A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity

TL;DR

This paper introduces a unified approach to establish boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity, bridging the gap between different cases of the sensitivity decay rate.

Contribution

It provides a new unified method to prove global boundedness in chemotaxis models, connecting previous separate results for different sensitivity decay rates.

Findings

Established global bounded solutions under a natural condition for hi

Unified the cases k=1 and k>1 for the sensitivity decay rate

Addressed gaps in previous proofs for boundedness conditions

Abstract

This paper deals with the Keller--Segel system with signal-dependent sensitivity \begin{equation*} u_t=\Delta u - \nabla \cdot (u \chi(v)\nabla v), \quad v_t=\Delta v + u - v, \quad x\in\Omega,\ t>0, \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$; $\chi$ is a function satisfying $\chi(s)\leq K(a+s)^{-k}$ for some $k\geq 1$ and $a\geq 0$. In the case that $k=1$, Fujie (J. Math. Anal. Appl.; 2015; 424; 675--684) established global existence of bounded solutions under the condition $K<\sqrt{\frac{2}{n}}$. On the other hand, when $k>1$, Winkler (Math. Nachr.; 2010; 283; 1664--1673) asserted global existence of bounded solutions for arbitrary $K>0$. However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary $K>0$. Moreover, the condition for $K$ when $k>1$ cannot connect to the condition when $k=1$. The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for $\chi$ and to build a mathematical bridge between the cases $k=1$ and $k>1$.

Equations

$$\{\begin{array}{ll}{u}_t=\Delta u-\nabla \cdot (u\chi (v)\nabla v),& x\in \Omega ,t>0,\\ v_t=\Delta v+u-v,& x\in \Omega ,t>0,\end{array}$$

$$\{\begin{array}{ll}{u}_t=\Delta u-\nabla \cdot (u\chi (v)\nabla v),& x\in \Omega ,t>0,\\ v_t=\Delta v+u-v,& x\in \Omega ,t>0,\\ \nabla u\cdot \nu =\nabla v\cdot \nu =0,& x\in \partial \Omega ,t>0,\\ u(x,0)=u_0(x),v(x,0)=v_0(x),& x\in \Omega ,\end{array}$$

$$\chi (s)=\frac{K}{s}\text{mbox}and\chi (s)=\frac{K}{(1+s{)}^2}(s>0).$$

$$K<\sqrt{\frac{2}{n}}.$$

$$\chi (s)\le \frac{K}{(a+s{)}^k}$$

$$K<k(a+\eta {)}^{k-1}\sqrt{\frac{2}{n}}.$$

$$\eta :=\underset{\tau >0}{\sup }\left(\min \left\{e^{-2\tau }\underset{x\in \overline{\Omega }}{\min }{v}_0(x),c_0\Vert u_0{\Vert }_{L^1(\Omega )}(1-e^{-\tau })\right\}\right),$$

$$0\le u_0\in C(\overline{\Omega })\setminus \{0\}\text{mbox}and\{\begin{array}{ll}0<v_0\in W^{1,q}(\Omega )(\exists q>n)& (a=0),\\ 0\le v_0\in W^{1.q}(\Omega )\setminus \{0\}(\exists q>n)& (a>0).\end{array}$$

$$u,v\in C(\overline{\Omega }\times [0,\infty ))\cap C^{2,1}(\overline{\Omega }\times (0,\infty ))$$

$$\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert v(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le C\text{mbox}forallt\ge 0.$$

$$K<{\left(k(a+\eta {)}^{k-1}\sqrt{\frac{2}{n}}\right)|}_{k=1}=\sqrt{\frac{2}{n}}.$$

$$\frac{d}{dt}{\int }_{\Omega }{u}^p\phi (v)$$

$$\phi (s):=\exp \left\{-r{\int }_0^s\frac{1}{(a+s{)}^k}d\tau \right\}$$

$$\{\begin{array}{ll}{v}_t=\Delta v-v+u,& x\in \Omega ,t\in (0,T),\\ \nabla v\cdot \nu =0& x\in \partial \Omega ,t\in (0,T),\\ v(x,0)=v_0(x)& x\in \Omega ,\end{array}$$

$$\underset{x\in \Omega }{\inf }v(x,t)\ge \eta >0,$$

$$\eta :=\underset{\tau >0}{\sup }\left(\min \left\{e^{-\tau }\underset{x\in \overline{\Omega }}{\min }{v}_0(x),\Vert u_0{\Vert }_{L^1(\Omega )}\cdot {\int }_0^{\tau }\frac{1}{(4\pi r{)}^{\frac{n}{2}}}{e}^{-\left(r+\frac{{\left({\max }_{x,y\in \overline{\Omega }}|x-y|\right)}^2}{4r}\right)}dr\right\}\right).$$

$$U(s+\tau ,x;s,y)=U(\tau ,x;0,y)\ge c_0>0\text{mbox}forallx,y\in \overline{\Omega },s>0.$$

$$u\in C(\overline{\Omega }\times [0,T_{\max }))\cap C^{2,1}(\overline{\Omega }\times (0,T_{\max })),$$

$$v\in C(\overline{\Omega }\times [0,T_{\max }))\cap C^{2,1}(\overline{\Omega }\times (0,T_{\max }))\cap L_{\mathrm{loc}}^{\infty }([0,T_{\max });W^{1,q}(\Omega ))$$

$${\int }_{\Omega }u(\cdot ,t)={\int }_{\Omega }{u}_0\text{mbox}forallt\in (0,T_{\max })$$

$$\underset{x\in \Omega }{\inf }v(x,t)\ge \eta \text{mbox}forallt\in (0,T_{\max }),$$

$$\underset{t\nearrow T_{\max }}{\lim }(\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}+\Vert v(\cdot ,t){\Vert }_{W^{1,\infty }(\Omega )})=\infty .$$

$$\Vert v(\cdot ,t){\Vert }_{L^{\mu }(\Omega )}\le C_1(\mu ,\theta )\left(1+\underset{s\in (0,\infty )}{\sup }\Vert u(\cdot ,s){\Vert }_{L^{\theta }(\Omega )}\right)\text{mbox}forallt\in (0,T).$$

$$\Vert u(\cdot ,t){\Vert }_{L^p(\Omega )}\le C(p)\text{mbox}forallt\in (0,T_{\max })$$

$$\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\le C^{\prime }\text{mbox}forallt\in (0,T_{\max }).$$

$$g(s):=-r{\int }_{\eta }^s\frac{1}{(a+\tau {)}^k}d\tau ,\phi (s):=\exp \{g(s)\}(s\ge \eta ),$$

$$\phi (s)=C_{\phi }\exp \left\{\frac{r}{(k-1)(a+s{)}^{k-1}}\right\}$$

$$\phi (s)=\frac{C_{\phi }}{(a+s{)}^r}$$

$$\frac{d}{dt}{\int }_{\Omega }{u}^p\phi (v)$$

$$\le {\int }_{\Omega }{u}^p{H}_{\epsilon }(v)\phi (v)|\nabla v{|}^2+c{\int }_{\Omega }{u}^p\phi (v)-r{\int }_{\Omega }{u}^{p+1}\frac{\phi (v)}{(a+v{)}^k},$$

$$H_{\epsilon }(s):=\frac{(\epsilon p+1-\epsilon )r^2}{(1-\epsilon )(p-1)(a+s{)}^{2k}}+\frac{\epsilon pKr}{(1-\epsilon )(a+s{)}^{2k}}-\frac{kr}{(a+s{)}^{k+1}}+\frac{p(p-1)K^2}{4(1-\epsilon )(a+s{)}^{2k}}.$$

Full text

0002010Mathematics Subject Classification. Primary: 35K51; Secondary: 35B45, 35A01, 92C17. 000*Key words and phrases: chemotaxis; sensitivity function; global existence; boundedness. *

A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity

Masaaki Mizukami

Department of Mathematics, Tokyo University of Science

1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

masaaki.mizukami.math@gmail.com

Tomomi Yokota***Corresponding author †††Partially supported by Grant-in-Aid for Scientific Research (C), No. 16K05182.

Department of Mathematics, Tokyo University of Science

1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

yokota@rs.kagu.tus.ac.jp

• Abstract. This paper deals with the Keller–Segel system

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$, $n\geq 2$; $\chi$ is a function satisfying $\chi(s)\leq K(a+s)^{-k}$ for some $k\geq 1$ and $a\geq 0$. In the case that $k=1$, Fujie (J. Math. Anal. Appl.; 2015; 424; 675–684) established global existence of bounded solutions under the condition $K<\sqrt{\frac{2}{n}}$. On the other hand, when $k>1$, Winkler (Math. Nachr.; 2010; 283; 1664–1673) asserted global existence of bounded solutions for arbitrary $K>0$. However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary $K>0$. Moreover, the condition for $K$ when $k>1$ cannot connect to the condition when $k=1$. The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for $\chi$ and to build a mathematical bridge between the cases $k=1$ and $k>1$.

1 Introduction

The chemotaxis system proposed by Keller and Segel in [9] describes a part of the life cycle of cellular slime molds with chemotaxis. After the pioneering work [9], a number of variations of the chemotaxis system are proposed and investigated (see e.g., [1, 6, 7]).

In this paper we consider the Keller–Segel system with signal-dependent sensitivity

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ ($n\geq 2$) with smooth boundary $\partial\Omega$ and $\nu$ is the outward normal vector to $\partial\Omega$. The initial data $u_{0}$ and $v_{0}$ are assumed to be nonnegative functions. The unknown function $u(x,t)$ represents the population density of species and $v(x,t)$ shows the concentration of the substance at place $x$ and time $t$. As to the sensitivity function $\chi$, we are interested in functions generalizing

[TABLE]

In a mathematical view, the difficulty caused by the sensitivity function is to deal with the additional term $u\chi^{\prime}(v)|\nabla v|^{2}$ which does not appear in the case that $\chi$ is a constant. Moreover, when $\chi$ is the singular sensitivity function, it is delicate to derive the estimate for $\frac{1}{v}$. In the case that $\chi(v)=\frac{K}{v}$, Winkler [14] first attained global existence of classical solutions when $K<\sqrt{\frac{2}{n}}$ and global existence of weak solutions when $K<\sqrt{\frac{n+2}{3n-4}}$. However, the result in [14] could not arrive at boundedness of solutions to (1.1). To overcome the difficulty in a singular sensitivity, Fujie [2] established the uniform-in-time lower estimate for $v$, and show global existence of classical bounded solutions to (1.1) in the case that $\chi(v)=\frac{K}{v}$ with

[TABLE]

As to the problem (1.1) with $\chi(v)=\frac{K}{v}$ in the 2-dimensional setting, Lankeit [10] obtained global existence of classical bounded solutions when $K<\chi_{0}$ ($\chi_{0}>1$). Futhermore Fujie–Senba [4] dealt with the 2-dimensional problem which was replaced $v_{t}$ with $\tau v_{t}$ in (1.1) and showed global existence and boundedness of radially symmetric solutions under the condition that $\chi(s)\to 0$ $(t\to\infty)$ and $\tau$ is sufficiently small. On the other hand, in the case that $\chi(v)\leq\frac{K}{(1+\alpha v)^{k}}$ $(k>1$, $\alpha>0$, $K>0$), Winkler [13] established global existence and boundedness in (1.1) without any restriction on $K>0$. The result in [13] affected the result in [5] which mentioned global existence of classical bounded solutions to (1.1) when $\chi$ is the strong singular sensitivity function such that $\chi(v)=\frac{K}{v^{k}}$ for $k>1$, and the method in [13] was used in Zhang–Li [15] and Zheng–Mu [16]. However, we cannot convince the result in [13] that global existence and boundedness hold for all $K>0$, because there is a gap in the proof. Recently, Fujie tried modifying it; nevertheless it also has a gap (cf. [3]). In general, it seems to be difficult to show global existence of bounded solutions for arbitrary $K>0$. Moreover, if $k\to 1$, then the condition “arbitrary $K>0$” cannot connect to (1.2).

The purpose of this paper is to obtain global existence and boundedness in (1.1) under a more natural and proper condition for $\chi$ and to build a mathematical bridge between the cases $k=1$ and $k>1$. We shall suppose that $\chi$ satisfies that

[TABLE]

with some $k\geq 1$, $a\geq 0$ and $K>0$ fulfiling

[TABLE]

Here

[TABLE]

where $c_{0}>0$ is a lower bound for the fundamental solution of $w_{t}=\Delta w-w$ with Neumann boundary condition (for more detail, see Remark 2.1). We suppose that

[TABLE]

Now the main results read as follows.

Theorem 1.1**.**

Let $n\geq 2$ and let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain with smooth boundary. Assume that $\chi$ satisfies (1.3) with some $k\geq 1$, $a\geq 0$, $K>0$ fulfiling (1.4). Then for any $u_{0},v_{0}$ satisfying (1.6) with some $q>n$, there exists an exactly one pair $(u,v)$ of functions

[TABLE]

which solves (1.1). Moreover, the solution $(u,v)$ is uniformly bounded, i.e., there exists a constant $C>0$ such that

[TABLE]

Remark 1.1**.**

The unified condition (1.3) with $K>0$ satisfying (1.4) may be a natural condition for $\chi$. Indeed, when $k=1$, (1.4) becomes $\eqref{condition;singular}$:

[TABLE]

The main theorem tells us the result in the typical case of singular sensitivity.

Corollary 1.2**.**

Let $\chi(s)=\frac{K}{s}$ with $K<\sqrt{\frac{2}{n}}$. Then for any $u_{0},v_{0}$ satisfying (1.6) with some $q>n$, (1.1) has a unique global bounded classical solution.

The strategy for the proof of Theorem 1.1 is to construct the estimate for $\int_{\Omega}u^{p}$ with some $p>\frac{n}{2}$. One of the keys for this strategy is to derive the unified inequality

[TABLE]

for some $\varepsilon\in[0,1)$ and $c>0$, where

[TABLE]

with $r>0$. This function $\varphi$ constructed in [11, 12] unifies mathematical structures in the cases $k>1$ and $k=1$.

This paper is organized as follows. In Section 2 we collect basic facts which will be used later. In Section 3 we give a unified view point in energy estimates. Section 4 is devoted to the proof of global existence and boundedness (Theorem 1.1).

2 Preliminaries

In this section we will collect elementary results. We first recall the uniform-in-time lower estimate for $v$ established by Fujie [2, 3].

Lemma 2.1**.**

Let $u_{0}\in C(\overline{\Omega})$ and $u\in C(\overline{\Omega}\times[0,T))$ be nonnegative functions such that $u_{0}\not\equiv 0$ and $\int_{\Omega}u(\cdot,t)=\int_{\Omega}u_{0}$ $(t\in[0,T))$. If $v_{0}\in C(\overline{\Omega})$ is a positive function in $\overline{\Omega}$ and $v\in C^{2,1}(\overline{\Omega}\times(0,T))\cap C(\overline{\Omega}\times[0,T))$ is a classical solution of

[TABLE]

then for all $t\in(0,T)$,

[TABLE]

where $\eta>0$ is defined as (1.5).

Remark 2.1**.**

When $\Omega$ is a convex bounded domain, the proof of this lemma is given in [2, Lemma 2.2] and the constant $\eta$ can be explicitly represented as

[TABLE]

On the other hand, if we do not assume the convexity of $\Omega$, then using the positivity of the fundamental solution $U(t,x;s,y)$ to $w_{t}=\Delta w-w$ in $\Omega\times(0,T)$ with $\nabla w\cdot\nu=0$ on $\partial\Omega\times(0,T)$ (see e.g., [8]), we have that there exists $c_{0}>0$ such that for all $\tau\in(0,\frac{T}{2})$,

[TABLE]

Then we can see the conclusion of Lemma 2.1 by a similar argument as in [2, Lemma 2.2].

We next recall the well-known result about local existence of solutions to (1.1) (see [10, Theorem 2.3], [4, Proposition 2.2] and [13, Lemma 2.1]).

Lemma 2.2**.**

Assume that $\chi$ satisfies (1.3) and the initial data $u_{0},v_{0}$ fulfil (1.6) for some $q>n$. Then there exist $T_{\rm max}\in(0,\infty]$ and exactly one pair $(u,v)$ of nonnegative functions

[TABLE]

which solves (1.1) in the classical sense and satisfies the mass conservation

[TABLE]

and the lower estimate

[TABLE]

where $\eta>0$ is defined as (1.5). Moreover, if $T_{\rm max}<\infty$, then

[TABLE]

At the end of this section we shall recall the result about the estimate for $v$ in dependence on boundedness features of $u$ derived by a straightforward application of well-known smoothing estimates for the heat semigroup under homogeneous Neumann boundary conditions (see [14, Lemma 2.4] and [2, Lemma 2.4]).

Lemma 2.3**.**

Let $T>0$ and $1\leq\theta,\mu\leq\infty$. If $\frac{n}{2}(\frac{1}{\theta}-\frac{1}{\mu})<1$, then there exists $C_{1}(\mu,\theta)>0$ such that

[TABLE]

3 A unified view point in energy estimates

Let $(u,v)$ be the solution of (1.1) on $[0,T_{\rm max})$ as in Lemma 2.2. For the proof of Theorem 1.1 we will recall an useful fact to derive the $L^{\infty}$-estimate for $u$.

Lemma 3.1**.**

Assume that the solution $u$ of (1.1) given in Lemma 2.2 satisfies

[TABLE]

with some $p>\frac{n}{2}$ and $C(p)>0$. Then there exists a constant $C^{\prime}>0$ such that

[TABLE]

Proof.

Combination of (2.2) and the same argument as in [14, Lemma 3.4] leads to the conclusion of Lemma 3.1. ∎

Unified test function. Thanks to Lemmas 2.2 and 3.1 we will only make sure that the $L^{p}$-estimate for $u$ with some $p>\frac{n}{2}$ to show global existence and boundedness of solutions to (1.1). To establish (3.1) we introduce the functions $g$ and $\varphi$ by

[TABLE]

where $r>0$ is a constant fixed later and $\eta$ is defined as (1.5). When $k>1$, by straightforward calculations we have

[TABLE]

with $C_{\varphi}=\exp\{-r(k-1)^{-1}(a+\eta)^{-k+1}\}>0$, which is a similar function used in [13]. On the other hand, when $k=1$, it follows that

[TABLE]

with some constant $C_{\varphi}=(a+\eta)^{r}>0$, which is a similar function used in [2]. Now we shall prove the following unified inequality by using the test function $\varphi(v)$.

Lemma 3.2**.**

Assume that $\chi$ satisfies (1.3). Then for all $\varepsilon\in[0,1)$ there exists $c>0$ such that

[TABLE]

where

[TABLE]

Proof.

We let $p\geq 1$. From (1.1) we have

[TABLE]

Integration by parts yields

[TABLE]

Invoking to the Young inequality, we infer that for all $\varepsilon\in[0,1)$,

[TABLE]

Thus combination of (3.4), (3.5) and (3.6) yields that

[TABLE]

where

[TABLE]

Noting that

[TABLE]

we can rewrite the function $F_{\varphi}(s)$ as

[TABLE]

Recalling by (3.2) that

[TABLE]

we see from (1.3) that

[TABLE]

where

[TABLE]

Therefore we obtain from (3.7) together with (3.8) that

[TABLE]

We finally confirm from the boundedness of $\frac{s}{(a+s)^{k}}$ $(k\geq 1)$ that there exists $c>0$ satisfying

[TABLE]

and thus we obtain (3.3). ∎

4 Global existence and boundedness

In this section we will show the $L^{p}$-estimate for $u$ with $p>\frac{n}{2}$ by using Lemma 3.2.

4.1 Energy estimate in the case $k>1$

We first derive the energy estimate in the case $k>1$. In this subsection we assume that $\chi$ satisfies (1.3) with some $k>1$. Now we shall show the following inequality by modifying the method in [11].

Lemma 4.1**.**

Assume that (1.3) and (1.4) are satisfied with some $k>1$, $a\geq 0$ and $K>0$. Then there exist $p>\frac{n}{2}$, $r>0$ and $\varepsilon\in(0,1)$ such that

[TABLE]

where $H_{\varepsilon}$ is defined in Lemma 3.2 and $\eta>0$ is defined as (1.5), which implies that

[TABLE]

Proof.

We take $p\geq 1$, $r>0$ and $\varepsilon\in[0,1)$ which will be fixed later. Due to the definition of $H_{\varepsilon}$ we write as

[TABLE]

where

[TABLE]

Noting from the condition (1.4) that there exist $p>\frac{n}{2}$ and $\varepsilon\in(0,1)$ satisfying

[TABLE]

we have that

[TABLE]

This implies that the discriminant

[TABLE]

is nonnegative for all $s\geq\eta$. Finally, we show that there exists $r>0$ such that

[TABLE]

for all $s\geq\eta$. Because $D_{r}(s)$ is nonnegative, we can define

[TABLE]

Then we see that $H_{\varepsilon}(s)\leq 0$ for each $s\geq\eta$ and all $r\in(r_{-}(s),r_{+}(s))$. Since the functions

[TABLE]

satisfy that

[TABLE]

for all $\tau\geq\frac{K\sqrt{p(\varepsilon p+1-\varepsilon)}}{1-\varepsilon}$, we obtain that

[TABLE]

holds for all $s\geq\eta$. Therefore if we put

[TABLE]

then $r=r_{0}\in(r_{-}(s),r_{+}(s))$ for all $s\geq\eta$, which means that (4.2) holds for all $s\geq\eta$. This implies the end of the proof. ∎

Now we are ready to show the $L^{p}$-estimate in the case $k>1$.

Lemma 4.2**.**

Assume that (1.3) and (1.4) are satisfied with some $k>1$, $a\geq 0$ and $K>0$. Then there exist $p>\frac{n}{2}$ and $C>0$ such that

[TABLE]

Proof.

The proof is similar as in [13]. From Lemmas 3.2 and 4.1 we obtain (4.1) with some $p>\frac{n}{2}$, $r>0$ and $\varepsilon\in(0,1)$. We shall show the $L^{p}$-estimate for $u$ by using (4.1). From the positivity of $u$, $v$ and $\varphi$ we have that

[TABLE]

We next deal with the term $\int_{\Omega}u^{p-1}\varphi(v)|\nabla u|^{2}$. Noting the boundedness of $\varphi$:

[TABLE]

and combining the Gagliardo–Nirenberg inequality with the mass conservation property (2.1), we deduce that there exists $c_{1}>0$ such that

[TABLE]

with $a=\frac{\frac{2n}{p}-\frac{n}{2}}{\frac{2n}{p}+1-\frac{n}{2}}\in(0,1)$. From (4.1), (4.3) and combination of (4.5) and

[TABLE]

we see that there exist $c_{2},c_{3}>0$ such that

[TABLE]

which implies that there exist $p>\frac{n}{2}$, $r>0$ (determined in Lemma 4.1) and $C>0$ satisfying

[TABLE]

Therefore we obtain from (4.4) that

[TABLE]

Thus we attain the goal of the proof. ∎

4.2 Energy estimates in the case $k=1$

In this section we assume that the sensitivity function $\chi$ satisfies (1.3) with $k=1$. We first show the following estimate for $H_{0}(s)$.

Lemma 4.3**.**

Assume that (1.3) and (1.4) are satisfied with $k=1$ and with some $a\geq 0$ and $K>0$. Then for all $p\in(1,\frac{1}{K^{2}})$ there exists an interval $I_{p}$ such that for all $r\in I_{p}$ and $s\geq\eta$,

[TABLE]

where $H_{0}$ is defined as in Lemma 3.2 with $\varepsilon=0$, which means

[TABLE]

Proof.

We pick $p\geq 1$ and $r>0$ which will be fixed later. Due to the definition of $H_{0}(s)$ we write as

[TABLE]

From (1.4) we note that $(1,\frac{1}{K^{2}})$ is not an empty set. If we choose $p\in(1,\frac{1}{K^{2}})$, then the discriminant of $b_{1}r^{2}+b_{2}r+b_{3}$ is nonnegative:

[TABLE]

Thus we can define the interval $I_{p}$ as

[TABLE]

for each $p\in(1,\frac{1}{K^{2}})$. Then we have from straightforward calculations that for each $p\in(1,\frac{1}{K^{2}})$ and all $r\in I_{p}$, $b_{1}r^{2}+b_{2}r+b_{3}\leq 0$ holds, which yields that for all $p\in(1,\frac{1}{K^{2}})$ there exists the interval $I_{p}$ such that

[TABLE]

for all $r\in I_{p}$. This implies the end of the proof. ∎

We next show the estimate for $\int_{\Omega}u^{p}\varphi(v)$. However we cannot easily obtain the estimate for $\int_{\Omega}u^{p}$ because $\varphi$ is not a bounded function. Therefore we will show the following lemma, which has an important role for obtaining the $L^{p}$-estimate.

Lemma 4.4**.**

Assume that (1.3) and (1.4) are satisfied with $k=1$ and with some $a\geq 0$ and $K>0$. Suppose that $p\in(1,\frac{1}{K^{2}})$, $r\in I_{p}$ such that $p-r\geq 1$. If there exists $C>0$ such that

[TABLE]

then there exists $C_{2}(p,r)>0$ satisfying

[TABLE]

Proof.

The proof is similar as in [2]. We let $p\in(1,\frac{1}{K^{2}})$. We denote by $I_{p}$ the interval defined in Lemma 4.3, and choose $r\in I_{p}$. We shall attain the conclusion from (4.6). By virtue of the Hölder inequality, we infer that

[TABLE]

Noting from (4.7) and the fact $p-r\geq 1$ that

[TABLE]

we obtain that there exists $c_{1}>0$ such that

[TABLE]

Plugging (4.8) into (4.6) yields that there exists $c_{2}>0$ such that

[TABLE]

Therefore we have from a standard ODE comparison argument that there exists $C>0$ such that

[TABLE]

which means the end of the proof. ∎

Lemma 4.5**.**

Assume that (1.3) and (1.4) are satisfied with $k=1$ and with some $a\geq 0$ and $K>0$. Then there exist $p>\frac{n}{2}$ and $C>0$ such that

[TABLE]

Proof.

We use a similar argument as in [2, Proof of Main Theorem (Step 1)]. First we choose a pair $(p_{0},r_{0})$ such that

[TABLE]

Then $p_{0}>r_{0}$, $r_{0}<\frac{n}{2}$, $r_{0}\in I_{p}$, $p_{0}-r_{0}\geq 1$ and $p_{0}-r_{0}<\frac{n}{n-2}$. Since $\frac{n}{2}(1-\frac{1}{p_{0}-r_{0}})<1$ from the inequality $p_{0}-r_{0}<\frac{n}{n-2}$, it follows from Lemma 2.3 and (2.1) that there exists $b_{0}>0$ such that

[TABLE]

Thus we have from the fact $p_{0}-r_{0}\geq 1$ and Lemma 4.4 that

[TABLE]

We will show that for all $q_{0}\in(\frac{2p_{0}}{p_{0}+1},\min\{p_{0},\frac{n(p_{0}-r_{0})}{n-2r_{0}}\})$ there exists $b_{0}^{\prime}>0$ such that

[TABLE]

We first note from the fact $p_{0}\not\in(\frac{n}{n+4},1)$ that

[TABLE]

Now we fix $q_{0}\in(\frac{2p_{0}}{p_{0}+1},\min\{p_{0},\frac{n(p_{0}-r_{0})}{n-2r_{0}}\})$. Applying the Hölder inequality, we infer that

[TABLE]

We can confirm from the fact $\frac{n}{2}(\frac{1}{q_{0}}-\frac{p_{0}-q_{0}}{q_{0}r_{0}})<1$ and Lemma 2.3 that

[TABLE]

Therefore plugging (4.11) and the inequality $\frac{q_{0}r_{0}}{p_{0}-r_{0}}\geq 1$ (from the fact $q_{0}\geq\frac{2p_{0}}{p_{0}+1}$) into (4.10) derives that there exists $b_{0}^{\prime\prime}>0$ satisfying

[TABLE]

Noting that $\frac{r_{0}}{p_{0}}<1$, we obtain (4.9). In the above argument, since $\frac{n}{2}<\min\{\frac{1}{K^{2}},n+1\}$, if $\frac{n+2}{n-2}>\frac{n}{2}$, then we can fix $p_{0}>\frac{n}{2}$. Thus we can find $q_{0}>\frac{n}{2}$ due to $p_{0}<\frac{n(p_{0}+1)}{2(n-p_{0}+1)}=\frac{n(p_{0}-r_{0})}{n-2r_{0}}$ if $\frac{n+2}{n-2}>\frac{n}{2}$. On the other hand, if $\frac{n+2}{n-2}\leq\frac{n}{2}$, then we proceed the iteration argument. For all $i\in\mathbb{N}$, we fix a pair $(p_{i},r_{i})$ defined inductively such as

[TABLE]

then we can see that $p_{i}>r_{i}$, $r_{i}<\frac{n}{2}$ and $r_{i}\in I_{p_{i}}$. Moreover, since the relation $p_{i}<\frac{p_{i-1}(n+2)}{n-2p_{i-1}}$ implies that

[TABLE]

we can find some $q_{i-1}\in(\frac{2p_{i-1}}{p_{i-1}+1},\min\{p_{i-1},\frac{n(p_{i-1}+1)}{2(n-p_{i-1}+1)}\})$ satisfying

[TABLE]

which means that $\frac{n}{2}(\frac{1}{q_{i-1}}-\frac{1}{p_{i}-r_{i}})<1$. Therefore we obtain

[TABLE]

According to (4.12) and $p_{i}-r_{i}\geq 1$ into Lemma 4.4, we have

[TABLE]

Due to a similar argument to the first iteration, we can show that

[TABLE]

with some constant $b_{i}^{\prime}(q_{i})>0$. Because the increasing function $f(x):=\frac{x(n+2)}{n-2x}$ satisfies $f(x)>1$ and $f(x)\to\infty$ as $x\to\frac{n}{2}$, we can find some $i_{0}\in\mathbb{N}$ such that $p_{i_{0}}>\frac{n}{2}$ and $p_{i_{0}-1}\leq\frac{n}{2}$, and hence $q_{i_{0}}>\frac{n}{2}$. Therefore we verify

[TABLE]

with some $p>\frac{n}{2}$ and some $C>0$, which completes the proof. ∎

4.3 Proof of Theorem 1.1

Combination of the $L^{p}$-estimate for $u$ (see Lemma 4.2 or Lemma 4.5) and Lemma 3.1 directly leads to Theorem 1.1. ∎