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Global higher integrability for very weak solutions to nonlinear subelliptic equations
Boundary Value Problems volume 2017, Article number: 93 (2017)
Abstract
In this paper we consider the following nonlinear subelliptic Dirichlet problem:
where \(X=\{X_{1},\ldots,X_{m}\}\) is a system of smooth vector fields defined in \(\mathbf{R}^{n}\) with globally Lipschitz coefficients satisfying Hörmander’s condition, and we prove the global higher integrability for the very weak solutions.
1 Introduction and main result
The theory of very weak solutions was introduced in the work of Iwaniec and Sbordone [1]. Iwaniec and Sbordone realized that the usual Sobolev assumption for weak solutions to p-harmonic equation can be relaxed to a slightly weaker Sobolev space and proved that very weak solutions are actually classical weak solutions by using the nonlinear Hodge decomposition to construct suitable test functions. Based on Whitney’s extension theorem and theory of \(A_{p}\) weights, Lewis [2] showed a completely different proof and obtained the same result to certain elliptic systems. After [1] and [2], many authors have devoted their energy to the study of the regularity of such solutions; see for example [3–5] and the references therein. We mention here that Xie and Fang [5] obtained the global higher integrability for very weak solutions to a class of nonlinear elliptic systems with Lipschitz boundary condition by using Hodge decomposition to construct a suitable test function. Recently the authors in [6] proved the global regularity result for a second-order degenerate elliptic systems of p-Laplacian type in the Euclidean setting.
In 2005, Zatorska-Goldstein [7] showed the local higher integrability of very weak solutions to the nonlinear subelliptic equations
where \(\Omega\subset\mathbf{R}^{n}\) is a bounded domain and \(X=\{ X_{1},\ldots,X_{m}\}\) (\(m\leq n\)) is a system of smooth vector fields in \(\mathbf{R}^{n}\) with globally Lipschitz coefficients satisfying the Hörmander’s condition and \(X^{*}=(X_{1}^{*},\ldots,X_{m}^{*})\) is a family of operators formal adjoint to \(X_{j}\) in \(L^{2}\).
In this work we are concerned with the boundary value problem for (1.1) with the boundary condition \(u-u_{0}\in W_{X,0}^{1,r}(\Omega)\), i.e.,
and establish the global higher integrability for very weak solutions. We assume that the functions \(A=(A_{1},\ldots,A_{m}):\mathbf {R}^{n}\times\mathbf{R}\times\mathbf{R}^{m}\rightarrow\mathbf {R}^{m}\) and \(B:\mathbf{R}^{n}\times\mathbf{R}\times\mathbf{R}^{m}\rightarrow \mathbf{R}\) are both Carathéodory functions satisfying
for a.e. \(x\in\mathbf{R}^{n}\), \(u\in\mathbf{R}\) and \(\xi\in\mathbf {R}^{m}\). Here \(p\geq2\), α, β are positive constants.
A function \(u\in W_{X}^{1,r}(\Omega)\) (\(r< p\)) is called a very weak solution to (1.1) if
holds for all \(\varphi\in C_{0}^{\infty}(\Omega)\).
In the above definition, the very weak means the integrable exponent is strictly lower than the natural exponent p and if \(r=p\), this is the classical definition of weak solution to (1.1).
To get our result, some regularity assumption introduced in [8] should be imposed on Ω. Let us first recall the notion of uniform \((X,p)\)-fatness which can be found in [9]: A set \(E\subset\mathbf{R}^{n}\) is called uniformly \((X,p)\) -fat if there exist constants \(C_{0},R_{0}>0\) such that
for all \(x\in\partial E\) and \(0< R< R_{0}\), where \(\operatorname{cap}_{p}\) is the variational p-capacity defined in Section 2.
We consider the following hypotheses on Ω:
- (\(H_{1}\)):
-
there exists a constant \(C_{1}\geq1\) such that, for all \(x\in\Omega\),
$$ \vert B_{\rho(x)} \vert \leq C_{1}\bigl\vert B_{\rho(x)}\cap\bigl( \mathbf{R}^{n}\setminus\Omega\bigr) \bigr\vert , $$where \(\rho(x)=2\operatorname{dist}(x,\mathbf{R}^{n}\setminus\Omega)\);
- (\(H_{2}\)):
-
the complement \(\mathbf{R}^{n}\setminus\Omega\) of Ω is uniformly \((X,p)\)-fat.
Under the hypotheses stated above, we prove the following.
Theorem 1.1
Assume that \(u_{0}\in W_{X}^{1,s}(\Omega)\), \(s>p\). Then there exists a \(\delta>0\) such that if \(u\in W_{X}^{1,p-\delta}(\Omega)\) is a very weak solution to the Dirichlet problem (1.2), we have \(u\in W_{X}^{1,p+\tilde{\delta}}(\Omega)\) for some \(\tilde{\delta}>0\).
The key technical tool in proving Theorem 1.1 is a Sobolev type inequality with a capacity term. With it we can prove a reverse Hölder inequality for the generalized gradient Xu of a very weak solution, which allows us to get the global higher integrability of Xu. This paper is organized as follows. In Section 2 we collect some known results on Carnot-Carathéodory spaces and prove a Sobolev type inequality characterized by capacity. Section 3 is devoted to the proof of Theorem 1.1.
2 Some known results and a Sobolev type inequality
Let \(\{X_{1},\ldots,X_{m}\}\) be a system of \(C^{\infty}\)-smooth vector fields in \(\mathbf{R}^{n} (n\geq3)\) satisfying Hörmander’s condition (see [10]):
The generalized gradient is denoted by \(Xu=(X_{1}u,\ldots,X_{m}u)\) and its length is given by
An absolutely continuous curve \(\gamma:[a,b]\rightarrow\mathbf {R}^{n}\) is said to be admissible with respect to the system \(\{X_{1},\ldots,X_{m}\}\), if there exist functions \(c_{i}(t), a\leq t\leq b\), satisfying
The Carnot-Carathéodory distance \(d(x,y)\) generated by \(\{ X_{1},\ldots,X_{m}\}\) is defined as the infimum of those \(T>0\) for which there exists an admissible path \(\gamma:[0, T]\rightarrow\mathbf{R}^{n}\) with \(\gamma(0)=x\), \(\gamma(T)=y\).
By the accessibility theorem of Chow [11], the distance d is a metric and therefore \((\mathbf{R}^{n},d)\) is a metric space which is called the Carnot-Carathéodory space associated with the system \(\{ X_{1},\ldots,X_{m}\}\). The ball is denoted by
For \(\sigma>0\) and \(B=B(x_{0}, R)\), we will write σB to indicate \(B(x_{0},\sigma R)\) and diamΩ the diameter of Ω with respect to d.
It was proved in [12] that the identity map is a homeomorphism of \((\mathbf{R}^{n},d)\) into \(\mathbf{R}^{n}\) with the usual Euclidean metric, and every set which is bounded with respect to the Euclidean metric is also bounded with respect to d. Moreover, by a result of Garofalo and Nhieu [13], Proposition 2.11, if the given vector fields have globally Lipschitz coefficients in addition, then a subset of \(\mathbf{R}^{n}\) is bounded with respect to d if and only if it is bounded with respect to the Euclidean metric.
Hereafter we assume that the vector fields \(X_{1},\ldots,X_{m}\) satisfy the Hörmander condition and have globally Lipschitz coefficients.
Lemma 2.1
For every bounded open set \(\Omega\subset\mathbf{R}^{n}\) there exists \(C_{d}\geq1\) such that
for any \(x\in\Omega\) and \(0< R\leq5\operatorname{diam}\Omega\).
Here, \(\vert B(x,R) \vert \) denotes the Lebesgue measure of \(B(x,R)\). The best constant \(C_{d}\) in (2.1) is called the doubling constant, the measure such that (2.1) holds is called a doubling measure and the homogeneous dimension relative to Ω is \(Q=\log_{2}C_{d}\).
Given \(1\leq p<\infty\), we define the Sobolev space \(W_{X}^{1,p}(\Omega)\) by
endowed with the norm
Here, \(X_{j}u\) is the distributional derivative of \(u\in L_{\operatorname{loc}}^{1}(\Omega)\) given by the identity
The space \(W_{X}^{1,p}(\Omega)\) is a Banach space which admits \(C^{\infty}(\Omega)\cap W_{X}^{1,p}(\Omega)\) as its dense subset. The completion of \(C_{0}^{\infty}(\Omega)\) under the norm \(\Vert \cdot \Vert _{W_{X}^{1,p}(\Omega)}\) is denoted by \(W_{X,0}^{1,p}(\Omega )\). The following Sobolev-Poincaré inequalities can be found in [14] and [15]:
Lemma 2.2
Let Q be the homogeneous dimension relative to Ω, \(B=B(x_{0},R)\subset\Omega, 0< R<\operatorname {diam}\Omega, 1\leq p<\infty\). There exists a constant \(C>0\) such that, for every \(u\in W^{1,p}_{X}(B)\),
where , and \(1\leq\kappa\leq{Q/(Q-p)}\), if \(1\leq p< Q\); \(1\leq \kappa<\infty\), if \(p\geq Q\). Moreover, for any \(u\in W^{1,p}_{X,0}(B)\),
Next we recall a Gehring lemma on the metric measure space \((Y,d,\mu )\), where d is a metric and μ is a doubling measure.
Lemma 2.3
[7]
Let \(q\in[q_{0},2Q]\), \(q_{0}>1\) is fixed. Assume that functions f, g are nonnegative and \(g\in L_{\operatorname{loc}}^{q}(Y,\mu)\), \(f\in L_{\operatorname {loc}}^{r_{0}}(Y,\mu)\), for some \(r_{0}>q\). If there exist constants \(b>1\) and θ such that for every ball \(B\subset\sigma B\subset Y\) the following inequality holds:
then there exist nonnegative constants \(\theta_{0}=\theta _{0}(q_{0},Q,C_{d},\sigma)\) and \(\varepsilon_{0}=\varepsilon _{0}(b,q_{0},Q,C_{d},\sigma)\) such that if \(0<\theta<\theta_{0}\) then \(g\in L_{\operatorname{loc}}^{p}(Y,\mu)\) for \(p\in [q,q+\varepsilon_{0})\).
For the Hardy-Littlewood maximal functions
and
we will use the following properties proved in [14] and [15].
Lemma 2.4
If \(f\in L^{p}(\Omega)\), \(1< p\leq\infty\), then \(M_{\Omega}f\in L^{p}(\Omega)\) and there exists a constant \(C=C(C_{d}, p)>0\) such that
Lemma 2.5
If \(u\in W^{1,p}_{X,\rm loc}(\Omega)\), \(1< p <\infty\), then there exists \(C>0\) such that, for a.e. \(x, y\in \Omega\),
Moreover, for any \(B=B(x_{0},R)\subset\Omega\) and \(u\in W^{1,p}_{X}(B)\), we have
It is worth noting that from Lemma 2.5 and Lemma 2.2 we can infer that, for a.e. \(x\in B\) and \(u\in W^{1,p}_{X,0}(B)\),
Let \(\omega(x)\geq0\) be a locally integrable function, we say that \(\omega\in A_{p}\), \(1< p <\infty\), if there exists some positive constant A such that
Lemma 2.6
Assume \(\omega\in L_{\operatorname {loc}}^{1}(\mathbf{R}^{n})\) is nonnegative and \(1< p <\infty\). Then \(\omega\in A_{p}\) if and only if there exists a constant \(C>0\) such that
for all \(f\in L^{p}(\omega(x)\,dx)\).
The \((X,p)\) -capacity of a compact set \(K\subset\Omega\) in Ω is defined by
and for an arbitrary set \(E\subset\Omega\), the \((X,p)\)-capacity of E is
We will use the following two-sided estimate of \((X,p)\)-capacity in [16]: For \(x\in\Omega\) and \(0< R<\operatorname{diam}\Omega\), there exist \(C_{1}, C_{2}>0\) such that
Lemma 2.7
[9]
If \(\mathbf {R}^{n}\backslash\Omega\) is uniformly \((X,p)\)-fat, then there exists \(1< q< p\) such that \(\mathbf {R}^{n}\backslash\Omega\) is also uniformly \((X,q)\)-fat.
The uniform \((X,q)\)-fatness also implies uniform \((X,p)\)-fatness for all \(p\geq q\), which is a simple consequence of Hölder’s and Young’s inequality.
At the end of this section we prove a Sobolev type inequality characterized by capacity. A similar inequality in the Euclidean setting can be found in [8].
Lemma 2.8
Let \(\Omega\subset\mathbf {R}^{n}\) be a bounded open set with the homogeneous dimension Q, \(1< q<\infty\) and \(0< R<\operatorname{diam}\Omega\). For any \(x\in \Omega\), denote \(B=B(x,R)\) and \(N(\varphi)=\{y\in\bar{B}:\varphi (y)=0\}\). Then there exists a constant \(C=C(Q,q)>0\) such that, for all \(\varphi\in C^{\infty}(2B)\cap W_{X}^{1,q}(2B)\),
where \(1\leq\kappa\leq{Q/(Q-q)}\) if \(1\leq q< Q\) and \(1\leq\kappa <\infty\) if \(q\geq Q\).
Proof
We always assume \(\varphi_{2B}\neq0\); otherwise, (2.5) follows immediately from Lemma 2.2 and (2.4). Let \(\eta\in C_{0}^{\infty}(2B), 0\leq\eta\leq1\) such that \(\eta =1\) on B̄ and \(\vert X\eta \vert \leq\frac{c}{R}\). Denoting \(v=\eta(\varphi_{2B}-\varphi)/\varphi_{2B}\), then \(v\in C_{0}^{\infty}(2B)\) and \(v=1\) in \(N(\varphi)\). It follows from Lemma 2.2 that
and then
Then Lemma 2.2 and (2.6) lead to
where in the last step we used the estimate
The proof is complete. □
3 Proof of Theorem 1.1
Assume that the function \(u\in W_{X}^{1,p-\delta}(\Omega)(\delta <\frac{1}{2})\) is a very weak solution to the Dirichlet problem (1.2). Choose a ball \(B_{0}\) such that \(\overline{\Omega}\subset\frac {1}{2}B_{0}\) and let B be a ball of radius R with \(3B\subset B_{0}\) for fixed \(0< R<1\). There are two cases: (i) \(3B\subset\Omega\) or (ii) \(3B\backslash\Omega\neq\emptyset\). In the case (i), the following estimate has been proved in [7]:
where θ small enough, \(b>1\), \(\max \{ 1,(p-\delta)_{*} \} < t< p-\delta\).
When \(3B\backslash\Omega\neq\emptyset\), a similar inequality (see (3.31) below) will be achieved.
Step 1. Let η be a smooth cut-off function on 2B, i.e. \(\eta\in C_{0}^{\infty}(2B)\) such that
Define \(\hat{u}=\eta(u-u_{0})\) and
We conclude from Lemma 2.5 and the assumption \((H_{1})\) that û is Lipschitz continuous on \(E_{\mu}\cup(\mathbf {R}^{n}\setminus\Omega)\).
Indeed, if \(x,y\in E_{\mu}\cap\Omega\), then Lemma 2.5 implies \(\vert \hat{u}(x)-\hat{u}(y) \vert \leq c\mu d(x,y)\); if \(x,y\in\mathbf{R}^{n}\setminus\Omega\), then \(\hat{u}(x)=\hat {u}(y)=0\). We set \(B_{\rho_{x}}=B(x,\rho_{x})\) with \(\rho _{x}=2\operatorname{dist}(x,\mathbf{R}^{n}\setminus\Omega)\) for the case \(x\in E_{\mu}\cap\Omega\) and \(y\in\mathbf{R}^{n}\setminus\Omega\). Since û is zero on \(\mathbf{R}^{n}\setminus\Omega\), it follows that
and then, from assumption \((H_{1})\) and Lemma 2.2,
Therefore, we have by (2.2) and (3.2)
It follows that û is a Lipschitz function on \(E_{\mu}\cup (\mathbf{R}^{n}\setminus\Omega)\) with the Lipschitz constant \(cC_{1}\mu\).
As in [7], we can use the Kirszbraun theorem (see e.g. [17]) to extend û to a Lipschitz function \(v_{\mu}\) defined on \(\mathbf{R}^{n}\) with the same Lipschitz constant. Moreover, there exists \(\mu_{0}\) such that, for every \(\mu\geq\mu_{0}\), \(\operatorname{supp}{v_{\mu}}\subset3B\cap\Omega\).
In fact, let \(D=2B\cap\Omega\) and \(x\in\mathbf{R}^{n}\backslash (3B\cap\Omega)\), we have by Lemma 2.1 that
where \(\vert B' \vert >\vert B \vert \), \(C_{d}\) is the doubling constant. Setting
then \(M\vert X\hat{u} \vert (x)\leq\mu, \mu\geq\mu_{0}\), which implies \(v_{\mu}(x)=\hat{u}(x)=0\) for \(x\in\mathbf {R}^{n}\backslash(3B\cap\Omega)\). So we can take the function \(v_{\mu}\) as a test function in (1.6).
Let \(\mu\geq\mu_{0}\) and take \(v_{\mu}\) as a test function in (1.6) to have
Noting that \({v_{\mu}}=\hat{u}\) on \((3B\cap\Omega)\cap{E_{\mu}}\) and that \(\operatorname{supp}\hat{u}\subset D\), we have by the structure conditions on \(A(x,u,\xi)\) and \(B(x,u,\xi)\)
where in the last inequality we use the fact that \(\vert Xv_{\mu} \vert \leq c\mu\), \(\vert v_{\mu} \vert \leq cR\mu\) (see [7]).
Multiplying both sides of (3.3) by \(\mu^{-(1+\delta)}\) and integrating over \((\mu_{0},\infty)\), we get
Interchanging the order of integration and applying (3.2), we have
Using Lemma 2.4 and Lemma 2.8, we have
where \(N(u-u_{0})= \{ x\in\bar{B}:u(x)=u_{0}(x) \} \). Since \(u-u_{0}\) vanishes outside Ω, we have \(\mathbf{R}^{n}\setminus\Omega \subset\{u-u_{0}=0\}\). On the other hand, by Lemma 2.7 and assumption \((H_{2})\), there exists \(\delta_{0}\) such that if \(0<\delta<\delta _{0}\), \(\mathbf{R}^{n}\setminus\Omega\) is uniformly \((X,p-\delta )\)-fat, and hence
From (3.6) and the doubling condition, we derive
and then (3.5) becomes
As regards the estimation of L, by changing the order of integration, we have
Since \(D\setminus E_{\mu_{0}}=D\setminus(D\cap E_{\mu_{0}})\), (1.3) and (1.4) imply
Step 2. Next, we will estimate \(I_{i}\) (\(i=1,2,3\)) one by one.
Now for estimation of \(I_{1}\). To this end, define the sets
and \(B_{\Omega}=B\cap\Omega\). Thus
Since \(( M\vert X\hat{u} \vert ) ^{-\delta}\le \vert X\hat{u} \vert ^{-\delta}\) a.e., it follows from (1.5) and (1.3) that
Since the function \((M\vert X\hat{u} \vert )^{-\delta}\) is an \(A_{p}\)-weight, we obtain from Lemma 2.6 that
By the doubling condition and Lemma 2.8 we see that, for \(x\in\frac{B}{2}\cap\Omega\),
where \(\max \{ 1,(p-\delta)_{*} \} < s'< p-\delta\) is such that \(\mathbf{R}^{n}\setminus\Omega\) is uniformly \((X,s')\)-fat and the last inequality comes from an argument similar to (3.6).
To continue, we define
So from (3.11) we see that \(M\vert X\hat{u} \vert \leq cM_{B_{\Omega}} \vert X(u-u_{0}) \vert \) on G, and then
Using the fact \(X\hat{u}=X(u-u_{0})\) on B and Young’s inequality, we have
Next from the definition of \(D_{2}\) and Lemma 2.4, we see
For \(I_{14}\), we have by using \(\vert X\eta(u-u_{0}) \vert \leq \vert X\hat{u} \vert +\vert Xu-Xu_{0} \vert \)
Using Young’s inequality and (3.7), we get
By the definition of \(D_{2}\) and noting that \(\vert X(u-u_{0}) \vert \le M_{D}\vert X(u-u_{0}) \vert \) a.e. D,
Finally, by Young’s inequality,
In order to estimate the second component of the right-hand side, we let \(s''=(p-\delta)(1-\vartheta)\), where \(0<\vartheta<\frac{p-\delta }{p-\delta+Q}\) if \(p-\delta\leq Q\) and \(0<\vartheta<\min \{ \frac {p-\delta-Q}{p-\delta},\frac{1}{2} \} \) if \(p-\delta> Q\). Denote
then \(\kappa s''\geq p-\delta\). Using Lemma 2.7 and Lemma 2.8, we derive
where the proof of the last inequality is similar to (3.6). Therefore,
Inserting (3.18) into (3.17), we have
A combination of (3.15), (3.16) and (3.19) implies
The definition of \(D_{1}\) and Lemma 2.4 give
The previous estimates show that
where \(t=\max\{s',s''\}< p-\delta\).
Now we address the estimation of \(I_{2}\). Using (3.7), we have
To estimate the last integral in (3.23), let \(0<\tau<\frac {1}{2}\) and \(x\in D\cap E_{\mu_{0}}\). If \(\vert Xu \vert \geq \tau^{-1}\mu_{0}\), then \(M\vert X\hat{u} \vert \leq\mu_{0}\leq \tau \vert Xu \vert \) and
if \(\vert Xu \vert <\tau^{-1}\mu_{0}\), then
By (3.24) and (3.25), we deduce that, for any \(x\in D\cap E_{\mu_{0}}\),
For the second term in (3.26), we first observe from the proof of (3.11) that
Noticing \(\mu_{0}=\frac{c}{\vert 2B \vert }\int_{D}\vert X\hat {u} \vert \,dx\), we have from Hölder’s inequality
By (3.26) and (3.27), it follows that
Taking (3.28) into (3.23), we have
For the estimation of \(I_{3}\): From (2.3), Lemma 2.8 and a similar process to the proof of (3.18), we have
where \(t=\max\{s',s''\}< p-\delta\).
Step 3. Taking into account (3.4), (3.8), substituting (3.22), (3.29) and (3.30) into (3.9), and letting \(\varepsilon=\tau^{1-\delta}\), it follows that
To sum up the cases \(3B\subset\Omega\) and \(3B\backslash\Omega\neq \emptyset\), we let
and
Thus we have from (3.1) and (3.31)
where \(q=\frac{p-\delta}{t}\), \(\theta=c ( \delta+\tau^{1-\delta} ) \) and \(b = c{\tau^{1 - p}}\). Choosing τ, δ small enough, we see by Lemma 2.3 that there exists \(t_{1}=p-\delta+\varepsilon_{0}\), for some \(\varepsilon_{0}>0\), such that \(\vert Xu \vert \in L^{t_{1}}(\Omega)\).
Furthermore, we will show that there exists \({t_{2}}>r=p-\delta\) such that \(u \in L^{t_{2}}(\Omega)\). Since \(u-u_{0}\in W_{X,0}^{1,r}(\Omega)\), we obtain from Lemma 2.2 that, for \(r< Q\), \(r^{*}=Qr/(Q-r)\),
Taking \(t_{2}=\min\{s,r^{*}\}>r\), we have
and then \(u\in L^{t_{2}}(\Omega)\) by \(u_{0}\in L^{s}(\Omega)\). If \(r\geq Q\) then we can apply the above reasoning for any \(r^{*}<\infty\) to obtain \(u\in L^{t_{2}}(\Omega)\).
We set \(\tilde{p}=\min\{t_{1},t_{2}\}>p-\delta\) and \(u\in W_{X}^{1,\tilde{p}}(\Omega)\). Repeating the preceding reasoning, we know that there exists \(\tilde{\delta}>0\) such that \(u\in W_{X}^{1,p+\tilde{\delta}}(\Omega)\) and the proof is complete.
4 Conclusions
In this paper, we obtained the global higher integrability for very weak solutions to the Dirichlet problem for a nonlinear subelliptic equation on Carnot-Carathéodory spaces which implies that such solutions are classical weak solutions. It is a generalization of the corresponding result in the classical Euclidean setting.
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Acknowledgements
The authors are grateful to anonymous reviewers for their careful reading of this paper and their insightful comments and suggestions, which improved the paper a lot. The current work is supported by the National Natural Science Foundation of China (No. 11271299).
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Du, G., Han, J. Global higher integrability for very weak solutions to nonlinear subelliptic equations. Bound Value Probl 2017, 93 (2017). https://doi.org/10.1186/s13661-017-0825-6
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DOI: https://doi.org/10.1186/s13661-017-0825-6
MSC
- 35H20
- 35J60
Keywords
- nonlinear subelliptic equations
- very weak solutions
- global higher integrability