Existence of nontrivial solutions for asymptotically linear periodic Schrödinger equations

References

• Willem M. Minimax theorems, progress in nonlinear differential equations and their applications. Vol. 24. Boston (MA): Birkhäuser Boston; 1996.
   Google Scholar
• Coti Zelati V, Rabinowitz P. Homoclinic type solutions for a semilinear elliptic PDE on ℝn. Comm. Pure Appl. Math. 1992;45:1217–1269.
   Web of Science ®Google Scholar
• Jeanjean L. On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on ℝN. Proc. Roy. Soc. Edinb. 1999;129A:787–809.
   Google Scholar
• Li YQ, Wang ZQ, Zeng J. Ground states of nonlinear Schrödinger equations with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire. 2006;23:829–837.
   Web of Science ®Google Scholar
• Ding YH, Lee C. Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms. J. Differ. Equ. 2006;222:137–163.
   Web of Science ®Google Scholar
• Kryszewski W, Szulkin A. Generalized linking theorem with an application to a semilinear Schrödinger equation. Adv. Differ. Equ. 1998;3:441–472.
   Google Scholar
• Schechter M, Zou WM. Weak linking theorems and Schrödinger equations with critical Sobolev exponent. ESAIM Control Optim. Calc. Var. 2003;9:601–619.
   Google Scholar
• Szulkin A, Zou W. Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 2001;187:25–41.
   Web of Science ®Google Scholar
• Liu S. On superlinear Schrödinger equations with periodic potential. Calc. Var. Par. Differ. Equ. 2012;45:1–9.
   Web of Science ®Google Scholar
• Kryszewski W, Szulkin A. Infinite-dimensional homology and multibump solutions. J. Fixed Point Theory Appl. 2009;5:1–35.
   Web of Science ®Google Scholar
• Schechter M. Superlinear Schrödinger operators. J. Funct. Anal. 2012;262:2677–2694.
   Web of Science ®Google Scholar
• Szulkin A, Weth T. Ground state solutions for some indefinite variational problems. J. Funct. Anal. 2009;257:3802–3822.
   Web of Science ®Google Scholar
• Yang M. Ground state solutions for a periodic Schrödinger equation with superlinear nonlinearities. Nonlinear Anal. 2010;72:2620–2627.
   Web of Science ®Google Scholar
• Yang M, Chen W, Ding YH. Solutions for periodic Schrödinger equation with spectrum zero and general superlinear nonlinearities. J. Math. Anal. Appl. 2010;364:404–413.
   Web of Science ®Google Scholar
• Li GB, Szulkin A. An asymptotically periodic Schrödinger equation with indefinite linear part. Commun. Contemp. Math. 2002;4:763–776.
   Web of Science ®Google Scholar
• Ackermann N. A superposition principle and multibump solutions of periodic Schrödinger equations. J. Func. Anal. 2006;234:277–320.
   Web of Science ®Google Scholar
• Alama S, Li YY. On multibump bound states for certain semilinear elliptic equations. Indiana Univ. Math. J. 1992;41:983–1026.
   Web of Science ®Google Scholar
• Chen S. Multi-bump solutions for a strongly indefinite semilinear Schrödinger equation without symmetry or convexity assumptions. Nonlinear Anal. 2008;68:3067–3102.
   Web of Science ®Google Scholar
• Heinz H-P, Küpper T, Stuart CA. Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation. J. Differ. Equ. 1992;100:341–354.
   Web of Science ®Google Scholar
• Pankov A, Pflüger K. On a semilinear Schrödinger equation with periodic potential. Nonlinear Anal. 1998;33:593–609.
   Web of Science ®Google Scholar
• Pankov A. Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 2005;73:259–287.
   Google Scholar
• Troestler C. Bifurcation into spectral gaps for a noncompact semilinear Schrödinger equation with nonconvex potential. Forthcoming. Available from: http://arxiv.org/pdf/1207.1052.pdf.
   Google Scholar
• Troestler C, Willem M. Nontrivial solution of a semilinear Schrödinger equation. Comm. Par. Differ. Equ. 1996;21:1431–1449.
   Web of Science ®Google Scholar
• Willem M, Zou W. On a Schrödinger equation with periodic potential and spectrum point zero. Indiana Univ. Math. J. 2003;52:109–132.
   Web of Science ®Google Scholar
• Ding YH. Variational methods for strongly indefinite problems. Vol. 7, Interdisciplinary mathematical sciences. Hackensack (NJ): World Scientific Publishing; 2007.
   Google Scholar
• Hempel R, Voigt J. The spectrum of a Schrödinger operator in Lp(ℝV) is p-independent. Comm. Math. Phys. 1986;104:243–250.
   Web of Science ®Google Scholar
• Gilbarg D, Trudinger N. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in mathematics. Berlin: Springer-Verlag; 1998.
   Google Scholar
• Adams R, Fournier JF. Sobolev spaces. 2nd ed. Vol. 140, Pure and applied mathematics (Amsterdam). Amsterdam: Elsevier/Academic Press; 2003.
   Google Scholar
• Rudin W. Functional analysis. 2nd ed., International series in pure and applied mathematics. New York (NY): McGraw-Hill; 1991.
   Google Scholar
• Jerison D, Kenig C. Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. of Math (2). 1985;121:463–494.
   Web of Science ®Google Scholar
• Rudin W. Real and complex analysis. 2nd ed., McGraw-Hill series in higher mathematics. New York (NY): McGraw-Hill Book; 1974.
   Google Scholar
• Simon B. Spectrum and continuum eigenfunctions of Schrödinger operators. J. Funct. Anal. 1981;42:347–355.
   Web of Science ®Google Scholar
• Simon B. Schrödinger semigroups. Bull. Amer. Math. Soc. (N.S.). 1982;7:447–526.
   Web of Science ®Google Scholar