I am interested in the following pde

$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball $B$ in $ R^N$ and where $p$ is large. Here we have $$ \epsilon(p):= \frac{(p-2)}{(p-1)^{p-1}}$$ and note this goes to zero when $ p \rightarrow \infty$. I have shown the existence of a positive solution $w_p$ of the above equation with Neumann boundary conditions. Additionally I have $w_p'(r) <0$ for $ r \in (0,1)$ and we have $ w'_p(0)=0$.

My interest is what happens when $ p \rightarrow \infty$. In particular I would like to show that $ \int_{B} w_p(x) dx \rightarrow 0$.

Any comments would be greatly appreciated. thanks

EDIT. I will rephrase question (and I am taking some limiting behaviour of $p$) which I assume should be easier. Let $W_p(r)$ denote a positive radial increasing smooth solution of $$-\Delta W_p + \frac{ | \nabla W_p|^2}{W_p} + (p-2) W_p = W_p^2$$ in $B_1$ (again the unit ball in $ R^N$) with $ W_p'(0)=W_p'(1)=0$ and $ W_p'(r)>0$ in $(0,1)$. We also have $ W_p(1)>p-1$. I would like to show that $W_p(0) \rightarrow \infty$ as $ p \rightarrow \infty$.