Journal
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Volume 334, Issue 2, Pages 775-786Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2007.01.020
Keywords
mountain pass; critical growth; standing waves; quasilinear schrodinger equations
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We consider the equation -Delta u + V(x)u - k/2(Delta(\u\(2)))u = g(x, u), u > 0, x epsilon R-2, where V:R-2 --> R and g:R-2 x R --> R are two continuous 1-periodic functions and k is a positive constant. Also, we assume g behaves like exp(beta\u\(4)) as \u\ --> infinity. Weprove the existence of at least one weak solution u epsilon H-1(R2) with u(2) epsilon H-1 (R-2). The mountain pass in a suitable Orlicz space together with the Trudinger-Moser inequality are employed to establish this result. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schrodinger equations. 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. (c) 2007 Elsevier lnc. All rights reserved.
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