Abstract
In this work, we prove the existence of a nontrivial solution to a quasilinear elliptic problem defined on the whole Euclidean space \( {\mathbb {R}}^N,\ N \ge 2, \) and involving a weighted 1-Laplacian operator. The nonlinear term has a singular behavior at the origin. This solution is obtained through an approximation technique, which consists in considering the problem with the 1-Laplacian operator as a limit of a family of problems with the p-Laplacian operators when \(p \rightarrow 1^+.\) For that aim, a new version of Anzellotti’s \( L^{\infty }-\)divergence−measure pairing theory is established and new arguments are used.
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1 Introduction and Statement of Main Results
In this work, we study the following quasilinear elliptic problem involving a weighted 1-Laplacian operator:
where \(\Delta _{1}\left( \textrm{au} \right) \) is the formal limit of the weighted p-Laplacian operator
as \(p \rightarrow 1^+, \) i.e., \( \Delta _{1}\left( \textrm{au} \right) =\text{ div }\left( a \frac{{\mathrm{{Du}}}}{|{\mathrm{{Du}}}|} \right) .\)
The problem (1.1) is taken under the following assumptions:
\((H_{1})\): \(a \in L^1({\mathbb {R}}^N)\cap L^{\infty }({\mathbb {R}}^N)\cap C^1({\mathbb {R}}^N)\) such that
We also assume that that \( b \in L^1({\mathbb {R}}^N)\cap L^\infty ({\mathbb {R}}^N)\) and there exists a constant \( C_0 > 0 \) such that \( a(x) \le C_0 b(x),\ \forall \ x \in {\mathbb {R}}^N. \) Thus,
\((H_{2})\): \(h : {\mathbb {R}}^N \rightarrow {\mathbb {R}}\) is a nonnegative function such that \(\frac{h}{b}\in L^{\infty }({\mathbb {R}}^N)\) and
Example: An example of functions a, b and h satisfying the hypotheses \( (H_1) \) and \( (H_2) \) is given by:
In the last years, problems involving the 1-Laplacian operator, formally defined by
caught the attention of so many specialists in partial differential equations, mainly because of its applications in image processing and in fracture mechanics. Concerning this applied aspect of such a problems, one can, for example, cite [8] where the authors showed the applications of minimizing functionals with linear growth arising from the 1-Laplacian operator in image restoration. In [25], the authors studied the inverse mean curvature flow problem, by studying problems involving the 1-Laplacian operator and containing some gradient terms.
By studying problems involving the 1-Laplacian operator in some domain \( \Omega \) of \( {\mathbb {R}}^N,\ N \ge 2, \) we could think that the natural space to solve them is \(X=W^{1,1}(\Omega );\) however, it lacks the lower semi-continuity property of the norm \(W^{1,1}(\Omega )\) with respect to the topology of \(L^1(\Omega ).\) This, in turn, makes it extremely difficult to prove the existence of solution to equations involving the 1-Laplacian operator. Nevertheless, there is another Banach space, which contains \(W^{1,1}(\Omega ),\) where the proper extension of the norm is lower semi-continuous in some sense. This space is the so-called space of bounded variation functions BV\((\Omega ).\) It is well known that this space is not easy to manipulate because its dual is not well known, and by consequence this space is not necessarily reflexive, making difficult to prove compactness conditions like Palais–Smale and Cerami conditions.
There are plenty of works dealing with this highly singular operator by using different approaches. Among them, we can, for example, cite [18,19,20,21, 39] , which applied the approach based on the variational method using nonsmooth critical point theory for energy functionals to get a solution, where by a solution is understood a function in the space of functions of bounded variation, in which the sub-differential of the energy functional vanishes. Problems involving such an operator can also be addressed using another method as it will be the case in the present work, that is the approximation technique. This technique is mainly based on the idea of working with a class of p-Laplacian problem and then taking the limit when \(p\rightarrow 1^+\) to finally obtain a solution to the 1-Laplacian problem (see, for instance, [1, 9, 22, 24, 29,30,31,32,33, 35,36,37]). By using this approach, the concept of solution to equations involving the 1-Laplacian was developed by F. Andreu et al. (see [1] and the book [2] ) and F. Demengel (see [15]). Indeed, in [1], the authors characterize the imprecise quotient \(\frac{\mathrm{{Du}}}{|\mathrm{{Du}}|}\) (where \(|\textrm{Du}|\) denotes the total variation of the Radon measure Du), by Anzellotti’s pairing theory (see [3] and also [2]). This theory allows them to introduce a vector field \(z\in L^\infty (\Omega , {\mathbb {R}}^N)\) such that \(\left( z, \mathrm{{Du}}\right) = \left| \textrm{Du}\right| \) in the sense of measures, what means that z somehow plays the role of \(\frac{\mathrm{{Du}}}{|\mathrm{{Du}}|},\) where \( (z, \textrm{Du}) \) is a Radon measure appropriately defined in such way that it can be considered as replacing the dot product in \( {\mathbb {R}}^N. \)
Recently, V. De Cicco, D. Giachetti and S. Segura de León in [14] used the approximation technique to deal with a problem similar to (1.1). More precisely, V. De Cicco, D. Giachetti and S. Segura de León studied the following Dirichlet problem for an equation involving the 1-Laplacian operator and a singular lower-order term
where \(\Omega \subset {\mathbb {R}}^N\) is a bounded open set with Lipschitz boundary \(\partial \Omega ,\) \(0< \gamma <1\) and f is a function belonging to \(L^N(\Omega ).\) In that work, through an approximation scheme, they proved the existence of a BV-solution. The boundedness of the domain played a crucial role to prove some a priori estimates and also in the process of the passage to the limit. A similar problem has been treated in [13]. In [29], the authors dealt with the case when the source f belongs to \( L^1( {\mathbb {R}}^N). \) Finally, we have also to cite a recently published article [17] where the authors investigated an equation similar to (1.2) but containing a gradient term.
In the present work, we firstly deal with the following approximated singular p-Laplacian equation related to (1.1):
The existence of a solution \( u_p \) to (1.3) is proved using variational tools. In order to overcome the difficulties due to the existence of the singularity as well as the unboundedness of the domain, some new sophisticated arguments have to be employed. Some a priori estimates are proved. Next, we study the behavior of \(u_p\) as \(p\rightarrow 1^+,\) proving that,
and finally, we prove that this function u is a solution to our initial problem (1.1). Due to the fact that the weights a and b are unbounded from below, many difficulties arise when we try to establish the a priori estimates for the family \( ( u_p)_{1< p < 2} \) and also when passing to the limit as \( p \rightarrow 1^+. \)
Before stating our main results, let us define the space in which we study the problem (1.1). Since in (1.1) we are dealing with the weighted 1-Laplacian operator, the standard BV\(( {\mathbb {R}}^N)\) space cannot be used to analyze this problem. In fact, we will consider a weighted space, denoted by \(\textrm{BV}^b_{a}({\mathbb {R}}^N), \) used as functional framework. Such kind of weighted space has been introduced in [5] and used later by J.C. Ortizo Chata, M.T.O. Pimenta and S.S. Léon in [6] to treat the following elliptic equation containing a unbounded weights:
where \( \Omega \) is a bounded regular open set of \( {\mathbb {R}}^N,\ N \ge 2, \) containing the origin and the parameters a and b satisfy \( 0< a < N-1 \) and \( 0< b < a+1, \) and f is some continuous function with polynomial behavior satisfying some appropriate conditions. The weighted space \(\textrm{BV}^{b}_{a}({\mathbb {R}}^{N^{}})\) is the space of functions \(u:{\mathbb {R}}^N \rightarrow {\mathbb {R}}\) measurable such that \(b(x)u\in L^1({\mathbb {R}}^N)\) and a(x)Du is a finite Radon measure, i.e.,
where \( {\mathcal {M}}( {\mathbb {R}}^N, {\mathbb {R}}^N) \) is the set of finite vectorial Radon measures.
See Sect. 2 for more details concerning the space \( \textrm{BV}_a^b( {\mathbb {R}}^N). \)
Definition 1
We say that \(u\in \textrm{BV}^{b}_{a}({\mathbb {R}}^N)\cap L^{\infty }({\mathbb {R}}^N) \) is a weak solution to problem (1.1) if \( u(x) \ge 0 \) a.e. \( x \in {\mathbb {R}}^N \) and there exist \(z\in {\mathcal {D}}{\mathcal {M}}^{\infty }_{a, \textrm{loc}}({\mathbb {R}}^N)\) with \(|z|_{L^{\infty }( {\mathbb {R}}^N, {\mathbb {R}}^N)}\le 1\) and \(\gamma \in L^ \infty ({\mathbb {R}}^N)\) with \(| \gamma |_{L^{\infty }( {\mathbb {R}}^N)}\le 1\) such that
where \(\chi _{\{u>0\}}\) is the characteristic function of the region \({\{u>0\}},\ \chi ^*_{\{u>0\}} \) its precise representative,
\({\mathcal {M}}_{\textrm{loc}}({\mathbb {R}}^N)\) is the space of Radon measures which are locally finite in \({\mathbb {R}}^N,\) and the pairing \( (\textrm{az}, \mathrm{{Du}}) \) is defined in Sect. 2.2 (it can be seen as replacing the dot product in \( {\mathbb {R}}^N). \)
Theorem 1.1
Under the assumptions \( (H_1) \) and \( (H_2), \) there exists a solution to the problem (1.1) in the sense of Definition 1.
Theorem 1.2
Suppose that \((H_1)\) and \((H_2)\) hold. We also assume that \(h(x)> 0, \) a.e. \( x \in {\mathbb {R}}^N. \) Then, the problem (1.1) has a solution \(u\in \textrm{BV}^{b}_{a}({\mathbb {R}}^N)\cap L^{\infty }({\mathbb {R}}^N) \) such that:
Moreover, u is unique.
This paper is organized as follows: in Sect. 2 we study the functional space \(W^{1,p}_{a,b}({\mathbb {R}}^{N})\) (the functional framework in which we will study the problem (1.3)). We will also define the weighted space \(\textrm{BV}^{b}_{a}({\mathbb {R}}^N)\) and try to establish some of its important properties. Finally, we extend the Anzellotti pairing theory to include unbounded vector fields. In Sect. 3, we prove the existence of a nontrivial solution \(u_p\) to equation (1.3) and prove some a priori estimates for the family \( (u_p)_{1< p < 2}. \) To finish, in Sect. 4 we will proceed to the passage to the limit as p tends to \( 1^+ \) and by consequence show that the limit of \(u_p\) is actually a solution to our problem (1.1). More precisely, we will prove Theorems (1.1) and (1.2).
2 Preliminaries
In this section, we provide some fundamental properties of the functional spaces \(\textrm{BV}^{b}_{a}({\mathbb {R}}^{N})\) and \( W^{1,p}_{a,b}( {\mathbb {R}}^N) \) which will be used in the proof of Theorems 1.1 and 1.2. A weighted version of the theory of \(L^{\infty }-\)divergence−measure vector fields of Anzellotti’s type is established.
2.1 The Space \(\textrm{BV}^{b}_{a}({\mathbb {R}}^N)\)
In this subsection, we provide the most important properties of our functional space \( \textrm{BV}_a^b( {\mathbb {R}}^N). \) First of all, for \( u \in \textrm{BV}_\textrm{loc}( {\mathbb {R}}^N) \) and \( 1 \le j \le N, \) one can define the application
Hence, we say that \( a \mathrm{{Du}} \in {\mathcal {M}}( {\mathbb {R}}^N, {\mathbb {R}}^N) \) if and only if, for all \( 1 \le j \le N, \) there exists a constant \( c_j > 0 \) such that
Theorem 2.1
The following statements are equivalent:
-
\(u\in \textrm{BV}^{b}_{a}({\mathbb {R}}^N);\)
-
\(bu\in L^1({\mathbb {R}}^N)\) and
$$\begin{aligned} |a\mathrm{{Du}}|({\mathbb {R}}^N)= \sup \left\{ \int _{{\mathbb {R}}^N}u\ \text{ div }\ \varphi ,\ \varphi \in C^{1}_c({\mathbb {R}}^N, {\mathbb {R}}^N),\ \left| \varphi (x)\right| \le a(x),\ \forall x\in {\mathbb {R}}^N \right\} < \infty . \end{aligned}$$
Proof
By the virtue of the Riesz representation theorem for Radon measures, we know that there exists a real regular Borel measure \( \mu =(\mu _1, \mu _2,\cdots ,\mu _N) \) such that
Thus, \( a\mathrm{{Du}} = \mu \) in the distributional sense. Moreover,
where \(\left| \mu \right| ( {\mathbb {R}}^N)\) denotes the total variation of the Borel measure \( \mu . \)
From the density of \( C_c^{1}( {\mathbb {R}}^N,{\mathbb {R}}^N) \) in \( C_c( {\mathbb {R}}^N, {\mathbb {R}}^N),\) it follows
If we denote by \( \left| \mu \right| = \left| a \mathrm{{Du}}\right| = a(x) \left| \mathrm{{Du}}\right| , \) i.e.
then we can see that \(u\in \textrm{BV}^{b}_{a}({\mathbb {R}}^N)\) if and only if \( bu\in L^1({\mathbb {R}}^N)\) and
\(\square \)
Naturally, we endow the space \(\textrm{BV}_a^b({\mathbb {R}}^N)\) with the norm
Equipped with that norm, \( \textrm{BV}_a^b( {\mathbb {R}}^N) \) is a Banach space but it is not necessarily reflexive. The following semi-continuity property is fundamental.
Lemma 1
Let \( (u_n)_{n\ge 0} \subset \textrm{BV}_{a}^{b}( {\mathbb {R}}^N) \) and \( u \in L^1_{\textrm{loc}}({\mathbb {R}}^N) \) be such that \( u_n \rightarrow u \) strongly in \( L^1(B)\) for all balls \(B\subset {\mathbb {R}}^N.\) Then,
Proof
Observe that if \( \displaystyle {\liminf _{n \rightarrow +\infty } \int _{{\mathbb {R}}^N}a |\mathrm{{Du}}_n| = + \infty ,} \) then the result is trivial. Hence, we can suppose that \( \displaystyle {\liminf _{n \rightarrow +\infty } \int _{{\mathbb {R}}^N}a |\mathrm{{Du}}_n| < + \infty .} \) Let B be a ball of \({\mathbb {R}}^N\) and \(\varphi \in C^{1}_{c}({\mathbb {R}}^N, {\mathbb {R}}^N)\) with compact support in B such that \(\left| \varphi (x)\right| \le 1,\ \forall x\in {\mathbb {R}}^N.\) We have
Now, we take the supremum over all such \( \varphi \), and the result immediately follows. \(\square \)
Proposition 1
Let B be an open ball of \( {\mathbb {R}}^N. \) The space \( \textrm{BV}^{b}_{a}( {\mathbb {R}}^N) \) is continuously embedded into the Lebesgue spaces \( L^t(B),\ \forall \ 1 \le t \le 1^*, \) where \(1^* =\frac{N}{N-1}. \)
Proof
Clearly, by \( (H_1), \) it yields
with continuous embedding. By [4, Theorem 10.1.3], we know that \( \textrm{BV}(B) \) is continuously embedded into \( L^t(B),\ \forall \ t \le 1 \le 1^*. \) This ends the proof of Proposition 1. \(\square \)
Remark 1
The embedding \(\textrm{BV}^{b}_{a}({\mathbb {R}}^N)\hookrightarrow L^t(B)\) is compact for \(1\le t < 1^*,\) for all balls \(B\subset {\mathbb {R}}^N.\) This result can be immediately deduced from [4, Theorem 10.1.4].
2.2 Weighted Anzellotti’s Theory
In the classical Anzellotti’s theory (see [3]), in order to define the pairing \( (z, \mathrm{{Du}}) \) as the distribution
where \( \Omega \) is an open set of \( {\mathbb {R}}^N,\ N \ge 2, \) and \( u^* \) is the precise representative of the function u (see [14]), we need some compatibility conditions such as \( u \in \textrm{BV}( \Omega ) \cap L^{\infty }( \Omega ) \) and \(\text{ div } z \in L^{1}( \Omega ) \) or \( u \in \textrm{BV}( \Omega ) \cap L^{\infty }( \Omega ) \cap C( \Omega ) \) and \( \text{ div } z \in {\mathcal {M}}( \Omega ). \) However, this definition has been extended in [7] to the case where \( u \in \textrm{BV}( \Omega ) \cap L^{\infty }( \Omega ) \) (i.e. the additional regularity condition \( u \in C( \Omega ) \) is removed) and \( \text{ div } z \in {\mathcal {M}}( \Omega ). \) Another further extension to the case where \( u \in \textrm{BV}_\textrm{loc}( \Omega ) \cap L^1( \Omega , \text{ div } z) \) and \(z \in \mathcal {D M}_{\textrm{loc}}^{\infty }( \Omega ) \) has been proved in [14]. We can also refer to [10,11,12].
In our present work, we need to establish a slightly further extension of the Anzellotti’s theory. For that aim, we define the space
and we reintroduce the space \({\mathcal {D}} {\mathcal {M}}^{\infty }_{a,\textrm{loc}}({\mathbb {R}}^N)\) defined as the space of all vector fields \(z\in L^{\infty }({\mathbb {R}}^N, {\mathbb {R}}^N)\) such that \(\text{ div }(\textrm{az})\in {\mathcal {M}}(B),\) for all balls \(B\subset {\mathbb {R}}^N.\)
Definition 2
Let \(z\in {\mathcal {D}}{\mathcal {M}}^{\infty }_{a, \textrm{loc}}({\mathbb {R}}^N)\) and \(u\in \textrm{BV}^{b}_{a}({\mathbb {R}}^N)\cap L^{\infty }({\mathbb {R}}^N).\) Then, we define the functional \(\Big ( \textrm{az},\mathrm{{Du}}\Big ):C^{\infty }_{c}({\mathbb {R}}^N)\rightarrow {\mathbb {R}}\) as
where \(\nu \) is the regular Borel measure that represents \( \text{ div }(a z). \)
Proposition 2
Let \(z\in {\mathcal {D}} {\mathcal {M}}^{\infty }_{a, \textrm{loc}}({\mathbb {R}}^N)\) and \(u\in \textrm{BV}_{a}^{b}({\mathbb {R}}^N)\cap L^{\infty }({\mathbb {R}}^N). \) Then,
Proof
Fix \(u \in \textrm{BV}_{a}^{b}({\mathbb {R}}^N)\cap L^{\infty }({\mathbb {R}}^N).\) Let \( (\rho _n)_n \) be a sequence of mollifiers and define the sequence \( (u_n)_n \subset C^{\infty }( {\mathbb {R}}^N) \cap L^{\infty }( {\mathbb {R}}^N) \) by \( u_n = \rho _n * u,\ n \ge 1. \) Plainly, \( u_n(x) \rightarrow u(x) \) a.e. \( x \in {\mathbb {R}}^N \) and \( u_n(x) \rightarrow u^*(x),\ {\mathcal {H}}^{N-1}-\)a.e. \( x \in {\mathbb {R}}^N, \) where \( {\mathcal {H}}^{N-1} \) stands for the \( (N-1)-\)dimensional Hausdorff measure in \( {\mathbb {R}}^N. \) Moreover, \( (u_n)_n \) is bounded in \( L^{\infty }( {\mathbb {R}}^N). \) It is easy to show that
and
as \( n \rightarrow + \infty . \) Taking into account that \( \left| \text{ div }(\textrm{az})\right|<< {\mathcal {H}}^{N-1} \) (see [7]), then one can apply the Lebesgue’s dominated convergence theorem with respect to the Radon measure \( \text{ div }(\textrm{az}), \) to obtain that
In a similar way, applying the Lebesgue’s dominated convergence Theorem with respect to the Lebesgue measure, we also get
Hence,
On the other hand, let B be a ball such that \( \text{ supp }( \varphi ) \subset B. \) Having in mind that,
we infer
Passing to the limit as \( n \rightarrow + \infty , \) in that last inequality, and taking (2.2) into account, we deduce that (2.1) holds true. \(\square \)
Corollary 1
The distribution \((a z,\mathrm{{Du}})\) is a Radon measure and its total variation \(|(a z,\mathrm{{Du}})|\) are absolutely continuous with respect to the measure \(a|\mathrm{{Du}}|\) and
holds for all Borel sets \(B \subseteq {\mathbb {R}}^N.\)
Next, according to Definition 2, we can easily see that the following proposition holds.
Proposition 3
Let \(z\in {\mathcal {D}} {\mathcal {M}}^{\infty }_{a,\textrm{loc}}({\mathbb {R}}^N)\) and \( u \in \textrm{BV}^{b}_{a}({\mathbb {R}}^N)\cap L^{\infty }({\mathbb {R}}^N).\) Then, \(\textrm{az} u\in {\mathcal {D}} {\mathcal {M}}^{\infty }_{a, \textrm{loc}}({\mathbb {R}}^N).\) Moreover the following formula holds in the sense of measures
Finally, we prove that the following Gauss–Green’s formula holds.
Theorem 2.2
Let \( u \in \textrm{BV}_a^b( {\mathbb {R}}^N) \cap L^{\infty }( {\mathbb {R}}^N) \) and \( z \in L^{\infty }( {\mathbb {R}}^N,{\mathbb {R}}^N) \) be such that \( \text{ div }(\textrm{az}) \in L^1( {\mathbb {R}}^N). \) Then, we have
Proof
Consider a cutoff function \( \theta \in C_c^{\infty }( {\mathbb {R}}^N)\) such that \( 0 \le \theta \le 1,\ \theta (x) = 1,\ \forall \ \left| x\right| \le 1, \) and \( \theta (x) = 0,\ \forall \ \left| x\right| \ge 2. \) For \( k=1,2,3,\cdots , \) define the function \( \theta _k: {\mathbb {R}}^N \rightarrow {\mathbb {R}} \) by
Since \( \text{ supp }( \theta _k) \) is compact, then \( \theta _k u \in \textrm{BV}( {\mathbb {R}}^N). \) Taking into account that \( \text{ div }(\textrm{az}) \in L^1( {\mathbb {R}}^N), \) one can apply the classical Gauss–Green’s formula ( [3]), that is
Having in mind that
it yields
Observing that \( \theta _k(x) \rightarrow 1,\ \forall \ x \in {\mathbb {R}}^N \) and \( 0 \le \theta _k(x) \le 1,\ \forall \ x \in {\mathbb {R}}^N,\ \forall \ k=1,2,3,\cdots , \) , then by the Lebesgue-dominated convergence theorem, it follows that
and
On the other hand, we have
Combining (2.7), (2.6) and (2.5) with (2.4), we obtain
\(\square \)
2.3 The Space \(W^{1,p}_{a,b}({\mathbb {R}}^{N})\)
Taking into account the fact that our existence result will be established through some approximation scheme, some properties of the weighted Sobolev space \( W^{1,p}_{a,b}( {\mathbb {R}}^N) \) have to be stated. This is the object of the present subsection.
Let \(1<p<2.\) Set,
This space is provided with the norm
Since \(a,\ b \in L^1_{\textrm{loc}}({\mathbb {R}}^{N}),\) and \(a^{-1/(p-1)} ,\ b^{-1/(p-1)}\in L^1_{\textrm{loc}}({\mathbb {R}}^{N}),\) then \(\Big (W^{1,p}_{a,b}({\mathbb {R}}^{N}), \Vert .\Vert _{W^{1,p}_{a,b}({\mathbb {R}}^{N})}\Big )\) becomes a Banach, reflexive and separable space (see [28, 29]).
Lemma 2
Let \( 1< p < 2. \) The embedding \(W^{1,p}_{a,b}({\mathbb {R}}^N)\hookrightarrow W^{1,1}_{a,b}({\mathbb {R}}^N)\) is continuous. Precisely, there exists a constant \(C> 0\) independent of p such that
where
and
Proof
Let \(u\in W^{1,p}_{a,b}({\mathbb {R}}^N).\) By \((H_1)\) and the Hölder’s inequality, we have
\(\square \)
Lemma 3
For all \(R>0,\) there exists a constant \(C_R>0\) independent of p such that
where \(B(0,R)=\{x\in {\mathbb {R}}^N,\ |x|<R\}.\)
Proof
Let \( R > 0 \) and \(u\in W_{a,b}^{1,p}({\mathbb {R}}^N).\) By \((H_{1}), \) we have
Therefore, inequality (2.9) can be deduced from Lemma 1. \(\square \)
Remark 2
One of the consequences of Lemma 3 is that the following continuous embedding holds
Moreover, by the virtue of the Rellich–Kondrachov compactness Theorem, we can immediately deduce that the embedding \(W^{1,p}_{a,b}({\mathbb {R}}^{N})\hookrightarrow L^t(B(0,R))\) is compact for all \(1\le t < 1^*,\ \forall \ R>0.\)
Now, we prove that \( C^{\infty }_c( {\mathbb {R}}^N)\) is dense in \(W^{1,1}_{a,b}( {\mathbb {R}}^N).\)
Proposition 4
The space \( C^{\infty }_c( {\mathbb {R}}^N) \) is dense in \( W^{1,1}_{a,b}( {\mathbb {R}}^N) \).
Proof
Let \( v\in W^{1,1}_{a,b}( {\mathbb {R}}^N) \) and \( \phi \in C_c^{\infty }( {\mathbb {R}}^N) \) be a cutoff function such that \( \phi (x) = 1,\ \forall \ \left| x\right| \le 1,\ \phi (x) = 0,\ \forall \ \left| x\right| \ge 2, \) and \( 0 \le \phi \le 1. \) For \( k \in {\mathbb {N}}^* \) and \( x \in {\mathbb {R}}^N, \) set \( \phi _k(x) = \phi \left( \frac{x}{k}\right) \) and \( v_k(x) = \phi _k(x) v(x). \) We claim that \(v_k \rightarrow v,\ k \rightarrow + \infty , \) strongly in \(W^{1,1}_{a,b}( {\mathbb {R}}^N). \) We have
Observing that \(|\phi _k(x) -1 | \rightarrow 0,\ \forall x\in {\mathbb {R}}^N, \) and that \(|\phi _k(x) -1|\le 2,\ \forall x\in {\mathbb {R}}^N, \) then by the Lebesgue dominated convergence Theorem, it follows that
On the other hand,
where we used the fact that \( a(x)\le C_0 b(x),\ \forall x\in {\mathbb {R}}^N. \) Plainly,
Again by the virtue of the Lebesgue-dominated convergence theorem, we get
By (2.13) and (2.12), we infer
Combining (2.14) and (2.10), we deduce that \(v_k \rightarrow v\) strongly in \(W^{1,1}_{a,b}( {\mathbb {R}}^N). \) Hence, for every \( \epsilon > 0, \) there exists \( n_0 = n_0(\epsilon ) \ge 1 \) such that
Now, recall that
Then,
Having in mind that \( v_{n_0} \in W_0^{1,1}(B(0,2n_0)) \) and that \( W_0^{1,1}(B(0,2n_0)) \hookrightarrow W^{1,1}_{a,b}( {\mathbb {R}}^N), \) we infer that there exists \( {\widetilde{v}} \in C_c^{\infty }(B(0,2n_0)) \subset C_c^{\infty }( {\mathbb {R}}^N) \) such that
Combining (2.15) and (2.16), we obtain the claimed result. \(\square \)
3 Weak Solution to the Approximated p-Laplacian Problem
For \(1< p < 2,\) let us consider the following problem
where \( u^+ = \max (u,0). \) We define the functional
It is well known that the singular term leads to the non-differentiability of the functional \(J_p\) on \(W^{1,p}_{a,b}({\mathbb {R}}^N), \) so \(J_p\) does not belong to \(C^1(W^{1,p}_{a,b}({\mathbb {R}}^N),\ {\mathbb {R}}). \) Therefore, the problem cannot be considered by using directly the classical critical point theory. Nevertheless, we shall prove that problem (3.1) still has a solution, which is a local minimizer of the energy functional \(J_p.\) For \( n=1,2,3,\cdots , \) let us consider the following approximated problem:
and its corresponding energy functional:
where
with \(s^-=\min \{s,0\}. \) By studying deeply the functional \(J_{n,p},\) we can deduce the existence of solution to our original problem (3.1) for \( 1< p < 2.\)
Definition 3
A function \(u_p\in W^{1,p}_{a,b}({\mathbb {R}}^N) \) is said to be a weak solution of (3.1) if it satisfies that \(u_p(x)> 0 \) a.e. \( x \in {\mathbb {R}}^N,\ \frac{h v}{u_p^{1-q}} \in L^1({\mathbb {R}}^N),\ \forall v\in W^{1,p}_{a,b}({\mathbb {R}}^N),\) and
Theorem 3.1
Assume that \((H_1)\) and \((H_2)\) hold. Then, for any fixed \( 1< p < 2, \) there exists a bounded weak solution to (3.1) in the sense of Definition (3).
The proof of Theorem (3.1) will be divided in several steps.
Lemma 4
For \(1<p<2,\) there exist r and \( \rho >0 \) independent of p such that
Moreover,
Proof
By Lemma 2, Hölder’s inequality and \((H_2),\) for \( u \in W^{1,p}_{a,b}( {\mathbb {R}}^N),\) we have
Recall that the constant C given in Lemma 2 is independent of p. It yields,
Choose \(r>0\) small enough such that \(\frac{r^{p-q}}{p} -C_1\ge \frac{r^{2-q}}{2} -C_1 > 0,\ \forall \ 1<p<2.\) By (3.3), we infer
Now, let \(\varphi \in W^{1,p}_{a,b}({\mathbb {R}}^N)\) be such that \(\varphi > 0\) and \(\Vert \varphi \Vert _{W^{1,p}_{a,b}({\mathbb {R}}^N)}\le r.\) For \(t>0, \) we get
By consequence, one can easily find \(0<t_0<1\) small enough such that \(\Vert t_0 \varphi \Vert _{W^{1,p}_{a,b}({\mathbb {R}}^N)}\le r\) and \(J_p(t_0\varphi )<0. \) We conclude that \(m_p\le J_p(t_0\varphi )<0.\) This ends the proof of Lemma (4). \(\square \)
Lemma 5
The problem (3.1) has a positive solution \(u_p \in W^{1,p}_{a,b}({\mathbb {R}}^N) \) satisfying that \(J_p(u_p)<0. \)
Proof
First, we claim that there exists \(u_p \in W^{1,p}_{a,b}({\mathbb {R}}^N)\) such that \(\Vert u_p\Vert _{W^{1,p}_{a,b}({\mathbb {R}}^N)}\le r\) and \(J_p(u_p)=m_p<0.\)
For that aim, fix \(n\ge 1. \) Observe that since \(0<q<1, \) then
By Lemma 4, it yields
Consequently,
Since \(J_{n,p}\) is of class \(C^1\) on \( W^{1,p}_{a,b}({\mathbb {R}}^N),\) then we can apply the Ekeland variational principle (see [16]) to construct a sequence \((u_{j,n,p})_j\subset W^{1,p}_{a,b}({\mathbb {R}}^N)\) such that \(\Vert u_{j,n,p}\Vert _{W^{1,p}_{a,b}( {\mathbb {R}}^N)}\le r,\ J_{n,p}(u_{j,n,p}) \rightarrow m_{n,p}= \inf \limits _{\Vert u\Vert _{W^{1,p}_{a,b}( {\mathbb {R}}^N)}\le r} J_{n,p}(u), \) and \( J'_{n,p}(u_{j,n,p}) \rightarrow 0,\) as \(j \rightarrow + \infty .\) Since \((u_{j,n,p})_j\) is bounded in \(W^{1,p}_{a,b}({\mathbb {R}}^N),\) then there exists \(u_{n,p}\in W^{1,p}_{a,b}({\mathbb {R}}^N)\) such that \(u_{j,n,p}\rightharpoonup u_{n,p}\) weakly in \(W^{1,p}_{a,b}({\mathbb {R}}^N)\) and \(u_{j,n,p}(x)\rightarrow u_{n,p}(x) \) a.e. \( x\in {\mathbb {R}}^N.\) This implies that \(b^{\frac{1}{p}}u_{j,n,p}\rightharpoonup b^{\frac{1}{p}} u_{n,p}\) weakly in \(L^p({\mathbb {R}}^N)\) and \(b^{\frac{1}{p}} (x)u_{j,n,p}(x)\rightarrow b^{\frac{1}{p}}(x)u_{n,p}(x), \) a.e. \( x\in {\mathbb {R}}^N.\) Set \(w_{j,n,p}=u_{j,n,p} -u_{n,p}. \) By Brezis–Lieb’s lemma ( see [26, Chapter 1, Lemma 4.6]), one has
where \(o_j(1)\) denotes a sequence of real numbers tending to 0 when j tends to \(+\infty .\)
Now, we claim that, for a.e. \( x \in {\mathbb {R}}^N,\ a(x)^{\frac{1}{p}}\nabla u_{j,n,p}(x) \rightarrow a(x)^{\frac{1}{p}}\nabla u_{n,p}(x),\) as \(j \rightarrow \infty .\) Since \(\left\langle J'_{n,p}(u_{j,n,p}), w_{j,n,p} \right\rangle \rightarrow 0,\) as \(j \rightarrow +\infty ,\) then
We have
By the weak convergence of \((w_{j,n,p})_j\) to 0 in \(W^{1,p}_{a,b}({\mathbb {R}}^N),\) we infer
That inequality together with (3.6) leads to
Similarly, we can show that
On the other hand, since \(0<q<1,\) we have
Observing that \(b^{\frac{1}{p}}|w_{j,n,p}|\rightharpoonup 0\) weakly in the Lebesgue space \(L^{p}({\mathbb {R}}^N)\) and that the function \(\frac{h}{b^{\frac{1}{p}}}\) belongs to the space \(L^{\frac{p}{p-1}}({\mathbb {R}}^N)\) which is the dual of \(L^{p}({\mathbb {R}}^N), \) then
It yields,
Combining (3.9), (3.8) and (3.7) with (3.5), we obtain
Let us now recall the following elementary inequality [38, formula 2.2](see also [23, p. 713])
for all \(x,y\in {\mathbb {R}}^N.\) Combining (3.10) and (3.11), we conclude that \(u_{j,n,p} \rightarrow u_{n,p}\) strongly in \(W^{1,p}_{a,b}({\mathbb {R}}^N). \) Therefore,
That last result allows us to use the Brezis–Lieb lemma to assert that
Next, using the fact that \(\chi _n( u_{j,n,p})(x) \rightarrow \chi _n( u_{n,p})(x), \) a.e. \( x \in {\mathbb {R}}^N,\) together with the inequality
we deduce that, up to a subsequence, \(b^{\frac{1}{p}}\chi _n( u_{j,n,p}) \rightharpoonup b^{\frac{1}{p}}\chi _n( u_{n,p})\) weakly in \(L^p({\mathbb {R}}^N)\) as \(j \rightarrow + \infty . \) Taking into account that \(h/b^{\frac{1}{p}}\in L^{\frac{p}{p-1}}({\mathbb {R}}^N),\) we infer
Combining (3.13) and (3.14), we deduce that
It yields,
i.e.
Tending \(j\rightarrow +\infty ,\) it follows
Taking into account that \(\Vert u_{n,p}\Vert _{W^{1,p}_{a,b}({\mathbb {R}}^N)}\le \liminf \limits _{j \rightarrow + \infty } \Vert u_{j,n,p}\Vert _{W^{1,p}_{a,b}({\mathbb {R}}^N)} \le r,\) we deduce that \(m_{n,p}= J_{n,p}(u_{n,p}). \) Since \(m_{n,p}\le J_{n,p}(0)=0 \) and \(J_{n,p}(v)>0,\ \forall \ v \in W^{1,p}_{a,b}({\mathbb {R}}^N),\ \Vert v\Vert _{W^{1,p}_{a,b}({\mathbb {R}}^N)}=r, \) we conclude that \(\Vert u_{n,p}\Vert _{W^{1,p}_{a,b}({\mathbb {R}}^N)}< r. \) Having in mind that \(J_{n,p}\) is of class \(C^1,\) it follows that \(J'_{n,p}(u_{n,p})=0,\) i.e., \(u_{n,p}\) is a critical point of \(J_{n,p}.\) It is easy to see that \( u_{n,p} \ge 0. \) By the strong maximum principle (see [34, Theorem 2]), we can finally show that \(u_{n,p}(x)>0\) a.e. \( x \in {\mathbb {R}}^N. \) Since \((u_{n,p})_n \) is bounded in \( W^{1,p}_{a,b}({\mathbb {R}}^N), \) then there exists \( u_p \in W^{1,p}_{a,b}({\mathbb {R}}^N) \) such that, up to a subsequence, \(u_{n,p}\rightharpoonup u_p \) weakly in \( W^{1,p}_{a,b}({\mathbb {R}}^N),\ u_{n,p}(x) \rightarrow u_p(x) \) a.e. \( x\in {\mathbb {R}}^N. \) We claim that \( J_{n,p}(u_{n,p}) \rightarrow m_p,\ n \rightarrow +\infty . \) First, since \( J_{n,p}(u_{n,p})\ge J_p(u_{n,p}) \) and \(\Vert u_{n,p}\Vert _{W^{1,p}_{a,b}({\mathbb {R}}^N)}\le r, \) it yields
By the definition of \(m_p, \) we know that there exists a minimizing sequence \(\{y_k\}\) such that \(\lim \limits _{k \rightarrow +\infty } J_p(y_k)=m_p<0.\) we have
Using the same arguments as previously, we can easily show that
Thus,
Tending n to \(+\infty , \) we obtain
Finally, tending \(k \rightarrow + \infty ,\) we get
Combining (3.16) with (3.17), we deduce that \( \lim \limits _{n \rightarrow +\infty } J_{n,p}(u_{n,p})= m_p. \) Next, taking into account that \(J'_{n,p}(u_{n,p})=0, \) after introducing the sequence \(w_{n,p}=u_{n,p}-u_p,\) we can proceed as for the sequence \(\left( w_{j,n,p}\right) _j \) used at the beginning of the proof to prove that
Tending \(n \rightarrow + \infty , \) we deduce that \(m_p\ge J_p(u_p).\) Hence, \(J_p(u_p)= m_p< 0. \) It follows that \(u_p\) is a local minimizer of \(J_p. \) The next step in the proof consists of proving that \(u_p\) is a solution of (3.1). Since \(u_p\) is a local minimizer of \(J_p, \) then for any \(\psi \in W^{1,p}_{a,b}({\mathbb {R}}^N),\ \psi \ge 0, \) and for all \( 0<t<1 \) small enough such that \(\Vert u_p +t \psi \Vert _{ W^{1,p}_{a,b}({\mathbb {R}}^N)}\le r,\) one has
Dividing by \(t>0 \) and passing to the limit as \(t \rightarrow 0^+,\) it gives
Since \(\left( u_p+t \psi \right) \ge u_p. \) Thus, by using Fatou’s lemma, we have
Putting that last inequality in (3.18), we infer
Since \(\Vert u_p\Vert _{ W^{1,p}_{a,b}({\mathbb {R}}^N)}< r,\) then \(\Vert u_p(1+t )\Vert _{ W^{1,p}_{a,b}({\mathbb {R}}^N)}< r,\ \forall \ -1<t<\frac{r}{\Vert u_p\Vert _{ W^{1,p}_{a,b}({\mathbb {R}}^N)}}-1.\) We define the function \(k:]-1,-1+\frac{r}{\Vert u_p\Vert _{ W^{1,p}_{a,b}({\mathbb {R}}^N)}}[ \rightarrow {\mathbb {R}} \) by \(k(t)=J_p((1+t)u_p).\) Clearly, k attains its minimum at \(t=0.\) Since k is derivable at \(t=0,\) then
Now, let \(\varphi \in W^{1,p}_{a,b}({\mathbb {R}}^N)\) and \(\epsilon >0\). Since \((u_p +\epsilon \varphi )^+\in W^{1,p}_{a,b}({\mathbb {R}}^N)\) and \((u_p +\epsilon \varphi )^+\ge 0,\) then by (3.19) we deduce that
Using (3.20) and dividing by \(\epsilon >0 \) in (3.21), it follows
Since \(1_{\{u_p ^+ +\epsilon \varphi < 0\}}(x) \rightarrow 0, \) a.e. \( x\in {\mathbb {R}}^N,\) as \( \epsilon \rightarrow 0^+, \) then we can use the Lebesgue-dominated convergence theorem to deduce that all the integrals over the set \(\{u_p ^+ +\epsilon \varphi <0\}\) appearing in (3.22) tend to zero as \(\epsilon \rightarrow 0^+. \) Hence,
By the arbitrariness of \(\varphi , \) this inequality also holds for \(-\varphi , \) i.e.
The strong maximum principle implies that \( u_p(x) > 0, \) a.e. \( x \in {\mathbb {R}}^N. \) \(\square \)
Now, let us show that \(u_p\in L^{\infty }({\mathbb {R}}^N).\) More precisely, we prove the following result.
Lemma 6
Let \(u_p\) be a solution of (3.1). Then, there exists \(k>0\) independent of p such that
Proof
Taking \(\left( u_p- k\right) ^+\) with \(k\in ]1, +\infty [\) as test function in (3.1), it yields
Since
we infer
Now, choosing \(k>1\) such that \(\left| \frac{h}{b}\right| _{L^{\infty }({\mathbb {R}}^N)}\le k^{1-q}\le k^{p-q},\ \forall \ 1<p<2,\) we deduce that
Therefore, \((u_p-k)^+=0\) and by consequence \(u_p(x)\le k,\) a.e. \(x \in {\mathbb {R}}^N.\) \(\square \)
To conclude the proof of Theorem 3.1, it remains to show that
For that aim, it suffices to use Young’s inequality and (3.23) to obtain
4 Completion of the Proof of Theorem 1.1: The Limit Problem
First, observe that the family \((u_p)_{1<p<2}\) is bounded in \(\textrm{BV}^{b}_{a}({\mathbb {R}}^N).\) Indeed, taking \(u_p\) as test function in problem (3.1) and by using the Hölder inequality and Lemma 2, we get
Thus,
The Young’s inequality together with (4.1) implies
where \({\hat{C}}\) is a positive constant independent of p. Having in mind that \(W^{1,1}_{a,b}({\mathbb {R}}^N) \) is continuously embedded into \( \textrm{BV}_{a}^{b}({\mathbb {R}}^N), \) then, from (4.2), we can assert that the family \( (u_p)_{1< p < 2} \) is bounded in \( \textrm{BV}_a^b( {\mathbb {R}}^N). \)
Hence, by the virtue of the local compact embeddings of \( \textrm{BV}_a^b( {\mathbb {R}}^N), \) we deduce the existence of a function \(u\in \textrm{BV}_{a}^{b}({\mathbb {R}}^N)\) such that
for all \( 1\le t< 1^* \) and B ball of \( {\mathbb {R}}^N, \) there exists \( g_t\in L^t( B) \)such that
Observe that, from Lemma 6, we immediately see that \(u\in L^{\infty }({\mathbb {R}}^N).\)
Now let us prove that \(\frac{h}{u^{1-q}}\in L^1_{\textrm{loc}}({\mathbb {R}}^N).\) For that aim, fix \(1<p <2 \) and let \(\varphi \in C^{\infty }_c({\mathbb {R}}^N).\) Applying Young’s inequality, by (3.1) and (4.1), it yields
Fatou’s lemma gives
This implies that \(\frac{h}{u^{1-q}}\in L^1_{\textrm{loc}}({\mathbb {R}}^N). \)
Remark 3
Let us point out that since
then
which means that
Lemma 7
There exist \(z\in L^{\infty }({\mathbb {R}}^N,{\mathbb {R}}^N)\) and \(\gamma \in L^{\infty }({\mathbb {R}}^N)\) such that
and, up to subsequences, the following convergences hold as \( p \rightarrow 1^+, \)
for all balls \(B\subset {\mathbb {R}}^N.\)
Proof
Let \(B=B(0, R)\) be a ball of \({\mathbb {R}}^N.\) Fix \(r\in (1,+\infty )\) and consider \(1< p < \frac{r}{r-1}.\) Using the continuous embedding \(W^{1,p}_{a,b}({\mathbb {R}}^N)\hookrightarrow W^{1,p}(B), \) it follows that
where \( C_R \) is a positive constant depending only on R. Thus, up to a subsequence, there exists \(z_r\in L^{r}(B, {\mathbb {R}}^N)\) such that
Now, through a standard diagonal argument, one can deduce the existence of a unique vector field z defined independently of r, such that
Moreover, letting \(p\rightarrow 1^+,\) one yields to
Letting \(r\rightarrow +\infty ,\) we also have \( z \in L^{\infty }(B, {\mathbb {R}}^N) \) and \( |z|_{L^{\infty }(B, {\mathbb {R}}^N)}\le 1.\) Since B is arbitrary, we can finally conclude that
On the other hand, observe that \( |\nabla u_{p}|^{p-2}\nabla u_{p}\rightharpoonup z\) weakly in \(L^{1}(B, {\mathbb {R}}^N).\) Since a is bounded from below on B, then we get
Hence,
Similarly, we can also show that there exists \( \gamma \in L^{\infty }( {\mathbb {R}}^N) \) such that \( \left| \gamma \right| _{L^{\infty }( {\mathbb {R}}^N)} \le 1 \) and
Moreover, we know that, if \(u(x)\ne 0,\) then we have
Combining (4.14) and (4.15), we deduce that
Now, using the Lebesgue-dominated convergence theorem, we can easily see that
It remains to verify that \(\text{ div }\left( a z\right) \in {\mathcal {M}}\left( B\right) ,\) for all balls \(B\subset {\mathbb {R}}^N. \) To do this, we take \(0\le \varphi \in C^{\infty }_c({\mathbb {R}}^N)\) with \( \text{ supp } (\varphi )\subset B \) as test function in (3.1) to obtain
By (4.3), (4.13) and (4.17), we infer
By the arbitrariness of \(\varphi \in C^{\infty }_c({\mathbb {R}}^N),\ \varphi \ge 0, \) we deduce that the total variation of \(-\text{ div }\left( a z\right) \) is locally finite, that is \( \text{ div }(a z) \in {\mathcal {M}}(B),\ \forall \ B \) balls of \( {\mathbb {R}}^N. \) \(\square \)
Combining (4.13), (4.16) and using Fatou’s lemma in (4.17), it follows that
Therefore,
Now, we claim that
Lemma 8
The vector field z satisfies (4.19).
Proof
Let B be a ball of \({\mathbb {R}}^N.\) Let \(0 \le \varphi \in C^{\infty }_{c}({\mathbb {R}}^N)\) be such that \( \text{ supp }( \varphi ) \subset B.\) If we consider \(u_{p}\varphi \in W^{1,p}_{a,b}({\mathbb {R}}^N)\) as test function in (3.1), we obtain
Applying Young’s inequality, it yields
Now, using again Young’s inequality and taking the lower semicontinuity of the map \(u\longmapsto \int _{{\mathbb {R}}^N}\varphi a(x)|\mathrm{{Du}}|\) with respect to the \(L^1 ({\mathbb {R}}^N)\) convergence into account, we obtain
Moreover, by (4.7) and (4.8), it follows that
and
Finally, we claim that
We have
On the other hand, by \((H_2)\), it yields
Having in mind that \(b^qu_{p}^q\rightharpoonup b^q u^q\) weakly in the Lebesgue space \(L^\frac{1}{q}({\mathbb {R}}^N)\) and that \(h /b^q \in L^\frac{1}{1-q}({\mathbb {R}}^N)\) which is the dual of \(L^\frac{1}{q}({\mathbb {R}}^N)\), we can assert that \(hu_{p}^q\rightarrow hu^q\) strongly in \(L^1({\mathbb {R}}^N)\). Consequently, there exists \(g\in L^1({\mathbb {R}}^N)\) such that, up to a subsequence,
Using (4.25) and (4.26), we can easily apply the Lebesgue-dominated convergence theorem to deduce (4.24). Combining (4.21), (4.22), (4.23), (4.24) and taking the limit in the inequality (4.20) as \(p\rightarrow 1^+\), it follows that
Now, let us show that
Let \(\varphi \in C^{\infty }_{c}({\mathbb {R}}^N)\) be such that \( \varphi \ge 0 \) and \((\rho _n)_n\) a standard sequence of mollifiers. Clearly, \((u*\rho _n)\varphi \in C_c^{\infty }( {\mathbb {R}}^N) \) and \( (u * \rho _n) \varphi \ge 0. \) Then, we can take it as test function (4.18) getting
We know that
and
Having in mind that \(|u*\rho _n (x)|\le |u|_{ L^{\infty }({\mathbb {R}}^N)},\ \forall \ x \in {\mathbb {R}}^N,\ \forall \ n \ge 1, \) and \(b \gamma \varphi \in L^1( {\mathbb {R}}^N),\ \frac{h}{u^{1-q}}\varphi \in L^1({\mathbb {R}}^N), \) we can pass to the limit in both sides of the inequality (4.28) using the Lebesgue’s dominated convergence theorem with respect to the measure \(\text{ div }(a z)\) (more precisely with respect to the regular measure that locally represents \( \text{ div }(a z \)) on the left-hand side and to the Lebesgue measure on the right-hand side. Hence,
Inserting that inequality in (4.27), we obtain
for all \( \varphi \in C_c^{\infty }( {\mathbb {R}}^N),\ \varphi \ge 0. \) Then,
as measures in \({\mathbb {R}}^N, \) which implies that in fact \( a|D u| = \left( \textrm{az},\mathrm{{Du}}\right) ,\) since the opposite inequality follows from Corollary 1. \(\square \)
To conclude the proof of Theorem 1.1, it remains to show that \(\chi _{\{u> 0 \}}\) belongs to \( \textrm{BV}_{\textrm{loc}}({\mathbb {R}}^N)\) and
Lemma 9
u solves the equation (4.30).
Proof
For \(\delta >0,\) we define the function \( S_{\delta } : {\mathbb {R}} \rightarrow {\mathbb {R}} \) by
Let \( \varphi \in C_c^{\infty }( {\mathbb {R}}^N) \) be such that \( \varphi \ge 0. \) Taking \(S_\delta (u_p)\varphi \) as test function in (3.1), we obtain
Moreover, from (4.31), using Young’s inequality, we have
We have
Since \((u_p)_{1< p < 2}\) is bounded in \( W^{1,1}_{a,b} ( {\mathbb {R}}^N) \) (and by consequence in \( \textrm{BV}^b_a({\mathbb {R}}^N)), \) then \( (S_\delta (u_p))_{1< p < 2}\) is a bounded family in \(\textrm{BV}^b_a({\mathbb {R}}^N). \) It follows that
Moreover, we have
Taking into account that \(\frac{h}{u_p^{1-q}}S_{\delta }(u_p)\varphi \rightarrow \frac{h}{u^{1-q}}S_{\delta }(u)\varphi \) a.e. in \( {\mathbb {R}}^ N,\) we can use the Lebesgue’s dominated convergence theorem to obtain
Finally, by (4.7) and (4.8), we get
Then, from (4.32), (4.33), (4.34), (4.35) and (4.36), it follows that
Thanks to the facts that \(z\in {\mathcal {M}}^{\infty }_{a, \textrm{loc}}({\mathbb {R}}^N), \gamma \in L^{\infty }({\mathbb {R}}^N)\) and \(\frac{h}{u^{1-q}}\in L^1_{\textrm{loc}}({\mathbb {R}}^N),\) we can see that all terms except the first one in the inequality (4.37) are uniformly bounded with respect to \(\delta .\) Hence, \( (S_\delta (u))_{\delta > 0}\) is bounded in \(\textrm{BV}_\textrm{loc}({\mathbb {R}}^N)\) and we are allowed to pass to the limit when \(\delta \) tends to 0 (using once again weak lower semicontinuity in the first term and Lebesgue’s theorem for the remaining terms), getting
It yields
and
From Corollary 1, we infer
Also, by Proposition 3, we have \(a z\chi _{\{u>0\}}\in {\mathcal {D}}{\mathcal {M}}_{a, \textrm{loc}}^{\infty }({\mathbb {R}}^N)\), and the following equality holds
Combining (4.41), (4.40) and (4.39), we deduce that
Next, we try to prove the reverse inequality. For that aim, we take \((\chi _{\{u>0\}}*\rho _n)\varphi \) as test function in (4.18) where \(0\le \varphi \in C^{\infty }_c({\mathbb {R}}^N)\) and \((\rho _n)_n\) is a sequence of mollifiers. By the virtue of the Lebesgue’s dominated convergence theorem and Fatou’s lemma, one can pass to the limit as n tends to \( + \infty \) obtaining
It yields
Combining (4.43) and (4.42), deduce that (4.30) holds true. \(\square \)
5 Proof of Theorem (1.2)
Let u be the solution given by Theorem 1.1. We already know that \((a(x)z,\mathrm{{Du}})=a(x)|\mathrm{{Du}}|\) as measures in \( {\mathbb {R}}^N.\) Moreover, in view of Remark 3, \(u(x)>0\) a.e. \( x \in {\mathbb {R}}^N. \) Consequently, \( \gamma (x) = 1, \) a.e. \( x \in {\mathbb {R}}^N, \) and
Plainly, it suffices to prove that \(\frac{h}{u^{1-q}} \in L^1({\mathbb {R}}^N).\) First, observe that
Given \(0\le v \in W^{1,1}_{a,b}({\mathbb {R}}^N). \) We claim that
According to Proposition 4, there exists a sequence \( (\varphi _k)_k \subset C^{\infty }_c({\mathbb {R}}^N)\) such that \( \varphi _k \ge 0 \) and
Consider now a sequence \( (\rho _n)_n \) of mollifiers. Set \( v \wedge \varphi _k:=\inf \left\{ v,\ \varphi _k\right\} . \) Taking \( \rho _n*(v \wedge \varphi _k) \) as test function in (5.1), we get
Clearly, \( v \wedge \varphi _k \in W_{a,b}^{1,1}( {\mathbb {R}}^N). \) Then, having the boundedness of a and b in mind, it follows that
Hence,
and
For the right-hand side of (5.3), note that there exists a compact \( K_k \subset {\mathbb {R}}^N \) independent of n such that \( \text{ supp }\left( \rho _n*(v \wedge \varphi _k)\right) \subset K_k,\ \forall \ n \ge 1. \) Moreover, \(\left| \rho _n*(v \wedge \varphi _k)\right| _{L^{\infty }({\mathbb {R}}^N)} \le \left| v \wedge \varphi _k\right| _{L^{\infty }({\mathbb {R}}^N)},\ \forall \ n \ge 1,\) and \( \rho _n*(v \wedge \varphi _k) (x)\) converges a.e. \( x \in \ {\mathbb {R}}^N\) to \( (v \wedge \varphi _k)(x)\) as \( n \rightarrow + \infty . \) Consequently, up to a subsequence, \( \rho _n*(v \wedge \varphi _k)\) is \( \text{ weakly}^* \) (weakly star) convergent to \(v \wedge \varphi _k\) in \(L^{\infty }({\mathbb {R}}^N) \) as \( n \rightarrow + \infty . \) Since \(\frac{h}{u^{1-q}} \in L^1(K_k), \) we deduce that,
Combining (5.4), (5.5) and (5.6), we obtain
Now, we are going to pass to the limit in (5.7) as \(k \rightarrow +\infty . \) Since \( v \wedge \varphi _k \rightarrow v\ \text{ in }\ W^{1,1}_{a,b}({\mathbb {R}}^N),\) then
and
We claim that
We have
and
On the other hand, using the strong convergence of \( (\varphi _k)_k \) to v in \( W^{1,1}_{a,b}( {\mathbb {R}}^N), \) it yields
where C is some positive constant independent of k. Then, an application of the Fatou’s lemma implies that \(\frac{h}{u^{1-q}}v\in L^1({\mathbb {R}}^N).\) Therefore, our claim (5.10) follows from the Lebesgue’s dominated convergence theorem. Combining (5.8), (5.9) and (5.10) and passing to the limit in (5.7) as \( k \rightarrow + \infty , \) we infer that the claim (5.2) holds true. In particular, choosing \(v =1, \) we obtain
This implies that \(\frac{h}{u^{1-q}}\in L^1({\mathbb {R}}^N).\) By consequence, \( \text{ div }(a z) \in L^1( {\mathbb {R}}^N). \)
Since \( \text{ div }(\textrm{az}) \in L^1( {\mathbb {R}}^N) \) and \( \frac{h}{u^{1-q}} \in L^1( {\mathbb {R}}^N), \) then we can easily see that u satisfies the more large following identity:
In order to complete the proof of Theorem 1.2, it remains to show that u is unique. Assume, for the sake of contradiction, that the problem (1.1) admits another positive solution \( u' \) in the sense of Definition (1.5). Thus, there exists a vector field \( z' \in L^{\infty }( {\mathbb {R}}^N, {\mathbb {R}}^N) \) satisfying all the requirements of Definition (1.5). Taking \( v = (u-u') \) as test function in (5.11) for both solutions u and \( u', \) we obtain
and
By the Gauss–Green’s formula given in Theorem 2.2, identities (5.12) and (5.13) become
and
Having in mind that \( (a z , \mathrm{{Du}}) = a \left| \mathrm{{Du}}\right| \) and \( (\textrm{az}',\mathrm{{Du}}') = a \left| \mathrm{{Du}}'\right| , \) subtracting (5.15) from (5.14), it follows
By Corollary 1, we know that
we deduce from (5.16) that
Since \( \left( u^{1-q} - (u')^{1-q}\right) (u-u') \ge 0, \) we immediately deduce that \( u(x) = u'(x) \) a.e. \( x \in {\mathbb {R}}^N. \) This ends the proof of Theorem 1.2.
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Aouaoui, S., Dhifet, M. On Some Weighted 1-Laplacian Problem on \( {\mathbb {R}}^N \) with Singular Behavior at the Origin. Bull. Malays. Math. Sci. Soc. 47, 20 (2024). https://doi.org/10.1007/s40840-023-01622-y
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DOI: https://doi.org/10.1007/s40840-023-01622-y
Keywords
- Unbounded domain
- Weighted 1-Laplacian
- Bounded variation
- Approximation technique
- A priori estimates
- Anzelotti’s pairing theory
- Variational method