We consider the following equation $$\Delta_p u(\textbf{x})+ f(u,|\textbf{x}|)=0,$$ where $\textbf{x} \in \mathbb{R}^n$, $n>p>1$, and we assume that $f$ is negative for $|u|$ small and $\lim_{u \to +\infty} \frac{f(u,0)}{u|u|^{q-2}}=a_0>0$ where $p_*=\frac{p(n-1)}{n-p} <q< p^*=\frac{np}{n-p}$, so $f(u,0)$ is subcritical and superlinear at infinity. In this paper we generalize the results we obtained in a previous paper, 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 $1<p \le 2$ and $q_1>2$. We manage to prove the existence of a radial ground state, for more generic functions $f(u,|\textbf{x}|)$ and also in the case $p>2$ and $1<q_1<2$. 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.

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

#### Abstract

We consider the following equation $$\Delta_p u(\textbf{x})+ f(u,|\textbf{x}|)=0,$$ where $\textbf{x} \in \mathbb{R}^n$, $n>p>1$, and we assume that $f$ is negative for $|u|$ small and $\lim_{u \to +\infty} \frac{f(u,0)}{u|u|^{q-2}}=a_0>0$ where $p_*=\frac{p(n-1)}{n-p} 2$. We manage to prove the existence of a radial ground state, for more generic functions $f(u,|\textbf{x}|)$ and also in the case $p>2$ and \$1
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2010
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/39509
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