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Applicable Analysis
An International Journal
Volume 96, 2017 - Issue 3
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Original Articles

Existence and multiplicity of solutions for nonlocal systems involving fractional Laplacian with non-differentiable terms

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Pages 528-548 | Received 13 May 2015, Accepted 15 Jan 2016, Published online: 07 Feb 2016

References

  • Clarke F. Optimization and nonsmooth analysis. New York (NY): Wiley; 1983.
  • Applebaum D. Lévy process-from probability to finance and quantum groups. Not. Am. Math. Soc. 2004;51:1336–1347.
  • Laskin N. Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A. 2000;268:298–305.
  • Valdinoci E. From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. SMA. 2009;49:33–44.
  • Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 2007;32:1245–1260.
  • Barrios B, Colorado E, de Pablo A, et al. On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 2012;252:6133–6162.
  • Cabré X, Tan J. Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 2010;224:2052–2093.
  • Tan J. The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differ. Equ. 2011;42:21–41.
  • Servadei R, Valdinoci E. Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 2012;389:887–898.
  • Servadei R, Valdinoci E. Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 2013;33:2105–2137.
  • Servadei R, Valdinoci E. Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators. Rev. Mat. Iberoam. 2013;29:1091–1126.
  • Teng K. Multiplicity results for hemivariational inequalities driven by nonlocal elliptic operators. J. Math. Anal. Appl. 2012;396:386–395.
  • Teng K. Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators. Nonlinear Anal. RWA. 2013;14:867–874.
  • Gasinki L, Papageorgiou N. Nonsmooth critical point theory and nonlinear boundary value problems. Boca Raton: Chapman Hall/CRC; 2005.
  • Boccardo L, de Figuriredo DG. Some remarks on a system of quasilinear elliptic equations. Nonlinear Differ. Equ. Appl. 2002;9:309–323.
  • Chang KC, Wang ZQ, Zhang T. On a new index theory and non semi-trivial solutions for elliptic systems. Discrete Contin. Dyn. Sys. 2010;28:809–826.
  • Costa DG, Magalhâes CA. A variational approach to subquadratic perturbations of elliptic systems. J. Differ. Equ. 1994;111:103–122.
  • Costa DG, Magalhâes CA. A variational approach to noncooperative elliptic systems. Nonlinear Anal. 1995;25:699–715.
  • Bartsch T, Ding YH. Homoclinic solutions of an infinite-dimensional Hamiltonian system. Math. Z. 2002;240:289–310.
  • Clément P, Felmer P, Mitidieri E. Homoclinic orbits for a class of infinite dimensional Hamiltonian systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 1987;24:367–393.
  • Brecknera BE, Horváth A, Varga C. A multiplicity result for a special class of gradient-type systems with nondifferentiable term. Nonlinear Anal. 2009;70:606–620.
  • Kristály A. An existence result for gradient-type systems with a nondifferentiable term on unbounded strips. J. Math. Anal. Appl. 2004;299:186–204.
  • Kristály A. Infinitely many solutions for a differential inclusion problem in ℝN. J. Differ. Equ. 2006;220:511–530.
  • Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973;14:349–381.
  • Szulkin A. Critical point theory of Ljusternik-Schnirelmann type and applications to partial differential equations. In: Part 2 of the proceedings: variational methods in nonlinear problems; Montréal; 1989.
  • Chang KC. Variational methods of the NATO ASI for non-differential functions and their applications to partial differential equations. J. Math. Anal. Appl. 1981;80:102–129.
  • Chang KC, Shi SZ. A local minimax theorem without compactness. Nonlinear and convex analysis proceedings in honor of Ky Fan. New York (NY): Marcel Dekker; 1987. p. 211–233.
  • Wu X. A new critical point theorem for locally Lipschitz functionals with applications to differential equations. Nonlinear Anal. TMA. 2007;66:624–638.
  • Kristly A, Motreanu V, Varga C. A minimax principle with general Palais-Smale conditions. Commun. Appl. Anal. 2005;9:285–299.
  • Zhong CK, Fan XL, Chen WY. Introduction of nonlinear functional analysis. Lanzhou: Lanzhou University Publishing House; 1998.
  • Wu X. Some critical point theorems on the product spaces and high energy solutions of systems of Kirchhoff-type equations in ℝN. J. Math. Phy. 2012;53:063508.
  • Kristály A. Existence of nonzero weak solutions for a class of elliptic variational inclusions systems in ℝN. Nonlinear Anal. 2006;65:1578–1594.
  • Zeidler E. Nonlinear functional analysis and its applications. Vol. II/B. Berlin-Heidelberg-New York (NY): Springer-Verlag; 1985.

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