MULTI-PEAK SOLUTIONS FOR LOGARITHMIC SCHRO?DINGER EQUATIONS WITH POTENTIALS UNBOUNDED BELOW

摘要

In this paper, we consider the following logarithmic Schro & BULL;dinger equation -& epsilon;2 & UDelta;u + V (x)u = u log u2 in RN, where & epsilon; > 0, N & GE; 1, V (x) & ISIN; C(RN, R) is a continuous potential which can be unbounded below. By variational methods and penalized idea, we show that the problem has a family of solutions u & epsilon; concentrating at any finite given local minima of V. Our results generalize the single peak case in [36] to the multi-peak case but the penalization in this paper is different.

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