Multiple solutions for a class of fractional quasi-linear equations with critical exponential growth in ℝ^N

References

• Nezza ED, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 2012;136:521–573.
   Web of Science ®Google Scholar
• Molica Bisci G, Rădulescu V, Servadei R. Variational methods for nonlocal fractional problems. Vol. 162, Encyclopedia of mathematics and its applications. Cambridge: Cambridge University Press; 2016. ISBN: 9781107111943.
   Google Scholar
• Laskin N. Fractional quantum mechanics and Levy path integrals. Phys. Lett. A. 2000;268:298–305.
   Web of Science ®Google Scholar
• Laskin N. Fractional Schrödinger equation. Phys. Rev. E. 2002;66:056108.
   Web of Science ®Google Scholar
• Iannizzotto A, Squassina M. 1/2-Laplacian problems with exponential nonlinearity. J. Math. Anal. Appl. 2014;414:372–385.
   Web of Science ®Google Scholar
• de Souza M. On a class of nonhomogeneous fractional quasilinear equations in ℝN with exponential growth. Nonlinear Differ. Equ. Appl. 2015;22:499–511.
   Web of Science ®Google Scholar
• Teng KM. Multiple solutions for a class of fractional Schrödinger equation in ℝN. Nonlinear Anal.: Real World Appl. 2015;21:76–86.
   Web of Science ®Google Scholar
• Felmer P, Torres C. Nonlinear Schrödinger equation with non-local regional diffusion. Calc. Var. 2015;54:75–98.
   Web of Science ®Google Scholar
• Barrios B, Colorado E, Servadei R, et al. A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincaré Anal. Non Linéaire. 2015;32:875–900.
   Web of Science ®Google Scholar
• 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.
   Web of Science ®Google Scholar
• Servadei R, Valdinoci E. Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 2012;389:887–898.
   Web of Science ®Google Scholar
• Gou TX, Sun HR. Solutions of nonlinear Schrödinger equation with fractional Laplacian without the Ambrosetti--Rabinowitz condition. Appl. Math. Comput. 2015;257:409–416.
   Web of Science ®Google Scholar
• Hua YX, Yu XH. On the ground state solution for a critical fractional Laplacian equation. Nonlinear Anal. 2013;87:116–125.
   Web of Science ®Google Scholar
• Xu JF, Wei ZL, Dong W. Existence of weak solutions for a fractional Schr\"{o}dinger equation. Commun. Nonlinear Sci. Numer. Simul. 2015;22:1215–1222.
   Web of Science ®Google Scholar
• Wei YH, Su XF. Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian. Calc. Var. 2015;52:95–124.
   Web of Science ®Google Scholar
• Secchi S. On fractional Schrödinger equations in ℝN without the Ambrosetti--Rabinowitz condition. arXiv:1210.0755 [math.AP].
   Google Scholar
• Molica Bisci G, Rădulescu V. Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. 2015;54:2985–3008.
   Web of Science ®Google Scholar
• Bisci GM, Rădulescu V. Multiplicity results for elliptic fractional equations with subcritical term. Nonlinear Differ. Equ. Appl. 2015;22:721–739.
   Web of Science ®Google Scholar
• Binlin Z, Molica Bisci G, Servadei R. Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity. 2015;28:2247–2264.
   Web of Science ®Google Scholar
• Molica Bisci G, Servadei R. A bifurcation result for non-local fractional equations. Anal. Appl. 2015;13:371–394.
   Web of Science ®Google Scholar
• Molica Bisci G, Repov D. On doubly nonlocal fractional elliptic equations. Rend. Lincei Mat. Appl. 2015;26:161–176.
   Google Scholar
• Moser J. A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 1971;20:1077–1092.
   Web of Science ®Google Scholar
• Ozawa T. On critical cases of Sobolev inequality. J. Funct. Anal. 1995;127:259–269.
   Web of Science ®Google Scholar
• Trudinger NS. On the imbedding into Orlicz spaces and some applications. J. Math. Mech. 1967;17:473–484.
   Web of Science ®Google Scholar
• Aouaoui S. On some semilinear elliptic equation involving exponential growth. Appl. Math. Lett. 2014;33:23–28.
   Web of Science ®Google Scholar
• de Freitas LR. Multiplicity of solutions for a class of quasilinear equations with exponential critical growth. Nonlinear Anal. 2014;95:607–624.
   Web of Science ®Google Scholar
• Wang Y, Yang J, Zhang Y. Quasilinear elliptic equations involving the N-Laplacian with critical exponential growth in ℝN. Nonlinear Anal. 2009;71:6157–6169.
   Web of Science ®Google Scholar
• Alves CO, de Freitas LR. Multiplicity of nonradial solutions for a class of quasilinear equations on annulus with exponential critical growth. Topol. Methods Nonlinear Anal. 2012;39:243–262.
   Web of Science ®Google Scholar
• Aouaoui S. Multiple solutions for some quasilinear equations of N-Laplacian type and containing a gradient term. Nonlinear Anal. 2015;116:64–74.
   Web of Science ®Google Scholar
• Adachi S, Watanabe T. Uniqueness and non-degeneracy of positive radial solutions for quasilinear elliptic equations with exponential nonlinearity. Nonlinear Anal. 2014;108:275–290.
   Web of Science ®Google Scholar
• Alves CO, Figueiredo GM. On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in ℝN. J. Differ. Equ. 2009;246:1288–1311.
   Web of Science ®Google Scholar
• Lam N, Lu G. Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in ℝN. J. Funct. Anal. 2012;262:1132–1165.
   Web of Science ®Google Scholar
• Li Q, Yang ZD. Multiple solutions for N-Kirchhoff type problems with critical exponential growth in ℝN. Nonlinear Anal. 2015;117:159–168.
   Web of Science ®Google Scholar
• de Souza M, do \’{O} JM. On a singular and nonhomogeneous N-Laplacian equation involving critical growth. J. Math. Anal. Appl. 2011;380:241–263.
   Web of Science ®Google Scholar
• do Ó JM. Semilinear Dirichlet problems for the N-Laplacian in ℝN with nonlinearities in critical growth range. Differ. Integral Equ. 1996;9:967–979.
   Google Scholar
• Kuzin I, Pohozaev S. Entire solutions of semilinear elliptic equations. Vol. 33, Progress in nonlinear differential equations and their applications. Boston (MA): Birkhäuser; 1997.
   Google Scholar
• Kozono H, Sato T, Wadade H. Upper bound of the best constant of a Trudinger-Moser inequality and its application to a Gagliardo--Nirenberg inequality. Indiana Univ. Math. J. 2006;55:1951–1974.
   Web of Science ®Google Scholar
• Struwe M. Variational methods, applications to nonlinear partial differential equations and Hamiltonian systems. 3rd ed. New York (NY): Springer-Verlag; 2000.
   Google Scholar
• Ambrosetti A, Rabinowitz PH. Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973;14:349–381.
   Google Scholar
• Castro AD, Kuusi T, Palatucci G. Nonlocal Harnack inequalities. J. Funct. Anal. 2014;267:1807–1836.
   Web of Science ®Google Scholar