I am going to find a solution for the following PDE: $\frac{\partial V}{\partial x}\frac{e^{-t}x}{1+x^{2n}}+\frac{\partial V}{\partial t}\le 0$
we should have $V(t,0)=0$ , $ V(t,x)\leq \eta_2(x)$ and $V(t,x)\ge {\eta }_1(x)\textit{}$. In addition $t>0$ and for all $x\neq0$ we should have $\eta_1>0$ and $\eta_2>0$. Applying the method of characteristic is failing. Morover, the method of sepration cannot give any solution. I will be grateful if someone help me.
Not so sure that separation of variables doesn't give a solution. Let $V(t,x)=X(x)\,T(t),$ and denote time derivatives with dots and spatial derivatives with primes. Then the PDE (using equality) becomes \begin{align*} X'T\frac{e^{-t}x}{1+x^{2n}}+X\dot{T}&=0 \\ X'T\frac{e^{-t}x}{1+x^{2n}}&=-X\dot{T} \\ \frac{xX'}{(1+x^{2n})X}&=-\frac{e^t\,\dot{T}}{T}=k\\ xX'&=k(1+x^{2n})X\\ \dot{T}&=-ke^{-t}\,T. \end{align*} The spatial equation yields $$X(x)= c_1\, x^k\,e^{\frac{kx^{2 n}}{2 n}},$$ and the temporal equation yields $$T(t)= c_2\,e^{k\,e^{-t}}.$$ The final solution, then, would be (combining the constants into one constant $C$) $$V(t,x)=C\,x^k\,e^{\frac{kx^{2 n}}{2 n}}\,e^{k\,e^{-t}}=C\,x^k\,\exp\left(\frac{kx^{2 n}}{2 n}+k\,e^{-t}\right). $$ We can see that $V(t,0)=0.$ As for $\eta_1$ and $\eta_2,$ you would need to type those up for us to see if this solution satisfies those constraints. Hopefully, this will get you started.