Multi-peak positive solutions for nonlinear Schr�dinger equations with critical frequency

Sato, Yohei
July 2007
Calculus of Variations & Partial Differential Equations;Jul2007, Vol. 29 Issue 3, p365
Academic Journal
We study the nonlinear Schr�dinger equations: $$-\epsilon^{2}\Delta u + V(x)u=u^p,\quad u > 0\quad \mbox{in } {\bf R}^{N},\quad u\in H^{1} ({\bf R}^{N}).$$ where p > 1 is a subcritical exponent and V( x) is nonnegative potential function which has �critical frequency� $$\inf_{x\in{\bf R}^{N}} V(x)=0$$ . We also assume that V( x) satisfies $$0 < \liminf_{|x|\to\infty}V(x)\le \sup_{x\in{\bf R}^{N}}V(x) < \infty$$ and V( x) has k local or global minima. In critical frequency cases, Byeon-Wang [5,6] showed the existence of single-peak solutions which concentrating around global minimum of V( x). Their limiting profiles�which depend on the local behavior of the potential V( x)�are quite different features from non-critical frequency case. We show the existence of multi-peak positive solutions joining single-peak solutions which concentrate around prescribed local or global minima of V( x). Moreover, under additional conditions on the behavior of V( x), we state the limiting profiles of peaks of solutions u e( x) as follows: rescaled function $$w_\epsilon(y)=\left(\frac{g(\epsilon)}{\epsilon}\right)^{\frac{2}{p-1}} u_\epsilon(g(\epsilon)y+x_\epsilon)$$ converges to a least energy solution of -? w + V 0( y) w = w p , w > 0 in O0, $$w\in H^{1}_0(\Omega_0)$$ . Here g(e), V 0( x) and O0 depend on the local behaviors of V( x).


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