American Institute of Mathematical Sciences

Advanced
October  2014, 19(8): 2483-2499. doi: 10.3934/dcdsb.2014.19.2483

Existence of weak solutions for non-local fractional problems via Morse theory

Massimiliano Ferrara 1, , Giovanni Molica Bisci 2, and  Binlin Zhang 3,

1. 

University of Reggio Calabria and CRIOS University Bocconi of Milan, Via dei Bianchi presso Palazzo Zani, 89127 Reggio Calabria, Italy
2. 

Dipartimento P.A.U., Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di Vito, 89100 Reggio Calabria
3. 

Department of Mathematics, Heilongjiang Institute of Technology, 150050 Harbin, China

Received  November 2013 Revised  February 2014 Published  August 2014

In this paper, we study the existence of non-trivial solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. Non-trivial solutions are obtained by computing the critical groups and Morse theory. Our results extend some classical theorems for semilinear elliptic equations to the non-local fractional setting.
Keywords: Fractional Laplacian, Morse theory., integro-differential operator, critical groups.
Mathematics Subject Classification: 35J60, 35J91, 58E0.
Citation: Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483
References:
[1]

C. O. Alves and S. B. Liu, On superlinear $p(x)$-Laplacian equations in $\mathbbR^N$, Nonlinear Anal., 73 (2010), 2566-2579. doi: 10.1016/j.na.2010.06.033.  Google Scholar

[2]

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

[3]

T. Bartsch, and S. J. 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

[4]

K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  Google Scholar

[5]

K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys., 271 (2007), 179-198. doi: 10.1007/s00220-006-0178-y.  Google Scholar

[6]

D. Bors, Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstr. Appl. Anal., (2013), Art. ID 240863, 10 pp.  Google Scholar

[7]

D. Bors, Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian, Scientific World Journal, (2014), Art. ID 920537, 10 pp. doi: 10.1155/2014/920537.  Google Scholar

[8]

C. Brändle, E. Colorado, A.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

[9]

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

[10]

L. Caffarelli, J. M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.} doi: 10.4171/JEMS/226.  Google Scholar

[11]

L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

S. Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.  Google Scholar

[16]

M. del Mar González, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differential Equations, 36 (2009), 173-210. doi: 10.1007/s00526-009-0225-6.  Google Scholar

[17]

F. Fang, and S. B. Liu, Nontrivial solutions of superlinear $p$-Laplacian equations, J. Math. Anal. Appl., 351 (2009), 138-146. doi: 10.1016/j.jmaa.2008.09.064.  Google Scholar

[18]

A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators, Z. Anal. Anwendungen, 32 (2013), 411-431. doi: 10.4171/ZAA/1492.  Google Scholar

[19]

D. Idczak and R. Kamocki, On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in $\mathbbR^n$, Fractional Calculus & Applied Analysis, 14 (2011), 538-553. doi: 10.2478/s13540-011-0033-5.  Google Scholar

[20]

D. Idczak and R. Kamocki, Fractional differential repetitive process with Riemann-Liouville and Caputo derivatives, Multidim. Syst. Sign Process, (2013). doi: 10.1007/s11045-013-0249-0.  Google Scholar

[21]

D. Idczak, R. Kamocki and M. Majewski, Fractional continuous Roesser model with Reimann-Liouville derivative, 8th Int. workshop on multidimensional Systems, (2013), 33-38. Google Scholar

[22]

D. Idczak, R. Kamocki and M. Majewski, On a fractional continuous counterpart of Fornasini-Marchesini model, 8th Int. Workshop on Multidimensional Systems, (2013), 45-49. Google Scholar

[23]

D. Idczak and M. Majewski, Fractional fundamental lemma of order $\alpha\in (n-\frac{1}{2}, n)$ with $n\in\mathbbN, n\geq2$, Dynamic System and Applications, 12 (2012), 251-268.  Google Scholar

[24]

D. Idczak and M. Majewski, Compactness of fractional embeddings for boundary value problems, IEEE, 7 (2013), 599-603. Google Scholar

[25]

D. Idczak, A. Skowron and S. Walczak, Sensitivity of a fractional integrodifferential Cauchy problem of Volterra type, Abstract and Applied Analysis, 2013 (2013), 1-8.  Google Scholar

[26]

D. Idczak, and S. Walczak, A fractional imbedding theorem, Fractional Calculus & Applied Analysis, 15 (2012), 418-425. doi: 10.2478/s13540-012-0030-3.  Google Scholar

[27]

D. Idczak and S. Walczak, Compactness of fractional imbeddings, IEEE, 2 (2012), 585-588. doi: 10.1109/MMAR.2012.6347820.  Google Scholar

[28]

D. Idczak and S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, Journal of Function Spaces and Applications, 2013 (2013), 1-15.  Google Scholar

[29]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.  Google Scholar

[30]

K. Kaleta and T. Kulczycki, Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians, Potential Anal., 33 (2010), 313-339. doi: 10.1007/s11118-010-9170-4.  Google Scholar

[31]

R. Kamocki and M. Majewski, On a fractional Dirichlet problem, IEEE, 2 (2012), 60-63. doi: 10.1109/MMAR.2012.6347911.  Google Scholar

[32]

A. Kristály, V. Rădulescu and Cs. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511760631.  Google Scholar

[33]

S. B. Liu, On ground states of superlinear $p$-Laplacian equations in $\mathbbR^N$, J. Math. Anal. Appl., 361 (2010), 48-58. doi: 10.1016/j.jmaa.2009.09.016.  Google Scholar

[34]

S. B. Liu, On superlinear problems without Ambrosetti-Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795. doi: 10.1016/j.na.2010.04.016.  Google Scholar

[35]

S. B. Liu, Existence of solutions to a superlinear $p$-Laplacian equation, Electron J. Differential Equations, 66 (2001), 1-6.  Google Scholar

[36]

J. Q. Liu and J. B. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222. doi: 10.1006/jmaa.2000.7374.  Google Scholar

[37]

Q. S. Liu and J. B. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601. Google Scholar

[38]

G. Molica Bisci, Fractional equations with bounded primitive, Appl. Math. Lett., 27 (2014), 53-58. doi: 10.1016/j.aml.2013.07.011.  Google Scholar

[39]

G. Molica Bisci, Sequences of weak solutions for fractional equations, Math. Res. Lett., 21 (2014), 1-13. Google Scholar

[40]

G. Molica Bisci and B. A. Pansera, Three weak solutions for nonlocal fractional equations, Adv. Nonlinear Stud., 14 (2014), 591-601. Google Scholar

[41]

G. Molica Bisci and D. Repovs, Higher nonlocal problems with bounded potential, J. Math. Anal. Appl., 420 (2014), 167-176. Google Scholar

[42]

G. Molica Bisci and R. Servadei, A bifurcation result for non-local fractional equations,, to appear in Analysis and Applications., ().  doi: 10.1142/S0219530514500067.  Google Scholar

[43]

K. Perera, R. P. Agarwal and D. O'Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operatiors, Mathematical Surveys and Monographs, 161, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/161.  Google Scholar

[44]

X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350 (2012), 505-508. doi: 10.1016/j.crma.2012.05.011.  Google Scholar

[45]

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

[46]

S. Serfaty and J. Luis Vazquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calc. Var. Partial Differential Equations, 49 (2014), 1091-1120. doi: 10.1007/s00526-013-0613-9.  Google Scholar

[47]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., 595 (2013), 317-340. doi: 10.1090/conm/595/11809.  Google Scholar

[48]

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

[49]

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

show all references

References:
[1]

C. O. Alves and S. B. Liu, On superlinear $p(x)$-Laplacian equations in $\mathbbR^N$, Nonlinear Anal., 73 (2010), 2566-2579. doi: 10.1016/j.na.2010.06.033.  Google Scholar

[2]

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

[3]

T. Bartsch, and S. J. 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

[4]

K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  Google Scholar

[5]

K. Bogdan and T. Jakubowski, Estimates of heat kernel of fractional Laplacian perturbed by gradient operators, Comm. Math. Phys., 271 (2007), 179-198. doi: 10.1007/s00220-006-0178-y.  Google Scholar

[6]

D. Bors, Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstr. Appl. Anal., (2013), Art. ID 240863, 10 pp.  Google Scholar

[7]

D. Bors, Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian, Scientific World Journal, (2014), Art. ID 920537, 10 pp. doi: 10.1155/2014/920537.  Google Scholar

[8]

C. Brändle, E. Colorado, A.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

[9]

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

[10]

L. Caffarelli, J. M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179.} doi: 10.4171/JEMS/226.  Google Scholar

[11]

L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

S. Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.  Google Scholar

[16]

M. del Mar González, Gamma convergence of an energy functional related to the fractional Laplacian, Calc. Var. Partial Differential Equations, 36 (2009), 173-210. doi: 10.1007/s00526-009-0225-6.  Google Scholar

[17]

F. Fang, and S. B. Liu, Nontrivial solutions of superlinear $p$-Laplacian equations, J. Math. Anal. Appl., 351 (2009), 138-146. doi: 10.1016/j.jmaa.2008.09.064.  Google Scholar

[18]

A. Fiscella, R. Servadei and E. Valdinoci, A resonance problem for non-local elliptic operators, Z. Anal. Anwendungen, 32 (2013), 411-431. doi: 10.4171/ZAA/1492.  Google Scholar

[19]

D. Idczak and R. Kamocki, On the existence and uniqueness and formula for the solution of R-L fractional Cauchy problem in $\mathbbR^n$, Fractional Calculus & Applied Analysis, 14 (2011), 538-553. doi: 10.2478/s13540-011-0033-5.  Google Scholar

[20]

D. Idczak and R. Kamocki, Fractional differential repetitive process with Riemann-Liouville and Caputo derivatives, Multidim. Syst. Sign Process, (2013). doi: 10.1007/s11045-013-0249-0.  Google Scholar

[21]

D. Idczak, R. Kamocki and M. Majewski, Fractional continuous Roesser model with Reimann-Liouville derivative, 8th Int. workshop on multidimensional Systems, (2013), 33-38. Google Scholar

[22]

D. Idczak, R. Kamocki and M. Majewski, On a fractional continuous counterpart of Fornasini-Marchesini model, 8th Int. Workshop on Multidimensional Systems, (2013), 45-49. Google Scholar

[23]

D. Idczak and M. Majewski, Fractional fundamental lemma of order $\alpha\in (n-\frac{1}{2}, n)$ with $n\in\mathbbN, n\geq2$, Dynamic System and Applications, 12 (2012), 251-268.  Google Scholar

[24]

D. Idczak and M. Majewski, Compactness of fractional embeddings for boundary value problems, IEEE, 7 (2013), 599-603. Google Scholar

[25]

D. Idczak, A. Skowron and S. Walczak, Sensitivity of a fractional integrodifferential Cauchy problem of Volterra type, Abstract and Applied Analysis, 2013 (2013), 1-8.  Google Scholar

[26]

D. Idczak, and S. Walczak, A fractional imbedding theorem, Fractional Calculus & Applied Analysis, 15 (2012), 418-425. doi: 10.2478/s13540-012-0030-3.  Google Scholar

[27]

D. Idczak and S. Walczak, Compactness of fractional imbeddings, IEEE, 2 (2012), 585-588. doi: 10.1109/MMAR.2012.6347820.  Google Scholar

[28]

D. Idczak and S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, Journal of Function Spaces and Applications, 2013 (2013), 1-15.  Google Scholar

[29]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc. Edinburgh, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.  Google Scholar

[30]

K. Kaleta and T. Kulczycki, Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians, Potential Anal., 33 (2010), 313-339. doi: 10.1007/s11118-010-9170-4.  Google Scholar

[31]

R. Kamocki and M. Majewski, On a fractional Dirichlet problem, IEEE, 2 (2012), 60-63. doi: 10.1109/MMAR.2012.6347911.  Google Scholar

[32]

A. Kristály, V. Rădulescu and Cs. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511760631.  Google Scholar

[33]

S. B. Liu, On ground states of superlinear $p$-Laplacian equations in $\mathbbR^N$, J. Math. Anal. Appl., 361 (2010), 48-58. doi: 10.1016/j.jmaa.2009.09.016.  Google Scholar

[34]

S. B. Liu, On superlinear problems without Ambrosetti-Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795. doi: 10.1016/j.na.2010.04.016.  Google Scholar

[35]

S. B. Liu, Existence of solutions to a superlinear $p$-Laplacian equation, Electron J. Differential Equations, 66 (2001), 1-6.  Google Scholar

[36]

J. Q. Liu and J. B. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258 (2001), 209-222. doi: 10.1006/jmaa.2000.7374.  Google Scholar

[37]

Q. S. Liu and J. B. Su, Existence and multiplicity results for Dirichlet problems with $p$-Laplacian, J. Math. Anal. Appl., 281 (2003), 587-601. Google Scholar

[38]

G. Molica Bisci, Fractional equations with bounded primitive, Appl. Math. Lett., 27 (2014), 53-58. doi: 10.1016/j.aml.2013.07.011.  Google Scholar

[39]

G. Molica Bisci, Sequences of weak solutions for fractional equations, Math. Res. Lett., 21 (2014), 1-13. Google Scholar

[40]

G. Molica Bisci and B. A. Pansera, Three weak solutions for nonlocal fractional equations, Adv. Nonlinear Stud., 14 (2014), 591-601. Google Scholar

[41]

G. Molica Bisci and D. Repovs, Higher nonlocal problems with bounded potential, J. Math. Anal. Appl., 420 (2014), 167-176. Google Scholar

[42]

G. Molica Bisci and R. Servadei, A bifurcation result for non-local fractional equations,, to appear in Analysis and Applications., ().  doi: 10.1142/S0219530514500067.  Google Scholar

[43]

K. Perera, R. P. Agarwal and D. O'Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operatiors, Mathematical Surveys and Monographs, 161, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/161.  Google Scholar

[44]

X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350 (2012), 505-508. doi: 10.1016/j.crma.2012.05.011.  Google Scholar

[45]

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

[46]

S. Serfaty and J. Luis Vazquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calc. Var. Partial Differential Equations, 49 (2014), 1091-1120. doi: 10.1007/s00526-013-0613-9.  Google Scholar

[47]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., 595 (2013), 317-340. doi: 10.1090/conm/595/11809.  Google Scholar

[48]

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

[49]

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

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