The positive solutions of quasilinear elliptic inequalities on Riemannian products

In this paper asymptotic behavior of positive solutions of quasilinear elliptic inequalities (1) on warped Riemannian products is researched. In particular, we find exact conditions under which Liouville theorems on no nontrivial solutions are satisfied, as well as the conditions of existence and cardinality of the set of positive solutions of the studied inequalities on the Riemannian manifolds. The results generalize similar results obtained previously by Naito. Y. and Usami H. for the Euclidean space $\mathrm{R}^n$ and results obtained previously by Losev A. and Mazepa E. on the model Riemannian manifolds.
We describe warped Riemannian products. Fix the origin $O \in \mathrm{R}^n$ and a smooth function $q_i > 0, i = 1, \ldots , k$ in the interval $(0, \infty)$. We define a Riemannian manifold $M$ as follows:
(1) the set of points $M$ is all $\mathrm{R}^n$;
(2) in coordinates $(r, {\theta}_1, \ldots , {\theta}_k)$ (where $r \in (0, \infty)$ and ${\theta}_i \in S^{n_i}$) Riemannian metric on $M \setminus \{ O\}$ defined as

$$ ds^2 = dr^2 + q_1^2 (r) d{{\theta}_1}^2 + \dots + q_k^2 (r) d{{\theta}_k}^2, $$

where $d {\theta}_i$ — the standard Riemannian metric on the sphere $S^{n_i}$, $n=n_1+\dots n_k+1$ — the dimension of $M$;
(3) Riemannian metric at $ O $ is a smooth continuation of the metric.
Will further assume that the function $ A $ in the inequality (1) satisfies the following conditions:
$$ \left\{ \begin{aligned} &A \in C(0, \infty), \quad A (p)> 0 \quad {\text for }\ p> 0, \\ &pA(| p |) \in C({\rm \bf R})\cap C^1(0, \infty), \\ &(p A (p)) '> 0 \; {\text for } \; p> 0, \\ \end{aligned} \right. $$

$ c (x)\equiv c(r) $ — continuous positive on $ \mathrm{R}_{+} $ function, and the function $ f \not\equiv 0 $ such that $ f(x, u) \in C (M)$, where $x=(r,\theta), f\not\equiv 0$ and $f(\cdot,0)=0$.
Introduce designations $\theta=(\theta_1,\dots,\theta_k)\ $, $K=S_{1}\times S_{2} \dots\times S_{k}\ $, $q(r)=\prod\limits_{i=1}^k q_i^{n_i}(r)$,
$$I(r)=\frac{1}{q(r)}\int\limits_{0}^{r}c(s)q(s)\,ds.$$

We also use the following assumption on the function $ f $:

$$ {\mathrm{(F)}} \quad \left\{ \begin{aligned} &{\text there \; are \; continuous \; functions \;} c(r)>0 \; {\text and \;} g(u)>0 \; {\text so \; that \;} \\ &g'(u)\geq 0, \; 0<c(r)g(u)\leq f(x,u) \; {\text for \;} u>0, \, r>0, \, \theta \in K, g(0)=0.\\ \end{aligned} \right. $$

First, consider the case where $ \lim\limits_ {p\rightarrow\infty} p A (p) <\infty. $
Theorem 1. Let $ \lim\limits_{p \rightarrow \infty}pA (p) <\infty$ and manifold $ M $ is such that $I(+0)=\lim\limits_{r \rightarrow +0}I (r)<\infty$ and $\limsup\limits_{r \rightarrow \infty} I (r) = \infty $. Then, if the condition (F), then positive integer solutions of the inequality (1) on $ M $ does not exist.
Next, consider the case where $ \lim\limits_{p \rightarrow \infty} pA (p) = \infty $. We prove a theorem on the non-existence of positive solutions of (1) and the conditions for the existence of a continuum of positive integer solutions of the inequality.