Asymptotically linear fractional Schrödinger equations

References

• Metzler R, Klafter J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 2000;339:1–77.
   Web of Science ®Google Scholar
• Metzler R, Klafter J. The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A. 2004;37:161–208.
   Google Scholar
• Laskin N. Fractional Schrödinger equations. Phys. Rev. E. 2002;66:056108.
   Web of Science ®Google Scholar
• Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 2012;136:512–573.
   Web of Science ®Google Scholar
• Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 2007;32:1245–1260.
   Web of Science ®Google Scholar
• Lopes O, Maris M. Symmetry of minimizers for some nonlocal variational problems. J. Funct. Anal. 2008;254:535–592.
   Web of Science ®Google Scholar
• Byeon J, Jeanjean L, Maris M. Symmetry and monotonicity of least energy solutions. Calc. Var. Partial Differ. Equ. 2009;36:481–492.
   Web of Science ®Google Scholar
• Lehrer R, Maia LA. Positive solutions to asymptotically linear equations via Pohǒzaev manifold. J. Funct. Anal. 2014;266:213–246.
   Web of Science ®Google Scholar
• Serrin J, Tang M. Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. J. Math. 2000;49:897–923.
   Web of Science ®Google Scholar
• Frank RL, Lenzmann E. Uniqueness and non degeneracy of ground states for (− Δ)s Q + Q −Qα + 1 = 0 in ℝ. Acta Math. 2013;210:261–318.
   Web of Science ®Google Scholar
• Frank RL, Lenzmann E, Silvestre L. Uniqueness of radial solutions for the fractional laplacian. Forthcoming
   Google Scholar
• Stuart CA, Zhou HS. Applying the mountain pass theorem to an asymptotically linear elliptic equation on ℝn. Comm. Partial Differ. Equ. 1999;9–10:1731–1758.
   Web of Science ®Google Scholar
• Stuart CA. Guidance properties of nonlinear planar waveguides. Arch. Rational Mech. Anal. 1993;125:145–200.
   Web of Science ®Google Scholar
• Chang X. Ground state solutions of asymptotically linear fractional Schrödinger equations. J. Math. Phys. 2013;54:061504.
   Web of Science ®Google Scholar
• Chang X, Wang Z-Q. Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity. 2013;26: 479–494.
   Web of Science ®Google Scholar
• Felmer P, Quaas A, Tan J. Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. Royal Soc. Edinburgh A. 2012;142:1237–1262.
   Web of Science ®Google Scholar
• Costa DG, Tehrani H. On a class of asymptotically linear elliptic problems in ℝn. J. Differ. Equ. 2001;173:470–494.
   Web of Science ®Google Scholar
• Azzollini A, Pomponio A. On the Schrödinger equation in ℝn under the effect of a general nonlinear term. Indiana Univ. Math. J. 2009;58:1361–1378.
   Web of Science ®Google Scholar
• Jeanjean L, Tanaka K. A remark on least energy solutions in ℝN. Proc. Am. Math. Soc. 2002;131:2399–2408.
   Web of Science ®Google Scholar
• Ambrosetti A, Rabinowitz PH. Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973;14:349–381.
   Google Scholar
• Lang S. Differential and riemannian manifolds. 3rd ed. Graduate Texts in mathematics. New York (NY): Springer-Verlag; 1995.
   Google Scholar
• Ambrosetti A, Cerami G, Ruiz D. Solitons of linearly coupled systems of semilinear non-autonomous equations on ℝn. J. Funct. Anal. 2008;254:2816–2845.
   Web of Science ®Google Scholar
• Spradlin GS. Existence of solutions to a hamiltonian system without convexity condition on the nonlinearity. Electron. J. Differ. Equ. 2004;21:1–13.
   Google Scholar
• Spradlin GS. Interacting near-solutions of a hamiltonian system. Calc. Var. Partial Differ. Equ. 2005;22:447–464.
   Web of Science ®Google Scholar
• Lions PL. The concentration-compactness principle in the calculus of variations. The locally compact case. Ann. Inst. Henri Poincarè Analyse Non Linèaire. 1984;1: 109–145 and 223–283.
   Web of Science ®Google Scholar
• Ekeland I. Convexity methods in Hamiltonian mechanics. New York (NY): Springer-Verlag; 1990.
   Google Scholar
• Ghoussoub N, Preiss D. A general mountain pass principle for locating and classifying critical points. Ann. Inst. Henri Poincarè Analyse Non Linèaire. 1989;6:321–330.
   Web of Science ®Google Scholar
• Willem M Minimax theorems. Vol. 24. Boston (MA): Birkhauser; 1996.
   Google Scholar
• Bartolo P, Benci V, Fortunato D. Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 1983;7:981–1012.
   Web of Science ®Google Scholar