Uniqueness of singular radial solutions for a class of quasilinear problems

Abstract

We establish the uniqueness and the blow-up rate of the large positive solution of the singular boundary problem $-\Delta_{p} u=\lambda u^{p-1}-b(x) u^q$ in $B_R(x_0)$, $u|_{\partial B_R(x_0)}=+\infty$, where $B_R(x_0)$ is a ball domain of radius $R$ centered at $x_0\in\mathbb{R}^N$, $N\geq3$, $\lambda>0$ and the potential function $b$ is a positive radially symmetric function. Our result extends the previous work by Ouyang and Xie from the case $p=2$ to the case $p>2$ and we prove that any large solution $u$ must satisfy $$\lim_{d(x)\rightarrow 0}\frac{u(x)}{K(b^{*}(\|x-x_{0}\|))^{-\beta}}=1,$$ where $d(x)= {\rm dist}(x, \partial B_{R}(x_{0}))$, $K$ is a constant defined by $$K:=\left[(p-1)[(\beta +1)C_{0}-1]\beta^{p-1}(C_{0}b_{0})^{(p-2)/2}\right]^{\frac{1}{q-p+1}},$$ with $$\beta:=\frac{p}{2(q-p+1)},\;q>p-1>1,\; b_{0}:=b(R)>0,\; C_{0}:=\lim_{r\rightarrow R}\frac{(B(r))^{2}}{b^{*}(r)b(r)}\geq 1$$ and $$B(r):=\int_{r}^{R} b(s)ds,\; b^{*}(r)=\int_{r}^{R}\int_{s}^{R} b(t)dt ds.$$