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Stationary waves of Schrödinger-type equations with variable exponent

Dušan Repovš

Faculty of Education, and Faculty of Mathematics and Physics, University of Ljubljana, Kardeljeva ploščad 16, SI-1000 Ljubljana, Slovenia

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https://doi.org/10.1142/S0219530514500420Cited by:61 (Source: Crossref)

Abstract

We are concerned with a class of nonlinear Schrödinger-type equations with a reaction term and a differential operator that involves a variable exponent. By using related variational methods, we establish several existence results.

Keywords:

AMSC: 35J62, 35J70, 46E35, 58E05, 35M86, 47J20

References

M. J. Ablowitz , B. Prinari and A. D. Trubatch , Discrete and Continuous Nonlinear Schrödinger Systems ( Cambridge University Press , Cambridge , 2004 ) . Google Scholar
G. Afrouzi , M. Mirzapour and V. Rădulescu , Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl. 25 , 91 ( 2014 ) . Crossref, Google Scholar
A. Ambrosetti and P. Rabinowitz , J. Funct. Anal. 14 , 349 ( 1973 ) . Crossref, Google Scholar
T. Bartsch , Nonlinear Anal. 20 , 1205 ( 1993 ) . Crossref, Web of Science, Google Scholar
Y. Chen , S. Levine and M. Rao , SIAM J. Appl. Math. 66 , 1383 ( 2006 ) . Crossref, Web of Science, Google Scholar
P. G. Ciarlet , Linear and Nonlinear Functional Analysis with Applications ( SIAM , Philadelphia, PA , 2013 ) . Crossref, Google Scholar
L. Diening et al. , Lebesgue and Sobolev Spaces with Variable Exponents ( Springer , Heidelberg , 2011 ) . Crossref, Google Scholar
D. E. Edmunds and J. Rákosník , Proc. Roy. Soc. London Ser. A 437 , 229 ( 1992 ) . Crossref, Web of Science, Google Scholar
D. E. Edmunds and J. Rákosník , Studia Math. 143 , 267 ( 2000 ) . Crossref, Web of Science, Google Scholar
X. L. Fan , J. Math. Anal. Appl. 312 , 464 ( 2005 ) . Crossref, Web of Science, Google Scholar
X. L. Fan and X. Y. Han , Nonlinear Anal. 59 , 173 ( 2004 ) . Web of Science, Google Scholar
X. L. Fan and Q. H. Zhang , Nonlinear Anal. 52 , 1843 ( 2003 ) . Crossref, Web of Science, Google Scholar
T. C. Halsey , Science 258 , 761 ( 1992 ) . Crossref, Web of Science, Google Scholar
O. Kováčik and J. Rákosník , Czechoslovak Math. J. 41 , 592 ( 1991 ) . Crossref, Web of Science, Google Scholar
N. Laskin , Phys. Lett. A 268 , 298 ( 2000 ) . Crossref, Web of Science, Google Scholar
M. Mihăilescu , P. Pucci and V. Rădulescu , J. Math. Anal. Appl. 340 , 687 ( 2008 ) . Crossref, Web of Science, Google Scholar
M. Mihăilescu and V. Rădulescu , Proc. Amer. Math. Soc. 135 , 2929 ( 2007 ) . Crossref, Web of Science, Google Scholar
J. Musielak , Orlicz Spaces and Modular Spaces ( Springer , Berlin , 1983 ) . Crossref, Google Scholar
H. Nakano , Modulared Semi-Ordered Linear Spaces ( Maruzen , Tokyo , 1950 ) . Google Scholar
W. Orlicz , Studia Math. 3 , 200 ( 1931 ) . Crossref, Google Scholar
M. Reed and B. Simon , Methods of Modern Mathematical Physics IV. Analysis of Operators ( Academic Press , New York , 1978 ) . Google Scholar
M. Ruzicka , Electrorheological Fluids. Modeling and Mathematical Theory , Lecture Notes in Mathematics 1748 ( Springer , Berlin , 2000 ) . Crossref, Google Scholar
J. Simon, Journées d'Analyse Non Linéaire, Lecture Notes in Mathematics 665, eds. P. Benilan and J. Robert (Springer, Berlin, 1978) pp. 205–227. Crossref, Google Scholar
W. A. Strauss , Comm. Math. Phys. 55 , 149 ( 1977 ) . Crossref, Web of Science, Google Scholar
C. Sulem and P. L. Sulem , The Nonlinear Schrödinger Equation. Self-Focusing and Wave Collapse , Applied Mathematical Sciences 139 ( Springer , New York , 1999 ) . Google Scholar
M. Willem , Minimax Theorems ( Birkhäuser , Boston , 1996 ) . Crossref, Google Scholar
V. Zhikov , Math. USSR-Izv. 29 , 675 ( 1987 ) . Crossref, Google Scholar
V. Zhikov , Rus. J. Math. Phys. 3 , 249 ( 1995 ) . Web of Science, Google Scholar