Monotonicity and symmetry of positive solution for 1-Laplace equation

1.   Introduction and main results

We are interested in the symmetry and monotonicity of solutions to the problem

where $\Delta_1u = \text{div}(\frac{D u}{|D u|})$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq 2$, and strictly convex. The purpose of the paper is to investigate a priori estimates and symmetric properties of the solutions when the domain is assumed to have symmetric properties and $f$ is supposed to satisfy the following conditions $(H_1)$, $(H_2)$ and $(H_4)$. We also assume that $f$ satisfies the following conditions $(H_3)$ and $(H_5)$ to use mountain pass lemma to get a nontrivial solution.

$(H_1)$: $f:[0, +\infty)$ is a locally Lipschitz continuous function and $f(s)\geq 0$ for $\forall$ $s\in [0, +\infty)$.

$(H_2)$: $f(s)\leq C_1(1+s^{1^\ast-1})$, for $\forall$ $s\in [0, +\infty)$, with $1^\ast = \frac{N}{N-1}$ and a constant $C_1 > 0$.

$(H_3)$: There exists $\theta > 1$, and $k_0 > 0$ such that

$(H_4)$: There exists a constant $C_2 > 0$ such that

where $F(s) = \int_0^sf(t)dt$.

$(H_5)$: There exists a constant $\alpha\in (0, \frac{1}{N-1})$ such that

We point out that the similar $p$-Laplace problems $(p > 1)$ have many applications and have been studied for a long time, more precisely, Dirichlet problems for the $p$-Laplace operator,

In the case $p = 2$, the problem (1.2) $-\Delta_p u = f(u)$ has been widely studied. Gidas and Spruck ^[27] prove a priori bounds for nonlinearities $f$ for $N\geq3$ behave as a subcritical power at infinity, introducing the blow up method together with Liouville type theorems for solutions in $\mathbb{R}^N$. Figneiredo, Lions and Nussbaum ^[19] consider the existence and a priori estimates of positive solutions of the problem (1.2) when $f$ satisfies the superlinear grow at infinity. They prove a priori bound for positive solutions of the problem (1.2) under the hypothesis $\lim_{s\rightarrow \infty}\frac{f(s)}{s^{\frac{N+2}{N-2}}} = 0$, together with the monotonic results by Gidas, Ni and Nirenberg ^[28] obtained by the Alexandrov-Serrin moving plane method ^[37]. The moving plane method has been improved and simplified by Beresticky and Nirenberg ^[7] with the aid of the maximum principle in small domain. With the help of the blow up procedure, Azizieh and Clément ^[5] prove a priori estimates for the problem (1.2) in the case of $\Omega$ being a strictly convex domain and $f$ satisfying some suitable assumption. Damascelli and Pacella ^[14,15] apply the moving plane method to prove some monotonic and symmetric results for the $p$-Laplace equation in the singular case $1 < p < 2$, also see ^[6,13]. The results are later extended to the case $p > 2$ in the papers ^[12,17,18]. Damascelli and Pardo ^[16] used the technique introduced in ^[19] that allowed to give the a priori estimates for solutions in case $1 < p < N$, case $p = N$, and case $p > N$. Esposito, Montoro and Sciunzi ^[24] study symmetric and monotonic properties of singular positive solutions to the problem (1.2) via moving plane method under suitable assumptions on $f$. However, all the above mentioned papers can not deal with the case $p = 1$. In this paper, we can extend the case $p > 1$ to the case $p = 1$.

Obviously, the problem of $\Delta_1$ is different from $\Delta_p$ ($p > 1$). When $p = 1$, it is necessary to replace $W^{1, 1}$ by $BV$, the space of functions of bounded variation. A function $u\in L^1(\Omega)$ is called a function of bounded variation, whose partial derivatives in the sense of distribution are Radon measures. We point out that the space $W^{1, p}(\Omega)$ is reflexive, however, the space $BV(\Omega)$ is not reflexive, so that we can not follow the arguments on $\Delta_p$. The 1-Laplace operator $\Delta_1$ introduces some extra difficulties and special features. The first difficulty occurs by defining the quotient $\frac{Du}{|Du|}$, $Du$ being just a Radon measure. To deal with the 1-Laplacian operator, we need the theory of pairing of $L^\infty$ divergence measure vector fields (see the pioneering works ^[3,4,8]).

Demengel ^[21] is concerned with existence of solution in $BV(\Omega)$ to the problem $-\text{div}z+z\text{sign}u = f|u|^{1^\ast-2}u$ with $z\cdot\nabla u = \nabla u$ in $\Omega$ and $-z\cdot \gamma = u$ on $\partial\Omega$. Demengel ^[22] is devoted to the elliptic equations with 1-Laplacian operator

and introduces the concept of locally almost 1-harmonic functions in $\Omega$. The comparison principle, the first eigenvalue and related eigenfunctions for the 1-Laplacian operator are established in ^[22]. Kawohl and Schuricht ^[30] consider a number of problems that are associated with the 1-Laplace operator $\Delta_1$, the formal limit of the $p$-Laplace operator as $p\rightarrow1$, by investigating the underlying variational problem. Since the corresponding solution typically belongs to $BV$ and not to $W^{1, 1}$, they have to study the minimizers of the functionals containing the total variation. In particular, they look for constrained minimizers subject to a prescribed $L^1$ norm which can be considered as an eigenvalue problem for the 1-Laplace operator. Degiovanni and Magrone ^[20] are concerned with the problem (1.3) with $f(x, u) = \lambda\frac{u}{|u|}+|u|^{1^\ast-2}u$. It is proved that for every $\lambda\geq \lambda_1$, the problem (1.3) admits a nontrivial solution by the non-standard linking methods. Salas and Segura de León ^[35] study the problem (1.3) with $f(x, u)$ satisfying subcritical growth; i.e., $|f(x, u)|\leq C(1+|u|^{q})$ with $0 < q < 1^{\ast}-1$. They prove that for the problem (1.3) there exists at least two nontrivial solutions, one nonnegative and one nonpositive, by using known existence results for the $p$-Laplacian $(p > 1)$ and considering the limit as $p\rightarrow1^+$. De Cicco, Giachetti, Oliva and Petitta ^[9] study the existence and regularity of special distributional nonnegative solutions to the boundary value singular problem (1.3) with $f(x, u) = h(u)g(x)$. They show existence of nonnegative solutions to (1.3) with $u^{\max\{1, \gamma\}}\in BV(\Omega)$. These solutions are obtained as a limit as $p\rightarrow1^+$ of nonnegative solutions of the $p$-Laplacian problems $-\Delta_p u_p = h(u_p)g$ with $u_p = 0$ on $\partial\Omega$. We also refer to ^{[33,34,35,36,38]} for the a priori estimates and gradient estimates of solutions. In this paper we can study the monotonicity and symmetry of positive solution to the 1-Laplace problem and show the a priori estimates for the solution.

By the theory of pairing of $L^\infty$ divergence measure vector fields, we introduce the following definition of solutions to the problem (1.1).

Definition 1.1. We say that $u\in BV_{loc}(\Omega)$, $u > 0$, is a solution to problem (1.1) if there exists a vector field $z\in \mathcal{DM}^\infty(\Omega)$ with $\|z\|_{L^\infty}\leq 1$ such that

where $\gamma$ is the unit exterior normal on $\partial\Omega$, and the spaces $BV_{loc}(\Omega)$ and $\mathcal{DM}^\infty(\Omega)$ are given in Section 2.

To state more precisely some known result about the monotonicity and symmetry of solutions of the problem (1.1), we need some notations. Let $\nu$ be a direction in $\mathbb{R}^N$. For a real number $\mu$ we define

and

If $\mu > a(\nu)$ then $\Omega_\mu^\nu$ is nonempty, thus we set

Following ^[6] and ^{[12,13,14,15,16,17,18]}, we observe that $\mu-a(\nu)$ small then $(\Omega_\mu^\nu)'$ is contained in $\Omega$ and will remain in it, at least until one of the following occurs:

$(A)$ $(\Omega_\mu^\nu)'$ becomes internally tangent to $\partial\Omega$.

$(B)$ $T_\mu^\nu$ is orthogonal to $\partial\Omega$.

Let $\Pi_1(\nu)$ be the set of those $\mu > a(\nu)$ such that for each $\eta < \mu$ none of the conditions $(A)$ and $(B)$ holds and define

Moreover, let

and

Since $\Omega$ is supposed to be smooth, note that neither $\Pi_1(\nu)$ nor $\Pi_2(\nu)$ are empty and $\Pi_1(\nu)\subset \Pi_2(\nu)$, so that $\mu_1(\nu)\leq\mu_2(\nu)$.

We deal with solutions to the problem (1.1) in the sense of Definition 1.1. Our main result is stated as follows.

Theorem 1.2. Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq2$, which is strictly convex. Assume the nonlinearity $f$ satisfies the conditions $(H_1)-(H_5)$. Then there exists a nontrivial positive solution $u$ to the problem (1.1) in the sense Definition 1.1, bounded in $L^\infty(\Omega)$ (i.e., $u\in L^\infty(\Omega)$), and for any direction $\nu$ and for $\mu$ in the interval $(a(\nu), \mu_1(\nu)]$,

where $a(\nu)$ and $\mu_1(\nu)$ are given by (1.7) and (1.8) respectively.

If $f$ is locally Lipschitz continuous in the closed interval $[0, +\infty)$, the condition (1.10) holds for any $\mu$ in the interval $(a(\nu), \mu_2(\nu)]$.

Corollary 1.3. Let the smooth bounded domain $\Omega\subset\mathbb{R}^N$, $N\geq 2$, be strictly convex with respect to a direction $\nu$ and symmetric with respect to the hyperplane $T_0^\nu = \{ x\in \mathbb{R}^N\mid x\cdot \nu = 0\}$. Assume that the nonlinearity $f$ satisfies the conditions $(H_1)-(H_5)$, which is locally Lipschitz continuous in the closed interval $[0, +\infty)$ and strictly positive in $(0, +\infty)$. Then there exists a nontrivial positive solution $u$ to the problem (1.1) in the sense Definition 1.1, bounded in $L^\infty(\Omega)$, almost everywhere symmetric, i.e., $u(x) = u(x_0^\nu)$ and nondecreasing in the $\nu$-direction a.e. in $\Omega_0^\nu$.

Remark 1.4. Since the moving plane procedure can be performed in the same way but in the opposite direction, then it is obvious that Corollary 1.3 is obtained by Theorem 1.2 (see Corollary 2.4 of ^[16]).

2.   Preliminaries on BV space

Throughout this paper, $\Omega$ denotes an bounded subset of $\mathbb{R}^N$ with Lipschitz boundary. The symbol $|\Omega|$ stands for its $N$ dimensional Lebesgue measure and $H^{N-1}(E)$ for the $N-1$ dimensional Hausdorff measure of a set $E\subset \mathbb{R}^N$. An outward normal with vector $\gamma = \gamma(x)$ is defined for $H^{N-1}$ a.e. $x\in \partial\Omega$. We will denote by $W_0^{1, p}(\Omega)$ the usual Sobolev space, of measureable functions having weak gradient in $L^p(\Omega; \mathbb{R}^N)$ and zero trace on $\partial\Omega$. If $1 < p < N$, denote by $p^\ast = \frac{Np}{N-p}$ its critical Sobolev exponent. $BV(\Omega)$ will denote the space of functions of bounded variation

where $Dv:\Omega\rightarrow \mathbb{R}^N$ is the distributional gradient of $u$. It is endowed with the norm by

where

$BV(\Omega)$ is a Banach space which is non-reflexive and non-separable. The notion of a trace on the boundary can be extended to functions $v\in BV(\Omega)$ and this fact allows us to write $v|_{\partial \Omega}$. Moreover, the trace defines a linear bounded operator $i:BV(\Omega)\hookrightarrow L^1(\partial \Omega)$ which is onto. By the trace, we have an equivalent norm on $BV(\Omega)$

where $H^{N-1}$ denotes the $N-1$ dimensional Hausdorff measure. We will often use this norm in what follows. In addition, the following continuous embeddings hold

which are compact for $1\leq m < \frac{N}{N-1}$ (see for instance ^[25,41]). We denote by $\mathcal{M}(\Omega)$ the space of Radon measures with finite total variation over $\Omega$, by

and by

The theory of $L^\infty$ divergence measure vector fields is due to Anzellotti ^[4] and Chen and Frid ^[8]. We define the following distribution $(z, Dv)$

for $\forall$ $\varphi\in C_c^1(\Omega)$. In Anzellotti's theory we need some compatibility conditions, such as $\text{div} z\in L^1(\Omega)$ and $v\in BV(\Omega)\cap L^\infty (\Omega)$ or $\text{div} z$ a Radon measure with finite total variation and $v\in BV(\Omega)\cap L^\infty (\Omega)\cap C(\Omega)$.

Lemma 2.1 (^[34,35]). Let $v\in BV_{loc}(\Omega)\cap L^1(\Omega, \mu)$ and $z\in \mathcal{DM}_{loc}^\infty (\Omega)$. Then the distribution $(z, Dv)$ defined in (2.1) previously satisfies

for all open set $U\subset\subset \Omega$ and all $\varphi \in C_c^1(U)$.

Lemma 2.2 (^[34,35]). The distribution $(z, Dv)$ is a Radon measure. It and its total variation $|(z, Dv)|$ are absolutely continuous with respect to the measure $|Dv|$ and

holds for all Borel sets $B$ and for all open sets $U$ such that $B\subset U\subset \Omega$.

Lemma 2.3 (^[10,11,34]). Let $z\in \mathcal{DM}_{loc}^\infty (\Omega)$ and let $v\in BV(\Omega)\cap L^\infty (\Omega)$. Then $zv\in \mathcal{DM}_{loc}^\infty (\Omega)$. Moreover, the following formula holds in the sense of measures

It follows from Anzellotti's theory that every $z\in \mathcal{DM}^\infty (\Omega)$ has a weak trace on $\partial \Omega$ of the normal component of $z$ which is denoted by $[z, \gamma]$ with $\gamma$ the unit exterior normal on $\partial\Omega$, which satisfies

and

for all $z\in \mathcal{DM}^\infty (\Omega)$ and $v\in BV(\Omega)\cap L^\infty(\Omega)$.

Lemma 2.4 (Green formula ^[10,11,34]). Let $z\in \mathcal{DM}_{loc}^\infty (\Omega)$, $\varpi = \text{div} z$ and $v\in BV(\Omega)$ and assume $v \in L^1(\Omega, \mu)$. Then $vz\in \mathcal{DM}^\infty (\Omega)$ and the following holds

Lemma 2.5 (^[34,35]). Let $z\in \mathcal{DM}_{loc}^\infty (\Omega)$ and $v\in BV(\Omega)\cap L^\infty(\Omega)$. If $vz\in \mathcal{DM}^\infty (\Omega)$, then

3.   Weak solution to p-Laplacian problem

Let $p_0: = \min\{\theta, \frac{N}{N-1}\}$, with $\theta > 1$ given by $(H_3)$. For each $1 < p < p_0$, let us consider the following problem

where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geq 2$, $1 < p < p_0$ and $f:[0, +\infty)\rightarrow \mathbb{R}$ satisfies the conditions $(H_1)-(H_5)$. We need the following propositions and a priori estimates of $p$-Laplace equation to prove Theorem 1.2.

Definition 3.1. We say $u_p\in W_0^{1, p}(\Omega)$, $u_p\geq0$, is a weak solution to the problem (3.1) in the sense that

for $\forall$ $\varphi\in W_0^{1, p}(\Omega)$.

If $u_p\in W^{1, p}(\Omega)$ is a weak solution of the problem (3.1) with $f$ satisfying the critical growth, then $u_p\in C^{1, \alpha}(\Omega)$ with $\alpha\in (0, 1)$ (see ^[23,31,40]), so that we suppose from the beginning a $C^1$ regularity for the solution. Next, we recall some results on the monotonicity and estimates of solutions for the $p$-Laplace equation. One can refer to ^{[1,16,19,29,32]} for the proof of the following Proposition 3.2-3.7.

Proposition 3.2 (^[16]). Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq2$, $1 < p < \infty$, $f:[0, \infty)\rightarrow \mathbb{R}$ a continuous function which is locally Lipschitz continuous in $(0, \infty)$ and strictly positive in $(0, \infty)$ if $p > 2$. Let $w\in C^1(\overline{\Omega})$ be a weak solution of (3.1). Then for any direction $\nu$ and for $\mu$ in the interval $(a(\nu), \mu_1(\nu)]$, we have

If $f$ is locally Lipschitz continuous in the closed interval $[0, +\infty)$, then (3.3) holds for any $\mu$ in the interval $(a(\nu), \mu_2(\nu)]$, where $a(\nu)$, $\mu_1(\nu)$ and $\mu_2(\nu)$ are given by (1.7), (1.8) and (1.9).

Proposition 3.3 (^[16,19]). Let $\Omega$ be a strictly convex bounded smooth domain, and define $\Omega_\delta = \{x\in \Omega\mid \text{dist}(x, \partial \Omega) > \delta\}$, for $\delta > 0$. Then the following result holds for a weak solution $w\in C^1(\Omega)$ of the problem (3.1) with $f$ satisfying the condition $(H_1)$

$I_x$ is a part of a cone $K_x$ with vertex in $x$, where all the $K_x$ are congruent to a fixed cone $K$, and if $x\in \Omega\setminus \Omega_\frac{\varepsilon}{2}$, then $I_x = K_x\cap \Omega_\frac{\varepsilon}{2}$.

Proposition 3.4 (^[32]). Let us define

where $\theta$ is given by $(H_3)$. Then, $\lambda_1$ is the first eigenvalue of the operator $-\Delta_{p_0}$ ($\lambda_1\leq \lambda$ for any eigenvalue $\lambda$), it is simple, i.e., there is only an eigenfunction up to multiplication by a constant, and it is isolated. Moreover a first eigenfunction does not change sign in $\Omega$ and by the strong maximum principle it is in fact either strictly positive or strictly negative in $\Omega$. So we can select a unique eigenfunction $\phi_1$ such that

The following extension of the Picone's identity for the $p$-Laplacian has been proved in ^[1].

Proposition 3.5 (Picone's identity ^[1]). Let $v_1, v_2\geq0$ be differentiable functions in an open set $\Omega$, with $v_2 > 0$ and $p > 1$. Set

and

Then $R(v_1, v_2) = L(v_1, v_2)\geq 0$.

As a consequence we have

The following extension of the Pohozaev's identity for the $p$-Laplacian has been given by ^[29].

Proposition 3.6 (Pohozaev's identity for $p$-Laplace ^[29]). Let $w\in W_0^{1, p}(\Omega)\cap L^\infty(\Omega)$, $p > 1$, be a weak solution of the problem

where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geq2$ and $f:[0, +\infty)\rightarrow \mathbb{R}$ is a continuous function. Denote $F(w) = \int_0^w f(s)ds$. Then

where $\gamma$ is the unit exterior normal on $\partial \Omega$.

We need also local $W^{1, \infty}(\Omega)$ result at the boundary. This result follows from the global estimates by Lieberman ^[31] extending the local interior estimates by Dibenedetto ^[23].

Proposition 3.7 (^[16]). Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq2$, and $w\in C^1(\overline{\Omega})$ be a solution of the problem

with $h\in L^{(p^\ast)'}(\Omega)$. For $\delta > 0$, let $\Omega_\delta = \{x\in \Omega \mid \text{dist}(x, \partial \Omega) > \delta\}$ and suppose that $w, h\in L^\infty (\Omega\setminus \Omega_\delta)$ with

Then there exists a constant $C > 0$ only depending on $M$ and $\delta$ such that

Next, we will give the estimate of the solution for the problem (3.1).

Theorem 3.8. If $u_p$ is a weak solution to the problem (3.1) and $f$ satisfies the conditions $(H_2)-(H_4)$, then $u_p$ satisfies

where the constant $C' > 0$ is not dependent on $p$.

Proof. By $1 < p < p_0$, Proposition 3.4, Proposition 3.5 with $v_2 = u_p$, $v_1 = \phi_1$ and Young's inequality, we have

By the condition $(H_3)$, there exists a constant $C_3 > 0$ such that

that is

where $k_1 = \max\{k_0, 1\}$ and $k_0$ is given by $(H_3)$.

Indeed, from $(H_3)$, it holds

Setting $k_1 = \max\{k_0, 1\}$ and integrating the above inequality (3.7) with respect to $t$ on the interval $[k_1, s]$, one has

That is

Setting $C_3: = \frac{k_1^\theta}{\theta F(k_1)}$ in (3.8), we get

Considering (3.9) and $sf(s)\geq \theta F(s)$, for $s\geq k_1$, one gets the inequality (3.6).

Now, taking into account (3.5), (3.6) with $s = u_p$ and Young's inequality, we get

where $\lambda_1+|\Omega|$ is given by (3.5) and $-C_3\int_{\{ 0 < u_p\leq k_0\}}\frac{f(u_p)}{u_p^{p-1}}\phi_1^p dx\leq 0$ is given by the condition $(H_1)$ ($f(s)\geq0$, for all $s\geq0$) respectively, and the last inequality is given by Proposition 3.4 with $\int_\Omega \phi_1^{p_0}dx = 1$ and $k_1 = \max\{k_0, 1\}\geq 1$. By Proposition 3.3 and (3.10), for any $x\in \Omega\setminus\Omega_\delta$, we have that

i.e.,

where the constant $C_5$ may be depend on $C_4$, $\sigma$, $\theta$, $p_0$ and $\phi_1$ by (3.11), but are independent of $p$. Estimate (3.11) gives the uniform $L^\infty$ bounds near the boundary: $\exists$ $\delta > 0$ and $C_5 > 0$ such that

for $\forall$ $u_p\in W_0^{1, p}(\Omega)$ satisfying the problem (3.1). On the other hand, from the condition $(H_2)$ and (3.12), we have

It is clear that $f(u_p(\cdot))\in L^{(p^\ast)'}$ is given by the condition $(H_2)$ and Sobolev embedding. By Proposition 3.7, (3.12) and (3.13), we get

where the constant $C_7 > 0$ is only depending on $C_5$, $C_6$ and $\delta$. By Proposition 3.6 (Pohozaev's identity)

$p^\ast = \frac{Np}{N-p} > \frac{N}{N-1} = 1^\ast$ and $(H_4)$, there exists a large enough constant $k_2 > 0$ such that

as $s\geq k_2$, so that by the condition $(H_2)$ and taking $s = u_p$ in (3.15)

That is

From the definitions of $C_5$, $C_6$, $C_7$ and $C_8$, i.e., (3.11)-(3.14) and (3.16), we obtain that the constant $C'$ is not dependent on $p$. The proof of Theorem 3.8 is completed.

The following existence result holds.

Theorem 3.9. Let $f$ satisfy the conditions $(H_1)$, $(H_2)$, $(H_3)$ and $(H_5)$. Then there exists a nontrivial positive solution $u_p$ to the problem (3.1).

Proof. By the conditions $(H_1)$, $(H_2)$, $(H_3)$ and $(H_5)$, it is well known that there exists a nontrivial solution $u_p\geq0$ to the problem (3.1). The positive solution $u_p$ is obtained using the mountain pass lemma by Ambrosetti and Rabinowitz ^[2] for the following truncated functional $J_p^+: W_0^{1, p}(\Omega)\rightarrow \mathbb{R}$ given by

where $F_+(s) = \int_0^sf_+(t)dt$ and

We claim that $J_p^+$ satisfies the structure of mountain pass lemma and the $(P-S)$ condition.

Indeed, by the condition $(H_5)$, $0$ is a local minimum of $J_p^+$. From the condition $(H_3)$, there exist two constants $\widetilde{C}$, $\widehat{C} > 0$, such that

for all $s\in [0, +\infty)$ with $\theta > 1$. This implies that

for $\forall$ $w\in W_0^{1, p}(\Omega)$. We can choose a $w_0\in W_0^{1, p}(\Omega)$ and $\|w_0\|_{W_0^{1, p}} = 1$ such that

as $t\rightarrow +\infty$, with $1 < p < p_0: = \min\{\theta, \frac{N}{N-1}\}$. Whence there exists a large number $t_0 > 0$ such that

We set $e: = t_0w_0\in W_0^{1, p}(\Omega)$. Since $(H_2)$ and the embedding $W_0^{1, p}(\Omega)\hookrightarrow L^{1^\ast}(\Omega)$, $1^\ast = \frac{N}{N-1} < \frac{Np}{N-p} = p^\ast$, is compact, we obtain that $f_+$ satisfies the subcritical grow, i.e.,

Considering (3.21) and $(H_3)$, $J_p^+$ satisfies the $(P-S)$ condition.

4.   The proof of Theorem 1.2

In this section we prove our main results concerning the case $p = 1$, namely Theorem 1.2. Under the same assumption of Theorem 1.2, we divide the proof into few steps.

Step 1. Existence of a solution $u$ and a field $z$.

Step 2. $(z, Du) = |Du|$ as measures in $\Omega$.

Step 3. $[z, \gamma]\in$ sign$(-u)$ on $\partial\Omega$.

Step 4. The monotonicity of solution $u$.

Step 5. $u\in L^\infty (\Omega)$.

Step 6. $u$ is nontrivial.

Step 1. Existence of a solution $u$ for the problem (1.1) and existence of a field $z\in \mathcal{DM}^\infty(\Omega)$ satisfying (1.4) and $\|z\|_{L^\infty}\leq1$.

Proof of Step 1: From Theorem 3.8, we obtain that $u_p$ is bounded in $W_0^{1, p}(\Omega)\hookrightarrow L^m(\Omega)$, with $1\leq m\leq \frac{N}{N-1} < p^\ast = \frac{Np}{N-p}$, $1 < p < p_0 < 2\leq N$.

as $p\rightarrow 1^+$.

Next, we will show that there exists a vector field $z$ satisfying (1.4). Recalling Theorem 3.8, we obtain that $\{u_p\}$ is bounded in $W_0^{1, p}(\Omega)\subset BV(\Omega)$. So that for $1\leq r < p' = \frac{p}{p-1}$, we have

and thus

where the constant $C_8$ is given by (3.16). This implies that $|\nabla u_p|^{p-2}\nabla u_p$ is bounded in $L^r(\Omega; \mathbb{R}^N)$ with respect to $p$. Then there exists $z_r\in L^r(\Omega; \mathbb{R}^N)$ such that

as $p\rightarrow 1^+$. A standard diagonal argument shows that there exists a unique vector field $z$ which is defined on $\Omega$ independently of $r$, such that

as $p\rightarrow 1^+$. By applying the semicontinuity of the $L^r$ norm the previous inequality (4.4) implies

so that, letting $r\rightarrow \infty$ we have $z\in L^\infty(\Omega; \mathbb{R}^N)$ and

Using $\varphi \in C_c^1(\Omega)$ with $\varphi\geq0$ as a test function in (3.1), we have

Taking $p\rightarrow 1^+$ in the left hand side of (4.7) and by (4.6), we get

for $\forall$ $\varphi \in C_c^1(\Omega)$. On the other hand, thanks to (4.2) and $f(s)$ a locally Lipschitz continuous function, we have

Moreover, we deduce from $(H_2)$ and (4.3) that

Consequently, by the Dominated Convergence Theorem, we get

for $\forall$ $\varphi\in C_c^1(\Omega)$. Therefore, (4.7), (4.8) and (4.9) imply that

Step 2. $(z, Du) = |Du|$ as measures in $\Omega$.

Before proving $(z, Du) = |Du|$, we need the following lemma for which one can refer to ^[9].

Lemma 4.1 (^[9]). Under the same assumptions of Theorem 1.2, the following identity holds

for $\forall$ $\varphi \in C_c^1(\Omega)$.

Proof of Step 2: We take $u_p\varphi \in W_0^{1, p}(\Omega)$ as a test function in (3.1) with $0\leq \varphi \in C_c^1(\Omega)$, $\max_{x\in\Omega}|\varphi(x)| = M_0$ and get

By Young's inequality and Fatou's Lemma, we estimate the first integral term in (4.12)

On the other hand, by (4.6) we have

From

and the Dominated Convergence Theorem, we obtain the right hand side of (4.12) is as follows

From (4.12)-(4.15), we have

By (4.16) and Lemma 4.1, we also have

Therefore, by (2.1), we get

The arbitrariness of $\varphi$ implies that

as measures in $\Omega$. On the other hand, since $\|z\|_{L^\infty}\leq1$, and

as measures in $\Omega$, we have

Step 3. The boundary condition $[z, \gamma]\in$ sign$(-u)$ on $\partial\Omega$.

Proof of Step 3: It is easy to check that this fact is equivalent to show

Choosing $u_p$ as a test function in (3.1), we have

Since $u_p\in W_0^{1, p}(\Omega)$ is bounded, by the fact that $u_p = 0$ on $\partial \Omega$ and Young's inequality, we get

We use the lower semicontinuity (4.18) to pass to the limit as $p\rightarrow 1^+$ and obtain

where the last equality is given by the Dominated Convergence Theorem and

Furthermore, by Lemma 2.3 and Lemma 2.4, we have

From $(z, Du) = |Du|$, (4.19) and (4.20), we have

The inequality (4.21) and $|u|\geq |u|\|z\|_{L^\infty}\geq |u[z, \gamma]|\geq -u[z, \gamma]$ give the desired equality (4.17) and we conclude that

Step 4. The monotonicity of the solution $u$ of problem (1.1).

Proof of Step 4: By Proposition 3.2, we obtain $u_p$ satisfies the following result. For any direction $\nu$ and $\mu$ in the interval $(a(\nu), \mu_1(\nu)]$, then

where $a(\nu)$ and $\mu_1(\nu)$ are given by (1.7) and (1.8). Considering this fact and $u_p(x)\rightarrow u(x)$ a.e. in $\Omega$, taking $p\rightarrow 1^+$ in (4.22), we have

We get the result of monotonicity for the solution $u$. Inequality (4.23) also holds for any $\mu\in (a(\nu), \mu_2(\nu)]$ by Proposition 3.2, if $f$ is locally Lipschitz continuous, and $a(\nu)$ and $\mu_2(\nu)$ are given by (1.7) and (1.9).

Step 5. The boundedness of the solution $u$, i.e., $u\in L^\infty(\Omega)$.

Before proving $u\in L^\infty(\Omega)$, we need to prove the following lemma.

Lemma 4.2. For every $\varepsilon > 0$ there exists $k_3 > 0$ which does not depend on $p$, such that

for every $k\geq k_3$ and $\forall$ $p\in (1, p_0)$, with $A_k = \{ x\in\Omega \mid u_p(x) > k\}$.

Proof of Lemma 4.2: Using Sobolev embedding $W_0^{1, p}(\Omega)\subset BV(\Omega)\hookrightarrow L^\frac{N}{N-1}(\Omega)$, Theorem 3.8 and Holder's inequality, we obtain that

where $S_1$ is given by the best Sobolev constant

see ^[26,39], and $|A_k|$ stands for its $N$ dimensional Lebesgue measure. Inequality (4.25) implies that $\lim_{k\rightarrow \infty}|A_k| = 0$. It holds that for $\forall$ $\varepsilon > 0$, there exists a large number $k_4 > 0$ such that

On the other hand, by Sobolev embedding $u_p\in W_0^{1, p}(\Omega)\subset BV(\Omega)\hookrightarrow L^\frac{N}{N-1}(\Omega)$, Theorem 3.8 and (4.3), we get

and

which implies that $u_p(x) < \infty$ a.e. in $\Omega$. Considering (4.27), $\lim_{k\rightarrow \infty}|A_k| = 0$ and by absolute continuity of integrable function, we have

From (4.28), for $\forall$ $\varepsilon > 0$, $\exists$ $k_5 > 0$ large enough (not depend on $p$) and $\delta > 0$ small enough such that as $k\geq k_5$, we have $|A_k| < \delta$ and

From (4.26) and (4.29), we obtain

for all $k\geq k_3: = \max\{ k_4, k_5\}$. The proof of Lemma 4.2 is completed.

Proof of Step 5: Next, we would like to use Stampacchia truncation ^[38] to prove the boundedness of the positive solution $u$. For every $k > 0$, we define the auxiliary function $G_k:[0, \infty)\rightarrow \mathbb{R}$ as

Then, choosing $G_k(u_p)$ as a test function in (3.1), we get

By (4.31), $(H_2)$, Sobolev embedding, Young's inequality and Holder's inequality, we have

By Lemma 4.2 and taking $\varepsilon = \frac{1}{(2C_1S_1)^N}$, there exists $k_3 > 0$ which does not depend on $p$, such that

for all $k\geq k_3$ and $p\in (1, p_0)$. Consequently, from (4.32) and (4.33) we obtain

Since $u_p(x)\rightarrow u(x)$ a.e. $x\in \Omega$ and Fatou's Lemma, we can pass to the limit on $p\rightarrow 1^+$ in (4.34), to conclude that

for $\forall$ $k\geq k_3 > 0$. Thus $u\in L^\infty(\Omega)$.

Step 6. $u$ is nontrivial.

Proof of Step 6: For $\forall$ $v\in BV(\Omega)$, we define the functional $J^+:BV(\Omega)\rightarrow \mathbb{R}$ as

where $F_+(s) = \int_0^s f_+(t)dt$ and $f_+$ is given by (3.18).

We will say that $v_0\in BV(\Omega)$ is a critical point of $J^+$ if there exists $z\in \mathcal{DM}^\infty(\Omega)$ with $\|z\|_{L^\infty}\leq 1$ such that

where $\gamma$ is the unit exterior normal on $\partial\Omega$. The critical points of $J^+$ coincide with solutions to the problem (1.1) in the sense Definition 1.1, for which one can refer to ^[9] or ^[35].

We shall show that $0$ is a local minimum of $J^+$.

Indeed, by the condition $(H_5)$, there exists small enough $\delta > 0$ such that

for $\forall$ $|s|\in (0, \delta)$ and for some constant $C_9 > 0$ with $\alpha\in (0, \frac{1}{N-1})$. Moreover, by the definition of $F_+(s)$, we have

for $\forall$ $|s|\in (0, \delta)$. By (4.35) and the norm $\|v\|_{BV} = \int_\Omega |Dv|+ \int_{\partial\Omega} |v|dH^{N-1}$, $v\in BV(\Omega)$, it holds

where the last inequality is given by the embedding $BV(\Omega)\hookrightarrow L^{1+\alpha}(\Omega)$, $\alpha\in (0, \frac{1}{N-1})$. Choosing a positive constant $\rho < \min\{\delta, (\frac{1}{2C_{10}})^\frac{1}{\alpha}\}$, we obtain

for $\forall$ $v\in BV(\Omega)$ and $\|v\|_{BV}\leq \rho$. This implies that $0$ is a local minimum of $J^+$.

Now, we introduce the auxiliary functional

where $J_p^+$ is given by (3.17). By Young's inequality and (4.18), we can fix $p\in (1, p_0)$ and obtain

for $\forall$ $w\in W_0^{1, p}(\Omega) \subset BV(\Omega)$, with $p_0 = \min\{\theta, \frac{N}{N-1}\}$. From (4.38) and (3.19), one gets

for all $w\in W_0^{1, p}(\Omega)$ with $p < p_0 < \theta$. Recalling the structure of mountain pass lemma in Theorem 3.9, we can deduce that there exists $e = t_0 w_0\in W_0^{1, p}(\Omega)\subset BV(\Omega)$ and $\|e\|_{BV} > \rho$ such that $J(e) < 0$ by (3.20).

Obviously, the critical points of $I_p$ are identical with the critical points of $J_p^+$. Then $u_p$ given by Theorem 3.9 is a critical point of $J_p^+$, and also a critical point of $I_p$, which implies that the critical point $u_p$ satisfies

where $\Gamma_p = \{\eta\in C([0, 1], W_0^{1, p}(\Omega)) \mid \eta(0) = 0, \eta(1) = e\}$. Considering any path $\eta \in \Gamma_p$ and the continuity of the map $t\rightarrow I_p(\eta(t))$, there exists $t_0 > 0$ such that $\|\eta (t_0)\|_{BV} = \rho$. From (4.36), (4.38), (4.40) and $\|\eta (t_0)\|_{BV} = \rho$, we obtain that

On the other hand, choosing $u_p$ as a test function in (3.1), by the Dominated Convergence Theorem, (4.2) and (4.20), we have

where the last equality is given by Step 2 and Step 3. From $(H_2)$, (4.2) and (4.3), we can apply the Dominated Convergence Theorem to obtain

By (4.37), (4.42) and (4.43), we can get

Summarizing (4.41) and (4.44) we obtain that

with $0 < \rho < \min\{ \delta, (\frac{1}{2C_{10}})^\frac{1}{\alpha}\}$, and then $u$ is nontrivial, because $J^+(0) = 0$.

The proof of Theorem 1.2 is completed.

Acknowledgments

The author sincerely thanks the editors and reviewers for their valuable suggestions and useful comments. This work was supported by the Natural Science Foundation of Jiangsu Province of China (BK20180638).

Conflict of interest

For the publication of this article, no conflict of interest among the authors is disclosed.