# American Institute of Mathematical Sciences

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

 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.
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. 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. 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. 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. 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. 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). Google Scholar [7] D. Bors, Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian,, Scientific World Journal, (2014). 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. 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. 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. 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. 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. 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. doi: 10.3934/cpaa.2011.10.1645. Google Scholar [14] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems,, Birkhäuser, (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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [24] D. Idczak and M. Majewski, Compactness of fractional embeddings for boundary value problems,, IEEE, 7 (2013), 599. 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. Google Scholar [26] D. Idczak, and S. Walczak, A fractional imbedding theorem,, Fractional Calculus & Applied Analysis, 15 (2012), 418. doi: 10.2478/s13540-012-0030-3. Google Scholar [27] D. Idczak and S. Walczak, Compactness of fractional imbeddings,, IEEE, 2 (2012), 585. 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. 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. 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. doi: 10.1007/s11118-010-9170-4. Google Scholar [31] R. Kamocki and M. Majewski, On a fractional Dirichlet problem,, IEEE, 2 (2012), 60. 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, (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. 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. 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. 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. 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. Google Scholar [38] G. Molica Bisci, Fractional equations with bounded primitive,, Appl. Math. Lett., 27 (2014), 53. 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. Google Scholar [40] G. Molica Bisci and B. A. Pansera, Three weak solutions for nonlocal fractional equations,, Adv. Nonlinear Stud., 14 (2014), 591. Google Scholar [41] G. Molica Bisci and D. Repovs, Higher nonlocal problems with bounded potential,, J. Math. Anal. Appl., 420 (2014), 167. 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, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). Google Scholar [7] D. Bors, Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian,, Scientific World Journal, (2014). 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. 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. 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. 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. 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. 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. doi: 10.3934/cpaa.2011.10.1645. Google Scholar [14] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems,, Birkhäuser, (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. 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. 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. 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. 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. 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. 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. 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. Google Scholar [24] D. Idczak and M. Majewski, Compactness of fractional embeddings for boundary value problems,, IEEE, 7 (2013), 599. 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. Google Scholar [26] D. Idczak, and S. Walczak, A fractional imbedding theorem,, Fractional Calculus & Applied Analysis, 15 (2012), 418. doi: 10.2478/s13540-012-0030-3. Google Scholar [27] D. Idczak and S. Walczak, Compactness of fractional imbeddings,, IEEE, 2 (2012), 585. 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. 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. 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. doi: 10.1007/s11118-010-9170-4. Google Scholar [31] R. Kamocki and M. Majewski, On a fractional Dirichlet problem,, IEEE, 2 (2012), 60. 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, (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. 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. 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. 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. 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. Google Scholar [38] G. Molica Bisci, Fractional equations with bounded primitive,, Appl. Math. Lett., 27 (2014), 53. 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. Google Scholar [40] G. Molica Bisci and B. A. Pansera, Three weak solutions for nonlocal fractional equations,, Adv. Nonlinear Stud., 14 (2014), 591. Google Scholar [41] G. Molica Bisci and D. Repovs, Higher nonlocal problems with bounded potential,, J. Math. Anal. Appl., 420 (2014), 167. 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, (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. 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. 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. 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. 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. 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. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar
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