September  2017, 16(5): 1741-1766. doi: 10.3934/cpaa.2017085

Semilinear nonlocal elliptic equations with critical and supercritical exponents

Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhaba Road, Pune-411008, India

* Corresponding author : Mousomi Bhakta

Received  October 2016 Revised  April 2017 Published  May 2017

Fund Project: The first author is supported by the INSPIRE research grant DST/INSPIRE 04/2013/000152 and the second author is supported by the NBHM grant 2/39(12)/2014/RD-Ⅱ.

We study the problem
$\left\{ \begin{align} &{{(-\Delta lta )}^{s}}u={{u}^{p}}-{{u}^{q}}\ \text{in}\ \text{ }{{\mathbb{R}}^{N}}, \\ &u\in {{{\dot{H}}}^{s}}({{\mathbb{R}}^{N}})\cap {{L}^{q+1}}({{\mathbb{R}}^{N}}), \\ &u>0\ \ \text{in}\ \ {{\mathbb{R}}^{N}}, \\ \end{align} \right.$
where
$s∈(0,1)$
is a fixed parameter,
$(-Δ)^s$
is the fractional Laplacian in
$\mathbb{R}^N$
,
$q>p≥q \frac{N+2s}{N-2s}$
and
$N>2s$
. For every
$s∈(0,1)$
, we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all
$s∈(0,1)$
. Using those decay estimates, we prove Pohozaev type identity in ${{\mathbb{R}}^{N}}$ and we show that the above problem does not have any solution when
$p=\frac{N+2s}{N-2s}$
. We also discuss radial symmetry and decreasing property of the solution and prove that when
$p>\frac{N+2s}{N-2s}$
, the above problem admits a solution. Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every
$p≥q \frac{N+2s}{N-2s}$
and every solution is a classical solution.
Citation: Mousomi Bhakta, Debangana Mukherjee. Semilinear nonlocal elliptic equations with critical and supercritical exponents. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1741-1766. doi: 10.3934/cpaa.2017085
References:
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D. Applebaum, Lévy processes from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

[2]

B. BarriosE. ColoradoA. De Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Diff. Eqns, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar

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M. Bhakta, D. Mukherjee and S. Santra, Profile of solutions for nonlocal equations with critical and supercritical nonlinearities, preprint, arXiv: 1612.01759. Google Scholar

[4]

M. Bhakta and S. Santra, On a singular equation with critical and supercritical exponents To appear in J. Differential Equations. Google Scholar

[5]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

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X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calc. Var. Partial Differential Equations, 49 (2014), 233-269.  doi: 10.1007/s00526-012-0580-6.  Google Scholar

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

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[15]

N. Ghoussoub and S. Shakerian, Borderline variational problems involving fractional Laplacians and critical singularities, Adv. Nonlinear Stud., 15 (2015), 527-555.  doi: 10.1515/ans-2015-0302.  Google Scholar

[16]

S. Jarohs and T. Weth, Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 2581-2615.  doi: 10.3934/dcds.2014.34.2581.  Google Scholar

[17]

T. JinY. Y. Li and J. Xiong, On a fractional Nirenberg problem, part Ⅰ: blow up analysis and compactness of solutions, J. Eur. Math. Soc.(JEMS), 16 (2014), 1111-1171.  doi: 10.4171/JEMS/456.  Google Scholar

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M. K. KwongJ. B. McleodL. A. Peletier and W. C. Troy, On ground state solutions of $-\Delta u = u^p - u^q$, J. Differential Equations, 95 (1992), 218-239.  doi: 10.1016/0022-0396(92)90030-Q.  Google Scholar

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F. Merle and L. Peletier, Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth, I. The radial case, Arch. Rational Mech. Anal., 112 (1990), 1-19.  doi: 10.1007/BF00431720.  Google Scholar

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F. Merle and L. Peletier, Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth, Ⅱ. The non-radial case, J. Funct. Anal, 105 (1992), 1-41.  doi: 10.1016/0022-1236(92)90070-Y.  Google Scholar

[21]

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226.  doi: 10.2307/1995882.  Google Scholar

[22]

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

[23]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y.  Google Scholar

[24]

Y. J. Park, Fractional Polya-Szego inequality, J. Chungcheong Math. Soc., 24 (2011), 267-271.   Google Scholar

[25]

X. Ros-Oton and J. Serra, Regularity theory for general stable operators, J. Differential Equations, 260 (2016), 8675-8715.  doi: 10.1016/j.jde.2016.02.033.  Google Scholar

[26]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl(9), 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[27]

X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 587-628.  doi: 10.1007/s00205-014-0740-2.  Google Scholar

[28]

X. Ros-Oton and J. Serra, Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Comm. Partial Differential Equations, 40 (2015), 115-133.  doi: 10.1080/03605302.2014.918144.  Google Scholar

[29]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.   Google Scholar

[30]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc, 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[31]

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.  Google Scholar

[32]

J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983.  doi: 10.3934/dcds.2011.31.975.  Google Scholar

[33]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA No., 49 (2009), 33-44.   Google Scholar

[34]

L. Vlahos, H. Isliker, K. Kominis and K. Hizonidis, Normal and anomalous diffusion: a tutorial, preprint, arXiv: 0805.0419. Google Scholar

show all references

References:
[1]

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

[2]

B. BarriosE. ColoradoA. De Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Diff. Eqns, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar

[3]

M. Bhakta, D. Mukherjee and S. Santra, Profile of solutions for nonlocal equations with critical and supercritical nonlinearities, preprint, arXiv: 1612.01759. Google Scholar

[4]

M. Bhakta and S. Santra, On a singular equation with critical and supercritical exponents To appear in J. Differential Equations. Google Scholar

[5]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

[6]

X. Cabré and E. Cinti, Sharp energy estimates for nonlinear fractional diffusion equations, Calc. Var. Partial Differential Equations, 49 (2014), 233-269.  doi: 10.1007/s00526-012-0580-6.  Google Scholar

[7]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

[8]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[9]

R. Cont and P. Tankov, Financial Modelling with Jump Processes Vol. 2. CRC press, 2003. doi: 1-5848-8413-4.  Google Scholar

[10]

J. Dávila, L. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc. . Google Scholar

[11]

S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of ${{\mathbb{R}}^{N}}$, preprint, arXiv: 1506.01748. Google Scholar

[12]

E. FabesC. Kenig and R. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116.  doi: 10.1080/03605308208820218.  Google Scholar

[13]

M. M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.  doi: 10.1016/j.jfa.2012.06.018.  Google Scholar

[14]

P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian Commun. Contemp. Math. , 16 (2014), 1350023, 24 pp. . doi: 10.1142/S0219199713500235.  Google Scholar

[15]

N. Ghoussoub and S. Shakerian, Borderline variational problems involving fractional Laplacians and critical singularities, Adv. Nonlinear Stud., 15 (2015), 527-555.  doi: 10.1515/ans-2015-0302.  Google Scholar

[16]

S. Jarohs and T. Weth, Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 2581-2615.  doi: 10.3934/dcds.2014.34.2581.  Google Scholar

[17]

T. JinY. Y. Li and J. Xiong, On a fractional Nirenberg problem, part Ⅰ: blow up analysis and compactness of solutions, J. Eur. Math. Soc.(JEMS), 16 (2014), 1111-1171.  doi: 10.4171/JEMS/456.  Google Scholar

[18]

M. K. KwongJ. B. McleodL. A. Peletier and W. C. Troy, On ground state solutions of $-\Delta u = u^p - u^q$, J. Differential Equations, 95 (1992), 218-239.  doi: 10.1016/0022-0396(92)90030-Q.  Google Scholar

[19]

F. Merle and L. Peletier, Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth, I. The radial case, Arch. Rational Mech. Anal., 112 (1990), 1-19.  doi: 10.1007/BF00431720.  Google Scholar

[20]

F. Merle and L. Peletier, Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth, Ⅱ. The non-radial case, J. Funct. Anal, 105 (1992), 1-41.  doi: 10.1016/0022-1236(92)90070-Y.  Google Scholar

[21]

B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226.  doi: 10.2307/1995882.  Google Scholar

[22]

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

[23]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y.  Google Scholar

[24]

Y. J. Park, Fractional Polya-Szego inequality, J. Chungcheong Math. Soc., 24 (2011), 267-271.   Google Scholar

[25]

X. Ros-Oton and J. Serra, Regularity theory for general stable operators, J. Differential Equations, 260 (2016), 8675-8715.  doi: 10.1016/j.jde.2016.02.033.  Google Scholar

[26]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl(9), 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[27]

X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 587-628.  doi: 10.1007/s00205-014-0740-2.  Google Scholar

[28]

X. Ros-Oton and J. Serra, Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Comm. Partial Differential Equations, 40 (2015), 115-133.  doi: 10.1080/03605302.2014.918144.  Google Scholar

[29]

R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.   Google Scholar

[30]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc, 367 (2015), 67-102.  doi: 10.1090/S0002-9947-2014-05884-4.  Google Scholar

[31]

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.  Google Scholar

[32]

J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst., 31 (2011), 975-983.  doi: 10.3934/dcds.2011.31.975.  Google Scholar

[33]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA No., 49 (2009), 33-44.   Google Scholar

[34]

L. Vlahos, H. Isliker, K. Kominis and K. Hizonidis, Normal and anomalous diffusion: a tutorial, preprint, arXiv: 0805.0419. Google Scholar

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