### On the boundary value problem for some quasilinear equations

#### 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).

#### Article information

Source
Adv. Differential Equations Volume 20, Number 1/2 (2015), 1-22.

Dates
First available in Project Euclid: 11 December 2014