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January  2017, 16(1): 163-188. doi: 10.3934/cpaa.2017008

Resonant problems for fractional Laplacian

Yutong Chen and  Jiabao Su

School of Mathematical Sciences, Capital Normal University, Beijing 100048, People's Republic of China

Received  February 2016 Revised  August 2016 Published  November 2016

Fund Project: Supported by KZ201510028032 and NSFC11601353,11671026.

In this paper we consider the following fractional Laplacian equation
$ \left\{\begin{array}{ll} (-\Delta).s u=g(x, u) & x\in\Omega,\\ u=0, & x \in \mathbb{R}.N\setminus\Omega,\end{array} \right. $
where $ s\in (0, 1)$ is fixed, $\Omega$ is an open bounded set of $\mathbb{R}.N$, $N > 2s$, with smooth boundary, $(-\Delta).s$ is the fractional Laplace operator. By Morse theory we obtain the existence of nontrivial weak solutions when the problem is resonant at both infinity and zero.
Keywords: Fractional Laplacian, resonance, angle condition, Palais-Smale condition, critical group, Morse theory.
Mathematics Subject Classification: 49J35, 35A15, 35S15, 45G25, 58E05.
Citation: Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008
References:
[1]

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

[2]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Springer, Berlin, 2011. doi: 10.1007/978-0-85729-227-8.  Google Scholar

[3]

B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, 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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B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concaveconvex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003.  Google Scholar

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P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to nonlinear problems with "strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

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T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28 (1997), 419-441.  doi: 10.1016/0362-546X(95)00167-T.  Google Scholar

[7]

G. M. Bisci and R. Servadei, A Brezis-Nirenberg splitting approach for nonlocal fractional equations, Nonlinear Anal., 119 (2015), 341-353.  doi: 10.1016/j.na.2014.10.025.  Google Scholar

[8]

G. M. Bisci, D. Mugnai and R. Servadei, On multiple solutions for nonlocal fractional problems via $\nabla $-theorems, arXiv: 1510.08701. Google Scholar

[9]

H. Brezis, Analyse fonctionelle, Theorie et applications, Masson, Paris, 1983.  Google Scholar

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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

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A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662.  doi: 10.3934/cpaa.2011.10.1645.  Google Scholar

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K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems, Birkhauser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

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D. G. De Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339-346.  doi: 10.1080/03605309208820844.  Google Scholar

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A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-Local elliptic operators, Zeitschrift für Analysis und ihre Anwendungen, 32 (2013), 411-431.  doi: 10.4171/ZAA/1492.  Google Scholar

[15]

A. Fiscella, Saddle point solutions for nonlocal elliptic operators, arXiv: 1210.8401. Google Scholar

[16]

A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by fractionl operators, Math. Methods Appl. Sci. , to appear. doi: 10.1002/mma.3438.  Google Scholar

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D. Gromoll and M. Meyer, On differential functions with isolated point, Topology, 8 (1969), 361-369.   Google Scholar

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D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math., 121 (1985), 463-494.  doi: 10.2307/1971205.  Google Scholar

[19]

E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623.   Google Scholar

[20]

S. Li and J. Su, Existence of multiple solutions of a two-point boundary value problem at resonance, Topo. Meth. Nonl. Anal., 10 (1997), 123-135.   Google Scholar

[21]

S. Li and J. Su, Existence of multiple critical points for asymptotically quadratic functional with applications, Abst. Appl. Anal., 1 (1996), 283-305.  doi: 10.1155/S1085337596000140.  Google Scholar

[22]

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

[23]

D. Mugnai and D. Pagliardini, Existence and multiplicity results for the fractional Laplacian in bounded domains, Adv. Calc. Var. , to appear. Google Scholar

[24]

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.  Google Scholar

[25]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. , 65, American Mathematical Society, Providence, RI 1986. doi: 10.1090/cbms/065.  Google Scholar

[26]

R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal., 43 (2014), 251-267.   Google Scholar

[27]

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

[28]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non) local operators, Rev. Mat. Iberoam, 29 (2013), 1091-1126.  doi: 10.4171/RMI/750.  Google Scholar

[29]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.  Google Scholar

[30]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[31]

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

[32]

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

[33]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[34]

J. Su, Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity, J. Differential Equations, 145 (1998), 252-273.  doi: 10.1006/jdeq.1997.3360.  Google Scholar

[35]

J. Su and C. Tang, Multiple resutlts for semilinear elliptic equations with resonance at higher eigenvalues, Nonlinear Anal., 44 (2001), 311-321.   Google Scholar

[36]

J. Su and L. Zhao, An elliptic resonance problem with multiple solutions, J. Math. Appl. Appl., 319 (2006), 604-616.  doi: 10.1016/j.jmaa.2005.10.059.  Google Scholar

[37]

Z. -Q. Wang, Multiple solutions for indefinite functionals and applications to asymptotically linear problems, Acta Math. Sinica(N.S.), 5 (1989), 101-113.  doi: 10.1007/BF02107664.  Google Scholar

show all references

References:
[1]

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

[2]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Springer, Berlin, 2011. doi: 10.1007/978-0-85729-227-8.  Google Scholar

[3]

B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, 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

[4]

B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concaveconvex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.  doi: 10.1016/j.anihpc.2014.04.003.  Google Scholar

[5]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to nonlinear problems with "strong" resonance at infinity, Nonlinear Anal., 7 (1983), 981-1012.  doi: 10.1016/0362-546X(83)90115-3.  Google Scholar

[6]

T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal., 28 (1997), 419-441.  doi: 10.1016/0362-546X(95)00167-T.  Google Scholar

[7]

G. M. Bisci and R. Servadei, A Brezis-Nirenberg splitting approach for nonlocal fractional equations, Nonlinear Anal., 119 (2015), 341-353.  doi: 10.1016/j.na.2014.10.025.  Google Scholar

[8]

G. M. Bisci, D. Mugnai and R. Servadei, On multiple solutions for nonlocal fractional problems via $\nabla $-theorems, arXiv: 1510.08701. Google Scholar

[9]

H. Brezis, Analyse fonctionelle, Theorie et applications, Masson, Paris, 1983.  Google Scholar

[10]

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

[11]

A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662.  doi: 10.3934/cpaa.2011.10.1645.  Google Scholar

[12]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems, Birkhauser, Boston, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[13]

D. G. De Figueiredo and J. P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations, 17 (1992), 339-346.  doi: 10.1080/03605309208820844.  Google Scholar

[14]

A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-Local elliptic operators, Zeitschrift für Analysis und ihre Anwendungen, 32 (2013), 411-431.  doi: 10.4171/ZAA/1492.  Google Scholar

[15]

A. Fiscella, Saddle point solutions for nonlocal elliptic operators, arXiv: 1210.8401. Google Scholar

[16]

A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by fractionl operators, Math. Methods Appl. Sci. , to appear. doi: 10.1002/mma.3438.  Google Scholar

[17]

D. Gromoll and M. Meyer, On differential functions with isolated point, Topology, 8 (1969), 361-369.   Google Scholar

[18]

D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math., 121 (1985), 463-494.  doi: 10.2307/1971205.  Google Scholar

[19]

E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623.   Google Scholar

[20]

S. Li and J. Su, Existence of multiple solutions of a two-point boundary value problem at resonance, Topo. Meth. Nonl. Anal., 10 (1997), 123-135.   Google Scholar

[21]

S. Li and J. Su, Existence of multiple critical points for asymptotically quadratic functional with applications, Abst. Appl. Anal., 1 (1996), 283-305.  doi: 10.1155/S1085337596000140.  Google Scholar

[22]

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

[23]

D. Mugnai and D. Pagliardini, Existence and multiplicity results for the fractional Laplacian in bounded domains, Adv. Calc. Var. , to appear. Google Scholar

[24]

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.  Google Scholar

[25]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. , 65, American Mathematical Society, Providence, RI 1986. doi: 10.1090/cbms/065.  Google Scholar

[26]

R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal., 43 (2014), 251-267.   Google Scholar

[27]

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

[28]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non) local operators, Rev. Mat. Iberoam, 29 (2013), 1091-1126.  doi: 10.4171/RMI/750.  Google Scholar

[29]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.  Google Scholar

[30]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.   Google Scholar

[31]

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

[32]

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

[33]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[34]

J. Su, Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity, J. Differential Equations, 145 (1998), 252-273.  doi: 10.1006/jdeq.1997.3360.  Google Scholar

[35]

J. Su and C. Tang, Multiple resutlts for semilinear elliptic equations with resonance at higher eigenvalues, Nonlinear Anal., 44 (2001), 311-321.   Google Scholar

[36]

J. Su and L. Zhao, An elliptic resonance problem with multiple solutions, J. Math. Appl. Appl., 319 (2006), 604-616.  doi: 10.1016/j.jmaa.2005.10.059.  Google Scholar

[37]

Z. -Q. Wang, Multiple solutions for indefinite functionals and applications to asymptotically linear problems, Acta Math. Sinica(N.S.), 5 (1989), 101-113.  doi: 10.1007/BF02107664.  Google Scholar

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