Existence and regularity of solutions to a quasilinear elliptic problem involving variable sources

Abstract

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation $-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=\frac{f(x)}{u ^{\alpha(x)}}$ with $f\in L^{m}(\Omega)$ ($m\geqslant1$) and $\alpha(x)>0$. Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent $\alpha(x)$, some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of $\alpha(x)$.

1 Introduction

In this paper, we study the existence of solutions for the following quasilinear elliptic problem with nonlinear singular terms and variable exponent:

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

(1.1)

where Ω is a bounded domain in $R^{N}$ ($N\geqslant p$) with smooth boundary ∂Ω, $p>2$, $\alpha(x)$ is a continuous function on Ω̅, $\alpha(x)>0$, $\alpha^{+}= \sup_{x\in\overline{\Omega}}\alpha(x)$, $\alpha^{-}=\inf_{x\in\overline {\Omega}}\alpha(x)$, f is a nonnegative function belonging to the Lebesgue space $L^{m}(\Omega)$ for some suitable $m\geqslant1$.

Problem (1.1) has been widely applied in many areas such as the contexts of chemical heterogeneous catalysts, non-Newtonian fluids and also the theory of heat conduction in electrically conducting materials, see [1–4] for detailed discussion.

For constant-exponent cases, Lazer and Mckenna in [5] discussed the case when $p=2$ and f is a positive regular function in Ω̅. They proved that the solution was in $H_{0}^{1}( \Omega)$ if and only if $\alpha<3$, while it was not in $C^{1}(\overline{ \Omega})$ if $\alpha>1$.

Lair and Shaker in [6] improved the results of [5]. More specially speaking, they proved that this problem with $0<\alpha<1$ has a unique weak positive solution in $H_{0}^{1}( \Omega)$ if $f(x)$ is a nonnegative nontrivial function in $L^{2}(\Omega)$.

In 2004, the results of Lair and Shaker were generalized by Zhang and Cheng (see [7]) to the following problem:

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

(1.2)

where $g(s)$ is singular near $s=0$. They proved the existence and uniqueness of classical solutions under the assumption that $f(x)\in C^{\alpha}(\Omega)$.

Recently, Boccardo and Orsina in [8] studied the existence, regularity and nonexistence of solutions for the following problem:

$$ \textstyle\begin{cases} -\operatorname {div}(M(x)\nabla u)= \frac{f(x)}{u^{\alpha}}, &x\in\Omega, \\ u=0, &x\in \partial\Omega. \end{cases} $$

(1.3)

They discussed how the summability of f and the value of α affected the existence and regularities of solutions to the above problems. For other results of the related problems, the interested readers may refer to [9, 10] and the references therein. Problem (1.3) was discussed and extended to the more general problem of which the right-hand side is $f(x)/u^{\alpha(x)}$ in [11]. For more related questions, refer to [12–14].

In this paper, we generalize the results in [11] to the case when the left-hand side is a p-Laplace operator. Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent $\alpha(x)$, some classical methods may not directly be applied to our problem. We apply the method of regularization and the Schauder fixed point theorem, construct a suitable test function as well as a necessary compactness argument to overcome the difficulties arising from a variable exponent and a nonlinear differential operator and give an almost complete classification of coefficient m and variable exponent $\alpha(x)$, then we prove the existence and regularity of solutions.

2 Preliminaries

Firstly, we give the definition of weak solutions to problem (1.1).

Definition 2.1

A function $u\in W^{1,p}_{0}(\Omega)$ is called a weak solution of problem (1.1) if the following identity holds:

$$ \int_{\Omega} \vert \nabla u\vert ^{p-2}\nabla u \nabla \varphi \,\mathrm{dx}= \int_{\Omega}\frac{f}{u^{\alpha(x)}}\varphi \,\mathrm{dx}, \quad\forall\varphi \in C_{0}^{\infty}(\Omega). $$

(2.1)

In order to prove our results, we will consider the following approximation problem:

$$ \textstyle\begin{cases} -\operatorname {div}(\vert \nabla u_{n}\vert ^{p-2}\nabla u_{n})= \frac{f_{n}}{(u_{n}+\frac{1}{n})^{\alpha(x)}}, &x\in\Omega, \\ u_{n}=0, &x\in \partial\Omega, \end{cases} $$

(2.2)

where $f_{n}(x)=\min\{f(x),n\}$, $n\in N$.

Lemma 2.1

Problem (2.2) has a nonnegative solution $u_{n}$ in $W_{0}^{1,p}( \Omega)\bigcap L^{\infty}(\Omega)$.

Proof

Let $n\in N$ be fixed, and ω be a function in $L^{p}(\Omega)$. It is not difficult to prove that the following problem has a unique solution $v\in W_{0}^{1,p}(\Omega)\bigcap L^{\infty}(\Omega)$ (see [15, 16]):

$$ \textstyle\begin{cases} -\operatorname {div}(\vert \nabla v\vert ^{p-2}\nabla v)= \frac{f_{n}}{(\vert \omega \vert +\frac{1}{n})^{\alpha(x)}}, &x\in\Omega, \\ v=0, &x\in \partial\Omega. \end{cases} $$

(2.3)

So, for any $\omega\in L^{p}(\Omega)$, we define the mapping $\Gamma: L^{p}(\Omega)\rightarrow L^{p}(\Omega)$ as $\Gamma( \omega)=v$. Taking v as a test function for (2.3), we have

$$ \int_{\Omega} \vert \nabla v\vert ^{p}\,\mathrm{dx}= \int_{\Omega}\frac{f_{n}}{(\vert \omega \vert +\frac{1}{n})^{\alpha(x)}}v\,\mathrm{dx}\leqslant \int_{ \Omega}\frac{n}{(\frac{1}{n})^{\alpha^{+}}}v\,\mathrm{dx}\leqslant n ^{\alpha^{+}+1} \int_{\Omega} \vert v\vert \,\mathrm{dx}. $$

By the Poincaré inequality (on the left-hand side) and the Sobolev embedding theorem on the right-hand side ($W^{1,p}(\Omega )\hookrightarrow L^{1}(\Omega)$), we get that

$$\begin{aligned} \Vert v\Vert ^{p}_{W^{1,p}}\leqslant Cn^{\alpha^{+}+1} \Vert v\Vert _{W^{1,p}}, \end{aligned}$$

this implies that

$$ \Vert v\Vert _{W^{1,p}}\leqslant Cn^{\frac{\alpha^{+}+1}{p-1}}. $$

Since the embedding $W^{1,p}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact, we obtain that Γ is a compact operator. Moreover, if $u=\lambda\Gamma u$ for some $0<\lambda\leqslant1$, then $\Gamma u=\frac{u}{\lambda}$ and hence $\Vert u\Vert _{L^{p}(\Omega)}\leqslant \Vert u\Vert _{W^{1,p}(\Omega)}\leqslant C$ for a constant C independent of λ. Then, by Schauder’s fixed point theorem, we know that there exists $u_{n}\in W_{0}^{1,p}(\Omega)$ such that $u_{n}=\Gamma(u_{n})$, i.e., problem (2.2) has a solution. Since $\frac{f_{n}}{(u_{n}+ \frac{1}{n})^{\alpha(x)}}\geqslant0$, the maximum principle in [17, 18] shows that $u_{n}\geqslant0$, $u _{n}\in L^{\infty}(\Omega)$. □

Lemma 2.2

The sequence $\{u_{n}\}$ is increasing with respect to n, $u_{n}>0$ in Ω, and for every $\Omega'\subset\subset\Omega$, there exists $C_{\Omega'}>0$ (independent of n) such that

$$ u_{n}(x)\geqslant C_{\Omega'}>0 \quad \textit{for every }x\in \Omega', \textit{for every }n\in N. $$

(2.4)

Proof

Due to $0\leqslant f_{n}\leqslant f_{n+1}$ and $\alpha(x)>0$, we have that

$$\begin{aligned}& -\operatorname {div}\bigl(\vert \nabla u_{n}\vert ^{p-2}\nabla u_{n}\bigr)=\frac {f_{n}}{(u_{n}+ \frac{1}{n})^{\alpha(x)}}\leqslant\frac{f_{n+1}}{(u_{n}+ \frac{1}{n+1})^{\alpha(x)}}, \\& -\operatorname {div}\bigl(\vert \nabla u_{n+1}\vert ^{p-2}\nabla u_{n+1}\bigr)=\frac{f_{n+1}}{(u _{n+1}+\frac{1}{n+1})^{\alpha(x)}}, \end{aligned}$$

so that

$$ -\operatorname {div}\bigl(\vert \nabla u_{n} \vert ^{p-2}\nabla u_{n}-\vert \nabla u_{n+1} \vert ^{p-2}u _{n+1}\bigr) =f_{n+1} \frac{(u_{n+1}+\frac{1}{n+1})^{\alpha(x)}-(u_{n}+ \frac{1}{n+1})^{\alpha(x)}}{(u_{n}+\frac{1}{n+1})^{\alpha(x)}(u_{n+1}+ \frac{1}{n+1})^{\alpha(x)}}. $$

Choosing $(u_{n}-u_{n+1})_{+}=\max\{u_{n}-u_{n+1},0\}$ as a test function and observing that

$$\begin{aligned} &\bigl(\vert \nabla u_{n}\vert ^{p-2}\nabla u_{n}-\vert \nabla u_{n+1}\vert ^{p-2}\nabla u_{n+1}\bigr) \nabla(u_{n}-u_{n+1})_{+} \geqslant0, \\ & \biggl(\biggl(u_{n+1}+\frac{1}{n+1}\biggr)^{\alpha(x)}- \biggl(u_{n}+\frac{1}{n+1}\biggr)^{ \alpha(x)} \biggr) (u_{n}-u_{n+1})_{+}\leqslant0, \end{aligned}$$

we get that

$$ 0\leqslant \int_{\Omega}\bigl(\vert \nabla u_{n}\vert ^{p-2}\nabla u_{n}-\vert \nabla u _{n+1}\vert ^{p-2}\nabla u_{n+1}\bigr)\nabla (u_{n}-u_{n+1})_{+} \,\mathrm{dx}\leqslant0, $$

which implies that $(u_{n}-u_{n+1})_{+}=0$ a.e. in Ω, that is, $u_{n}\leqslant u_{n+1}$ for every $n\in N$. Since the sequence $\{u_{n}\}$ is increasing with respect to n, we only need to prove that (2.4) holds for $u_{1}$. Using Lemma 2.1, we know that $u_{1}\in L^{\infty}(\Omega)$, i.e., there exists a constant C (depending only on Ω and N) such that

$$ \Vert u_{1}\Vert _{L^{\infty}(\Omega)}\leqslant C\Vert f_{1}\Vert _{L^{\infty}( \Omega)}\leqslant C, $$

then

$$ -\operatorname {div}\bigl(\vert \nabla u_{1}\vert ^{p-2}\nabla u_{1}\bigr)=\frac {f_{1}}{(u_{1}+1)^{ \alpha(x)}}\geqslant\frac{f_{1}}{(C+1)^{\alpha(x)}}. $$

Since $\frac{f_{1}}{(C+1)^{\alpha(x)}}\geqslant0$, $\frac{f_{1}}{(C+1)^{ \alpha(x)}}\not\equiv0$, the strong maximum principle implies that $u_{1}>0$ in Ω and (2.4) holds for $u_{1}$. Because of the monotonicity of $u_{n}$, (2.4) holds for $u_{n}$. □

Remark 2.1

If $u_{n}$ and $v_{n}$ are two solutions of (2.2), following the lines of the proof of the first part in Lemma 2.2, we may show that $u_{n}\leqslant v_{n}$. By symmetry, this implies that the solution of (2.2) is unique.

Lemma 2.3

The solution $u_{1}$ to problem (2.2) with $n=1$ satisfies

$$ \int_{\Omega}u_{1}^{-r}\,\mathrm{dx}< \infty, \quad \forall r< 1. $$

(2.5)

Proof

By $\frac{\min\{f(x), 1\}}{(u_{1}+1)^{\alpha(x)}}\leqslant1$ and Lemma 2.2 in [17], we know that there exists $0<\beta<1$ such that $u_{1}\in C^{1,\beta}(\overline{\Omega})$ and $\Vert u_{1}\Vert _{C ^{1,\beta}}\leqslant C$, which implies that the gradient of $u_{1}$ exists everywhere, then the Hopf lemma in [19] shows that $\frac{\partial u_{1}(x)}{\partial\nu}>0$ in Ω̅, where ν is the outward unit normal vector of ∂Ω at x. Moreover, following the lines of the proof of lemma in [5], we get that

$$ \int_{\Omega}u_{1}^{r}\,\mathrm{dx}< \infty \quad \mbox{if and only if} \quad r>-1. $$

 □

We know clearly that the estimates on $u_{n}$ depend on f and $\alpha(x)$, we will discuss this in different cases.

3 The case $0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1$

In this case, we obtain a priori estimates on $u_{n}$ in $H_{0}^{1}( \Omega)$ only if f is more regular than $L^{1}(\Omega)$. We have the following results.

Lemma 3.1

([20])

Let $u_{n}$ be the solution of (2.2) with $0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1$, and suppose that $f\in L^{m}(\Omega)$ with $m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}= (\frac{p^{*}}{1-\alpha^{-}} )^{\prime}$. Then the sequence $\{u_{n}\}$ is bounded in $W_{0}^{1,p}(\Omega)$.

Once we have the boundedness of $u_{n}$, we can prove an existence result for (1.1).

Theorem 3.1

([20])

Suppose that f is a nonnegative function in $L^{m}(\Omega)$ ($f\not\equiv0$), with $m=\frac{Np}{Np-N+p+(N-p)\alpha ^{-}}= (\frac{p^{*}}{1-\alpha^{-}} )'$, $f\not\equiv0$, and let $0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1$. Then problem (1.1) has a solution $u\in W_{0}^{1,p}(\Omega)$ satisfying (2.1).

The summability of u depends on the summability of f, which is proved in the next lemma.

Lemma 3.2

Suppose that $f\in L^{m}(\Omega)$, $m\geqslant\frac{Np}{Np-N+p+(N-p) \alpha^{-}}$, and let $0<\alpha^{-}\leqslant\alpha(x)\leqslant \alpha^{+}<1$. Then the solution u of (1.1) given by Theorem  3.1 is such that:

• (i) if $m>\frac{N}{p}$, then $u\in L^{\infty}(\Omega)$;

• (ii) if $\frac{Np}{Np-N+p+(N-p)\alpha^{-}}\leqslant m<\frac {N}{p}$, then $u\in L^{s}(\Omega)$, $s=\frac{Nm(\alpha^{-}+p-1)}{N-pm}$.

Proof

To prove (i), let $k>1$ and define $G_{k}(s)=(s-k)_{+}$. Taking $G_{k}(u_{n})$ as a test function in (2.2), we obtain

$$ \int_{\Omega}\bigl\vert \nabla G_{k}(u_{n}) \bigr\vert ^{p}\,\mathrm{dx}\leqslant \int_{ \Omega}\bigl(\bigl\vert \nabla G_{k}(u_{n}) \bigr\vert ^{p-2}\nabla G_{k}(u_{n})\bigr)\cdot \nabla G_{k}(u_{n})\,\mathrm{dx}= \int_{\Omega}\frac{f_{n}G_{k}(u_{n})}{(u _{n}+\frac{1}{n})^{\alpha(x)}}\,\mathrm{dx}. $$

Since $G_{k}(u_{n})\neq0$, it implies that

$$ \int_{\Omega}\bigl\vert \nabla G_{k}(u_{n}) \bigr\vert ^{p}\,\mathrm{dx}\leqslant \int_{ \Omega}fG_{k}(u_{n})\,\mathrm{dx}. $$

(3.1)

Starting from inequality (3.1), Theorem 4.2 in [21] shows that there exists a constant C (independent of n) such that

$$ \Vert u_{n}\Vert _{L^{\infty}(\Omega)}\leqslant C\Vert f\Vert _{L^{m}(\Omega)}, $$

which implies that u belongs to $L^{\infty}(\Omega)$.

To prove (ii), noting that if $m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}$, $s=\frac{Np}{N-p}=p^{*}$, since $u\in W_{0}^{1,p}(\Omega)$, the result when $m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}$ is true by the Sobolev embedding theorem. If $\frac{Np}{Np-N+p+(N-p)\alpha^{-}}< m< \frac{N}{p}$, letting $\delta>1$ and choosing $u_{n}^{p\delta-p+1}$ as a test function in (2.2), using Hölder’s inequality, we get that

$$ \begin{aligned}[b] &(p\delta-p+1) \int_{\Omega} \vert \nabla u_{n}\vert ^{p}u_{n}^{p\delta-p} \,\mathrm{dx}\\ &\quad \leqslant \int_{\{x\in\Omega, u_{n}\geqslant1\}}\frac{fu_{n}^{p \delta-p+1}}{u_{n}^{\alpha^{-}}}\,\mathrm{dx}+ \int_{\{x\in\Omega, u_{n}< 1\}}\frac{fu_{n}^{p\delta-p+1}}{u_{n} ^{\alpha^{+}}}\,\mathrm{dx}\\ &\quad = \Vert f\Vert _{L^{m}(\Omega)} \biggl( \biggl( \int_{\Omega }u_{n}^{(p\delta-p+1- \alpha^{-})m'}\,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad\quad{} + \vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}} \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'}\,\mathrm{dx}\biggr)^{\frac{p \delta-p+1-\alpha^{+}}{(p\delta-p+1-\alpha^{-})m'}} \biggr). \end{aligned} $$

(3.2)

By the Sobolev inequality (on the left-hand side), we have that

$$ \int_{\Omega} \vert \nabla u_{n}\vert ^{p}u_{n}^{p\delta -p}\,\mathrm{dx}=\frac{1}{ \delta^{p}} \int_{\Omega}\bigl\vert \nabla u_{n}^{\delta} \bigr\vert ^{p}\,\mathrm{dx}\geqslant\frac{S}{\delta^{p}} \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}}, $$

(3.3)

where S is the constant of the Sobolev embedding theorem. Combining with (3.2) and (3.3), we have that

$$ \begin{aligned}[b] & \frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega}u_{n}^{p^{*} \delta} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \\ & \quad \leqslant \Vert f\Vert _{L^{m}( \Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad\quad {} + \vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}} \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'}\,\mathrm{dx}\biggr)^{\frac{p \delta-p+1-\alpha^{+}}{(p\delta-p+1-\alpha^{-})m'}} \biggr). \end{aligned} $$

(3.4)

We choose δ in such a way that $p^{*}\delta=(p\delta-p+1- \alpha^{-})m'$, i.e.,

$$ \delta=\frac{(\alpha^{-}+p-1)m(N-p)}{p(N-mp)}, $$

which yields that $\delta>1$ if and only if $\frac{Np}{Np-N+p+(N-p) \alpha^{-}}< m<\frac{N}{p}$, and that $p^{*}\delta=\frac{Nm(\alpha ^{-}+p-1)}{N-pm}=s$. Therefore, (3.4) becomes

$$ \begin{aligned} \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} &\leqslant \frac{\delta^{p}}{S(p\delta-p+1)}\Vert f\Vert _{L^{m}(\Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad{} +\vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}} \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p\delta-p+1-\alpha ^{+}}{(p\delta-p+1-\alpha^{-})m'}} \biggr), \end{aligned} $$

which implies that

$$ \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p-1+\alpha^{+}}{p ^{*}\delta}} \leqslant\frac{\delta^{p}}{S(p\delta-p+1)}\Vert f\Vert _{L ^{m}(\Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{ \alpha^{+}-\alpha^{-}}{p^{*}\delta}} +\vert \Omega \vert ^{\frac{\alpha^{+}- \alpha^{-}}{p^{*}\delta}} \biggr). $$

(3.5)

Using Young’s inequality on the right-hand side in (3.5), we have that

$$ \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p-1+\alpha^{+}}{p ^{*}\delta}} \leqslant\frac{\delta^{p}}{S(p\delta-p+1)}\Vert f\Vert _{L ^{m}(\Omega)} \biggl(\varepsilon \biggl( \int_{\Omega}u_{n}^{s} \,\mathrm{dx}\biggr)^{\frac{p-1+\alpha^{+}}{p^{*}\delta}} + \varepsilon^{-\frac{\alpha^{+}-\alpha^{-}}{p-1+\alpha^{-}}}+\vert \Omega \vert ^{\frac{ \alpha^{+}-\alpha^{-}}{p^{*}\delta}} \biggr), $$

where $\varepsilon=\frac{S(p\delta-p+1)}{2\delta^{p}\Vert f\Vert _{L^{m}( \Omega)}}$. Thus, we get that

$$ \biggl( \int_{\Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p-1+\alpha^{+}}{p ^{*}\delta}} \leqslant\frac{2\delta^{p}\Vert f\Vert _{L^{m}(\Omega)}}{S(p \delta-p+1)} \biggl( \biggl( \frac{2\delta^{p}\Vert f\Vert _{L^{m}(\Omega)}}{S(p \delta-p+1)} \biggr)^{\frac{\alpha^{+}-\alpha^{-}}{p-1+\alpha^{-}}} +\vert \Omega \vert ^{\frac{\alpha^{+}-\alpha^{-}}{p^{*}\delta}} \biggr). $$

(3.6)

Therefore, we know that $u_{n}$ is bounded in $L^{s}(\Omega)$, so is $u\in L^{s}(\Omega)$. □

Theorem 3.2

Suppose that $f\in L^{m}(\Omega)$, $\frac{(p-1+\alpha^{+})N}{(\alpha ^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}\leqslant m<\frac{Np}{Np-N+p+(N-p) \alpha^{-}}$, and $0<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha ^{+}<1$. Then problem (1.1) has a solution u in $W_{0}^{1,q}( \Omega)$, $q=\frac{Nm(\alpha^{-}+p-1)}{N-m(1-\alpha^{-})}$.

Proof

The lines of our proof are that if we can prove that $u_{n}$ is bounded in $W_{0}^{1,q}(\Omega)$ (with q as in the statement), the existence of a solution u in $W_{0}^{1,q}(\Omega)$ of (1.1) will be proved by passing to the limit in (2.2) as in the proof of Theorem 3.1. To prove that $u_{n}$ is bounded in $W_{0}^{1,q}( \Omega)$, we begin by proving that it is bounded in $L^{s}(\Omega)$, with $s=\frac{Nm(\alpha^{-}+p-1)}{N-pm}$. To attain this goal, we choose $u_{n}^{p\delta-p+1}$ as a test function in (2.2) as in the statement of Lemma 3.2, where $\frac{p-1+\alpha^{+}}{p} \leqslant\delta<1$; however, $\nabla u_{n}^{p\delta-p+1}$ will be singular at $u_{n}=0$. Therefore, we choose $(u_{n}+\varepsilon)^{p \delta-p+1}-\varepsilon^{p\delta-p+1}$ as a test function in (2.2), where $\varepsilon<\frac{1}{n}$ for n fixed. We have that

$$ (p\delta-p+1) \int_{\Omega} \vert \nabla u_{n}\vert ^{p}(u_{n}+\varepsilon)^{p \delta-p}\,\mathrm{dx}\leqslant \int_{\Omega}\frac{f_{n}(u_{n}+ \varepsilon)^{p\delta-p+1}}{(u_{n}+\varepsilon)^{\alpha(x)}} \,\mathrm{dx}. $$

Since $f_{n}\leqslant f$, we have that

$$ \begin{aligned}[b] &(p\delta-p+1) \int_{\Omega} \vert \nabla u_{n}\vert ^{p}(u_{n}+\varepsilon)^{p \delta-p}\,\mathrm{dx}\\ &\quad \leqslant \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1-\alpha ^{-}} \,\mathrm{dx}+ \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1- \alpha^{+}} \,\mathrm{dx}. \end{aligned} $$

(3.7)

By the Sobolev embedding theorem ($W_{0}^{1,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega)$) on the left-hand side, it follows that

$$ \begin{aligned}[b] \int_{\Omega} \vert \nabla u_{n}\vert ^{p}(u_{n}+\varepsilon)^{p\delta-p} \,\mathrm{dx}&= \int_{\Omega}\frac{\vert \nabla ((u_{n}+\varepsilon)^{ \delta}-\varepsilon^{\delta} )\vert ^{p}}{\delta^{p}}\,\mathrm{dx}\\ &\geqslant \frac{S}{\delta^{p}} \biggl( \int_{\Omega} \bigl((u_{n}+\varepsilon )^{\delta}-\varepsilon^{\delta} \bigr)^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}}, \end{aligned} $$

(3.8)

where S is the best constant of the Sobolev embedding theorem. Combining (3.7) with (3.8), we have that

$$ \begin{aligned}[b] &\frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega} \bigl((u_{n}+ \varepsilon)^{\delta}- \varepsilon^{\delta} \bigr)^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \\ &\quad \leqslant \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1-\alpha ^{-}} \,\mathrm{dx}+ \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1- \alpha^{+}} \,\mathrm{dx}. \end{aligned} $$

(3.9)

Using Hölder’s inequality on the right-hand side, we get

$$ \begin{aligned} &\frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega} \bigl((u_{n}+ \varepsilon)^{\delta}- \varepsilon^{\delta} \bigr)^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \\ &\quad \leqslant \Vert f\Vert _{L^{m}(\Omega)} \biggl( \int_{\Omega}(u_{n}+\varepsilon)^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad\quad {} + \vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}}\Vert f\Vert _{L ^{m}(\Omega)} \biggl( \int_{\Omega}(u_{n}+\varepsilon)^{(p\delta-p+1- \alpha^{-})m'}\,\mathrm{dx}\biggr) ^{\frac{p\delta-p+1-\alpha^{+}}{(p \delta-p+1-\alpha^{-})m'}}. \end{aligned} $$

Letting $\varepsilon\rightarrow0$, we get (3.4), i.e.,

$$ \begin{aligned} \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}} &\leqslant \frac{\delta^{p}}{S(p\delta-p+1)}\Vert f\Vert _{L^{m}( \Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad{} + \vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}} \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'}\,\mathrm{dx}\biggr)^{\frac{p \delta-p+1-\alpha^{+}}{(p\delta-p+1-\alpha^{-})m'}} \biggr), \end{aligned} $$

where δ is chosen in such a way that $p^{*}\delta=(p\delta-p+1- \alpha^{-})m'$, i.e.,

$$ \delta=\frac{(\alpha^{-}+p-1)(N-p)m}{p(N-mp)}. $$

If $m= \frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}$, we choose $\delta=\frac{p-1+\alpha^{+}}{p}$ in (3.9), and letting $\varepsilon\rightarrow0$, we have that

$$ \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}}\leqslant\frac{\delta^{p}}{S(p\delta-p+1)} \biggl( \int_{\Omega }fu_{n}^{p\delta-p+1-\alpha^{-}}\,\mathrm{dx}+ \int_{\Omega}f \,\mathrm{dx}\biggr). $$

Using Hölder’s inequality and Young’s inequality, we get that

$$ \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}} \leqslant \frac{\delta^{p}\Vert f\Vert _{L^{m}(\Omega)}}{S(p\delta -p+1)} \biggl(\varepsilon \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}}+ \varepsilon^{- \frac{p^{*}}{pm'-p^{*}}}+\vert \Omega \vert ^{\frac{1}{m'}} \biggr), $$

where $\varepsilon=\frac{S(p\delta-p+1)}{2\delta^{p}\Vert f\Vert _{L^{m}( \Omega)}}$. Thus we have that

$$ \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}} \leqslant \frac{2\delta^{p}\Vert f\Vert _{L^{m}(\Omega)}}{S(p\delta -p+1)} \biggl( \biggl( \frac{2 \delta^{p}\Vert f\Vert _{L^{m}(\Omega)}}{S(p\delta -p+1)} \biggr)^{\frac{p^{*}}{pm'-p ^{*}}}+\vert \Omega \vert ^{\frac{1}{m'}} \biggr). $$

Therefore we obtain that $u_{n}$ is bounded in $L^{\frac{N(p-1+\alpha ^{+})}{N-p}}(\Omega)$, where $\frac{N(p-1+\alpha^{+})}{N-p}$ is the value of s for $m=\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+ \alpha^{+})}$.

If $\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}< m<\frac{Np}{Np-N+p+(N-p) \alpha^{-}}$, it is clear that the inequality on m holds true if and only if $\frac{p-1+\alpha^{+}}{p}<\delta<1$, starting from (3.4) and arguing as in the proof of Lemma 3.2, we also get that $u_{n}$ is bounded in $L^{s}(\Omega)$ with $s=\frac{Nm( \alpha^{-}+p-1)}{N-pm}$.

The right-hand side of (3.7) is bounded with respect to n (and ε, which we take smaller than 1) by using the estimate on $u_{n}$ in $L^{s}(\Omega)$ and the choice of δ.

Since $\delta<1$,

$$ \int_{\Omega}\frac{\vert \nabla u_{n}\vert ^{p}}{(u_{n}+\varepsilon)^{p-p \delta}} \,\mathrm{dx}= \int_{\Omega} \vert \nabla u_{n}\vert ^{p}(u_{n}+\varepsilon )^{p\delta-p}\,\mathrm{dx}\leqslant C. $$

If $q=\frac{Nm(\alpha^{-}+p-1)}{N-m(1-\alpha^{-})}< p$, by Hölder’s inequality, we have that

$$ \begin{aligned}[b] \int_{\Omega} \vert \nabla u_{n}\vert ^{q}\,\mathrm{dx}&= \int _{\Omega}\frac{\vert \nabla u_{n}\vert ^{q}}{(u_{n}+\varepsilon)^{(1-\delta )q}}(u_{n}+\varepsilon )^{(1-\delta)q}\,\mathrm{dx}\\ &\leqslant \biggl( \int_{\Omega}\frac{\vert \nabla u_{n}\vert ^{p}}{(u_{n}+ \varepsilon)^{p(1-\delta)}}\,\mathrm{dx}\biggr)^{\frac{q}{p}} \biggl( \int_{\Omega}(u_{n}+\varepsilon)^{\frac{pq(1-\delta)}{p-q}} \,\mathrm{dx}\biggr)^{1-\frac{q}{p}} \\ &\leqslant C \biggl( \int_{\Omega}(u_{n}+\varepsilon)^{\frac{pq(1- \delta)}{p-q}}\,\mathrm{dx}\biggr)^{1-\frac{q}{p}}. \end{aligned} $$

(3.10)

The choice of δ and the value of q are such that $\frac{pq(1- \delta)}{p-q}=s$, so that the right-hand side of (3.10) is bounded with respect to n and ε. Hence, $u_{n}$ is bounded in $W_{0}^{1,q}(\Omega)$. □

Theorem 3.3

Suppose that $f\in L^{m}(\Omega)$, $\frac{1}{2-p-\alpha^{+}+p\delta }< m<\frac{(p-1+ \alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}$ ($\frac{p-1+ \alpha^{-}}{p}<\delta<\frac{p-1+\alpha^{+}}{p}$), and $0<\alpha ^{-}\leqslant\alpha(x)\leqslant\alpha^{+}<1$. Then problem (1.1) has a solution u in $W_{0}^{1,q}(\Omega)$, $q=\frac{Nm( \alpha^{-}+p-1)}{N-m(1-\alpha^{-})}$.

Proof

The lines of our proof are similar to those in the proof of Theorem 3.2. We also begin by proving that $u_{n}$ is bounded in $L^{s}(\Omega)$, with $s=\frac{Nm(\alpha^{-}+p-1)}{N-pm}$. To this aim, we also choose $(u_{n}+\varepsilon)^{p\delta-p+1}- \varepsilon^{p\delta-p+1}$ as a test function in (2.2), where $\frac{p-1+\alpha^{-}}{p}<\delta<\frac{p-1+\alpha^{+}}{p}$, $\varepsilon<\frac{1}{n}$ for n fixed. Since $f_{n}\leqslant f$, using the Sobolev embedding theorem ($W_{0}^{1,p}(\Omega )\hookrightarrow L^{p^{*}}(\Omega)$) on the left-hand side again, we have that

$$ \begin{aligned} &\frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega} \bigl((u_{n}+ \varepsilon)^{\delta}- \varepsilon^{\delta} \bigr)^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \\ &\quad \leqslant \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1-\alpha ^{-}} \,\mathrm{dx}+ \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1- \alpha^{+}} \,\mathrm{dx}, \end{aligned} $$

where S is the best constant of the Sobolev embedding theorem.

Using Hölder’s inequality and Lemma 2.3 on the right-hand side, we get that

$$ \begin{aligned} &\frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega} \bigl((u_{n}+ \varepsilon)^{\delta}- \varepsilon^{\delta} \bigr)^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \\ &\quad \leqslant \int_{\Omega}f(u_{n}+\varepsilon)^{p\delta-p+1-\alpha ^{-}} \,\mathrm{dx}+ \int_{\Omega}\frac{f}{u_{1}^{p-1+\alpha^{+}-p\delta }}\,\mathrm{dx}\\ &\quad \leqslant \Vert f\Vert _{L^{m}(\Omega)} \biggl( \int_{\Omega }(u_{n}+\varepsilon )^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}}+C\Vert f\Vert _{L^{m}(\Omega)}. \end{aligned} $$

Letting $\varepsilon\rightarrow0$, we have that

$$ \biggl( \int_{\Omega}u_{n}^{p^{*}\delta} \,\mathrm{dx}\biggr)^{\frac{p}{p ^{*}}} \leqslant\frac{\delta^{p}}{S(p\delta-p+1)}\Vert f\Vert _{L^{m}( \Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}}+C \biggr), $$

(3.11)

where δ is chosen in such a way that $p^{*}\delta=(p\delta-p+1- \alpha^{-})m'$, i.e.,

$$ \delta=\frac{(\alpha^{-}+p-1)(N-p)m}{p(N-mp)}. $$

If $1< m<\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}$, it is clear that the inequality on m holds true if and only if $\frac{p-1+\alpha^{-}}{p}<\delta<\frac{p-1+\alpha^{+}}{p}$, and arguing as to the case $m=\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+ \alpha^{+})}$ in the proof of Theorem 3.2, we also obtain that $u_{n}$ is bounded in $L^{s}(\Omega)$, with $s=\frac{Nm(\alpha ^{-}+p-1)}{N-pm}$.

Since $\delta<1$,

$$ \int_{\Omega}\frac{\vert \nabla u_{n}\vert ^{p}}{(u_{n}+\varepsilon)^{p-p \delta}} \,\mathrm{dx}= \int_{\Omega} \vert \nabla u_{n}\vert ^{p}(u_{n}+\varepsilon )^{p\delta-p}\,\mathrm{dx}\leqslant C. $$

If $q=\frac{Nm(\alpha^{-}+p-1)}{N-m(1-\alpha^{-})}< p$, similarly to the proof of Theorem 3.2, we have by Hölder’s inequality that

$$ \int_{\Omega} \vert \nabla u_{n}\vert ^{q}\,\mathrm{dx}\leqslant C \biggl( \int_{ \Omega}(u_{n}+\varepsilon)^{\frac{pq(1-\delta)}{p-q}}\,\mathrm{dx}\biggr)^{1-\frac{q}{p}}. $$

Due to the choice of δ and the value of q, the right-hand side of the above inequality is bounded with respect to n and ε. Hence, $u_{n}$ is bounded in $W_{0}^{1,q}(\Omega)$. □

4 The case $1<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}$

The case $1<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}$ has many analogies with the case $0<\alpha^{-}<\alpha^{+}<1$. In this case, we can also prove that $u_{n}$ is bounded in $W_{0}^{1,p}(\Omega)$ only if f is more regular than $L^{1}(\Omega)$ and $\alpha^{+}$ and $\alpha^{-}$ is close to 1. Hence we obtain the existence of problem (1.1).

Lemma 4.1

Suppose that $f\in L^{m} (\Omega)$ ($m>1$), let $u_{n}$ be the solution of (2.2) with $1<\alpha^{-}<\alpha^{+}<2-\frac{1}{m}$. Then $u_{n}$ is bounded in $W_{0}^{1,p}(\Omega)$.

Proof

Taking $u_{n}$ as a test function in (2.2), we obtain that

$$ \int_{\Omega} \vert \nabla u_{n}\vert ^{p}\,\mathrm{dx}\leqslant \int_{\Omega}\frac{f}{u _{n}^{\alpha(x)-1}}\,\mathrm{dx}. $$

Using Lemma 2.1 and Lemma 2.2, we know that $u_{n}\geqslant u_{1}$ and there exists a constant $M>0$ s.t. $u_{1}\leqslant M$. Hence $(\frac{M}{u_{1}})^{\alpha(x)-1}\leqslant (\frac{M}{u_{1}})^{\alpha^{+}-1}$, and we have that

$$ \int_{\Omega} \vert \nabla u_{n}\vert ^{p}\,\mathrm{dx}\leqslant \int_{\Omega}\frac{f}{u _{1}^{\alpha(x)-1}}\,\mathrm{dx}\leqslant \bigl(1+M^{\alpha^{+}-\alpha^{-}}\bigr) \int_{\Omega}\frac{f}{u_{1}^{\alpha^{+}-1}}\,\mathrm{dx}. $$

Using Hölder’s inequality on the right-hand side and Lemma 2.3, we obtain

$$ \begin{aligned} \int_{\Omega} \vert \nabla u_{n}\vert ^{p}\,\mathrm{dx}\leqslant C\bigl(1+M^{\alpha ^{+}-\alpha^{-}}\bigr)\Vert f\Vert _{L^{m}(\Omega)}. \end{aligned} $$

Therefore, $u_{n}$ is bounded in $W_{0}^{1,p}(\Omega)$. □

Once we have the boundedness of $u_{n}$, we can prove the following existence theorem along the lines of Theorem 3.1.

Theorem 4.1

Suppose that $f\in L^{m}(\Omega)$ ($m>1$), $f\not\equiv0$ and $1<\alpha^{-}<\alpha^{+}<2-\frac{1}{m}$. Then problem (1.1) has a solution u in $W_{0}^{1,p}(\Omega)$.

The summability of u can be proved along the lines of Lemma 3.2 with small changes.

Lemma 4.2

Suppose that $f\in L^{m}(\Omega)$ ($m>1$) and $1<\alpha^{-}<\alpha^{+}<2- \frac{1}{m}$. Then the solution u of (1.1) given by Theorem  4.1 is such that:

• (i) if $m>\frac{N}{p}$, then $u\in L^{\infty}(\Omega)$;

• (ii) if $\frac{(p-1+\alpha^{+})N}{(\alpha ^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}\leqslant m<\frac{N}{p}$, then $u\in L^{s}(\Omega)$, $s=\frac{Nm( \alpha^{-}+p-1)}{N-pm}$.

Proof

The proof of (i) is similar to the proof of Lemma 3.2(i), we omit the details here.

To prove (ii), we choose $u_{n}^{p\delta-p+1}$ as a test function with $\delta\geqslant\frac{p-1+\alpha^{+}}{p}$ in (2.2). Similarly to the proof of Lemma 3.2, we obtain that

$$ \frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega}u_{n}^{p^{*} \delta} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \leqslant \int_{\Omega}fu _{n}^{p\delta-p+1-\alpha^{-}}\,\mathrm{dx}+ \int_{\Omega}fu_{n}^{p \delta-p+1-\alpha^{+}}\,\mathrm{dx}. $$

(4.1)

If $m= \frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}$, choosing $\delta=\frac{p-1+\alpha^{+}}{p}$ in (4.1), by Hölder’s inequality, we get that

$$ \frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega}u_{n}^{p^{*} \delta} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \leqslant \Vert f\Vert _{L^{m}( \Omega)} \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}}+\vert \Omega \vert ^{1-\frac {1}{m}}\Vert f\Vert _{L^{m}( \Omega)}. $$

We choose δ in such a way that $p^{*}\delta=(p\delta-p+1- \alpha^{-})m'$, i.e., $\delta=\frac{(\alpha^{-}+p-1)m(N-p)}{p(N-mp)}$. Since $m=\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}$, we get that $\frac{p}{p^{*}}>\frac{1}{m'}$. Because $s=p^{*}\delta$, we have the boundedness of $u_{n}$ in $L^{\frac{N(p-1+ \alpha^{+})}{N-p}}(\Omega)$, which is the value of s for $m=\frac{N(p-1+\alpha^{+})}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}$.

If $\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}< m< \frac{N}{p}$, starting from inequality (4.1), using Hölder’s inequality, we get that

$$ \begin{aligned} \frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega}u_{n}^{p^{*} \delta} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} &\leqslant \Vert f\Vert _{L^{m}( \Omega)} \biggl( \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'} \,\mathrm{dx}\biggr)^{\frac{1}{m'}} \\ &\quad{} + \vert \Omega \vert ^{ \frac{\alpha^{+}-\alpha^{-}}{(p\delta-p+1-\alpha^{-})m'}} \biggl( \int_{\Omega}u_{n}^{(p\delta-p+1-\alpha^{-})m'}\,\mathrm{dx}\biggr)^{\frac{p \delta-p+1-\alpha^{+}}{(p\delta-p+1-\alpha^{-})m'}} \biggr). \end{aligned} $$

We also choose δ in such a way that $p^{*}\delta=(p\delta-p+1- \alpha^{-})m'$, which yields that $\delta>\frac{p-1+\alpha^{+}}{p}$ if and only if $m>\frac{(p-1+\alpha^{+})N}{(\alpha^{-}+p-1)(N-p)+p(p-1+ \alpha^{+})}$, and that $p^{*}\delta=s$. So, since $\frac{p}{p^{*}}> \frac{1}{m'}$ being $m<\frac{N}{p}$, we have the boundedness of $u_{n}$ in $L^{s}(\Omega)$, so does $u\in L^{s}(\Omega)$. □

Moreover, we can prove that a positive power of $u_{n}$ is bounded in $W_{0}^{1,p}(\Omega)$ only if f is more regular than $L^{1}(\Omega )$ and $\alpha^{+}$ is close to $\alpha^{-}$, and we only have the boundedness of $u_{n}$ in $W_{\mathrm{loc}}^{1,p}(\Omega)$.

Lemma 4.3

Suppose that $f\in L^{m}(\Omega)$ ($m>1$), let $u_{n}$ be the solution of (2.2) with $1<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha ^{+}$ and $\alpha^{+}-\alpha^{-}<1-\frac{1}{m}$. Then $u_{n}^{\frac{p-1+ \alpha^{-}}{p}}$ is bounded in $W_{0}^{1,p}(\Omega)$, and $u_{n}$ is bounded in $W_{\mathrm{loc}}^{1,p}(\Omega)$ and in $L^{s}(\Omega)$, with $s=\frac{N(\alpha^{-}+p-1)}{N-p}$.

Proof

Taking $u_{n}^{\alpha^{-}}$ as a test function in (2.2), since $\frac{u_{n}^{\alpha^{-}}}{(u_{n}+\frac{1}{n})^{\alpha^{-}}}\leqslant 1$ and $f_{n}\leqslant f$, by Hölder’s inequality and Lemma 2.3, we get that

$$ \begin{aligned} \alpha^{-} \int_{\Omega} \vert \nabla u_{n}\vert ^{p}u_{n}^{\alpha^{-}-1} \,\mathrm{dx}&\leqslant \int_{\Omega}\frac{fu_{n}^{\alpha^{-}}}{(u_{n}+ \frac{1}{n})^{\alpha^{-}}}\,\mathrm{dx}+ \int_{\Omega}\frac{fu_{n}^{ \alpha^{-}}}{u_{n}^{\alpha^{+}}}\,\mathrm{dx}\\ &\leqslant \int_{\Omega}f \,\mathrm{dx}+ \int_{\Omega}\frac{f}{u_{n} ^{\alpha^{+}-\alpha^{-}}}\,\mathrm{dx}\leqslant \vert \Omega \vert ^{1- \frac{1}{m}}\Vert f\Vert _{L^{m}(\Omega)}+C\Vert f\Vert _{L^{m}(\Omega)}. \end{aligned} $$

Since

$$ \int_{\Omega} \vert \nabla u_{n}\vert ^{p}u_{n}^{\alpha ^{-}-1}\,\mathrm{dx}=\frac{p ^{p}}{(\alpha^{-}+p-1)^{p}} \int_{\Omega}\bigl\vert \nabla u_{n}^{\frac {\alpha ^{-}+p-1}{p}} \bigr\vert ^{p}\,\mathrm{dx}, $$

we have that

$$ \frac{p^{p}\alpha^{-}}{(\alpha^{-}+p-1)^{p}} \int_{\Omega}\bigl\vert \nabla u _{n}^{\frac{\alpha^{-}+p-1}{p}} \bigr\vert ^{p}\,\mathrm{dx}\leqslant\bigl(C+\vert \Omega \vert ^{1-\frac{1}{m}}\bigr)\Vert f\Vert _{L^{m}(\Omega)}. $$

Thus, we have that $u_{n}^{\frac{\alpha^{-}+p-1}{p}}$ is bounded in $W_{0}^{1,p}(\Omega)$.

Applying the Sobolev embedding theorem to $u_{n}^{ \frac{\alpha^{-}+p-1}{p}}(W_{0}^{1,p}(\Omega)\hookrightarrow L^{p ^{*}}(\Omega))$, we get that

$$ S \biggl( \int_{\Omega}\bigl\vert u_{n}^{\frac{\alpha^{-}+p-1}{p}}\bigr\vert ^{p^{*}} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \leqslant \int_{\Omega}\bigl\vert \nabla u _{n}^{\frac{\alpha^{-}+p-1}{p}} \bigr\vert ^{p}\,\mathrm{dx}, $$

where S is the best constant of the Sobolev embedding theorem. Since the boundedness of $u_{n}^{\frac{\alpha^{-}+p-1}{p}}$ in $W_{0}^{1,p}( \Omega)$, we thus have the boundedness of $u_{n}$ in $L^{s}(\Omega)$.

To prove the boundedness of $u_{n}$ in $W_{\mathrm{loc}}^{1,p}(\Omega)$, we choose $u_{n}\varphi^{p}$ as a test function in (2.2), where $\varphi\in C_{0}^{\infty}(\Omega)$, $\Omega'=\{x\in\Omega, \varphi\neq0\}$. By (2.4), we have that

$$ \begin{aligned} & \int_{\Omega} \vert \nabla u_{n}\vert ^{p}\varphi ^{p}\,\mathrm{dx}+p \int_{\Omega }\bigl(\vert \nabla u_{n}\vert ^{p-2}\nabla u_{n}\bigr)\cdot\nabla \varphi u_{n} \varphi ^{p-1}\,\mathrm{dx}\\ &\quad \leqslant \int_{\Omega} \frac{f_{n}\varphi^{p}}{u_{n}^{\alpha(x)-1}}\,\mathrm{dx}\leqslant \int_{\Omega}\frac{f_{n}\varphi^{p}}{C_{\Omega'}^{\alpha(x)-1}} \,\mathrm{dx}\leqslant\frac{1}{\min\{C_{\Omega'}^{\alpha^{+}-1}, C _{\Omega'}^{\alpha^{-}-1}\}} \int_{\Omega}f_{n}\varphi^{p} \,\mathrm{dx}. \end{aligned} $$

By Young’s inequality, we have that

$$ \begin{aligned} &p \int_{\Omega}\bigl(\vert \nabla u_{n}\vert ^{p-2}\nabla u_{n}\bigr)\cdot\nabla\varphi u_{n} \varphi^{p-1}\,\mathrm{dx}\\ &\quad \leqslant \frac{p\frac{1}{2(p-1)}}{\frac{p}{p-1}} \int_{\Omega} \bigl\vert \vert \nabla u_{n}\vert ^{p-2}\nabla u_{n} \bigr\vert ^{\frac{p}{p-1}} \varphi^{p} \,\mathrm{dx}+\frac{p(2(p-1))^{p-1}}{p} \int_{\Omega} \vert \nabla\varphi \vert ^{p}u _{n}^{p}\,\mathrm{dx}\\ &\quad \leqslant \frac{1}{2} \int_{\Omega} \vert \nabla u_{n}\vert ^{p}\varphi^{p} \,\mathrm{dx}+ \bigl(2(p-1) \bigr)^{p-1} \int_{\Omega} \vert \nabla\varphi \vert ^{p}u _{n}^{p}\,\mathrm{dx}. \end{aligned} $$

Since $u_{n}$ is bounded in $L^{s}(\Omega)$ (where $s\geqslant p$), by Hölder’s inequality, we obtain that

$$ \begin{aligned} &\frac{1}{2} \int_{\Omega} \vert \nabla u_{n}\vert ^{p}\varphi ^{p}\,\mathrm{dx}\\ & \quad \leqslant\frac{1}{\min\{C_{\Omega'}^{\alpha^{+}-1}, C_{\Omega'} ^{\alpha^{-}-1}\}} \int_{\Omega}f_{n}\varphi^{p}\,\mathrm{dx}+ \bigl(2(p-1)\bigr)^{p-1} \int_{\Omega} \vert \nabla\varphi \vert ^{p}u_{n}^{p} \,\mathrm{dx}\\ & \quad \leqslant \frac{\Vert \varphi \Vert _{L^{\infty}(\Omega)}^{p}}{\min \{C _{\Omega'}^{\alpha^{+}-1}, C_{\Omega'}^{\alpha^{-}-1}\}} \int_{ \Omega}f \,\mathrm{dx}+\bigl(2(p-1)\bigr)^{p-1}\Vert \nabla\varphi \Vert _{L^{\infty}( \Omega)}^{p} \int_{\Omega}u_{n}^{p}\,\mathrm{dx}\\ & \quad \leqslant \frac{\vert \Omega \vert ^{1-\frac{1}{m}}\Vert \varphi \Vert _{L^{\infty}( \Omega)}^{p}\Vert f\Vert _{L^{m}(\Omega)}}{\min\{ C_{\Omega'}^{\alpha^{+}-1}, C_{\Omega'}^{\alpha^{-}-1}\}} +\bigl(2(p-1)\bigr)^{p-1}\vert \Omega \vert ^{1- \frac{p}{s}}\Vert \nabla\varphi \Vert _{L^{\infty}(\Omega )}^{p} \biggl( \int_{ \Omega}u_{n}^{s}\,\mathrm{dx}\biggr)^{\frac{p}{s}}, \end{aligned} $$

and hence $u_{n}$ is bounded in $W_{\mathrm{loc}}^{1,p}(\Omega)$. □

Once we have the boundedness of $u_{n}$, we can prove the following existence theorem along the lines of Theorem 3.1.

Theorem 4.2

Suppose that f is a nonnegative function in $L^{m}(\Omega)$ ($m>1$), ($f \not\equiv0$), $1<\alpha^{-}\leqslant\alpha(x)\leqslant\alpha^{+}$ and $\alpha^{+}-\alpha^{-}<1-\frac{1}{m}$. Then problem (1.1) has a solution u in $W_{\mathrm{loc}}^{1,p}(\Omega)$. Furthermore, $u^{\frac{ \alpha^{-}+p-1}{p}}$ belongs to $W_{0}^{1,p}(\Omega)$.

The summability of u can be proved as the following lemma, the proof is similar to the proof of Lemma 4.2

Lemma 4.4

Suppose that $f\in L^{m}(\Omega)$, $1<\alpha^{-}\leqslant\alpha(x) \leqslant\alpha^{+}$ and $\alpha^{+}-\alpha^{-}<1-\frac{1}{m}$. Then the solution u of (1.1) given by Theorem  4.2 is such that:

• (i) if $m>\frac{N}{p}$, then $u\in L^{\infty}(\Omega)$;

• (ii) if $\frac{N(p-1+\alpha^{+})}{(\alpha ^{-}+p-1)(N-p)+p(p-1+\alpha ^{+})}\leqslant m<\frac{N}{p}$, then $u\in L^{s}(\Omega)$, $s=\frac{Nm( \alpha^{-}+p-1)}{N-mp}$.

5 The case $0<\alpha^{-}<1<\alpha^{+}$

If $0<\alpha^{-}<1<\alpha^{+}$, the boundedness of $u_{n}$ in $W_{0}^{1,p}(\Omega)$ can also be obtained only if f is more regular than $L^{1}(\Omega)$. Furthermore, the existence of problem (1.1) is obtained, the proof has many analogies with the case $0<\alpha^{-}<\alpha^{+}<1$. We have the following results.

Lemma 5.1

Suppose that $f\in L^{m}(\Omega)$, with $m=\frac{Np}{Np-N+p+(N-p) \alpha^{-}}$, and let $u_{n}$ be the solution of (2.2) with $0<\alpha^{-}<1<\alpha^{+}<2-\frac{1}{m}$. Then the sequence $\{u_{n}\}$ is bounded in $W_{0}^{1,p}(\Omega)$.

Proof

We choose $u_{n}$ as a test function in (2.2), by Hölder’s inequality and Lemma 2.3, since $f_{n}\leqslant f$, we have that

$$ \begin{aligned}[b] \int_{\Omega} \vert \nabla u_{n}\vert ^{p}\,\mathrm{dx}&\leqslant \int_{\Omega}fu _{n}^{1-\alpha^{-}}\,\mathrm{dx}+ \int_{\Omega}\frac{f}{u_{1}^{\alpha ^{+}-1}}\,\mathrm{dx}\\ & \leqslant \Vert f\Vert _{L^{m}(\Omega)} \biggl( \int_{\Omega }u_{n}^{(1-\alpha ^{-})m'}\,\mathrm{dx}\biggr)^{\frac{1}{m'}} +C\Vert f\Vert _{L^{m}(\Omega)}. \end{aligned} $$

(5.1)

Applying the Sobolev embedding theorem on the left-hand side, we get

$$ S \biggl( \int_{\Omega}u_{n}^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \leqslant \int_{\Omega} \vert \nabla u_{n}\vert ^{p}\,\mathrm{dx}. $$

(5.2)

Combining (5.1) with (5.2) implies that

$$ S \biggl( \int_{\Omega}u_{n}^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \leqslant \Vert f\Vert _{L^{m}(\Omega)} \biggl( \int_{\Omega }u_{n}^{(1-\alpha ^{-})m'}\,\mathrm{dx}\biggr)^{\frac{1}{m'}}+C\Vert f\Vert _{L^{m}(\Omega)}. $$

Let $p^{*}=(1-\alpha^{-})m'$, it follows that

$$ S \biggl( \int_{\Omega}u_{n}^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \leqslant \Vert f\Vert _{L^{m}(\Omega)} \biggl( \int_{\Omega }u_{n}^{p^{*}} \,\mathrm{dx}\biggr)^{\frac{1}{m'}}+C\Vert f\Vert _{L^{m}(\Omega)}. $$

By Young’s inequality, we get that

$$ S \biggl( \int_{\Omega}u_{n}^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \leqslant \Vert f\Vert _{L^{m}(\Omega)} \biggl( \varepsilon \biggl( \int_{\Omega}u _{n}^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} + \varepsilon^{-\frac{p^{*}}{pm'-p^{*}}} \biggr)+C\Vert f\Vert _{L^{m}(\Omega)}. $$

Thus, we have that

$$ \biggl( \int_{\Omega}u_{n}^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} \leqslant\frac{\Vert f\Vert _{L^{m}(\Omega )}}{S-\varepsilon \Vert f\Vert _{L^{m}( \Omega)}} \bigl(\varepsilon^{-\frac{p^{*}}{pm'-p^{*}}}+C \bigr). $$

We choose $\varepsilon=\frac{S}{2\Vert f\Vert _{L^{m}(\Omega)}}$ to get

$$ \biggl( \int_{\Omega}u_{n}^{p^{*}}\,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} =\frac{2 \Vert f\Vert _{L^{m}(\Omega)}}{S} \biggl( \biggl(\frac{2\Vert f\Vert _{L^{m}(\Omega)}}{S} \biggr)^{\frac{p^{*}}{pm'-p^{*}}}+C \biggr). $$

So the boundedness of $u_{n}$ in $L^{p^{*}}(\Omega)$ is obtained. Using the estimate and (5.1) again, we have the estimate of $u_{n}$ in $W_{0}^{1,p}(\Omega)$. □

Once the boundedness of $u_{n}$ in $W_{0}^{1,p}(\Omega)$ is obtained, we can prove the following existence theorem.

Theorem 5.1

Suppose that $f\in L^{m}(\Omega)$ with $m=\frac{Np}{Np-N+p+(N-p) \alpha^{-}}$, $f\not\equiv0$, and $0<\alpha^{-}<1<\alpha^{+}<2- \frac{1}{m}$. Then problem (1.1) has a solution u in $W_{0}^{1,p}(\Omega)$.

Lemma 5.2

Suppose that $f\in L^{m}(\Omega)$ with $m\geqslant\frac{Np}{Np-N+p+(N-p) \alpha^{-}}$, and $0<\alpha^{-}<1<\alpha^{+}<2-\frac{1}{m}$. Then the solution u of (1.1) given by Theorem  5.1 is such that:

• (i) if $m>\frac{N}{p}$, then $u\in L^{\infty}(\Omega)$;

• (ii) if $\frac{Np}{Np-N+p+(N-p)\alpha^{-}}\leqslant m<\frac {N}{p}$, then $u\in L^{s}(\Omega)$, $s=\frac{Nm(\alpha^{-}+p-1)}{N-mp}$.

Proof

The proof of (i) is similar to that for Lemma 3.2(i), we omit the details here.

To prove (ii), if $\frac{N(p-1+\alpha^{+})}{(\alpha^{-}+p-1)(N-p)+p(p-1+ \alpha^{+})}\leqslant m<\frac{N}{p}$, the proof is identical to that for Lemma 4.2, we also omit it here.

If $m=\frac{Np}{Np-N+p+(N-p)\alpha^{-}}$, we can prove the results by the Sobolev embedding theorem.

If $\frac{Np}{Np-N+p+(N-p)\alpha^{-}}< m<\frac{N(p-1+\alpha^{+})}{( \alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}$, we choose $1<\delta<\frac{p-1+ \alpha^{+}}{p}$, and use once again $u_{n}^{p\delta-p+1}$ as a test function in (2.2). Using $\delta>1>\frac{p-1+\alpha^{-}}{p}$, as well as Hölder’s inequality, the Sobolev embedding theorem, Lemma 2.3, we get that

$$ \begin{aligned} \frac{S(p\delta-p+1)}{\delta^{p}} \biggl( \int_{\Omega}u_{n}^{p^{*} \delta} \,\mathrm{dx}\biggr)^{\frac{p}{p^{*}}} &\leqslant \int_{\Omega}fu _{n}^{p\delta-p+1-\alpha^{-}}\,\mathrm{dx}+ \int_{\Omega}\frac{f}{u _{1}^{p-1+\alpha^{+}-p\delta}} \,\mathrm{dx}\\ &\leqslant \Vert f\Vert _{L^{m}(\Omega)} \biggl( \int_{\Omega }u_{n}^{(p\delta -p+1-\alpha^{-})m'}\,\mathrm{dx}\biggr)^{\frac{1}{m'}}+C\Vert f\Vert _{L^{m}( \Omega)}. \end{aligned} $$

The choice of δ in such a way that $p^{*}\delta=(p\delta-p+1- \alpha^{-})m'$ yields that $1<\delta<\frac{p-1+\alpha^{+}}{p}$ if and only if $\frac{Np}{Np-N+p+(N-p)\alpha^{-}}< m<\frac{N(p-1+\alpha^{+})}{( \alpha^{-}+p-1)(N-p)+p(p-1+\alpha^{+})}$, and that $p^{*}\delta=s$. The choice of $m<\frac{N}{p}$ implies that $\frac{p}{p^{*}}>\frac{1}{m'}$. Thus we have the boundedness of $u_{n}$ in $L^{s}(\Omega)$, and so does the limit u in $L^{s}(\Omega)$. □

6 Conclusions

In this paper, we study the existence and regularity of solutions to the quasilinear elliptic problem with nonlinear singular terms and variable exponent. Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent $\alpha(x)$, some classical methods may not directly be applied to our problem. We construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we prove that the existence and regularity of solutions depend on the summability of f and the value of $\alpha(x)$.