Abstract
In this paper, we consider the boundary value problem for the following quasilinear Schrödinger equation: \begin{align*} & \int_\Omega\sum_{ij=1}^N a_{ij}(x, u)D_i uD_j\varphi dx+\frac{1}{2}\int_\Omega\sum_{ij=1}^ND_sa_{ij}(x, u)D_iuD_ju\varphi dx\\ & \notag \quad +\displaystyle\int_{\partial\Omega}g(x)u\varphi dx=\int_{\partial\Omega}f(x, u)\varphi dx, \forall \varphi\in C^\infty(\bar\Omega), \tag*{(P)} \end{align*} where $\Omega\subset\mathbb{R}^N (N\geq 3)$ is a smooth bounded domain, $D_i=\frac{\partial}{\partial x_i},$ $ D_sa_{ij}(x, s)=\frac{\partial}{\partial s}a_{ij}(x, s). $ These kind of equations include the so-called Modified Nonlinear Schrödinger Equation (MNLS). By using a perturbation method, we prove the existence of infinitely many solutions for the problem (P).
Citation
Yuxia Guo. Xiangqing Liu. "On the boundary value problem for some quasilinear equations." Adv. Differential Equations 20 (1/2) 1 - 22, January/February 2015. https://doi.org/10.57262/ade/1418310441
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