Consider the following gradient system on a doubly connected planar domain $\Omega$: $$\begin{cases} \frac{dx}{dt}&= - u_x;\\ \frac{dy}{dt}&= - u_y, \end{cases}$$ where $u$ is the first eigenfunction of Dirichlet laplacian. i.e. if $\Gamma_0,\Gamma_1$ are respectively inner and outer boundary of $\Omega$, $u$ satisfies the following: $$\begin{cases}-\Delta u=\lambda_1 u & \text{ in }\Omega, \\ u=0 & \text{ on }\partial \Omega=\Gamma_0\cup\Gamma_1;\end{cases}$$ where $\lambda_1$ is the first eigenvalue. Define, $$G=\{x\in \Omega: \text{the solution curve of the gradient system starting from $x$ touches $\Gamma_0$} \}$$
Why $G$ is non-empty?
$\textbf{Note:}$ It is well known that $u>0$ inside $\Omega$ and $\frac{\partial u}{\partial\eta}<0$ on $\Gamma_0$.