  doi: 10.1007/s00526-014-0778-x.  Google Scholar

[17]

P. Felmer and C. Torres, Radial symmetry of ground states for a regional fractional nonlinear Schrödinger equation, Comm. Pure Appl. Math., 13 (2014), 2395-2406.  doi: 10.3934/cpaa.2014.13.2395.  Google Scholar

[18]

G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 315-328.   Google Scholar

[19]

A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385.  doi: 10.1016/j.jmaa.2013.12.059.  Google Scholar

[20]

A. Iannizzotto and M. Squassina, Weyl-type laws for fractional p-eigenvalue problems, Asymptotic Anal., 88 (2014), 233-245.   Google Scholar

[21]

A. IannizzottoS. LiuK. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.  doi: 10.1515/acv-2014-0024.  Google Scholar

[22]

T. KuusiG. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368.  doi: 10.1007/s00220-015-2356-2.  Google Scholar

[23]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[24]

E. Lieb and M. Loss, Analysis, Grad. Stud. Math. , vol. 14, Amer. Math. Soc. , Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[25]

J. Mawhin and M. Willen, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74, Springer, Berlin, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[26]

R. Metzler and J. Klafter, The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), 161-208.  doi: 10.1088/0305-4470/37/31/R01.  Google Scholar

[27]

P. Rabinowitz, Minimax Method in Critical Point Theory with Applications to Differential Equations, CBMS Amer. Math. Soc. , No 65,1986. doi: 10.1090/cbms/065.  Google Scholar

[28]

K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. RWA, 14 (2013), 867-874.  doi: 10.1016/j.nonrwa.2012.08.008.  Google Scholar

[29]

C. Torres, Multiplicity and symmetry results for a nonlinear Schrödinger equation with nonlocal regional diffusion, Math. Meth. Appl. Sci., 39 (2016), 2808-2820.  doi: 10.1002/mma.3731.  Google Scholar

[30]

C. Torres, Symmetric ground state solution for a non-linear Schrödinger equation with nonlocal regional diffusion, Complex Variables and Elliptic equations, (2016). doi: 10.1080/17476933.2016.1178730.  Google Scholar

[31]

C. Torres, Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well, Comm. Pure Appl. Math., 15 (2016), 353-547.  doi: 10.3934/cpaa.2016.15.535.  Google Scholar

[32]

M. Willen, Minimax Theorems, Birkhäuser, Boston, Basel, Berlin, 1996 doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[33]

J. Wloka, Partial Differential Equations, Cambridge University Press, 1987. doi: 10.1017/CBO9781139171755.  Google Scholar

[34]

W. Zou and M. Schechter, Critical Point Theory and Its Applications, Springer, New York, 2006.  Google Scholar

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