  In this paper, we study the nonexistence of positive solutions for the
following two mixed boundary value problems. The first problem is the mixed
nonlinear-Neumann boundary value problem $$ \left\{
  \begin{array}{ll}
  \displaystyle -\Delta u=f(u) &{\rm in}\quad \R, \\ \displaystyle \\
\frac{\partial u}{\partial \nu}=g(u) &{\rm on}\quad \Gamma_1,\\ \displaystyle
\\ \frac{\partial u}{\partial \nu}=0 &{\rm on}\quad \Gamma_0 \end{array}
\right. $$ and the second is the nonlinear-Dirichlet boundary value problem $$
\left\{
  \begin{array}{ll}
  \displaystyle -\Delta u=f(u) &{\rm in}\quad \R, \\ \displaystyle \\
\frac{\partial u}{\partial \nu}=g(u) &{\rm on}\quad \Gamma_1,\\ \displaystyle
\\ u=0 &{\rm on}\quad \Gamma_0, \end{array} \right. $$ where $\R=\{x\in \mathbb
R^N:x_N>0\}$, $\Gamma_1=\{x\in \mathbb R^N:x_N=0,x_1<0\}$ and $\Gamma_0=\{x\in
\mathbb R^N:x_N=0,x_1>0\}$. We will prove that these problems possess no
positive solution under some assumptions on the nonlinear terms. The main
technique we use is the moving plane method in an integral form.

Analysis of PDEs