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Resonant problems for fractional Laplacian
School of Mathematical Sciences, Capital Normal University, Beijing 100048, People's Republic of China |
$ \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. $ |
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. 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. |
[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. |
[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. |
[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. 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. |
[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 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. |
[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. 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. |
[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. |
[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. |
[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. 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. |
[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 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. |
[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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