Radial ground states and singular ground states for a spatial dependent p-Laplace equation

We consider the following equation $$Delta_p u( extbf{x})+ f(u,| extbf{x}|)=0,$$ where $ extbf{x} in mathbb{R}^n$, $n>p>1$, and we assume that $f$ is negative for $|u|$ small and $lim_{u o +infty} rac{f(u,0)}{u|u|^{q-2}}=a_0>0$ where $ p_*=rac{p(n-1)}{n-p} < p^*=rac{np}{n-p}$, so $f(u,0)$ is subcritical and superlinear at infinity. In this paper we generalize the results obtained in a previous paper, [11], where the prototypical nonlinearity $$f(u,r)=-k_1(r) u|u|^{q_1-2}+k_2(r) u|u|^{q_2-2},$$ is considered, with the further restriction $12$. We manage to prove the existence of a radial ground state, for more generic functions $f(u,| extbf{x}|)$ and also in the case $p>2$ and $1<2$. We also prove the existence of uncountably many radial singular ground states under very weak hypotheses. The proofs combine an energy analysis and a shooting method. We also make use of Wazewski's principle to overcome some difficulties deriving from the lack of regularity.