In the following, we introduce a preliminary lemma which will be used in the sequel.
Lemma 1
Assume that (M,g)is a connected C∞−Riemannian manifold without boundary, with metric g and has bounded geometry. Assume that there exist a constant γ such that 0<γ≤q∈C1(M), and
$$ \left\vert \triangle_{M}q(x)\right\vert \leq \sigma q^{\frac{3}{2} }(x),~for~all~x\in M, $$
(7)
where \(0<\sigma <\frac {2}{\sqrt {p-1}},~1< p<\infty \), and |△Mq(x)| denotes the norm of \(\triangle _{M}q(x)\in T_{x}^{\ast }M\) with respect to the inner product in \(T_{x}^{\ast }M\) induced by the metric g. Assume that f∈Lp(M) and that u∈Lp(M)∩C1(M) is a solution of the equation
$$ \triangle_{M}^{2}u+qu=f. $$
(8)
Additionally assume that for all \(k\in \left [ -\frac {1}{2},p-1\right ],\)
$$ \left\vert u\right\vert^{p}q^{k+\frac{1}{2}}\in L^{1}\left(M\right) \text{ and }{\lim}_{j\rightarrow \infty }\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) =0. $$
(9)
Then, the following properties hold:
$$ {\lim}_{j\rightarrow \infty }\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) =0, $$
(10)
and
$$ q^{k+1}\left\vert u\right\vert^{p}\in L^{1}\left(M\right),\text{ and } \int_{M}q^{k+1}\left\vert u\right\vert^{p}~~d\mu \leq C_{1}\left\Vert f\right\Vert_{p}^{p}, $$
(11)
for all \(k\in \left [ -\frac {1}{2},p-1\right ] \), where {ϕj} is as in (i-iv) and C1≥0 is a constant independent of u.
Proof
We first prove (10): Since u∈Lp(M)∩C1(M), using integration by parts, product rule of d, the definition of ΔM=d∗d, and the formula \(d(u_{\epsilon })=\frac {udu}{ u_{\epsilon }},\) we have
$$\begin{array}{@{}rcl@{}} \left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u,q^{k}u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(du,d\left(q^{k}u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(dudq^{k},u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(duq^{k}du,(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \\ &&+(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(duq^{k}du,u^{2}(u_{\epsilon })^{p-4}\triangle_{M}\phi_{j}\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(du,q^{k}u(u_{\epsilon })^{p-2}d(\triangle_{M}\phi_{j})\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(du,dq^{k}u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right)\\&&\quad +{\lim}_{\epsilon \rightarrow 0^{+}}\left(du,q^{k}u(u_{\epsilon })^{p-2}d(\triangle_{M}\phi_{j})\right) \\ &&+(p-1)\left(du,q^{k}du\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,d^{\ast }\left(dq^{k}u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,d^{\ast }\left(q^{k}u(u_{\epsilon })^{p-2}d(\triangle_{M}\phi_{j})\right) \right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(u,d^{\ast }\left(q^{k}du(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \right), \end{array} $$
using the product rule (4) of d∗, we get
$$\begin{array}{@{}rcl@{}} \left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) &=&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(ud\left(u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right),dq^{k}\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\triangle_{M}q^{k}\right) \\ &&-(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(ud\left(q^{k}(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right),du\right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(u,q^{k}(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\triangle_{M}u\right) \\ &&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(ud\left(q^{k}u(u_{\epsilon })^{p-2}\right),d(\triangle_{M}\phi_{j})\right) \\&&\quad+{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,q^{k}u(u_{\epsilon })^{p-2}\triangle_{M}^{2}\phi_{j}\right), \end{array} $$
using the product rule of d again, we get
$$\begin{array}{@{}rcl@{}} \left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) &=&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(ud\left(\triangle_{M}\phi_{j}\right),u(u_{\epsilon })^{p-2}dq^{k}\right) \\ &&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(u\triangle_{M}\phi_{j}du,(u_{\epsilon })^{p-2}dq^{k}\right) \\ &&-(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(u\triangle_{M}\phi_{j}du,u^{2}(u_{\epsilon })^{p-4}dq^{k}\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\triangle_{M}q^{k}\right) \\&&\quad+{\lim}_{\epsilon \rightarrow 0^{+}}\left(u,q^{k}u(u_{\epsilon })^{p-2}\triangle_{M}^{2}\phi_{j}\right) \end{array} $$
$$\begin{array}{@{}rcl@{}} &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(udq^{k},(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}du\right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(uq^{k}d\left(\triangle_{M}\phi_{j}\right),(u_{\epsilon })^{p-2}du\right) \\ &&-(p-1)(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(uq^{k}du,u(u_{\epsilon })^{p-4}\triangle_{M}\phi_{j}du\right) \\ &&-(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(u,q^{k}(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\triangle_{M}u\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(udq^{k},u(u_{\epsilon })^{p-2}d(\triangle_{M}\phi_{j})\right) \\ &&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(udq^{k}du,(u_{\epsilon })^{p-2}d(\triangle_{M}\phi_{j})\right) \\ &&-(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(udq^{k}du,u^{2}(u_{\epsilon })^{p-4}d(\triangle_{M}\phi_{j})\right). \end{array} $$
Hence, we obtain
$$\begin{array}{@{}rcl@{}} p\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) &=&\left(u\triangle_{M}q^{k},u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +\left(u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}^{2}\phi_{j}\right) \\ &&-(p-1)(p-2)\left(uq^{k}du,u\left\vert u\right\vert^{p-4}\triangle_{M}\phi_{j}du\right). \end{array} $$
Taking the limit as j→∞, we get
$$\begin{array}{@{}rcl@{}} p{\lim}_{j\rightarrow \infty }\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) &=&{\lim}_{j\rightarrow \infty }\left(u\triangle_{M}q^{k},u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) \\&&\quad+{\lim}_{j\rightarrow \infty }\left(u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}^{2}\phi_{j}\right) \\ &&-(p-1)(p-2){\lim}_{j\rightarrow \infty }\left(uq^{k}du,u\left\vert u\right\vert^{p-4}\triangle_{M}\phi_{j}du\right). \end{array} $$
By properties of {ϕj}, it follows that for all x∈M,ϕj(x)→1, dϕj(x)→0, △Mϕj(x)→0 and \(\triangle _{M}^{2}\phi _{j}(x)\rightarrow 0\) as j→∞, we apply dominated convergence theorem by using the assumption (7), the assumption \(\left \vert u\right \vert ^{p}q^{k+\frac {1}{2}}\in L^{1}\left (M\right) \) and the condition (iv), we obtain (10).
We now prove (11): Since u∈Lp(M)∩C1(M), using (8), integration by parts, product rule of d, the definition of ΔM=d∗d, and the formula \(d(u_{\epsilon })= \frac {udu}{u_{\epsilon }}\), we have
$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &=&\left(\triangle_{M}^{2}u,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}^{2}u,q^{k}u(u_{\epsilon })^{p-2}\phi_{j}\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(d\left(\triangle_{M}u\right),d\left(q^{k}u(u_{\epsilon })^{p-2}\phi_{j}\right) \right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \end{array} $$
$$\begin{array}{@{}rcl@{}} &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(d\left(\triangle_{M}u\right),q^{k}u(u_{\epsilon })^{p-2}d\phi_{j}\right) +{\lim}_{\epsilon \rightarrow 0^{+}}\left(d\left(\triangle_{M}u\right),q^{k}(u_{\epsilon })^{p-2}\phi_{j}du\right) \\ &&+(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(d\left(\triangle_{M}u\right),q^{k}u^{2}(u_{\epsilon })^{p-4}\phi_{j}du\right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(d\left(\triangle_{M}u\right),u(u_{\epsilon })^{p-2}\phi_{j}dq^{k}\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \\ &=&\left(d\left(\triangle_{M}u\right),q^{k}u\left\vert u\right\vert^{p-2}d\phi_{j}\right) +\left(d\left(\triangle_{M}u\right),u\left\vert u\right\vert^{p-2}\phi_{j}dq^{k}\right) \\ &&+(p-1)\left(d\left(\triangle_{M}u\right),q^{k}\left\vert u\right\vert^{p-2}\phi_{j}du\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \\ &=&{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u,d^{\ast }\left(q^{k}u(u_{\epsilon })^{p-2}d\phi_{j}\right) \right) +(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u,d^{\ast }\left(q^{k}(u_{\epsilon })^{p-2}\phi_{j}du\right) \right) \\ &&+{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u,d^{\ast }\left(u(u_{\epsilon })^{p-2}\phi_{j}dq^{k}\right) \right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right), \end{array} $$
using the product rule (4) of d∗, we get
$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &\,=\,&-\!{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}ud\left(q^{k}u(u_{\epsilon })^{p-2}\right),d\phi_{j}\right) \,+\,{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u,q^{k}u(u_{\epsilon })^{p-2}\triangle_{M}\phi_{j}\right) \\ &&\,-\,{\lim}_{\epsilon \rightarrow 0^{+}}\left(\! \triangle_{M}ud\left(u(u_{\epsilon })^{p-2}\phi_{j}\right),dq^{k}\right) \,+\,{\lim}_{\epsilon \rightarrow 0^{+}}\left((\triangle_{M}u)u(u_{\epsilon })^{p-2}\phi_{j},\triangle_{M}q^{k}\!\right) \\ &&-(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}ud\left(q^{k}(u_{\epsilon })^{p-2}\phi_{j}\right),du\right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}uq^{k}(u_{\epsilon })^{p-2}\phi_{j},\triangle_{M}u\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right), \end{array} $$
using the product rule of d again, we get
$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &=&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}udq^{k},u(u_{\epsilon })^{p-2}d\phi_{j}\right) -{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}uq^{k}du,(u_{\epsilon })^{p-2}d\phi_{j}\right) \\ &&-(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}uq^{k}du,u^{2}(u_{\epsilon })^{p-4}d\phi_{j}\right) \\ &&+\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}ud\phi_{j},u(u_{\epsilon })^{p-2}dq^{k}\right) \\ &&-{\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u\phi_{j}du,(u_{\epsilon })^{p-2}dq^{k}\right) +\left((\triangle_{M}u)u\left\vert u\right\vert^{p-2}\phi_{j},\triangle_{M}q^{k}\right) \\ &&-(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}u\phi_{j}du,u^{2}(u_{\epsilon })^{p-4}dq^{k}\right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}udq^{k},(u_{\epsilon })^{p-2}\phi_{j}du\right) \\ &&+(p-1){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}uq^{k},(u_{\epsilon })^{p-2}d\phi_{j}du\right) \\ &&-(p-1)(p-2){\lim}_{\epsilon \rightarrow 0^{+}}\left(\triangle_{M}uq^{k}du,u(u_{\epsilon })^{p-4}\phi_{j}du\right) \\ &&+(p-1)\left(\triangle_{M}uq^{k}\left\vert u\right\vert^{p-2}\phi_{j},\triangle_{M}u\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right). \end{array} $$
Hence, we obtain
$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &=&-(p-1)(p-2)\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) \notag \\ &&+\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +(p-1)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) \notag \\ &&+\left(\triangle_{M}u,u\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}q^{k}\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right). \end{array} $$
(12)
We now estimate the term (△Mu,u|u|p−2ϕj△Mqk).
Using the assumption (7), we get
$$ \left\vert \triangle_{M}q^{k}\right\vert \leq \sigma \left\vert k\right\vert q^{k+\frac{1}{2}}. $$
(13)
Using (13) and the inequality \(ab\leq (p-1)a^{2}+\frac {b^{2}}{4(p-1)},\) for all 0≤a,b∈R, we have
$$\left\vert \left(\triangle_{M}u,u\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}q^{k}\right) \right\vert \leq \int\limits_{M}\left\vert \triangle_{M}u\right\vert \left\vert \triangle_{M}q^{k}\right\vert \left\vert u\right\vert^{p-1}\phi_{j}~d\mu $$
$$\begin{array}{@{}rcl@{}} &\leq &\int\limits_{M}\sigma \left\vert \triangle_{M}u\right\vert \left\vert k\right\vert q^{k+\frac{1}{2}}\left\vert u\right\vert^{p-1}\phi_{j}~d\mu \notag \\ &=&\int\limits_{M}\left(\left\vert \triangle_{M}u\right\vert \left\vert u\right\vert^{\frac{p-2}{2}}\phi_{j}^{\frac{1}{2}}q^{\frac{k}{2}}\right) \left(\sigma \left\vert k\right\vert q^{\frac{k+1}{2}}\phi_{j}^{\frac{1}{2} }\left\vert u\right\vert^{\frac{p}{2}}\right) ~d\mu \notag \\ &\leq &(p-1)\int\limits_{M}\left\vert \triangle_{M}u\right\vert^{2}\left\vert u\right\vert^{p-2}\phi_{j}q^{k}~d\mu +\frac{\sigma^{2}k^{2} }{4(p-1)}\int\limits_{M}q^{k+1}\phi_{j}\left\vert u\right\vert^{p}~d\mu \notag \\ &=&(p-1)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) +\frac{\sigma^{2}k^{2}}{4(p-1)}\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \notag \\ &=&(p-1)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) +(1-\alpha)\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right), \end{array} $$
(14)
where \(\alpha =1-\frac {\sigma ^{2}k^{2}}{4(p-1)}\), and α∈(0,1].
From (14), we get
$$\left(\triangle_{M}u,u\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}q^{k}\right) \geq -\left\vert \left(\triangle_{M}u,u\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}q^{k}\right) \right\vert $$
$$ \geq (1-p)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) +(\alpha -1)\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right). $$
(15)
From (15) into (12), we obtain
$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &\geq &-(p-1)(p-2)\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) \notag \\ &&+\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +(p-1)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) \notag \\ &&+(1-p)\left(\triangle_{M}u,q^{k}\left\vert u\right\vert^{p-2}\phi_{j}\triangle_{M}u\right) \notag \\ &&+(\alpha -1)\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) +\left(qu,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \notag \\ &=&-(p-1)(p-2)\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) \notag \\ &&+\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +\alpha \left(u,q^{k+1}u\left\vert u\right\vert^{p-2}\phi_{j}\right). \end{array} $$
(16)
Now, we use the inequality:
$$ \left\vert ab\right\vert \leq \frac{\left\vert a\right\vert^{p}}{\lambda^{p}}+\lambda \left\vert b\right\vert^{t},\text{ } $$
(17)
where \(\frac {1}{p}+\frac {1}{t}=1, a,~b\in R\), and λ∈(0,1). Since ϕj≤1 and \(t=\frac {p}{p-1}>1,\) this implies (ϕj)t≤ϕj.
Using this and (17), we have
$$\begin{array}{@{}rcl@{}} \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) &\leq &\left\vert \left(f,q^{k}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \right\vert \notag \\ &\leq &\frac{1}{\lambda^{p}}\int\limits_{M}\left\vert f\right\vert^{p}d\mu +\lambda \int\limits_{M}(\phi_{j})^{t}q^{kt}\left\vert u\right\vert^{t}\left\vert u\right\vert^{(p-2)t}~d\mu \notag \\ &\leq &\lambda^{-p}\left\Vert f\right\Vert_{p}^{p}+\lambda \int\limits_{M}\phi_{j}q^{kt}\left\vert u\right\vert^{t}\left\vert u\right\vert^{(p-2)t}~d\mu \notag \\ &=&\lambda^{-p}\left\Vert f\right\Vert_{p}^{p}+\lambda \left(q^{\frac{kp}{ p-1}}\left\vert u\right\vert,\phi_{j}\left\vert u\right\vert^{p-1}\right). \end{array} $$
(18)
From (18) into (16), we get
$$\begin{array}{@{}rcl@{}} &&\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +\alpha \left(u,q^{k+1}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \\ &&-(p-1)(p-2)\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) \leq \lambda^{-p}\left\Vert f\right\Vert_{p}^{p}+\lambda \left(q^{\frac{kp}{p-1}}\left\vert u\right\vert,\phi_{j}\left\vert u\right\vert^{p-1}\right). \end{array} $$
Since k≤p−1 and λ∈(0,1) is arbitrary, we can choose a sufficiently small λ>0 such that
$$\begin{array}{@{}rcl@{}} &&-(p-1)(p-2)\left(\triangle_{M}uq^{k}du,u\left\vert u\right\vert^{p-4}\phi_{j}du\right) \notag \\ &&+\left(\triangle_{M}u,q^{k}u\left\vert u\right\vert^{p-2}\triangle_{M}\phi_{j}\right) +\frac{\alpha }{2}\left(u,q^{k+1}u\left\vert u\right\vert^{p-2}\phi_{j}\right) \leq \lambda^{-p}\left\Vert f\right\Vert_{p}^{p}. \end{array} $$
(19)
By Fatou’s lemma, we have
$$ \int\limits_{M}q^{k+1}\left\vert u\right\vert^{p}~d\mu \leq {\lim}_{j\rightarrow \infty }\inf \left(u,q^{k+1}u\left\vert u\right\vert^{p-2}\phi_{j}\right). $$
(20)
Combining (19) and (20) and using (9) and (10), we obtain \(\int \limits _{M}q^{k+1} \left \vert u\right \vert ^{p}~d\mu \leq C_{1}\left \Vert f\right \Vert _{p}^{p}, \) where C1≥0 is a constant independent of u, which is the proof of (11) and the lemma. □