Journal
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
Volume 11, Issue 4, Pages 547-583Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S021919970900348X
Keywords
Schrodinger equations; solitary waves; variational methods; Orlicz spaces; Trudinger-Moser inequality
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Funding
- Millennium Institute for the Global Advancement of Brazilian Mathematics IM-AGIMB
- CNPq/Brazil [200648/2006-3, 473929/2006-6]
- CAPES/MEC/Brazil.
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In this paper, the existence and qualitative properties of positive ground state solutions for the following class of Schrodinger equations -epsilon(2)Delta u+V(x)u-epsilon(2) [Delta(u(2))]u=f(u) in the whole two-dimensional space are established. We develop a variational method based on a penalization technique and Trudinger-Moser inequality, in a nonstandard Orlicz space context, to build up a one parameter family of classical ground state solutions which concentrates, as the parameter approaches zero, around some point at which the solutions will be localized. The main feature of this paper is that the nonlinearity f is allowed to enjoy the critical exponential growth and also the presence of the second order nonhomogeneous term -epsilon(2)[Delta(u(2))]u which prevents us from working in a classical Sobolev space. Our analysis shows the importance of the role played by the parameter epsilon for which is motivated by mathematical models in physics. Schrodinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.
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