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

Resonant problems for fractional Laplacian

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.
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.

[2]

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

[3]

B. BarriosE. ColoradoA. 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.

[4]

B. BarriosE. ColoradoR. 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.

[5]

P. BartoloV. 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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[14]

A. FiscellaR. 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.

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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.

[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.

[26]

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

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[35]

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

[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.

[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.

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.

[2]

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

[3]

B. BarriosE. ColoradoA. 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.

[4]

B. BarriosE. ColoradoR. 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.

[5]

P. BartoloV. 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.

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[12]

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

[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.

[14]

A. FiscellaR. 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.

[15]

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

[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.

[17]

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

[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.

[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.

[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.

[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.

[22]

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

[23]

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

[24]

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.

[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.

[26]

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

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[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.

[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.

[35]

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

[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.

[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.

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