Pohozaev-type inequalities and their applications for elliptic equations

Abstract

In this paper we derive the Pohozaev-type inequalities for p-Laplacian equations and weighted quasi-linear equations and then prove some non-existence results for the positive solutions of these equations in a class of domains that are more general than star-shaped ones.

1 Introduction

This paper is mainly concerned with the elliptic equation

$$ \textstyle\begin{cases} -\Delta_{p} u=f(x,u), & \mbox{in } \Omega, \\ u=0, & \mbox{on } \partial\Omega, \end{cases} $$

(1.1)

where $\Delta_{p}=\operatorname{div}(\vert \nabla u\vert ^{p-2} \nabla u)$, $f(x,u): R^{N} \times R^{1}\rightarrow R^{1}$ is continuous, and $\Omega\subset \mathbb {R}^{n}$ ($n\ge3$) is a bounded domain with smooth boundary. We will establish the Pohozaev-type inequality for the solutions of (1.1) and then discuss the non-existence of positive solutions of the problem in the non-star-shaped domains. We also discuss a similar topic for the following weighted quasi-linear elliptic equation:

$$ \textstyle\begin{cases} -\operatorname{div}(\vert x\vert ^{-ap}\vert \nabla u\vert ^{p-2}\nabla u)=f(x,u)& \mbox{in } \Omega, \\ u=0 & \mbox{on } \partial\Omega. \end{cases} $$

(1.2)

Recall that in the famous paper [1], Pohozaev considered the following elliptic boundary value problem:

$$ \textstyle\begin{cases} -\Delta u=f(u)& \mbox{in } \Omega, \\ u=0 & \mbox{on } \partial\Omega, \end{cases} $$

(1.3)

where $f\in C(R^{1}, R^{1})$, $\Omega\subset \mathbb {R}^{n}$ ($n\ge3$) is a domain with smooth boundary. Let $F(x)=\int_{0}^{x}f(s)\,ds$, and let $\nu(x)$ be the unit outward normal to ∂Ω at x. He proved the following famous identity.

Theorem A

Pohozaev identity, [1]

Suppose that $u\in C^{2}(\Omega)\cap C^{1}(\bar{\Omega})$ is a solution of (1.3). Then

$$\begin{aligned}& (2-n) \int_{\Omega}u f(u)\,dx+2n \int_{\Omega}F(u)\,dx \\& \quad = \int_{\partial\Omega}\bigl\langle x, \nu(x)\bigr\rangle \biggl\vert \frac{\partial u }{\partial\nu}\biggr\vert ^{2}\,ds. \end{aligned}$$

(1.4)

Based on this identity, Pohozaev obtained a remarkable non-existence result for the following elliptic boundary value problem under the conditions that Ω is star-shaped and $\alpha\ge \frac{n+2}{n-2}$, $\lambda\le0$.

$$ \textstyle\begin{cases} -\Delta u=\vert u\vert ^{\alpha-1}u+\lambda u,\quad u>0,& \mbox{in } \Omega, \\ u=0, & \mbox{on } \partial\Omega. \end{cases} $$

(1.5)

Since then, many new results on this topic have appeared. Some of them generalized Pohozaev’s results to the more general equations, such as quasi-linear elliptic, polyharmonic equations, fractional differential equations. Others considered the case of domains more general than star-shaped ones. See [2–9] and the references therein. In [10], Bahri and Coron proved that if Ω is a smooth domain with non-trivial topology, equation (1.5) may have solutions. Dancer [11] and Ding [12] constructed the examples of contractible domains on which (1.5) has a solution. Therefore, it is interesting to discuss the problem on non-star-shaped contractible domains.

In 1989, Guedda and Veron [13] established the Pohozaev identity of the solutions of (1.1) and got the non-existence results. In [14], Isaia generalized the Pohozaev identity to a non-existence result of higher-order regular strong solutions of (1.1). In [15], Takáč and Il’yasov improved the well-known regularity results of the weak solutions of p-Laplacian equation from [16, 17] and, using the new regularity results for the Dirichlet and Neumann problem, established and proved the Pohozaev-type identity. In [18], Bartsch, Peng and Zhang generalized the non-existence result to the more general problem (1.2). It is also interesting to discuss some special cases of (1.1), such as $f(x,u)=\lambda u^{q-1}+u^{s-1}$, $f(x,u)=p(x)u ^{\alpha}+q(x)u^{\beta}$, etc. See, for example, [19–22].

In this paper we also discuss the non-existence of the positive solution of (1.1) and (1.2). However, our method is different from all of the above work. Instead of the Pohozaev identities, we establish a kind of inequalities, named Pohozaev-type inequalities, which have the same effects as Pohozaev identities, and then prove some non-existence results for the positive solution of (1.1) and (1.2) on non-star-shaped domains.

2 The p-Laplacian equations

In this section, we consider the p-Laplacian equations (1.1). Firstly we give a lemma.

Lemma 2.1

Assume that $V(x)=(V_{1}(x),\ldots,V_{n}(x))$ is a $C^{1}$ vector field on $\mathbb {R}^{n}$ and $u\in W^{1,p}_{0}(\Omega)\cap C^{1}(\bar{\Omega})$ is a solution of (1.1). Then

$$\int_{\Omega}u \operatorname{div}\bigl(\vert \nabla u\vert ^{p-2}V(x)\bigr)\,dx=- \int_{ \Omega} \vert \nabla u\vert ^{p-2}\bigl\langle V(x), \nabla u\bigr\rangle \, dx, $$

and

$$\int_{\Omega}F(x,u)\operatorname{div}V(x)\,dx+ \int_{\Omega}F_{1}(x,u)\,dx=- \int_{\Omega}f(u)\bigl\langle V(x), \nabla u\bigr\rangle \,dx, $$

where $F(x,t)=\int_{0}^{t}f(x,s)\,ds$, $F_{1}(x,t)=\sum_{i=1}^{n}V_{i}\frac{ \partial F(x,t)}{\partial x_{i}}$.

Proof

By the divergence theorem and the fact $u(x)=0$ and $F(x,u)=0$ for $x\in\partial\Omega$, we have the following results:

$$\begin{aligned} 0 =& \int_{\partial\Omega}\bigl\langle u(x)\vert \nabla u\vert ^{p-2}V(x), \nu(x) \bigr\rangle \,ds = \int_{\Omega} \operatorname{div}\bigl(u(x)\vert \nabla u\vert ^{p-2}V(x)\bigr)\,dx \\ =& \int_{\Omega}u \operatorname{div}\bigl(\vert \nabla u\vert ^{p-2}V(x)\bigr)\,dx+ \int_{\Omega}\bigl\langle \vert \nabla u\vert ^{p-2} V(x), \nabla u\bigr\rangle \,dx \\ =& \int_{\Omega}u \operatorname{div}\bigl(\vert \nabla u\vert ^{p-2}V(x)\bigr)\,dx+ \int_{\Omega} \vert \nabla u\vert ^{p-2}\bigl\langle V(x), \nabla u\bigr\rangle \,dx, \end{aligned}$$

and because $V_{i}f(x,u)\frac{\partial u}{\partial x_{i}}=V_{i}(\frac{ \partial F}{\partial x_{i}}-\int_{0}^{u} \frac{\partial f(x,s)}{ \partial x_{i}}\,ds)$, we have

$$\begin{aligned}& -\sum_{i=1}^{n} \int_{\Omega}V_{i}f\frac{\partial u}{\partial x_{i}}\,dx \\& \quad =- \sum _{i=1}^{n}\biggl[ \int_{\Omega} \biggl(\frac{\partial}{\partial x_{i}}(V_{i}F)- \frac{ \partial V_{i}}{\partial x_{i}}F\biggr)\,dx- \int_{\Omega}V_{i} \frac{\partial F}{\partial x_{i}}\,dx\biggr] \\& \quad =- \int_{\partial\Omega}F(x,u) \bigl\langle V(x), \nu(x)\bigr\rangle \,ds+ \int_{\Omega}F(x,u)\operatorname{div}\bigl(V(x)\bigr)\,dx+ \int_{\Omega}F_{1}(x,u)x \\& \quad = \int_{\Omega}F(x,u)\operatorname{div}\bigl(V(x)\bigr)\,dx+ \int_{\Omega}F_{1}(x,u)x. \end{aligned}$$

The proof is complete. □

Based on Lemma 2.1, we can derive a Pohozaev-type inequality for the solutions of (1.1).

Theorem 2.2

Pohozaev-type inequality

Let $V(x)$ be a linear vector field on $\mathbb {R}^{n}$ with the form

$$V(x)= \left( \begin{matrix}a_{11} & \cdots& a_{1n} \\ \vdots& \vdots& \vdots \\ a_{n1} & \cdots& a_{nn} \end{matrix}\right) x. $$

Suppose that $V(x)$ satisfies ${\operatorname{div}V(x)=n}$ and $\langle V(x),x \rangle>0$ for $\forall x\in\mathbb{R}^{n}\setminus\{0\}$. If $u\in W^{1,p}_{0}(\Omega)\cap C^{1}(\bar{\Omega})$ is a solution of (1.1), then

$$\begin{aligned}& (p\mu-n) \int_{\Omega} uf(x,u)\,dx+pn \int_{\Omega}F(x,u)\,dx+p \int_{\Omega}F_{1}(x,u)\,dx \\& \quad \geq(p-1) \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \biggl\vert \frac{\partial u}{\partial\nu}\biggr\vert ^{p}\,ds, \end{aligned}$$

(2.1)

where $\mu=\sup_{\vert x\vert \neq0}\frac{\langle V(x),x\rangle}{ \vert x\vert ^{2}}$.

Proof

It is easy to see that

$$0< \bigl\langle V(x),x\bigr\rangle < \mu \vert x\vert ^{2},\quad \forall x \in \mathbb{R}^{n}\setminus\{0\}. $$

We multiply the equation $-\Delta_{p} u=f(x,u)$ by $\langle V(x), \nabla u\rangle$, and then we integrate in Ω. By the divergence theorem and Lemma 2.1, the right-hand side is

$$\begin{aligned}& - \int_{\Omega}f(x,u)\bigl\langle V(x),\nabla u\bigr\rangle \,dx \\& \quad =- \int_{\Omega}\sum_{i=1}^{n}f(x,u)V_{i}(x) \frac{\partial u}{ \partial x_{i}} \\& \quad = \int_{\Omega}F(x,u) \operatorname{div}V(x)\,dx+ \int_{\Omega}F_{1}(x,u)\,dx. \end{aligned}$$

The left-hand side is

$$\begin{aligned}& \int_{\Omega} \operatorname{div}\bigl(\vert \nabla u\vert ^{p-2} \nabla u\bigr) \bigl\langle V(x), \nabla u\bigr\rangle \,dx \\& \quad =\sum_{j=1}^{n} \int_{\Omega}\frac{\partial}{\partial x_{j}}\biggl(\vert \nabla u\vert ^{p-2} \frac{\partial u}{\partial x_{j}}\biggr) \Biggl(\sum _{i=1}^{n} V _{i}(x) \frac{\partial u}{\partial x_{i}} \Biggr)\,dx \\& \quad =\sum_{j=1}^{n} \int_{\Omega}\frac{\partial}{\partial x_{j}}\Biggl(\vert \nabla u\vert ^{p-2} \frac{\partial u}{\partial x_{j}}\sum_{i=1}^{n} V_{i}(x) \frac{\partial u}{\partial x_{i}}\Biggr)-\vert \nabla u\vert ^{p-2} \frac{\partial u}{ \partial x_{j}}\frac{\partial}{\partial x_{j}}\Biggl(\sum _{i=1}^{n} V_{i}(x) \frac{\partial u}{\partial x_{i}} \Biggr)\,dx. \end{aligned}$$

Following this, we have

$$\begin{aligned}& = \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert \nabla u \vert ^{p}\,ds -\sum_{j=1}^{n} \int_{\Omega} \vert \nabla u\vert ^{p-2} \frac{\partial u}{\partial x_{j}}\Biggl(\sum_{i=1}^{n}a_{ij} \frac{\partial u}{\partial x_{i}}+\sum_{i=1} ^{n} V_{i}(x)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\Biggr) \\& = \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert \nabla u \vert ^{p}\,ds+ \int_{\Omega}u \operatorname{div}\bigl(\vert \nabla u\vert ^{p-2}V(\nabla u)\bigr) \\& \quad {}-\sum_{j=1} ^{n} \int_{\Omega} \vert \nabla u\vert ^{p-2} \frac{\partial u}{\partial x_{j}}\Biggl( \sum_{i=1}^{n}V_{i} \frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\Biggr) \\& = \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert \nabla u \vert ^{p}\,ds- \int_{\Omega} \vert \nabla u\vert ^{p-2}\bigl\langle V( \nabla u), \nabla u\bigr\rangle \,dx- \frac{1}{p}\sum _{i=1}^{n} \int_{\Omega}V_{i}(x)\frac{\partial}{ \partial x_{i}}\bigl(\vert \nabla u\vert ^{p}\bigr)\,dx \\& = \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert \nabla u \vert ^{p}\,ds- \int_{\Omega} \vert \nabla u\vert ^{p-2}\bigl\langle V( \nabla u), \nabla u\bigr\rangle \,dx \\& \quad {}-\frac{1}{p} \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert \nabla u \vert ^{p}\,ds+\frac{1}{p}\sum_{i=1}^{n}a_{ii} \int_{\Omega} \vert \nabla u\vert ^{p}\,dx. \end{aligned}$$

Comparing the left- and right-hand sides, we get the following identity:

$$\begin{aligned}& \biggl(1-\frac{1}{p}\biggr) \int_{\partial\Omega} \vert \nabla u\vert ^{p}\bigl\langle V(x), \nu(x)\bigr\rangle \,ds \\& \quad =-\frac{1}{p}\sum_{i=1}^{n}a_{ii} \int_{\Omega} \vert \nabla u\vert ^{p}\,dx+ \int_{\Omega} \vert \nabla u\vert ^{p-2}\bigl\langle V( \nabla u),\nabla u\bigr\rangle \,dx + \int_{\Omega}F \operatorname{div}V(x)\,dx \\& \quad =-\frac{1}{p}\sum_{i=1}^{n}a_{ii} \int_{\Omega}uf(x,u)\,dx+ \int_{ \Omega} \vert \nabla u\vert ^{p-2}\bigl\langle V( \nabla u),\nabla u\bigr\rangle \,dx+\sum_{i=1}^{n}a_{ii} \int_{\Omega}F(x,u)\,dx \\& \qquad {}+ \int_{\Omega}F_{1}(x,u)\,dx. \end{aligned}$$

Because $0<\langle V(x),x\rangle\leq\mu \vert x\vert ^{2}$, we know that

$$0< \bigl\langle V(\nabla u),\nabla u\bigr\rangle \leq\mu \vert \nabla u\vert ^{2}. $$

Thus,

$$\int_{\Omega} \vert \nabla u\vert ^{p-2}\bigl\langle V( \nabla u),\nabla u\bigr\rangle \,dx \leq\mu \int_{\Omega} \vert \nabla u\vert ^{p}\,dx=\mu \int_{\Omega}uf(u)\,dx. $$

By $\operatorname{div}V(x)=\sum_{i=1}^{n}a_{ii}=n$, we obtain the following inequality:

$$\begin{aligned}& (p\mu-n) \int_{\Omega} uf(x,u)\,dx+pn \int_{\Omega}F(x,u)\,dx+p \int_{ \Omega}F_{1}(x,u)\,dx \\& \quad \geq(p-1) \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \biggl\vert \frac{\partial u}{\partial\nu}\biggr\vert ^{p}\,ds. \end{aligned}$$

The proof is complete. □

Based on the above Pohozaev-type inequality, we discuss the non-existence of a positive solution of the following boundary value problem:

$$ \textstyle\begin{cases} -\Delta_{p} u=\lambda_{1}\vert u\vert ^{r-2}u+\lambda_{2}\vert u\vert ^{s-2}u, & \mbox{in } \Omega, \\ u=0, & \mbox{on } \partial\Omega, \end{cases} $$

(2.2)

where $1< r\leq s<+\infty$.

Theorem 2.3

Suppose that there exists a vector field ${V(x)}$ which satisfies the conditions of Theorem  2.2 and is transverse to ∂Ω. Then (2.2) has no positive solution in the following cases respectively:

($D_{1}$):

$p_{1}^{*}>r$, $\lambda_{1}>0$, $\lambda _{2}>0$, $p_{1}^{*}>s$;

($D_{2}$):

$p_{1}^{*}>r$, $\lambda_{1}>0$, $\lambda _{2}<0$, $p_{1}^{*}< s$;

($D_{3}$):

$p_{1}^{*}< r$, $\lambda_{1}<0$, $\lambda _{2}<0$, $p_{1}^{*}< s$,

where $p_{1}^{*}=\frac{np}{n-p\mu}$.

Proof

It is easy to see that

$$f(x,u)=\lambda_{1}\vert u\vert ^{r-2}u+ \lambda_{2}\vert u\vert ^{s-2}u, \qquad F(x,u)= \int_{0}^{u}f(x,s)\,ds=\frac{\lambda_{1}}{r}\vert u \vert ^{r} +\frac{ \lambda_{2}}{s}\vert u\vert ^{s}, $$

and then we have

$$uf(x,u)=\lambda_{1}\vert u\vert ^{r}+ \lambda_{2}\vert u\vert ^{s},\qquad \frac{ \partial F(x,u)}{\partial x_{i}}=0,\qquad F_{1}=0. $$

Suppose that u is a positive solution of (2.2), according to Theorem 2.2, inequality (2.1) holds, then we have

$$(p\mu-n) \int_{\Omega} \bigl(\lambda_{1}\vert u\vert ^{r}+\lambda_{2} \vert u\vert ^{s}\bigr)\,dx+pn \int_{\Omega}\frac{\lambda_{1}}{r}\vert u\vert ^{r} + \frac{\lambda_{2}}{s}\vert u\vert ^{s}\,dx \geq0. $$

This yields

$$ \lambda_{1}\Vert u\Vert _{r}^{r} \biggl(\frac{r-p_{1}^{*}}{r}\biggr)\geq\lambda_{2}\Vert u\Vert _{s}^{s}\biggl(\frac{p_{1}^{*}-s}{s}\biggr). $$

(2.3)

By ($D_{1}$)-($D_{3}$), the left-hand side of (2.3) is negative, and the right-hand side is positive, which is a contradiction. Then the proof is complete. □

Especially, we consider the problem in the case of $p=2$:

$$ \textstyle\begin{cases} -\Delta u=\lambda_{1}\vert u\vert ^{r-2}u+\lambda_{2}\vert u\vert ^{s-2}u, & \mbox{in } \Omega, \\ u=0, & \mbox{on } \partial\Omega. \end{cases} $$

(2.4)

Corollary 2.4

Suppose that there exists a vector field ${V(x)}$ which satisfies the conditions of Theorem  2.2 and is transverse to ∂Ω. Then (2.4) has no positive solution in the following cases respectively:

($D_{1}^{\prime}$):

$2_{1}^{*}>r$, $\lambda_{1}>0$, $\lambda_{2}>0$, $2_{1}^{*}>s$;

($D_{2}^{\prime}$):

$2_{1}^{*}>r$, $\lambda_{1}>0$, $\lambda_{2}<0$, $2_{1}^{*}< s$;

($D_{3}^{\prime}$):

$2_{1}^{*}< r$, $\lambda_{1}<0$, $\lambda_{2}<0$, $2_{1}^{*}< s$,

where $2_{1}^{*}=\frac{2n}{n-2\mu}$.

3 The weighted quasi-linear elliptic boundary value problem

This section is devoted to the weighted quasi-linear elliptic problem (1.2).

Theorem 3.1

Pohozaev-type inequality

Let $V(x)$ be a linear vector field on $\mathbb {R}^{n}$ with the form

$$V(x)= \left( \begin{matrix} a_{11} & \cdots& a_{1n} \\ \vdots& \vdots& \vdots \\ a_{n1} & \cdots& a_{nn} \end{matrix}\right) x. $$

Suppose that $V(x)$ satisfies ${\operatorname{div}V(x)=n}$ and $\langle V(x),x \rangle>0$ for $\forall x\in\mathbb{R}^{n}\setminus\{0\}$. If $u\in W^{1,p}_{0}(\Omega)\cap C^{1}(\bar{\Omega})$ is a solution of (1.2), then

$$\begin{aligned}& \bigl((a+1)p\mu-n\bigr) \int_{\Omega} uf(x,u)\,dx+pn \int_{\Omega}F(x,u)\,dx+p \int_{\Omega}F_{1}(x,u)\,dx \\& \quad \geq(p-1) \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \biggl\vert \frac{\partial u}{\partial\nu}\biggr\vert ^{p}\,ds, \end{aligned}$$

(3.1)

where $\mu=\sup_{\vert x\vert \neq0}\frac{\langle V(x),x\rangle}{ \vert x\vert ^{2}}$.

Proof

We multiply equation (1.2) by $\langle V(x), \nabla u\rangle$, and then we integrate in Ω. Similar to the proof of Theorem 2.2, by the divergence theorem and Lemma 2.1, we have

$$\begin{aligned} - \int_{\Omega}f(x,u)\bigl\langle V(x),\nabla u\bigr\rangle \,dx =&- \int_{\Omega} \sum_{i=1}^{n}f(x,u)V_{i}(x) \frac{\partial u}{\partial x_{i}} \\ =& \int_{\Omega}F(x,u) \operatorname{div}V(x)\,dx+ \int_{\Omega}F_{1}(x,u)\,dx, \end{aligned}$$

then

$$\begin{aligned}& =\sum_{j=1}^{n} \int_{\Omega}\frac{\partial}{\partial x_{j}}\biggl(\vert x\vert ^{-ap}\vert \nabla u\vert ^{p-2} \frac{\partial u}{\partial x_{j}}\biggr) \Biggl(\sum_{i=1}^{n} V _{i}(x) \frac{\partial u}{\partial x_{i}}\Biggr)\,dx \\& =\sum_{j=1}^{n} \int_{\Omega}\frac{\partial}{\partial x_{j}}\Biggl(\vert x\vert ^{-ap}\vert \nabla u\vert ^{p-2} \frac{\partial u}{\partial x_{j}}\sum _{i=1}^{n} V_{i}(x) \frac{\partial u}{\partial x_{i}}\Biggr) \\& \quad {}-\vert x\vert ^{-ap}\vert \nabla u\vert ^{p-2} \frac{ \partial u}{\partial x_{j}}\frac{\partial}{\partial x_{j}}\Biggl(\sum _{i=1} ^{n} V_{i}(x) \frac{\partial u}{\partial x_{i}} \Biggr)\,dx \\& = \int_{\partial\Omega} \vert x\vert ^{-ap}\vert \nabla u \vert ^{p} \bigl\langle V(x), \nu(x) \bigr\rangle \,ds \\& \quad {} -\sum _{j=1}^{n} \int_{\Omega} \vert x\vert ^{-ap}\vert \nabla u\vert ^{p-2}\frac{ \partial u}{\partial x_{j}}\Biggl(\sum_{i=1}^{n}a_{ij} \frac{\partial u}{ \partial x_{i}}+\sum_{i=1}^{n} V_{i}(x)\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\Biggr)\,dx \\& = \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert x\vert ^{-ap}\vert \vert \nabla u\vert ^{p}\,ds+ \int_{\Omega}u \operatorname{div}\bigl(\vert x\vert ^{-ap} \vert \nabla u\vert ^{p-2}V( \nabla u)\bigr)\,dx \\& \quad {}-\sum_{j=1}^{n} \int_{\Omega} \vert x\vert ^{-ap}\vert \vert \nabla u \vert ^{p-2}\frac {\partial u}{\partial x_{j}}\Biggl(\sum_{i=1}^{n} V_{i}(x)\frac{\partial ^{2}u}{\partial x_{i}\partial x_{j}}\Biggr)\,dx \\& = \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert x\vert ^{-ap}\vert \nabla u\vert ^{p}\,ds- \int_{\Omega} \vert x\vert ^{-ap}\vert \nabla u\vert ^{p-2}\bigl\langle V( \nabla u),\nabla u\bigr\rangle \,dx \\& \quad {}-\frac{1}{p}\sum_{i=1}^{n} \int_{\Omega} \vert x\vert ^{-ap} V_{i}(x) \frac{ \partial}{\partial x_{i}}\bigl(\vert \nabla u\vert ^{p}\bigr)\,dx \\& = \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert x\vert ^{-ap}\vert \nabla u\vert ^{p}\,ds- \int_{\Omega} \vert x\vert ^{-ap}\vert \nabla u\vert ^{p-2} \bigl\langle V( \nabla u),\nabla u\bigr\rangle \,dx \\& \quad {}-\frac{1}{p} \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert x\vert ^{-ap}\vert \nabla u\vert ^{p}\,ds+\frac{1}{p}\sum _{i=1}^{n}a_{ii} \int_{\Omega} \vert x\vert ^{-ap}\vert \nabla u\vert ^{p}\,dx \\& \quad {}+ \frac{1}{p}\sum_{i=1}^{n} \int_{\Omega} \vert \nabla u\vert ^{p}V _{i}\frac{\partial}{\partial x_{i}}\vert x\vert ^{-ap}\,dx \\& = \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert x\vert ^{-ap}\vert \nabla u\vert ^{p}\,ds- \int_{\Omega} \vert x\vert ^{-ap} \vert \nabla u \vert ^{p-2}\bigl\langle V( \nabla u),\nabla u\bigr\rangle \,dx \\& \quad {}-\frac{1}{p} \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \vert x\vert ^{-ap}\vert \nabla u\vert ^{p}\,ds +\frac{n}{p} \int_{\Omega} \vert x\vert ^{-ap}\vert \nabla u\vert ^{p}\,dx \\& \quad {}-a \int_{\Omega} \vert x\vert ^{-ap-2}\vert \nabla u \vert ^{p}\bigl\langle V(x),x\bigr\rangle \,dx. \end{aligned}$$

Since $0<\langle V(x),x\rangle\leq\mu \vert x\vert ^{2}$, and $0<\langle V( \nabla u),\nabla u\rangle\leq\mu \vert \nabla u\vert ^{2}$, we have

$$\begin{aligned} \int_{\Omega} \vert \nabla u\vert ^{p-2}\bigl\langle V( \nabla u),\nabla u\bigr\rangle \,dx & \leq\mu \int_{\Omega} \vert \nabla u\vert ^{p}\,dx=\mu \int_{\Omega}uf(u)\,dx. \end{aligned} $$

We obtain the following inequality:

$$\begin{aligned}& \bigl((a+1)p\mu-n\bigr) \int_{\Omega} uf(x,u)\,dx+pn \int_{\Omega}F(x,u)\,dx+p \int_{\Omega}F_{1}(x,u)\,dx \\& \quad \geq(p-1) \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \biggl\vert \frac{\partial u}{\partial\nu}\biggr\vert ^{p}\,ds. \end{aligned}$$

The proof is complete. □

Theorem 3.2

Suppose that there exists a vector field ${V(x)}$ which satisfies the conditions of Theorem  3.1 and is transverse to ∂Ω. Then (1.2) has no positive solution if

$$\bigl((a+1)p\mu-n\bigr) \int_{\Omega} uf(x,u)\,dx+pn \int_{\Omega}F(x,u)\,dx+p \int_{\Omega}F_{1}(x,u)\,dx< 0. $$

Now we consider two special but important cases as follows.

1. A weighted quasi-linear problem

$$ \textstyle\begin{cases} -\operatorname{div}(\vert x\vert ^{-ap}\vert \nabla u\vert ^{p-2}\nabla u)=\lambda_{1}\vert u\vert ^{r-2}u+\lambda _{2}\vert u\vert ^{s-2u}, & \mbox{in } \Omega, \\ u=0, & \mbox{on } \partial\Omega. \end{cases} $$

(3.2)

By Theorem 3.2, problem (3.2) has no solution in the following cases.

Theorem 3.3

Suppose that the vector field ${V(x)}$ of Theorem  3.1 is transverse to ∂Ω. Then (3.2) has no positive solution in the following cases respectively:

($E_{1}$):

$p_{2}^{*}>r$, $\lambda_{1}>0$, $\lambda_{2}>0$, $p_{2}^{*}>s$;

($E_{2}$):

$p_{2}^{*}>r$, $\lambda_{1}>0$, $\lambda_{2}<0$, $p_{2}^{*}< s$;

($E_{3}$):

$p_{2}^{*}< r$, $\lambda_{1}<0$, $\lambda_{2}<0$, $p_{2}^{*}< s$,

where $p_{2}^{*}=\frac{np}{n-(a+1)p\mu}$.

2. A non-autonomous weighted quasi-linear problem:

$$ \textstyle\begin{cases} -\operatorname{div}(\vert x\vert ^{-ap}\vert \nabla u\vert ^{p-2}\nabla u)=\lambda_{1}\vert x\vert ^{-\alpha} \vert u\vert ^{r-2}u+ \lambda_{2}\vert x\vert ^{-\beta} \vert u\vert ^{s-2}u, & \mbox{in } \Omega, \\ u=0, & \mbox{on } \partial\Omega. \end{cases} $$

(3.3)

By Theorem 3.2, we have the next theorem of problem (3.3).

Theorem 3.4

Suppose that there exists a vector field ${V(x)}$ which satisfies the conditions of Theorem  2.3 and is transverse to ∂Ω. If

$$ \biggl(\alpha_{0}-\frac{\alpha-n}{r}\biggr) \lambda_{1}< 0,\quad \textit{and}\quad \biggl(\frac{ \beta-n}{s}-\alpha_{0} \biggr)\lambda_{2}>0, $$

(3.4)

where $\alpha_{0}=(a+1)\mu-\frac{n}{p}$, then (3.3) has no positive solution.

Proof

It is easy to see that

$$\begin{aligned}& f(x,u)=\lambda_{1}\vert x\vert ^{-\alpha} \vert u\vert ^{r-2}u+\lambda_{2}\vert x\vert ^{-\beta }\vert u \vert ^{s-2}u, \\& F(x,u)= \int_{0}^{u}f(x,s)\,ds=\frac{\lambda_{1}}{r}\vert x \vert ^{-\alpha }\vert u\vert ^{r}+\frac{ \lambda_{2}}{s}\vert x\vert ^{-\beta} \vert u\vert ^{s}, \end{aligned}$$

and then we have

$$uf(x,u)=\lambda_{1}\vert x\vert ^{-\alpha} \vert u\vert ^{r}+\lambda_{2}\vert x\vert ^{-\beta} \vert u \vert ^{s}, $$

and

$$F_{1}(x,u)=\sum_{i=1}^{n}x_{i} \frac{\partial}{\partial x_{i}}F(x,u) =-\frac{ \alpha\lambda_{1}}{r}\vert x\vert ^{-\alpha} \vert u\vert ^{r}- \frac{\beta\lambda_{2}}{s}\vert x\vert ^{-\beta} \vert u\vert ^{s}. $$

Suppose that u is a positive solution of (3.3), then inequality (3.1) holds, that is,

$$\begin{aligned}& \bigl((a+1)p\mu-n\bigr) \int_{\Omega} \bigl(\lambda_{1}\vert x\vert ^{-\alpha} \vert u\vert ^{r}+ \lambda_{2} \vert x \vert ^{-\beta} \vert u\vert ^{s}\bigr)\,dx +pn \int_{\Omega}F(x,u)\,dx+p \int_{\Omega}F_{1}(x,u)\,dx \\& \quad \geq(p-1) \int_{\partial\Omega}\bigl\langle V(x),\nu(x)\bigr\rangle \biggl\vert \frac{\partial u}{\partial\nu}\biggr\vert ^{p}\,ds\geq0. \end{aligned}$$

This implies

$$\begin{aligned}& (p\mu-n) \int_{\Omega} \bigl(\lambda_{1}\vert x\vert ^{-\alpha} \vert u\vert ^{r}+\lambda _{2}\vert x \vert ^{-\beta} \vert u\vert ^{s}\bigr)\,dx +pn \int_{\Omega}\frac{\lambda_{1}}{r}\vert x\vert ^{- \alpha} \vert u\vert ^{r}+\frac{\lambda_{2}}{s}\vert x\vert ^{-\beta} \vert u\vert ^{s}\,dx \\& \quad \geq p \int_{\Omega}\frac{\alpha\lambda_{1}}{r}\vert x\vert ^{-\alpha } \vert u\vert ^{r}+\frac{ \beta\lambda_{2}}{s}\vert x\vert ^{-\beta} \vert u\vert ^{s}\,dx. \end{aligned}$$

Hence, we get

$$ \lambda_{1}\biggl(\alpha_{0}- \frac{\alpha-n}{r}\biggr) \int_{\Omega }\vert x\vert ^{-\alpha }\vert u\vert ^{r}\,dx\geq\lambda_{2} \biggl(\frac{\beta-n}{s}- \alpha_{0}\biggr) \int_{ \Omega} \vert x\vert ^{-\beta} \vert u\vert ^{s}, $$

(3.5)

which leads to a contradiction, because of (3.4). □

Remark

Let $\lambda_{1}=\lambda_{2}=-1$, $\alpha=br$, $\beta=cs$, $r=\frac{np}{n-p(a+1-b)}$, $s=\frac{np}{n-p(a+1-c)}$, problem (3.3) is the equation that is discussed in [23].