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Subelliptic equations with singular nonlinearities on the Heisenberg group
Boundary Value Problems volume 2018, Article number: 7 (2018)
Abstract
In this paper, we consider the Dirichlet boundary value problem to singular semilinear subelliptic equation on the Heisenberg group
We prove the positivity and continuity up to the boundary for the weak solutions. We also conclude monotonicity of cylindrical solutions to the problem based on a study of the equation \(-\Delta_{\mathbb {H}}u_{0}=\frac{1}{u_{0}^{\gamma}}\). The main technique is a generalization of the moving plane method to the Heisenberg group.
1 Introduction
Crandall, Rabinowitz, and Tartar [1] considered the Dirichlet problem to elliptic equation with singular nonlinearity and deduced the existence and continuity of solutions. The Dirichlet problem on the sufficiently regular bounded domain to the elliptic equation
was treated by Lazer and Mckenna [2], and the existence of solutions with smoothing in the domain and continuity up to the boundary was given under the assumption \(p(x)>0\). Boccardo and Orsina [3] derived the existence, regularity, and nonexistence for the Dirichlet problem to the model
where the results depend on γ and the summability of \(f(x)\) in some Lebesgue spaces.
Canino and Degiovanni [4] applied a variational approach to the singular semilinear elliptic equation
and obtained some properties of solutions u and \(u_{0}\), where \(u_{0}\) is the solution of the Dirichlet problem to the equation
Based on the approach in [4] and the moving plane method, the monotonicity properties of weak solutions of the Dirichlet problem to the singular elliptic equation
were proved by Canino, Grandinetti, and Sciunzi [5].
Along with the development of theory of degenerate elliptic partial differential equations, there were many attempts to extend various results of elliptic equations to subelliptic equations. Birindelli and Prajapat [6] derived a Liouville theorem for the nonnegative cylindrical solutions of the equation
in the Heisenberg group \({{\mathbb{H}}^{n}}=({{\mathbb {R}}^{2n+1}},\circ)\), where \({\Delta}_{\mathbb{H}}\) is the sub-Laplacian on \({\mathbb{H}}^{n}\), \(0< p<\frac{Q+2}{Q-2}\), by using the CR inversion to a regular function u. Xu [7] determined that there exist \({{C}^{\infty}}(\overline{\Omega})\) solutions of the Dirichlet problem
where f and φ satisfy proper smooth conditions, \(\{X_{j}\}\) (\(j=1,\ldots , m\)) is a system of vector fields satisfying Hörmander’s condition, Ω is a bounded domain, and its boundary ∂Ω is smooth and noncharacteristic for \(\{X_{j}\}\). By extending the Perron method in the Euclidean space to the Carnot group and constructing barrier functions, the existence and uniqueness of solutions for the Dirichlet problem
where L is the sub-Laplacian on the Carnot group, were established by Yuan and Yuan [8]. We also refer the readers to [9] and [10] for the related results.
The purpose of this paper is to consider the Dirichlet problem in \({\mathbb{H}}^{n}\) to the subelliptic equation with singular nonlinearities
where Ω is a bounded convex domain (see [11, 12]) in \({\mathbb{H}}^{n}\) with boundary ∂Ω satisfying the Wiener criterion (for any \(a\in(0,1)\), let the \(C_{j}\) be the capacity of the set \({{B}_{\mathbb{H}}}({{\xi}_{0}},a^{j})\cap\Omega^{C}\); then \({{\xi }_{0}}\in\partial\Omega\) is a regular point if and only if \(\sum_{j=0}^{\infty}{{{{C}_{j}}}/{{{a }^{j(Q-2)}}}}\) diverges; see [13, 14] etc.), \(\gamma>0\), and f satisfies the following condition:
-
(F)
f is a nonnegative nondecreasing locally Lipschitz function, \(f(s)>0\) for any \(s>0\), and \(f(0)\geq0\).
We say that \(u\in H^{1}(\Omega)\) is a weak solution to (1.1) if for any \(\varphi\in C_{0}^{\infty}(\Omega)\),
We first consider the Dirichlet problem
By borrowing the method in [3] to our setting we prove that existence, uniqueness, and boundedness of weak positive solutions \(u_{n}\) for the approximation problem
for each fixed n (the bound of \(u_{n}\) in \(H_{0}^{1}(\Omega)\) is local when \(\gamma>1\)), and then we obtain the existence and uniqueness of a weak solution u of (1.2) as the limit of an increasing sequence of positive solutions \(\{u_{n}\}\). Since ∂Ω satisfies the Wiener criterion, we use the sub-super solution method similarly to [4] and get that the positive solution \(u_{0}\) of (1.2) is continuous up to ∂Ω. The monotonicity of \(u_{0}\) is shown via the moving plane method.
Similarly to the method of energy functional in [4], we define the functional \(F_{f}:L^{2}(\Omega)\rightarrow(-\infty,+\infty]\) to problem (1.1) by
where \(G_{0}:\Omega\times\mathbb{R}\rightarrow[0,+\infty]\) is the Borel function
with
It follows that \(G_{0}(\xi,0)=0\) and \(G_{0}(\xi,\cdot)\) is convex and lower semicontinuous for any \(\xi\in\Omega\) and that \(F_{f}\) is convex, lower semicontinuous, and coercive with \(F_{f}(u_{0})=0\). The effective domain of \(F_{f}\) is \(\{u\in L^{2}(\Omega): F_{f}(u)<+\infty\}\), which is independent of f. Moreover, if \(u_{0}\) is the minimum of \(F_{0}\), then the effective domain of \(F_{f}\) is
and we see that \(u\in H_{\mathrm{loc}}^{1}(\Omega)\) is a weak solution to (1.1). Motivated by the idea in [5], we use the decomposition
and demonstrate two maximum principles for w satisfying
Then the monotonicity of w is derived. Based on this, we arrive at the monotonicity of u.
To obtain the monotonicity of u, we apply the moving plane method on \({\mathbb{H}}^{n}\) in [6] (see also [15–18]) by adopting the simpler way of [15] in the proof. Since \({\Delta}_{\mathbb{H}}\) is not invariant with respect to the general reflection, Birindelli and Prajapat [6] introduced the so-called H-reflection and pointed out that \({\Delta }_{\mathbb{H}}\) is invariant with respect to the H-reflection. Applying the H-reflection and the moving plane method, we prove that the cylindrical solution of (1.1) is monotone.
Our main result is the following:
Theorem 1.1
Let Ω be a bounded convex domain symmetric with respect to \(T_{\lambda_{1}}\) about H-reflection in \({\mathbb{H}}^{n}\) such that ∂Ω satisfies the Wiener criterion. If \(u\in H_{0}^{1}(\Omega)\) (\(\gamma\leq1\)) or \(u\in H_{\mathrm{loc}}^{1}(\Omega)\) (\(\gamma> 1\)) is a weak solution to (1.1), then u is positive, and \(u\in C(\overline{\Omega})\). If u is a positive cylindrical solution to (1.1), then for any \(\lambda_{0}<\lambda<\lambda_{1}\) (see Section 2),
with respect to t (the definitions of \(T_{\lambda_{1}}\), \(u_{\lambda }(\xi)\), and \(\Sigma_{\lambda}\) will be given in Section 2).
Since \({{\Delta}_{\mathbb{H}}}\) on \({{\mathbb{H}}^{n}}\) is essentially different from Δ on \({{\mathbb{R}}^{n}}\), the proof of Theorem 1.1 is also essentially different from those for Δ on \({{\mathbb{R}}^{n}}\). Unlike the Euclidean space case, there is no Poincaré’s inequality for the narrow region in \({{\mathbb{H}}^{n}}\), so we utilize the fact that the volume of the narrow region is small enough to prove the maximum principle on the narrow region. Because the invariance of \({{\Delta}_{\mathbb{H}}}\) does not hold for the reflection \({{u}_{\lambda}}(x,y,t)=u(y,2\lambda -x,t)\) or \({{u}_{\lambda}}(x,y,t)=u(2\lambda-y,x,t)\), the monotonicity of solutions to x or y cannot be established. The assumption that the boundary satisfies the Wiener criterion is weaker than the outer sphere or cone condition.
The paper is organized as follows. In Section 2, we collect some well-known facts about \({\mathbb{H}}^{n}\) and the subelliptic operator \({{\Delta}_{\mathbb{H}}}\). In Section 3, we give the existence, regularity, and monotonicity of \(u_{n}\) to (1.3). Section 4 is devoted to the regularity and monotonicity of \(u_{0}\) to (1.2) and to the interior gradient estimate for \(u_{0}\in C^{3}(\Omega)\). Finally, in Section 5, we derive the maximum principles of w and prove Theorem 1.1.
2 Preliminaries
In this section, we list some facts related to the Heisenberg group and sub-Laplacian \({{\Delta}_{\mathbb{H}}}\). For proofs and more details, we refer, for example, to [19–22].
The Heisenberg group \({\mathbb{H}}^{n}\) is the Euclidean space \({{\mathbb{R}}^{2n+1}}\) (\(n\ge1\)) endowed with the group action ∘ defined by
where \(\xi=({{x}_{1}},\ldots,{{x}_{n}},{{y}_{1}},\ldots ,{{y}_{n}},t):=(x,y,t)\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}\times {\mathbb{R}}\), \({{\xi}_{0}}=({{x}_{0}},{{y}_{0}},{{t}_{0}})\). We denote by \(\delta_{\lambda}\) the dilations on \({{\mathbb{R}}^{2n+1}}\), that is,
which satisfies \({{\delta}_{\lambda}}({{\xi}_{0}}\circ\xi )={{\delta}_{\lambda}}({{\xi}_{0}})\circ{{\delta}_{\lambda}}(\xi)\).
The left-invariant vector fields corresponding to \({\mathbb{H}}^{n}\) are of the form
The Heisenberg gradient of a function u is defined as
The sub-Laplacian \({{\Delta}_{\mathbb{H}}}\) on \({\mathbb{H}}^{n}\) is
with the expansion
It is easy to check that
and \(\{ {{X}_{1}},\ldots,{{X}_{n}},{{Y}_{1}},\ldots,{{Y}_{n}} \}\) satisfies Hörmander’s rank condition (see [22]). In particular, this implies that \({{\Delta}_{\mathbb{H}}}\) is hypoelliptic (see [22]) and the solution of equation including \({{\Delta}_{\mathbb{H}}}\) satisfies the maximum principle (see [20, 23]).
Denote by \(Q=2n+2\) the homogeneous dimension of \({\mathbb{H}}^{n}\). The norm \({{ \vert \xi \vert }_{\mathbb{H}}}\) is the distance of ξ to the origin (see [21]):
Afterward, we also denote \({{r}}=({{ \vert x \vert }^{2}}+{{ \vert y \vert }^{2}})^{\frac{1}{2}}\). Using this norm, we can define the distance between two points in \({\mathbb{H}}^{n}\) in the natural way:
where \(\eta^{-1}\) denotes the inverse of η with respect to the group action ∘, that is, \(\eta^{-1}=-\eta\).
The open ball of radius \(R>0\) centered at \(\xi_{0}\) is the set
By the dilation of the group, \(\xi\to{{ \vert \xi \vert }_{\mathbb {H}}}\) is homogeneous of degree one with respect to \(\delta_{\lambda}\), and
where \(\vert \cdot \vert \) denotes the Lebesgue measure. Noting that \(X_{i}\) and \(Y_{i}\) are homogeneous of degree minus one with respect to \(\delta_{\lambda}\), that is,
we see that \({{\Delta}_{\mathbb{H}}}\) is homogeneous of degree minus two and left invariant.
Define the Sobolev space
which is a Banach space with the norm
Denote by \(H_{0}^{1}(\Omega)\) the closure of \(C_{0}^{\infty}(\Omega )\) in \(H^{1}(\Omega)\).
If \(u(x,y,t)=u(r,t)\), then we say that u is cylindrical. It is easy to see that
If the weak solution \(u\in H_{0}^{1}(\Omega)\) to (1.1) is cylindrical, then u is said to be the cylindrical solution to (1.1).
For any \(\xi=(x,y,t)\in{\mathbb{H}}^{n}\) and \(\lambda\in\mathbb{R}\), we set
and
Let \({{\xi}_{\lambda}}=(y,x,2\lambda-t)\) be the H-reflection of ξ about \(T_{\lambda}\), and let
Denote
and
Similarly to [6], we denote by \(u_{\lambda}\) the H-reflection of u about \(T_{\lambda}\):
The following result is from [24].
Lemma 2.1
If \(f\in{{L}^{\infty}}(\Omega)\) and \(u\in H^{1}_{0}(\Omega)\) is a solution to the subelliptic equation
then
where C is a positive constant depending only on n and Ω.
Let us state Sobolev’s and Poincaré’s inequalities in \({{\mathbb {H}}^{n}}\); see [25, 26].
Lemma 2.2
Let U be a bounded domain in \({{\mathbb{H}}^{n}}\), and let \(\Omega \Subset U\). If \(1< p< Q\) and \(u\in W^{1,p}_{0}(\Omega)\), then there exists \(C>0\) depending on n, p, and Ω, such that for any \(1\le q\le\frac{pQ}{Q-p}\),
If \(1\leq p<\infty\) and \(u\in W^{1,p}_{0}(\Omega)\), then
Lemma 2.3
Let Ω be a bounded domain, and let \(K(\xi )>0\). If u satisfies
then \(u\geq0\) on Ω. Furthermore, \(u>0\) on Ω unless \(u\equiv0\).
3 Solutions \(u_{n}\) to (1.3)
Let Ω be a bounded domain of \(\mathbb{H}^{n}\), and let \(\gamma >0\). In this section, we prove the existence, uniqueness, and boundedness of solutions to the Dirichlet problem (1.3). We will work by approximation similarly to [3].
Lemma 3.1
Problem (1.3) has a nonnegative solution \({{u}_{n}}\in H_{0}^{1}(\Omega)\cap{{L}^{\infty}}(\Omega)\) for any given n.
Proof
Let \(v\in L^{2}(\Omega)\). It is well known that the Dirichlet problem to the subelliptic equation
has a unique solution \(v^{*}=S(v)\). Multiplying by \(v^{*}\) the equation and integrating over Ω, we have
By (2.8) in Lemma 2.2 and Hölder’s inequality we obtain
and
Then the ball in \({{L}^{2}}(\Omega)\) of radius \(C{{n}^{\gamma}}\) is invariant for \(S(v)= \vert v \vert \). Using the Sobolev embedding [25], we see that S is continuous and compact on \({{L}^{2}}(\Omega )\), and by Schauder’s fixed point theorem there exists \({{u}_{n}}\in H_{0}^{1}(\Omega)\) such that \({{u}_{n}}=S({{u}_{n}})\), that is, \({{u}_{n}}\) is a solution to the problem
Noting that \(\frac{1}{({ \vert {u}_{n} \vert +\frac{1}{n}})^{\gamma }}\geq0\), Lemma 2.3 gives \({u}_{n}\geq0\), and then \({u}_{n}\) solves (1.3). Since \(\frac{1}{({{u}_{n}+\frac {1}{n}})^{\gamma}}\) in (1.3) belongs to \({{L}^{\infty}}(\Omega )\), Lemma 2.1 implies that \({u}_{n}\in{{L}^{\infty}}(\Omega)\). □
Lemma 3.2
A solution of (1.3) is unique.
Proof
Since \(v_{n}\) satisfies
in the distribution sense, it follows that
Taking \({{({{u}_{n}}-{{v}_{n}})}^{+}}\) as the test function and observing that
it follows by (3.2) that
and therefore \({{({{u}_{n}}-{{v}_{n}})}^{+}}=0\) a.e. in Ω, which implies \({{u}_{n}}\leq{{v}_{n}}\). In the same way, taking \({{({{v}_{n}}-{{u}_{n}})}^{+}}\) as the test function, we can prove that \({{u}_{n}}\geq{{v}_{n}}\). This proves the uniqueness. □
Lemma 3.3
The nonnegative solution sequence \(\{{u}_{n}\}\) from (1.3) is nondecreasing with respect to n, and for every \(\omega\Subset\Omega\), there exists a constant \(c_{\omega}>0\), independent on n, such that, for any \(\xi\in\omega\),
Proof
For \(\gamma>0\), we have that
in the distribution sense, and then
Taking \({{({{u}_{n}}-{{u}_{n+1}})}^{+}}\) as the test function in (3.5) and using
we have
which implies \({({{u}_{n}}-{{u}_{n+1}})^{+}}=0\) a.e. in Ω, that is, \({u}_{n}\leq{u}_{n+1}\), so \(\{u_{n}\}\) is nondecreasing.
Noting that \({{u}_{1}}\in{{L}^{\infty}}(\Omega)\) by Lemma 3.1 and using Lemma 2.1 to the equation \(-{{\Delta }_{\mathbb{H}}}{{u}_{1}}=\frac{1}{{{({{u}_{1}}+1)}^{\gamma}}}\le1\), we get that there exists a constant \(C>0\), only depending on Ω and Q, such that \({{ \Vert {{u}_{1}} \Vert }_{{{L}^{\infty }}(\Omega)}}\le C\) and
Lemma 2.3 implies that \(u_{1}>0\) in Ω, and then (3.4) also holds for \(u_{1}\). By the monotonicity of \(u_{n}\), (3.4) holds for \(u_{n}\). □
Remark 3.1
Using (3.4), it follows that
and Lemma 2.1 yields
Lemma 3.4
Let \({{u}_{n}}\in H_{0}^{1}(\Omega)\) be the solution to (1.3). Then \({{u}_{n}}\) is bounded if \(0<\gamma\leq1\) and locally bounded if \(\gamma>1\).
Proof
If \(\gamma>1\), then by taking \({{u}_{n}}\) as the test function in (1.3) and using Lemma 3.3 we have
This shows that \({{u}_{n}}\) is locally bounded. For \(0<\gamma\leq1\), choosing \({{u}_{n}}\) as the test function in (1.3) and using Hölder’s inequality, we get that
If \(\gamma=1\), then
If \(0<\gamma<1\), the taking m in (3.7) satisfying \({{2}^{*}}=(1-\gamma)m\) and applying (2.7), we arrive at
Since \(\frac{1}{{{2}^{*}}}>\frac{1}{2m}\), this inequality implies \({{u}_{n}}\in{{L}^{{{2}^{*}}}}(\Omega)\). Using this estimate again together with (3.7), we have
so that \({{u}_{n}}\) is bounded in \(H_{0}^{1}(\Omega)\). □
Remark 3.2
Note that \({{u}_{n}}\in C(\overline{\Omega})\) since the right-hand side in (1.3) is nonsingular.
Lemma 3.5
Let \({{u}_{n}}\in{C}(\overline{\Omega} )\cap H_{0}^{1}(\Omega)\) (\(\gamma\leq1\)) and \(u_{n}\in{C}(\overline{\Omega} )\cap H_{\mathrm{loc}}^{1}(\Omega)\) (\(\gamma> 1\)) be solutions to (1.3). If \({{u}_{n}}\leq({u}_{{{n}}})_{\lambda}\) on ∂Ω, then \({{u}_{n}}\leq({u}_{{{n}}})_{\lambda}\) in Ω with respect to t, and then \({{u}_{n}}<({u}_{{{n}}})_{\lambda}\) in Ω unless \({{u}_{n}}\equiv({u}_{{{n}}})_{\lambda}\).
Proof
Since \({{u}_{n}}\) satisfies (1.3) and \({\Delta}_{\mathbb{H}}\) is invariant under the H-reflection, we have
Take positive functions \({{\phi}_{i}}\in C_{0}^{\infty}(\Omega)\), \(\operatorname{supp}{{\phi}_{i}}\subseteq\operatorname {supp}{{(u_{n}-({u}_{n})_{\lambda})}^{+}}\), \(i=1,2,3,\ldots \) , such that \({{\phi}_{i}}\) converges to \({{(u_{n}-({u}_{n})_{\lambda })}^{+}}\) in \({C}(\overline{\Omega} )\cap H_{0}^{1}(\Omega)\). Using \({{\phi}_{i}}\) as the test function in (3.8), we have by combining (1.3) that
Then \({{({{u}_{n}}-{{({{u}_{n}})}_{\lambda}})}^{+}}=0\) a.e. in Ω, which implies by Lemma 2.3 that \({{u}_{n}}\leq ({u}_{{{n}}})_{\lambda}\) in Ω.
Assume that there exists \(\xi_{0}\in\Omega\) such that \({{u}_{n}}({{\xi }_{0}})={{({{u}_{n}})}_{\lambda}}({{\xi}_{0}})\) and let \(R=R({{\xi}_{0}})>0\) be such that \(B({{\xi}_{0}},R)\Subset\Omega\). We have that
in the distribution sense. Since \({{u}_{n}}({{\xi}_{0}})>0\), we can assume that \({{u}_{n}}>0\) in \(B({{\xi}_{0}},R)\) for small R. By Lemma 3.3, \({{u}_{n}}\ge{{c}_{B({{\xi}_{0}},R)}}>0\). Therefore,
where \(c(\zeta)=\frac{-\gamma}{{{\zeta}^{\gamma+1}}}\) with ζ between \({{u}_{n}}+\frac{1}{n}\) and \({{({{u}_{n}})}_{\lambda}}+\frac {1}{n}\). We see that \(c(\zeta)\) is bounded by using \({{u}_{n}}\ge {{c}_{B({{\xi}_{0}},R)}}\). Thus there exists a positive constant \(\Lambda\ge \vert c(\zeta) \vert \) such that
Putting this into (3.9), we have
By Lemma 2.3 this implies that \({{u}_{n}}\equiv ({u}_{{{n}}})_{\lambda}\) in \(B({{\xi}_{0}},R)\). The usual covering argument shows that \({{u}_{n}}\equiv({u}_{{{n}}})_{\lambda}\) in Ω. □
4 Solutions to (1.2)
In this section, we get the regularity of solutions to (1.2) by using results to (1.3). Also, we obtain the monotonicity of cylindrical solutions to (1.2). The gradient estimate and Harnack inequality for classical solutions to (1.2) are provided by calculating the Harnack quantity, which is similar to Li [27] and Ma and Wei [28].
4.1 Regularity and symmetry of solutions to (1.2)
Lemma 4.1
Problem (1.2) allows a unique nonnegative solution \(u\in H_{0}^{1}(\Omega)\) (\(\gamma\leq1\)) or \(u\in H_{\mathrm{loc}}^{1}(\Omega)\) (\(\gamma> 1\)), and (3.4) is true for \(u_{0}\), that is, \(u_{0}(\xi)\geq c_{\omega}\) for \(\xi\in\omega\subset\subset\Omega\).
Proof
For every n, since the solution \({{u}_{n}}\in H_{0}^{1}(\Omega)\) to (1.3) is bounded by Lemma 3.4 and \({{u}_{n}}\) converges pointwise to \({{u}_{0}}\) in Ω by Remark 3.1, it follows that \({{u}_{n}}\) converges weakly to \({{u}_{0}}\) in \(H_{0}^{1}(\omega )\), \(\omega\Subset\Omega\), that is, for any \(\varphi\in C_{0}^{1}(\Omega )\) such that \(\operatorname{supp}\varphi=\omega\),
Noting that \({{u}_{n}}\) satisfies (3.4), we have that, for any \(\varphi\in C_{0}^{1}(\Omega)\),
By Lebesgue’s dominated convergence theorem,
This implies by (4.1) and (4.2) that problem (1.2) has a solution \({{u}_{0}}\).
Using \({{u}_{n}}\ge{{c}_{\omega}}\) on \(\omega\Subset\Omega\), this yields \({{u}_{0}}\ge0\) in Ω. The uniqueness can be proved as in the proof of Lemma 3.2. □
For the equation \({{\Delta}_{\mathbb{H}}}z=0\) in Ω, by [29] we have \(z\in {{C}^{2}}(\Omega)\).
Lemma 4.2
If ∂Ω satisfies the Wiener criterion and \({{z}_{1}}\in H_{0}^{1}(\Omega)\cap{{C}^{2}}(\Omega)\) solves
then \({{z}_{1}}\in{C}(\overline{\Omega} )\) is positive with \({{ {{z}_{1}}|}_{\partial\Omega}}=0\).
Proof
Let \({{z}_{2}}(\xi)=\frac{{x}_{1}^{2}}{2}\). Then \({{\Delta}_{\mathbb {H}}}{{z}_{2}}=1\). Since ∂Ω satisfies the Wiener criterion, it follows by [13] (p. 1158) that there exists \(z\in C(\overline{\Omega} )\cap{{C}^{2}}(\Omega)\) such that
We easily see that \(z-{{z}_{2}}\in H_{0}^{1}(\Omega)\cap {{C}^{2}}(\Omega)\) satisfies
and therefore \({{z}_{1}}=z-{{z}_{2}}\in{C}(\overline{\Omega} )\) is the required function with \(z_{1}>0\) and \({{ {{z}} |}_{\partial \Omega}}=0\). □
Lemma 4.3
If ∂Ω satisfies the Wiener criterion, then there exists a solution \({{u}_{0}}\) to problem (1.2), and \({{u}_{0}}\in {C}(\overline{\Omega} )\).
Proof
Taking \({u}_{0}^{*}={((\gamma+1){{z}_{1}})^{\frac{1}{\gamma+1}}}\), we have by Lemma 4.2 that
Noting that \({{u}_{0}}\) is nonnegative, we know by (4.3) that
that is, \({{u}_{0}^{*}}\) and 0 are the super- and subsolutions of (1.2), respectively. This implies the existence of a solution \({{u}_{0}}\) to (1.2). Using \({{z}_{1}}\in{C}(\overline{\Omega} )\) in Lemma 4.2, we have \({{u}_{0}}\in{C}(\overline{\Omega} )\). □
Lemma 4.4
If ∂Ω satisfies the Wiener criterion, \({{u}_{0}}\in {C}(\overline{\Omega} )\cap H_{0}^{1}(\Omega)\) (\(\gamma\leq1\)) or \(u_{0}\in{C}(\overline{\Omega} )\cap H_{\mathrm{loc}}^{1}(\Omega)\) (\(\gamma> 1\)) is a cylindrical solution to problem (1.2), then for any \({{\lambda}_{0}}<\lambda<{{\lambda}_{1}}\), we have
with respect to t.
Proof
Step 1. Suppose that λ is sufficiently close to \(\lambda_{0}\). Then by Lemma 3.3 we have that \(u_{n}>0\) in \({\Sigma}_{\lambda}\), and then \(u_{n}\leq({u_{n}})_{\lambda}\) on \(\partial{\Sigma }_{\lambda}\). Since \(u_{n}=({u_{n}})_{\lambda}\) on the hyperplane \(T_{\lambda}\), it follows that \(u_{n}\leq ({u_{n}})_{\lambda}\) in \({{\Sigma}_{\lambda}}\), and then \(u_{n}<({u_{n}})_{\lambda}\) on \({{\Sigma}_{\lambda}}\) by Lemma 3.5.
Step 2. Since Step 1 provides a starting point, we can move the plane \(T_{\lambda}\) to the right keeping \(u_{n}\leq ({u_{n}})_{\lambda}\) on \({{\Sigma}_{\lambda}}\) until its limiting position \(\lambda_{1}\). Denoting
we claim that \(\overline{\lambda}=\lambda_{1}\). Indeed, suppose \(\overline{\lambda}<\lambda_{1}\); we will show that the plane \(T_{\lambda}\) can be moved further to the right or, to be more rigorous, there exists \(\varepsilon>0\) such that, for any \(\lambda\in (\overline{\lambda},\overline{\lambda}+\varepsilon)\),
This is a contradiction to the definition of λ̅. Hence \(\overline{\lambda}=\lambda_{1}\).
To prove (4.5), we separate \({{\Sigma}_{\overline{\lambda }+\varepsilon}}\) into two parts, \({{\Sigma}_{\overline{\lambda }-\varepsilon}}\) and \({{\Sigma}_{\overline{\lambda}+\varepsilon }}\setminus {{\Sigma}_{\overline{\lambda}-\varepsilon}}\) for \(\varepsilon>0\).
Clearly, \({{({{u}_{n}})}_{{\bar{\lambda}}}}-{{u}_{n}}\ge {{c}_{\varepsilon}}>0\) in \({{\Sigma}_{\overline{\lambda}-\varepsilon }}\) for any \(\varepsilon>0\). Since \({{({{u}_{n}})}_{\lambda }}-{{u}_{n}}\) depends continuously on λ, there exists \(\varepsilon>0\) sufficiently small such that, for all \(\lambda\in (\overline{\lambda},\overline{\lambda}+\varepsilon)\), we have
On the other hand, since \({{({{u}_{n}})}_{\lambda}}-{{u}_{n}}\ge0\) on \(\partial({{\Sigma}_{\overline{\lambda}+\varepsilon}} \setminus {{\Sigma}_{\overline{\lambda}-\varepsilon}})\) for fixed \(\varepsilon >0\), it follows by Lemma 3.5 that
Combining (4.6) and (4.7), we prove (4.5).
Using the equality \(\overline{\lambda}=\lambda_{1}\) and the convergence of \(u_{n}\) to \(u_{0}\) a.e., we get that \({{u}_{0}}\le {{({{u}_{0}})}_{\lambda}}\) on \({{\Sigma}_{\lambda}}\). So (4.4) is proved by Lemma 2.3. □
4.2 Harnack inequality for a solution to (1.2)
Results in the subsection are not used in the sequel, but they are of independent interest.
Lemma 4.5
Let \({{u}_{0}}\in{C}^{3}(\Omega)\) be a positive solution to (1.2). Then, for any \(R=R({{\xi}_{0}})>0\), \({{\xi}_{0}}\in\Omega\), and \(B({{\xi}_{0}},2R)\subset\Omega\), there exists a positive constant \(C=C(R)\) such that
Proof
Let \(w_{0}=\log{{u}_{0}}\). Then
and
Taking a smooth cut-off function ϕ with \(\phi=1\) in \(B(\xi_{0},R)\) and \(\phi=0\) outside \(B(\xi_{0},2R)\), define
which as usual is called the Harnack quantity.
At the maximum point of P, we have \({\nabla}_{\mathbb{H}}P=0\) and \({\Delta}_{\mathbb{H}}P\leq0\). Then
Using the basic formula
for any smooth function v in \(\mathbb{H}^{n}\), it follows
Let us estimate the right-hand side in (4.14). For the first term, we have
Due to (4.10), the second term of (4.14) satisfies
Substituting (4.15) and (4.16) into (4.14), we obtain
From (4.12), \({{ \vert {{\nabla}_{\mathbb{H}}}w_{0} \vert }^{2}}=\frac{P}{\phi}\), (4.11), and (4.17) we have that
and so
This gives
Using the bounds \(\vert {{\nabla}_{\mathbb{H}}}\phi \vert \le C{{R}^{-1}}\) and \(\vert {{\Delta}_{\mathbb{H}}}\phi \vert \le C{{R}^{-2}}\), we get
Since \(\phi=1\) on \(B({{\xi}_{0}},R)\), by (4.19) we have that
which implies (4.8). □
Theorem 4.6
Let \({{u}_{0}}\in{C}^{3}(\Omega)\) be a positive solution to (1.2). Then, for any \(R=R({{\xi}_{0}})>0\), \({{\xi}_{0}}\in\Omega\), and \(B({{\xi}_{0}},2R)\subset\Omega\), there exists \(C=C(R)>0\) such that, for all \(\xi,\eta\in B({{\xi}_{0}},\frac{R}{2})\),
Proof
Let I be the shortest curve in \(B({{\xi}_{0}},R)\) joining ξ to η. Then the length of I is clearly at most 2R. Integrating the quantity \(\vert {{\nabla}_{\mathbb{H}}}\log{{u}_{0}} \vert \) along I yields
Applying (4.20), we get
5 Properties of w and proof of Theorem 1.1
In this section, we prove several properties of \(w=u-u_{0}\). Similarly to [5], we provide two maximum principles; see Lemmas 5.2 and 5.4. The idea to check monotonicity is inspired by [5, 15].
Lemma 5.1
If ∂Ω satisfies the Wiener criterion, then the weak solution \(u\in H_{0}^{1}(\Omega)\) (\(\gamma\leq1\)) or \(u\in H_{\mathrm{loc}}^{1}(\Omega)\) (\(\gamma> 1\)) to problem (1.1) is positive, and \({u}\in{C}(\overline{\Omega} )\).
Proof
By the hypothesis \(f\geq0\) we sees that u is a supersolution to the equation
Therefore, \(u\ge{{u}_{0}}>0\) in Ω.
Since \(u={{u}_{0}}+w\), it follows that \(w\geq0\) in Ω. Due to (1.4), we have that w satisfies
Since the equation in (5.1) has no any singularity with respect to w, the continuity in Ω of w is clearly obtained; refer to [7, 8, 10] etc. Now we only need to check the continuity of w near the boundary ∂Ω.
Indeed, by the nonnegativity of f we easily get that \({{w}_{*}}=0\) is a subsolution to (5.1). Also, note that \({{w}^{{*}}}={{((\gamma+1){{z}_{1}})}^{\frac{1}{\gamma+1}}}\) satisfies
that is,
near the boundary (\({{z}_{1}}\to0\) at this moment), which yields
near the boundary, and this shows that w is continuous near ∂Ω and then \(w\in C(\overline{\Omega})\). Hence \(u={{u}_{0}}+w\in C(\overline{\Omega})\). □
Lemma 5.2
Suppose that ∂Ω satisfies the Wiener criterion, \({u}\in {C}(\overline{\Omega} )\cap H_{0}^{1}(\Omega)\) (\(\gamma\leq1\)) or \(u\in{C}(\overline{\Omega} )\cap H_{\mathrm{loc}}^{1}(\Omega)\) (\(\gamma> 1\)) is a solution to problem (1.1), f is locally Lipschitz continuous, \(f(s)>0\) for any \(s>0\), and \(f(0)\geq0\). Then \(u>u_{0}\) in Ω, and \(w>0\) in Ω.
Proof
By contradiction assume that there exists \({{\xi}_{0}}\in\Omega\) such that
Take \(R=R({{\xi}_{0}})>0\) such that \(B({{\xi}_{0}},R)\subset\subset \Omega\). Then
Since \({{u}_{0}}({{\xi}_{0}})>0\) by Lemma 4.1, we can assume that \({{u}_{0}}>0\) in \(B({{\xi}_{0}},R)\). Since
and \(\vert c(\xi) \vert \leq\Lambda\) for a positive constant Λ, it follows that
Combining this with (5.3), we get
By Lemma 2.3 we have \(w\equiv0\) in \(B({{\xi}_{0}},R)\). The covering argument shows that \(w\equiv0\) in Ω. Therefore \(f\equiv 0\) by (5.3), which is impossible. □
Lemma 5.3
([5])
For \(\gamma>0\), consider the function
where \((a,b,c,d)\in D:= \{ (a,b,c,d)| 0\le a\le c;0\le d\le b \}\subset\mathbb{R}^{4}\). Then \(g_{\gamma}\leq0\) in D.
Lemma 5.4
(Narrow region maximum principle)
Assume that ∂Ω satisfies the Wiener criterion and T is a hyperplane in \(\mathbb{H}^{n}\). For \(\lambda\in\mathbb{R}\), denote
Let \(\Omega'\) be a bounded narrow region in H contained in \(\{ \xi\in\mathbb{H}^{n} | \lambda-l< t<\lambda \}\) with small \(l>0\). Suppose that \({u}\in{C}(\overline{\Omega} )\cap H_{0}^{1}(\Omega)\) (\(\gamma\leq1\)) or \(u\in{C}(\overline{\Omega} )\cap H_{\mathrm{loc}}^{1}(\Omega)\) (\(\gamma> 1\)) is a cylindrical solution to problem (1.1) and that f satisfies (F). If \(w\leq w_{\lambda }\) on \(\partial\Omega'\), then for a sufficiently small l, we have \(w\leq w_{\lambda}\) in \(\Omega'\) with respect to t. Furthermore, \(w< w_{\lambda}\) in \(\Omega'\), unless \(w\equiv w_{\lambda}\).
Proof
Since u satisfies problem (1.1), we have in the distribution sense that
and
Also,
and
Consider positive functions \({{\psi}_{i}}\in C_{0}^{\infty}({\Omega }')\) with \(\operatorname{supp}{{\psi}_{i}}\subseteq\operatorname {supp}{{(w-{{w}_{\lambda}})}^{+}}\) and \({{\psi}_{i}}\rightarrow {{(w-{{w}_{\lambda}})}^{+}}\in H_{0}^{1}({\Omega}')\) in \(H_{0}^{1}({\Omega}')\). Taking \({{\psi}_{i}}\) as the test function, by (5.1) and (5.7) we get that
Note that \({{u}_{0}}\le{({u}_{{{0}}})}_{\lambda}\) in \(\Omega'\cap H\) by Lemma 4.4 and that \(w\geq w_{\lambda}\) on \(\operatorname {supp}{{\psi}_{i}}\). Applying Lemma 5.3 with \(a={{u}_{0}}\), \(c={({u}_{{{0}}})}_{\lambda}\), \(b=w\), and \(d=w_{\lambda}\), we have
Since f is locally Lipschitz continuous and nondecreasing, it follows that (5.8) and (5.9) imply
Since \({\psi}_{i}\) converges to \({{(w-{{w}_{\lambda}})}^{+}}\), by Hölder’s inequality and Lemma 2.2 we have
Since \(C \vert {\Omega}' \vert ^{\frac{Q}{2}}<1\) for sufficiently small l, it follows that
and then \({{(w-{{w}_{\lambda}})}^{+}}=0\), that is, \(w\leq w_{\lambda}\) in \(\Omega'\).
Assume that there exists \({{\xi}_{0}}\in{\Omega}'\) such that \(w({{\xi}_{0}})={{w}_{\lambda}}({{\xi}_{0}})\). We have from (5.1) and (5.7) that
Noting that \({{u}_{0}}\le{({u}_{{{0}}})}_{\lambda}\) in \(\Omega'\) by Lemma 4.4 and \(w\leq w_{\lambda}\) in \(\Omega'\) from the previous result,by the monotonicity of f we obtain that
Applying Lemma 5.3 yields
Putting (5.14) and (5.15) into (5.13) shows that
Now similarly to the proof of Lemma 5.2, we derive that \({{w}_{\lambda}}-w\equiv0\) in \(\Omega'\). □
Remark 5.1
In proving (5.11) and (5.12), although there is no Poincaré’s inequality on the narrow domain in \({\mathbb{H}}^{n}\), Sobolev’s inequality (2.7) and the fact that \(\vert \Omega' \vert \) is small helps us to derive (5.12).
Lemma 5.5
If ∂Ω satisfies the Wiener criterion and \({u}\in H_{0}^{1}(\Omega)\cap{C}(\overline{\Omega} )\) (\(\gamma\leq1\)) or \(u\in H_{\mathrm{loc}}^{1}(\Omega)\cap{C}(\overline{\Omega} )\) (\(\gamma> 1\)) is a cylindrical solution of problem (1.1), then, for any \({{\lambda}_{0}}<\lambda<{{\lambda}_{1}}\), we have
for \(\xi\in{{\Sigma}_{\lambda}}\) with respect to t.
Proof
Step 1. If λ is sufficiently close to \(\lambda_{0}\), we have \(w>0\) in \({\Sigma}_{\lambda}\) by Lemma 5.2 and \(w\leq{{w}_{\lambda}}\) on \(\partial{\Sigma }_{\lambda}\). Since \(w={{w}_{\lambda}}\) on the hyperplane \(T_{\lambda}\), we get that \(w\leq {{w}_{\lambda}}\) in \({{\Sigma}_{\lambda}}\), and then \(w<{{w}_{\lambda}}\) on \({{\Sigma}_{\lambda}}\) by Lemma 5.4.
Step 2. Since Step 1 provides a starting point, we can move the plane \(T_{\lambda}\) to the right keeping \(w\leq{{w}_{\lambda}}\) on \({{\Sigma}_{\lambda}}\) to its limiting position \(\lambda_{1}\). We let
and claim that \(\overline{\lambda}=\lambda_{1}\). Indeed, suppose \(\overline{\lambda}<\lambda_{1}\); we will show that \(T_{\lambda}\) can further be moved to the right, that is, there exists \(\varepsilon>0\) such that, for any \(\lambda\in(\overline{\lambda},\overline{\lambda }+\varepsilon)\),
This is a contradiction to the definition of λ̅. Hence \(\overline{\lambda}=\lambda_{1}\).
Now we will prove that (5.18) holds on \({{\Sigma}_{\overline {\lambda}+\varepsilon}}\). Let us divide \({{\Sigma}_{\overline{\lambda }+\varepsilon}}\) into two parts: \({{\Sigma}_{\overline{\lambda }-\varepsilon}}\) and \({{\Sigma}_{\overline{\lambda}+\varepsilon }}\setminus {{\Sigma}_{\overline{\lambda}-\varepsilon}}\), where \(\varepsilon>0\) is to be chosen.
Since \({{w}_{\lambda}}-w\ge{{c}_{\varepsilon}}>0\) in \({{\Sigma }_{\overline{\lambda}-\varepsilon}}\) and \({{w}_{\lambda}}-w\) depends continuously on λ, there exists \(\varepsilon>0\) sufficiently small such that, for all \(\lambda\in(\overline{\lambda},\overline {\lambda}+\varepsilon)\),
On the other hand, since \({{w}_{\lambda}}-w\ge0\) on \(\partial({{\Sigma }_{\overline{\lambda}+\varepsilon}} \setminus {{\Sigma}_{\overline {\lambda}-\varepsilon}})\) for \(\varepsilon>0\) as before, we have by Lemma 5.4 that
Combining (5.19) and (5.20), there follows (5.18).
Then \(w\leq{{w}_{\lambda}}\) on \({{\Sigma}_{\lambda}}\). Using Lemma 5.4, we obtain \(w< {{w}_{\lambda}}\) on \({{\Sigma}_{\lambda }}\), and now (5.17) is proved. □
Proof of Theorem 1.1
6 Conclusions
In this paper, we obtained the monotonicity of cylindrical solutions to the Dirichlet boundary value problem to singular semilinear subelliptic equation on the Heisenberg group. It is a generalization of the corresponding result in the classical Euclidean setting.
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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.
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This work is supported by the Natural Science Basic Research plan in Shaanxi Province of China (Grant No. 2016JM1023). The first author partially supported by NSFC (Grant No. 11471188) and the National Science Foundation for Young Scientists of China (Grant No. 11601427).
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Wang, X., Wang, Y. Subelliptic equations with singular nonlinearities on the Heisenberg group. Bound Value Probl 2018, 7 (2018). https://doi.org/10.1186/s13661-018-0925-y
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DOI: https://doi.org/10.1186/s13661-018-0925-y