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PAPER: Existence of stable standing waves and instability of standing waves to a class of quasilinear Schrodinger equations with potential

REF. DATE:9 February 2011 CREATED/MODIFIED: 14 January 2016 380 views No Comment

Existence of stable standing waves and instability of standing waves to a class of quasilinear Schrödinger equations with potential (with J. Chen), Dynamics of PDEs 8:2(2011), 89-112.


ABSTRACT: We consider the following quasilinear Schrödinger equation with harmonic potential

(1)   \begin{equation*} i\varphi_t = -\triangle\varphi + |x|^2\varphi -|\varphi|^{p-1}\varphi - 2(\triangle|\varphi|^2)\varphi,\quad t\geq 0,\ x\in \mathbb{R}^N, \end{equation*}

where i^2 = 1, \varphi\equiv\varphi(t,x):\ \mathbb{R}_+\times\mathbb{R}^N\to \mathbb{C} is a complex-valued function and \triangle=\sum_{j=1}^N{{\partial^2}\over{\partial x_j^2}} is the standard Laplacian operator. We are concerned with stability and instability of standing wave solutions for (1). We will prove the existence of stable standing waves for 1 < p < 3 + {4\over N} and the existence of unstable standing waves for 3 + {4\over N}\leq p < 2\cdot 2^\ast - 1 (here and after, 2^\ast denotes the critical exponent, i.e. 2^\ast = {{2N}\over{N-2}} for N \geq 3 and 2^\ast = +\infty for N =1,\ 2). Our result indicates that the quasilinear term (\triangle|\varphi|^2)\varphi makes the standing wave more stable than their counterpart in the semilinear case, which is consistent with the physical phenomena and is in striking contrast with the classical semilinear Schrödinger equation.


Download: Preprint version (RIA |  download) / Publisher version (doi:10.4310/DPDE.2011.v8.n2.a2)
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