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The obstacle problem for non-coercive equations with lower order term and \(L^{1}\)-data
Journal of Inequalities and Applications volume 2019, Article number: 205 (2019)
Abstract
The aim of this paper is to study the obstacle problem associated with an elliptic operator having degenerate coercivity, a low order term, and \(L^{1}\)-data. We prove the existence of an entropy solution to the obstacle problem and show its continuous dependence on the \(L^{1}\)-data in \(W^{1,q}(\varOmega )\) with some \(q>1\).
1 Introduction
1.1 Problem setting and main result
Let Ω be a bounded domain in \(\mathbb{R}^{N}\) (\(N\geq 2\)), \(1< p<+\infty \), and \(\theta \geq 0\). Given functions \(g, \psi \in W ^{1,p}(\varOmega )\cap L^{\infty }(\varOmega )\) and data \(f\in L^{1}(\varOmega )\), the aim of this paper is to study the obstacle problem for nonlinear non-coercive elliptic equations with lower order term, governed by the operator
where \(b>0\) is a constant, and \(a:\varOmega \times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}\) is a Carathéodory function, satisfying the following conditions:
for almost every x in Ω and for every ξ, η, ζ in \(\mathbb{R}^{N}\) with \(\xi \neq \eta \), where \(\alpha ,\beta ,\gamma >0\) are constants, and j is a nonnegative function in \(L^{p'}( \varOmega )\).
If f has a fine regularity, e.g., \(f\in W^{-1,p'}(\varOmega )\), the obstacle problem corresponding to \((f,\psi ,g) \) can be formulated in terms of the inequality
whenever \(1\leq r< p\) and the convex subset
is nonempty. However, if \(f\in L^{1}(\varOmega )\), (6) is not well-defined. Following [1, 3, 5, 19] etc., we are led to the more general definition of a solution to the obstacle problem, using the truncation function
Definition 1
An entropy solution of the obstacle problem associated with operator A and functions \((f,\psi ,g)\) with \(f\in L^{1}(\varOmega )\) is a measurable function u such that \(u\geq \psi \) a.e. in Ω, \(\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\in (L^{1}( \varOmega ))^{N}\), \(\vert u \vert ^{r-1}\in L^{1}(\varOmega )\), and, for every \(s>0\), \(T_{s}(u)-T_{s}(g)\in W_{0}^{1,p}(\varOmega )\) and
Observe that no global integrability condition is required on u nor on its gradient in the definition. As pointed out in [3, 8], if \(T_{s} (u) \in W^{1,p}(\varOmega )\) for all \(s > 0\), then there exists a unique measurable vector field \(U:\varOmega \rightarrow \mathbb{R}^{N}\) such that \(\nabla (T_{s} (u)) = \chi _{\{ \vert u \vert < s\}}U\) a.e. in Ω, \(s > 0\), which, in fact, coincides with the standard distributional gradient of ∇u whenever \(u \in W^{1,1}(\varOmega )\).
Before stating the main result, we make some basic assumptions throughout this paper, i.e., without special statements, we always assume that
and \(\psi ,g\in W^{1,p}(\varOmega )\cap L^{\infty }(\varOmega )\) with \((\psi -g)^{+}\in W^{1,p}_{0}(\varOmega )\) such that \(K_{g,\psi }\neq \emptyset \). The following theorem is the main result obtained in this paper.
Theorem 1
Let \(f\in L^{1}(\varOmega )\). Then there exists at least one entropy solution u of the obstacle problem associated with \((f,\psi ,g) \). In addition, u depends continuously on f, i.e., if \(f_{n}\rightarrow f\) in \(L^{1}(\varOmega )\) and \(u_{n}\) is a solution to the obstacle problem associated with \((f_{n},\psi ,g)\), then
1.2 Some comments and remarks
Consider the Dirichlet boundary value problem having a form
with \(p>1\), \(\theta \in (0,1]\), \(b\geq 0\), \(f\in L^{1}(\varOmega )\). The item \(-\operatorname{div}\frac{ \vert \nabla u \vert ^{p-2}\nabla u}{(1+ \vert u \vert )^{\theta (p-1)}}\) may not be coercive when u tends to infinity. Due to this fact, the classical methods used to prove the existence of a solution for elliptic equations, e.g., [14], cannot be applied even if \(b=0\) and the data f is regular. Moreover, \(\frac{ \vert \nabla u \vert ^{p-2}\nabla u}{(1+ \vert u \vert )^{\theta (p-1)}}\), u and f are only in \(L^{1}(\varOmega )\), not in \(W^{-1,p'}(\varOmega )\). All these characteristics prevent us from employing the duality argument [17] or nonlinear monotone operator theory [18] directly.
To overcome these difficulties, “cutting” the non-coercivity term and using the technique of approximation, a pseudomonotone and coercive differential operator on \(W^{1,p}_{0}(\varOmega )\) can be applied to establish a priori estimates on approximating solutions. As a result, existence of solutions, or entropy solutions, can be obtained by taking limitation for \(f\in L^{m}(\varOmega )\), \(m\geq 1\), and \(b> 0\) due to the almost everywhere convergence of gradients of the approximating solutions, see, e.g., [4, 6, 9,10,11, 15] (see also [1, 2, 7, 12, 13, 16] for \(b=0\)). However, there is little literature that considers regularities for entropy solutions of obstacle problems governed by (1) and functions \((f,\psi ,g) \) with \(f\in L^{1}(\varOmega )\), except [19], in which the authors considered the obstacle problem (7) with \(b=0\) and \(L^{1}\)-data.
Motivated by the study on the non-coercive elliptic equations (9) and the problem considered in [19], in this paper, we consider the obstacle problem governed by (1) and functions \((f,\psi ,g) \) with \(f\in L^{1}(\varOmega )\). By the truncation method used in [8] and [19], we prove the existence of an entropy solution and show its continuous dependence on the \(L^{1}\)-data in \(W^{1,q}(\varOmega )\) with some \(q\in (1,p)\).
In the following, we give some remarks on our main result and inequalities that will be needed in the proofs. Some notations are provided at the end of this subsection.
Remark 1
-
(i)
\(0\leq \theta < \min \{\frac{N}{N-1}-\frac{1}{p-1}, \frac{p-r}{p-1} \}\Rightarrow r-1<(1-\theta )(p-1)< \frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)}\). Therefore Theorem 1 guarantees \(\vert u \vert ^{r-1}\in L^{1}(\varOmega )\), and the second integration in (7) makes sense.
-
(ii)
We will show that \(\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\in (L^{1}(\varOmega ))^{N} \) in Proposition 4. Therefore, the first integration in (7) makes sense.
-
(iii)
\(( \frac{N(r-1)}{N+r-1}, \frac{N(p-1)(1-\theta )}{N-1- \theta (p-1)} )\subset (1,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\) if \(\frac{2N-1}{N-1}\leq r< p\). Indeed, \(\theta < \frac{p-r}{p-1}+\frac{p(r-1)}{N(p-1)}\Leftrightarrow \frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)}>\frac{N(r-1)}{N+r-1}\), while \(\frac{2N-1}{N-1}\leq r \) gives \(\frac{N(r-1)}{N+r-1}\geq 1 \). Thus \(u_{n}\rightarrow u\) in \(W^{1,q}(\varOmega )\) for all \(q\in (1,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\).
-
(iv)
\(r-1<\frac{Nq}{N-q}\). Indeed, by \(1\leq r<\frac{2N-1}{N-1}\), there holds \(r-1<\frac{N}{N-1}< \frac{Nq}{N-q}\) for any \(q>1\), particularly, for \(q\in (1, \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\). For \(r\geq \frac{2N-1}{N-1}\), it suffices to note that \(q> \frac{N(r-1)}{N+r-1} \Leftrightarrow r-1< \frac{Nq}{N-q}\).
-
(v)
\(q< p\). Indeed, \(0\leq \theta <\frac{N}{N-1}-\frac{1}{p-1}< \frac{N-1}{p-1} \Rightarrow \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)}<p \).
Remark 2
Checking proofs in this paper (e.g., setting \(r=1\)), one may find that, for \(b=0\), (8) holds with
which is the same as [19, Theorem 1]. Thus, Theorem 1 can be seen as an extension of [19, Theorem 1].
Notations
\(\Vert u \Vert _{p}:= \Vert u \Vert _{L^{p}(\varOmega )}\) is the norm of u in \(L^{p}(\varOmega )\), where \(1\leq p\leq \infty \). \(\Vert u \Vert _{1,p}:= \Vert u \Vert _{W^{1,p}(\varOmega )}\) is the norm of u in \(W^{1,p}(\varOmega )\), where \(1< p<\infty \). \(p':=\frac{p}{p-1}\) with \(1< p<\infty \). \(\{u>s\}:=\{x\in \varOmega ;u(x)>s \}\). \(\{u\leq s\}:=\varOmega \setminus \{u>s\}\). \(\{u< s\}:=\{x\in \varOmega ;u(x)< s\}\). \(\{u\geq s\}:=\varOmega \setminus \{u< s\}\). \(\{u=s\}:=\{x \in \varOmega ;u(x)=s\}\). \(\{t\leq u < s\}:=\{u\geq t\}\cap \{u<s\} \). For a measurable set E in \(\mathbb{R}^{N}\), \(\vert E \vert :=\mathcal{L}^{N}(E)\), where \(\mathcal{L}^{N}\) is the Lebesgue measure of \(\mathbb{R}^{N}\). For a real-valued function u, \(u^{+}=\max \{u,0\}\), \(u^{-}=(-u)^{+}\). Without special statements, positive integers are denoted by n, h, k, \(k_{0}\), K. C is a positive constant, which may be different from each other.
2 Lemmas on entropy solutions
It is worthy to note that, for any smooth function \(f_{n}\), there exists at least one solution to the obstacle problem (6). Indeed, one can proceed exactly as in [1, 11] to obtain \(W^{1,p}\)-solutions due to assumptions (2)–(4) on a and \(r-1< p\). These solutions, in particular, are also entropy solutions. In this section, using the method of [8] and [19], we establish several auxiliary results on convergence of sequences of entropy solutions when \(f_{n}\rightarrow f\) in \(L^{1}(\varOmega )\).
Lemma 2
Let \(v_{0}\in K_{g,\psi }\cap L^{\infty }(\varOmega )\), and let u be an entropy solution of the obstacle problem associated with \((f,\psi ,g)\). Then we have
where C is a positive constant depending only on α, β, p, r, b, \(\Vert j \Vert _{p'}\), \(\Vert \nabla v_{0} \Vert _{p}\), \(\Vert v_{0} \Vert _{\infty }\), and \(\Vert f \Vert _{1}\).
Proof
Take \(v_{0}\) as a test function in (7). For t large enough such that \(t- \Vert v_{0} \Vert _{\infty }>0\), we get
We estimate each integration in the right-hand side of (11). It follows from (3) and Young’s inequality with \(\varepsilon >0\) that
Note that on the set \(\{ \vert u-v_{0} \vert \leq t\}\),
where C is a constant depending only on r, \(\Vert v_{0} \Vert _{\infty }\).
On the set \(\{ \vert u-v_{0} \vert > t\}\), we have \(\vert u \vert \geq t- \Vert v_{0} \Vert _{\infty } >0\), thus u and \(T_{t}(u-v_{0})\) have the same sign. It follows
Replacing t with \(t+ \Vert v_{0} \Vert _{\infty }\) in (17) and noting that \(\{ \vert u \vert < t\}\subset \{ \vert v_{0}-u \vert < t+ \Vert v_{0} \Vert _{\infty }\}\), one may obtain the desired result. □
In the rest of this section, let \(\{u_{n}\}\) be a sequence of entropy solutions of the obstacle problem associated with \((f_{n},\psi ,g)\) and assume that
Lemma 3
There exists a measurable function u such that \(u_{n}\rightarrow u \) in measure, and \(T_{k}(u_{n})\rightharpoonup T_{k}(u)\) weakly in \(W^{1,p}(\varOmega )\) for any \(k>0\). Thus \(T_{k}(u_{n})\rightarrow T _{k}(u) \) strongly in \(L^{p}(\varOmega )\) and a.e. in Ω.
Proof
Let s, t, and ε be positive numbers. One may verify that, for every \(m,n\geq 1\),
and
Due to \(v_{0}=g+(\psi -g)^{+}\in K_{g,\psi }\cap L^{\infty }(\varOmega )\), by Lemma 2, one has
Note that \(T_{t}(u_{n})-T_{t}(g)\in W^{1,p}_{0}(\varOmega )\). By (19), (20), and Poincaré’s inequality, for every \(t> \Vert g \Vert _{\infty }\) and for some positive constant C independent of n and t, there holds
Since \(0\leq \theta <\frac{p-r}{p-1}\), there exists \(t_{\varepsilon }>0\) such that
Now we have as in (19)
Using (20) and the fact that \(T_{t}(u_{n})-T_{t}(g)\in W^{1,p} _{0}(\varOmega )\) again, we see that \(\{T_{t_{\varepsilon }}(u_{n})\}\) is a bounded sequence in \(W^{1,p}(\varOmega )\). Thus, up to a subsequence, \(\{T_{t_{\varepsilon }}(u_{n})\}\) converges strongly in \(L^{p}( \varOmega )\). Taking into account (22), there exists \(n_{0}=n_{0}(t _{\varepsilon },s)\geq 1\) such that
Combining (18), (21), and (23), we obtain
Hence \(\{u_{n}\}\) is a Cauchy sequence in measure, and therefore there exists a measurable function u such that \(u_{n}\rightarrow u\) in measure. The remainder of the lemma is a consequence of the fact that \(\{T_{k}(u_{n})\}\) is a bounded sequence in \(W^{1,p}(\varOmega )\). □
Proposition 4
There exist a subsequence of \(\{u_{n}\}\) and a measurable function u such that, for each q given in (8), we have
Furthermore, if \(0\leq \theta < \min \{ \frac{1}{N-p+1},\frac{N}{N-1}-\frac{1}{p-1},\frac{p-r}{p-1} \}\), then
To prove Proposition 4, we need two preliminary lemmas.
Lemma 5
There exist a subsequence of \(\{u_{n}\}\) and a measurable function u such that, for each q given in (8), we have \(u_{n}\rightharpoonup u \) weakly in \(W^{1,q}(\varOmega )\), and \(u_{n}\rightarrow u \) strongly in \(L^{q}(\varOmega ) \).
Proof
Let \(k>0\) and \(n\geq 1\). Define \(D_{k}=\{ \vert u_{n} \vert \leq k\}\) and \(B_{k}=\{k\leq \vert u_{n} \vert < k+1\}\). Using Lemma 2 with \(v_{0}=g+(\psi -g)^{+}\), we get
where C is a positive constant depending only on α, β, b, p, r, \(\Vert j \Vert _{p'}\), \(\Vert f \Vert _{1}\), \(\Vert \nabla v_{0} \Vert _{p}\), and \(\Vert v_{0} \Vert _{\infty }\).
Using the function \(T_{k}(u_{n})\) for \(k> \{ \Vert g \Vert _{\infty }, \Vert \psi \Vert _{\infty }\}\), as a test function for the problem associated with \((f_{n},\psi ,g)\), we obtain
which and (2) give
Note that on the set \(\{ \vert u_{n} \vert \geq k+1\} \), \(u_{n}\) and \(T_{1}(u_{n}-T _{k}(u_{n}))\) have the same sign. Then
Thus we have
where q is given in (8) and \(q^{*}=\frac{Nq}{N-q} \).
Let \(s=\frac{q\theta (p-1)}{p}\). Note that \(q< p\) and \(\frac{ps}{p-q}< q ^{*}\). For \(\forall k>0\), we estimate \(\int _{B_{k}} \vert \nabla u_{n} \vert ^{q} \,\mathrm{d}x\) as follows:
where \(p_{1}=\frac{q}{p}\frac{r-1}{q^{*}}\), \(p_{2}=\frac{s}{q^{*}}+ \frac{q}{p}\frac{r-1}{q^{*}}\), \(C_{1}\) and \(C_{2}\) are positive constants to be chosen later.
Note that \(\theta <\frac{p-r}{p-1}\), it follows
Thus
Note that \(p_{1}< p_{2}<1\). Then, for \(i=1,2\), we always have
From this, we may find positive \(C_{i}\) (\(i=1,2\)) such that
It follows
which implies
with \(\alpha _{i}=\frac{1}{1-p_{i}-C_{i}}>1\), \(i=1,2\). Let \(\beta _{i}=\frac{1}{p _{i}+C_{i}}>1\), \(i=1,2\). Then we have \(\frac{1}{\alpha _{i}}+\frac{1}{ \beta _{i}}=1\) (\(i=1,2\)).
Since \(\vert B_{k} \vert \leq \frac{1}{k^{q^{*}}}\int _{B_{k}} \vert u_{n} \vert ^{q^{*}} \,\mathrm{d}x \), \(\vert B_{k} \vert ^{1-p_{1}-C_{1}} \leq \vert \varOmega \vert ^{1-p_{1}-C_{1}}\), and \(\vert B_{k} \vert ^{1-p_{2}-C_{2}} \leq \vert \varOmega \vert ^{1-p_{2}-C_{2}}\), we have, for \(k\geq k_{0}\geq 1\),
Summing up from \(k=k_{0}\) to \(k=K\) and using Hölder’s inequality, one has
Note that
To estimate the first integral in the right-hand side of (29), we compute by using Hölder’s inequality and (24), obtaining
where C depends only on α, β, b, p, θ, \(\Vert j \Vert _{p'}\), \(\Vert f \Vert _{1}\), \(\Vert \nabla v_{0} \Vert _{p}\), \(\Vert v_{0} \Vert _{\infty }\), and \(k_{0}\).
Note that \(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}(\frac{p-q}{p}-\frac{s}{q ^{*}})\frac{p}{q}}}\) and \(\sum_{k=k_{0}}^{K} \frac{1}{k^{q^{*}C _{i}\alpha _{i}}}\) converge as \(K\rightarrow \infty \) due to the fact that \(q^{*}(\frac{p-q}{p}-\frac{s}{q^{*}})\frac{p}{q}>1\) and \(q^{*}C_{i} \alpha _{i}>1\) by (27), respectively. Combining (28)–(30), we get for \(k_{0}\) large enough
Since \(p>q\), \(T_{K}(u_{n})\in W^{1,q}(\varOmega )\), \(T_{K}(g)=g\in W^{1,q}( \varOmega )\) for \(K> \Vert g \Vert _{\infty }\). Hence \(T_{K}(u_{n})-g\in W^{1,q} _{0}(\varOmega )\). Using the Sobolev embedding \(W^{1,q}_{0}(\varOmega ) \subset L^{q^{*}}(\varOmega )\) and Poincaré’s inequality, we obtain
Using the fact that
Note that \(p< N\Leftrightarrow \frac{q^{*}}{q}\frac{p-q}{p}<1\) and \((p_{i}+C_{i})\frac{q^{*}}{q}<1 \) by (26). It follows from (34) that, for \(k_{0}\) large enough, \(\int _{\{ \vert u_{n} \vert \leq K\}} \vert \nabla u_{n} \vert ^{q}\,\mathrm{d}x \) is bounded independently of n and K. Using (32) and (33), we deduce that \(\int _{\{ \vert u_{n} \vert \leq K\}} \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x\) is also bounded independently of n and K. Letting \(K\rightarrow \infty \), we deduce that \(\Vert \nabla u_{n} \Vert _{q}\) and \(\Vert u_{n} \Vert _{q^{*}}\) are uniformly bounded independently of n. Particularly, \(u_{n}\) is bounded in \(W^{1,q}(\varOmega )\). Therefore, there exist a subsequence of \(\{u_{n}\}\) and a function \(v\in W^{1,q}(\varOmega )\) such that \(u_{n}\rightharpoonup v\) weakly in \(W^{1,q}(\varOmega )\), \(u_{n}\rightarrow v\) strongly in \(L^{q}(\varOmega )\) and a.e. in Ω. By Lemma 3, \(u_{n}\rightarrow u\) in measure in Ω, we conclude that \(u=v\) and \(u\in W^{1,q}(\varOmega )\). □
Lemma 6
There exist a subsequence of \(\{u_{n}\}\) and a measurable function u such that \(\nabla u_{n}\) converges almost everywhere in Ω to ∇u.
Proof
Define \(A(x,u,\xi )=\frac{a(x,\xi )}{(1+ \vert u \vert )^{\theta (p-1)}}\) (for the sake of simplicity, we omit the dependence of \(A(x,u,\xi )\) on x). Let \(h>0\), \(k> \max \{ \Vert g \Vert _{\infty }, \Vert \psi \Vert _{\infty }\}\), and \(n\geq h+k\). Take \(T_{k}(u)\) as a test function for (7), obtaining
where
Note that \(r-1< q^{*}\), and \(\int _{\varOmega } \vert u_{n} \vert ^{q^{*}}\,\mathrm{d}x\) is uniformly bounded (see the proof of Lemma 5), thus \(\vert u_{n} \vert \) converges strongly in \(L^{1}(\varOmega )\). Therefore we have
Then, using the strong convergence of \(f_{n}\) in \(L^{1}(\varOmega )\), one has
It follows
Thanks to Lemma 3 and Lemma 5, we can proceed exactly as [19, Lemma 6] to conclude that, up to subsequence, \(\nabla u_{n}\rightarrow \nabla u\) a.e. □
Proof of Proposition 4
We shall prove that \(\nabla u_{n}\) converges strongly to ∇u in \(L^{q}(\varOmega )\) for each q being given by (8). To do that,we will apply Vitali’s theorem, using the fact that by Lemma 5, \(\nabla u_{n}\) is bounded in \(L^{q}(\varOmega )\) for each q given by (8). So let \(s\in (q,\frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)})\) and \(E\subset \varOmega \) be a measurable set. Then we have by Hölder’s inequality
uniformly in n, as \(\vert E \vert \rightarrow 0\). From this and from Lemma 6, we deduce that \(\nabla u_{n}\) converges strongly to ∇u in \(L^{q}(\varOmega )\).
Now assume that \(0\leq \theta < \min \{\frac{1}{N-p+1},\frac{N}{N-1}- \frac{1}{p-1},\frac{p-r}{p-1}\}\). Note that since \(\nabla u_{n}\) converges to ∇u a.e. in Ω, to prove the convergence
it suffices, thanks to Vitali’s theorem, to show that, for every measurable subset \(E\subset \varOmega \), \(\int _{E} \vert \frac{a(x, \nabla u_{n})}{(1+ \vert u_{n} \vert )^{\theta (p-1)}} \vert \,\mathrm{d}x\) converges to 0 uniformly in n, as \(\vert E \vert \rightarrow 0\). Note that \(p-1<\frac{N(p-1)(1- \theta )}{N-1-\theta (p-1)})\) by assumptions. For any \(q\in (p-1, \frac{N(p-1)(1-\theta )}{N-1-\theta (p-1)} )\), we deduce by Hölder’s inequality
□
Lemma 7
There exists a subsequence of \(\{u_{n}\}\) such that, for all \(k>0\),
Proof
See the proof of [19, Lemma 7]. □
3 Proof of the main result
Now we have gathered all the lemmas needed to prove the existence of an entropy solution to the obstacle problem associated with \((f,\psi ,g)\). In this part, let \(f_{n}\) be a sequence of smooth functions converging strongly to f in \(L^{1}(\varOmega )\), with \(\Vert f_{n} \Vert _{1}\leq \Vert f \Vert _{1}+1\). We consider the sequence of approximated obstacle problems associated with \((f_{n},\psi ,g)\). The proof can be proceeded in the same way as in [8] and [19]. We provide details for readers’ convenience.
Proof of Theorem 1
Let \(v\in K_{g, \psi }\cap L^{\infty }(\varOmega )\). Taking v as a test function in (7) associated with \((f_{n},\psi ,g)\), we get
Since \(\{ \vert u_{n}-v \vert < t\}\subset \{ \vert u_{n} \vert < s\}\) with \(s=t+ \Vert v \Vert _{\infty }\), the previous inequality can be written as
where \(\chi _{n}=\chi _{\{ \vert u_{n}-v \vert < t\}}\) and \(\nabla _{A}u=\frac{a(x, \nabla u)}{(1+ \vert u \vert )^{\theta (p-1)}}\). It is clear that \(\chi _{n} \rightharpoonup \chi \) weakly* in \(L^{\infty }(\varOmega )\). Moreover, \(\chi _{n}\) converges a.e. to \(\chi _{\{ \vert u-v \vert < t\}}\) in \(\varOmega \setminus \{ \vert u-v \vert =t\}\). It follows that
Note that we have \(\mathcal{L}^{N}(\{ \vert u-v \vert =t\})=0\) for a.e. \(t\in (0,\infty )\). So there exists a measurable set \(\mathcal{O} \subset (0,\infty )\) such that \(\mathcal{L}^{N}(\{ \vert u-v \vert =t\})=0\) for all \(t\in (0,\infty )\setminus \mathcal{O}\). Assume that \(t\in (0,\infty )\setminus \mathcal{O}\). Then \(\chi _{n}\) converges weakly* in \(L^{\infty }(\varOmega )\) and a.e. in Ω to \(\chi =\chi _{\{ \vert u-v \vert < t \}}\). Since \(\nabla T_{s}(u_{n})\) converges a.e. to \(\nabla T_{s}(u)\) in Ω (Proposition 4), we obtain by Fatou’s lemma
Using the strong convergence of \(\nabla _{A} T_{s}(u_{n})\) to \(\nabla _{A} T_{s}(u)\) in \(L^{1}(\varOmega )\) (Lemma 7) and the weak* convergence of \(\chi _{n}\) to χ in \(L^{\infty }(\varOmega )\), we obtain
Moreover, due to the strong convergence of \(f_{n}\) to f and \(\vert u_{n} \vert ^{r-2}u_{n} \) to \(\vert u \vert ^{r-2}u\) (by \(r-1< q^{*}\) and the boundedness of \(\Vert u_{n} \Vert _{q^{*}}\)) in \(L^{1}(\varOmega )\), and the weak* convergence of \(T_{t}(u_{n}-v)\) to \(T_{t}(u-v)\) in \(L^{\infty }( \varOmega )\), by passing to the limit in (35) and taking into account (36)–(37), we obtain
which can be written as
or since \(\chi =\chi _{\{ \vert u-v \vert < t\}}\) and \(\nabla (T_{t}(u-v))= \chi _{\{ \vert u-v \vert < t\}}\nabla (u-v)\)
For \(t\in \mathcal{O}\), we know that there exists a sequence \(\{t_{k}\}\) of numbers in \((0,\infty )\setminus \mathcal{O}\) such that \(t_{k}\rightarrow t\) due to \(\vert \mathcal{O} \vert =0\). Therefore, we have
Since \(\nabla (u-v)=0\) a.e. in \(\{ \vert u-v \vert =t\} \), the left-hand side of (38) can be written as
The sequence \(\chi _{\{ \vert u-v \vert < t_{k}\}}\) converges to \(\chi _{\{ \vert u-v \vert < t \}}\) a.e. in \(\varOmega \setminus \{ \vert u-v \vert =t\} \) and therefore converges weakly* in \(L^{\infty }(\varOmega \setminus \{ \vert u-v \vert =t\})\). We obtain
For the right-hand side of (38), we have
Similarly, we have
It follows from (38)–(41) that we have the inequality
Hence, u is an entropy solution of the obstacle problem associated with \((f,\psi ,g)\). The dependence of the entropy solution on the data \(f\in L^{1}(\varOmega )\) is guaranteed by Proposition 4. □
References
Alvino, A., Boccardo, L., Ferone, V., Orsina, L., Trombetti, G.: Existence results for nonlinear elliptic equations with degenerate coercivity. Ann. Mat. Pura Appl. 182(1), 53–79 (2003)
Alvino, A., Ferone, V., Trombetti, G.: A priori estimates for a class of non uniformly elliptic equations. Atti Semin. Mat. Fis. Univ. Modena 46(suppl.), 381–391 (1998)
Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vazquez, J.L.: An \(L^{1}\) theory of existence and uniqueness of nonlinear elliptic equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 22(2), 240–273 (1995)
Boccardo, L., Brezis, H.: Some remarks on a class of elliptic equations with degenerate coercivity. Boll. Unione Mat. Ital., B 6(3), 521–530 (2003)
Boccardo, L., Cirmi, G.R.: Existence and uniqueness of solution of unilateral problems with \(L^{1}\) data. J. Convex Anal. 6(1), 195–206 (1999)
Boccardo, L., Croce, G., Orsina, L.: Existence of solutions for some noncoercive elliptic problems involving deriratives of nonlinear terms. Differ. Equ. Appl. 4(1), 3–9 (2012)
Boccardo, L., Dall’Aglio, A., Orsina, L.: Existence and regularity results for some elliptic equations with degenerate coercivity. Atti Semin. Mat. Fis. Univ. Modena 46(suppl.), 51–81 (1998)
Challal, S., Lyaghfouri, A., Rodrigues, J.F.: On the A-obstacle problem and the Hausdorff measure of its free boundary. Ann. Mat. Pura Appl. 191(1), 113–165 (2012)
Chen, G.: Nonlinear elliptic equation with lower order term and degenerate coercivity. Math. Notes 93(1–2), 224–237 (2013)
Croce, G.: The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity. Rend. Mat. Appl. 27(7), 299–314 (2007)
Della Pietra, F., di Blasio, G.: Comparison, existence and regularity results for a class of nonuniformly elliptic equations. Differ. Equ. Appl. 2(1), 79–103 (2010)
Giachetti, D., Porzio, M.M.: Existence results for some non uniformly elliptic equations with irregular data. J. Math. Anal. Appl. 257(1), 100–130 (2001)
Giachetti, D., Porzio, M.M.: Elliptic equations with degenerate coercivity, gradient regularity. Acta Math. Sin. 19(2), 349–370 (2003)
Leray, J., Lions, J.L.: Quelques résultats de Vis̆ik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty–Browder. Bull. Soc. Math. Fr. 93, 97–107 (1965)
Li, Z., Gao, W.: Existence results to a nonlinear \(p(x)\)-Laplace equation with degenerate coercivity and zero-order term: renormalized and entropy solutions. Appl. Anal. 95(2), 373–389 (2016)
Porretta, A.: Uniqueness and homogenization for a class of noncoercive operators in divergence form. Atti Semin. Mat. Fis. Univ. Modena 46, 915–936 (1998)
Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15(1), 189–258 (1965)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II, B: Nonlinear Monotone Operators. Springer, New York (1990) (translated from the German by the author and F. Leo)
Zheng, J., Tavares, L.S.: The obstacle problem for nonlinear noncoercive elliptic equations with \(L^{1}\)-data. Bound. Value Probl. 2019, 53 (2019). https://doi.org/10.1186/s13661-019-1168-2
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Zheng, J. The obstacle problem for non-coercive equations with lower order term and \(L^{1}\)-data. J Inequal Appl 2019, 205 (2019). https://doi.org/10.1186/s13660-019-2157-9
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DOI: https://doi.org/10.1186/s13660-019-2157-9