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Multiplicity and boundedness of solutions for quasilinear elliptic equations on Heisenberg group
Boundary Value Problems volume 2015, Article number: 131 (2015)
Abstract
In this paper, we study a class of quasilinear elliptic equations on Heisenberg group by using the nonsmooth critical point theory. Under some weaker assumptions, the multiplicity and boundedness of solutions for these equations are obtained.
1 Introduction
Let \(\mathbb{H}^{N}\) be the space \(\mathbb{R}^{N}\times \mathbb{R}^{N}\times\mathbb{R}\). Then \(\mathbb{H}^{N}\) is a Lie group by the following group operation:
where ‘⋅’ represents the usual inner-product in \(\mathbb{R}^{N}\). The vector fields \(X_{1},\ldots,X_{N}\), \(Y_{1},\ldots ,Y_{N}\), and T given by
form a basis for the tangent space at \(\eta=(x,y,t)\).
Definition 1.1
The Heisenberg Laplacian is defined by
Denote \(\nabla_{\mathbb{H}}u\) as the 2N-vector \((X_{1}u,\ldots ,X_{N}u,Y_{1}u,\ldots,Y_{N}u)\), and then \(\operatorname{div}_{\mathbb{H}}\vec{F}=X_{1}F_{1}+\cdots +X_{N}F_{N}+Y_{1}G_{1}+\cdots+Y_{N}G_{N}\), where \(\vec {F}=(F_{1}, \ldots, F_{N},G_{1},\ldots,G_{N})\).
In this paper, we will study the multiplicity and boundedness of solutions for the equation
where \(N>2\), \(\lambda\in\mathbb{R}\), and \(b(\cdot)\) is a continuous function, satisfying \(b(\eta)\geq0 \) for all \(\eta\in\mathbb{H}^{N}\) and \(\lim_{|\eta|_{\mathbb{H}^{N}}\rightarrow\infty}b(\eta )=+\infty\).
There have been a number of papers concerned with the existence and multiplicity of solutions for nonlinear equations or systems, such as [1–12].
In [1], Aouaoui established the existence of infinitely many solutions for the problem
In [13], Pellacci and Squassina studied the quasilinear elliptic problem
with homogeneous boundary and bounded open set \(\Omega\subset\mathbb{R}^{N}\).
Set
We will use the variational methods to solve the problem of (1.1). Explicitly, we will look for critical points of the functional \(I:E\rightarrow\mathbb{R}\),
where \(F(\eta,\xi)=\int_{0}^{\xi}f(\eta,t)\, dt\). The main difficulty in this problem is that the functional is continuous but not differentiable in whole space E. Nevertheless, the derivatives of I exist along the directions of \(E\cap L^{\infty}(\mathbb{H}^{N})\).
Remark 1.1
\(E\hookrightarrow L^{p}(\mathbb{H}^{N})\), when \(2\leq p\leq2^{*}\); \(E\hookrightarrow\hookrightarrow L^{p}(\mathbb {H}^{N})\), when \(2\leq p<2^{*}\), where \({{2^{*}=\frac{2Q}{Q-2}}}\), \(Q=2N+2\).
Proof
When \(2\leq p\leq2^{*}\), it is obvious that \(E\hookrightarrow L^{p}(\mathbb{H}^{N})\) by Folland-Stein embedding theorem [11].
It is sufficient to prove the conclusion when \(p=2\). Let \(\{u_{n}\}\) be a weakly convergent sequence to zero in E. Since \(\lim_{|\eta|_{\mathbb{H}^{N}}\rightarrow\infty }b(\eta )=+\infty\), for any \(\varepsilon>0\) there exists \(M_{\varepsilon}>0\), such that \({\frac{1}{b(\eta)}}<\varepsilon\) for any \(|\eta |_{\mathbb{H}^{N}}>M_{\varepsilon}\). Thus, we have
As \(\{u_{n}\}\) is bounded in E, \(\{u_{n}\}\) possess a subsequence strongly converging to zero in \(L^{2}(\{|\eta|_{\mathbb{H}^{N}}\leq M_{\varepsilon}\})\) by the Folland-Stein embedding theorem [11]. The proof is completed. □
Firstly, we introduce the eigenvalue problem. Let \(\mathfrak {L}u=-\Delta _{\mathbb{H}}u+b(\eta)u\). We consider the following eigenvalue problem:
By virtue of the spectral theory for compact operators, we get a sequence of eigenvalues
with \(\lambda_{n}\rightarrow+\infty\) as \(n\rightarrow\infty\) and the first eigenvalue \(\lambda_{1}\) has the variational characterization
Definition 1.2
A critical point u of the functional I is defined to be a function \(u\in E\) such that \(\langle I^{\prime}(u), h\rangle=0\), \(\forall h\in E\cap L^{\infty}(\mathbb{H}^{N})\).
Next we state the assumptions and main results of this paper. We make the following hypotheses:
- (J1):
-
\(J(\cdot,\cdot,\cdot):\mathbb{H}^{N}\times\mathbb {R}\times \mathbb{R}^{2N}\rightarrow\mathbb{R}\) satisfies:
-
for each \((s,\xi)\in\mathbb{R}\times\mathbb{R}^{2N}\), \(J(\eta ,s,\xi )\) is measurable with respect to η;
-
for a.e. \(\eta\in\mathbb{H}^{N}\), \(J(\eta,s,\xi)\) is of class \(C^{1}\) with respect to \((s,\xi)\);
-
\(J(\eta,s,\xi)\) is convex with respect to ξ.
-
- (J2):
-
There exist \(0< \alpha< \beta< +\infty\) such that
$$\begin{aligned}& \alpha|\xi|^{2}\leq J(\eta,s,\xi)\leq\beta|\xi|^{2} \quad \text{a.e. } \eta\in\mathbb{H}^{N} \text{ and } \forall(s,\xi)\in \mathbb {R}\times\mathbb{R}^{2N}, \\& \bigl\vert J_{s}(\eta,s,\xi)\bigr\vert \leq\beta| \xi|^{2} \quad \text{a.e. } \eta\in \mathbb {H}^{N} \text{ and } \forall(s,\xi)\in\mathbb{R}\times\mathbb {R}^{2N}. \end{aligned}$$ - (J3):
-
There exist \(R>0\), \(\theta>2\), \(1<\gamma<\frac{\theta }{2}\), and \(\alpha_{1}>0\) such that
$$\begin{aligned}& J_{s}(x,s,\xi)s\geq0, \quad |s|>R, \\& \theta J(x,s,\xi)-\gamma J_{s}(x,s,\xi)s-\gamma J_{\xi}(x,s, \xi )\cdot\xi \geq\alpha_{1}|\xi|^{2}. \end{aligned}$$ - (J4):
-
\(J(\eta,s,\xi)=J(\eta,-s,-\xi)\) a.e. \(\eta\in\mathbb{H}^{N}\) and \(\forall(s,\xi)\in\mathbb{R}\times\mathbb{R}^{2N}\).
- (f1):
-
We assume that \(f(\cdot,\cdot):\mathbb{H}^{N}\times \mathbb {R}\rightarrow\mathbb{R}\) is a Carathéodory function such that
$$\begin{aligned}& \theta F(\eta,s)\leq f(\eta,s)s+a_{0}(\eta)+b_{0}(\eta)|s| \quad \text{a.e. } \eta\in\mathbb{H}^{N}, \\& F(\eta,s)\geq k|s|^{\theta}-\bar{a}(\eta)-\bar {b}(\eta)|s| \quad \text{a.e. } \eta\in\mathbb{H}^{N}, \end{aligned}$$where θ is as in (J3), k is a positive constant, \(\bar{a}(\eta),a_{0}(\eta)\in L^{1}(\mathbb{H}^{N})\), and \(b_{0}(\eta ), \bar{b}(\eta)\in L^{\frac{2Q}{Q+2}}(\mathbb{H}^{N})\).
- (f2):
-
\(|f(\eta,s)|\leq a_{\varepsilon}(\eta)+\varepsilon |s|^{2^{*}-1}\) a.e. \(\eta\in\mathbb{H}^{N}\) and \(\forall s\in\mathbb{R}\).
- (f3):
-
\(f(\eta,-s)=-f(\eta,s)\) a.e. \(\eta\in\mathbb {H}^{N}\) and \(\forall s\in\mathbb{R}\).
Remark 1.2
Under assumptions (J1) and (J2), we have
-
(1)
\(|J_{\xi}(\eta,s,\xi)|\leq4\beta|\xi|\),
-
(2)
\(J_{\xi}(\eta,s,\xi)\cdot\xi\geq\alpha|\xi|^{2}\).
Proof
\(J(\eta,s,\xi)\) is convex with respect to ξ, which means
If \(J_{\xi}(\eta,s,\xi)=0\), then (1) holds obviously. If \(J_{\xi }(\eta ,s,\xi)\neq0\), by taking \(\zeta={\frac{J_{\xi}(\eta,s,\xi)|\xi |}{|J_{\xi}(\eta,s,\xi)|}}\), then (1) holds by using (J2).
On the other hand,
by virtue of assumption (J2), one has (2). □
Remark 1.3
Under assumptions (J1)-(J4) and (f1)-(f3), for the functional I, we have the following assertions:
-
(1)
\(I:E\rightarrow\mathbb{R}\) is continuous.
-
(2)
For any \(u\in E\) and \(h\in E\cap L^{\infty}(\mathbb{H}^{N})\), we have
$$\begin{aligned} \bigl\langle I^{\prime}(u),h\bigr\rangle =& \int_{\mathbb{H}^{N}}J_{\xi }( \eta ,u,\nabla_{\mathbb{H}}u)\nabla_{\mathbb{H}}h+ \int_{\mathbb {H}^{N}}J_{s}( \eta,u,\nabla_{\mathbb{H}}u)h \\ &{}+\int_{\mathbb{H}^{N}}\bigl(b(\eta)-\lambda\bigr)uh-\int _{\mathbb {H}^{N}}f(\eta, u)h. \end{aligned}$$
Moreover, for any \(h\in E\cap L^{\infty}(\mathbb{H}^{N})\), the map \(u\mapsto\langle I^{\prime}(u),h\rangle\) is continuous.
Thirdly, we recall some definitions and properties of nonsmooth critical theory (see [6, 14–19]).
Definition 1.3
Let \(f:X\rightarrow\mathbb{R}\) be a continuous functional and \(u\in X\). We denote by \(|df|(u)\) the supremum of the \(\sigma^{\prime}\) in \([0,+\infty)\) such that there exist \(\delta>0\) and a continuous map \(\mathcal{H}:B(u,\delta)\times[0,\delta ]\rightarrow X\) such that for all \((v,t)\in B(u,\delta)\times[0,\delta]\),
The extended real number \(|df|(u)\) is called the weak slope of f at u.
Remark 1.4
For any \(u\in E\),
Proof
If \(\sup\{\langle I^{\prime}(u),h\rangle;h\in E\cap L^{\infty}(\mathbb{H}^{N}),\Vert h \Vert \leq1 \}=0\), then the conclusion holds.
Otherwise, for a given σ with
there exists \(h\in E\cap L^{\infty}(\mathbb{H}^{N})\) such that \(\Vert h \Vert \leq1\) and \(\langle I^{\prime}(u),h\rangle>\sigma\). Since \(\langle I^{\prime}(u),h\rangle\) is continuous with respect to u, there exists \(\delta_{1}>0\) such that
for any \(v\in B(u,\delta_{1})\). Define a continuous map:
by \(\mathcal{H}(v,t)=v-th\). It is trivial that \(\Vert \mathcal {H}(v,t)-v \Vert \leq t\). On the other hand, by Lagrange mean value theorem, it is easy to see that
It follows that \(|dI|(u)\geq\sigma\), and we complete the proof by the arbitrariness of σ. □
Definition 1.4
Let X be a metric space and \(f:X\rightarrow \mathbb{R} \) be a continuous functional. For a \(c\in\mathbb{R}\), we say that f satisfies the Palais-Smale condition at level c, denoted by \((\mathit{PS})_{c}\), if every sequence \(\{u_{n}\}\) in X with \(|df|(u_{n})\rightarrow0\) and \(f(u_{n})\rightarrow c\) admits a strongly convergent subsequence.
The main result of this paper is the following theorem.
Theorem 1.1
Assume (J1)-(J4) and (f1)-(f3) hold. Then there exists a sequence \(\{u_{n}\} \subset E\cap L^{\infty}(\mathbb{H}^{N})\) of weak solutions of problem (1.1) with \(I(u_{n})\rightarrow+\infty\).
The paper is organized as follows. In Section 2, we introduce and establish some lemmas for Theorem 1.1. In Section 3, we will prove the main theorem. In the last section, we obtain boundedness of critical points (Theorem 4.1).
2 Preliminaries and fundamental lemmas
First we introduce the following fundamental theorem (see Theorem 1.4 of [15]), which is an extension of a well-known result for \(C^{1}\) functionals (see Theorem 9.12 of [20]).
Lemma 2.1
Let X be an infinite-dimensional Banach space and \(f:X\rightarrow\mathbb{R}\) be continuous, even and satisfy \((\mathit{PS})_{c}\) for any \(c\in\mathbb{R}\). Assume, in addition, that:
-
(1)
there exist \(\rho>0\), \(\alpha>f(0)\) and a subspace \(V\subset X\) of finite codimension such that
$$\forall u\in V\mbox{:} \quad \Vert u \Vert =\rho\quad \Rightarrow\quad f(u) \geq \alpha; $$ -
(2)
for every finite-dimensional subspace \(W\subset E\), there exists \(R>0\) such that
$$\forall u\in W\mbox{:}\quad \Vert u \Vert =R\quad \Rightarrow \quad f(u) \leq f(0). $$
Then there exists a sequence \(\{c_{n}\}\) of critical values of f with \(c_{n}\rightarrow+\infty\).
Now, in order to prove that the functional I satisfies the Palais-Smale condition, we will introduce an auxiliary notion.
Definition 2.1
Let c be a real number. We say that functional I satisfies the concrete Palais-Smale condition at level c (denoted by \((\mathit{CPS})_{c}\)), if every sequence \(\{u_{n}\}\subset E\) satisfies
where \(\{\omega_{n}\}\) is a sequence converging to zero in \(E^{*}\), which is possible to extract a strongly convergent subsequence in E.
Lemma 2.2
Let c be a real number. If I satisfies \((\mathit{CPS})_{c}\), then I satisfies \((\mathit{PS})_{c}\).
By Remark 1.4, the proof of this lemma is standard, and we omit it here.
Lemma 2.3
Let \(\{u_{n}\}\) be a bounded sequence in E, satisfying
where \(\{\omega_{n}\}\) is a sequence converging to zero in \(E^{*}\). Then there exists \(u\in E\) such that \(\nabla_{\mathbb {H}}u_{n}\rightarrow\nabla_{\mathbb{H}}u\) a.e. in \(\mathbb{H}^{N}\) and, up to a subsequence, \(\{u_{n}\}\) is weakly convergent to u in E. Moreover, we have
i.e., u is a critical point of I.
Proof
By the argument as [21], we get \(\nabla_{\mathbb {H}}u_{n}(\eta)\rightarrow\nabla_{\mathbb{H}}u(\eta)\) a.e. \(\eta\in \mathbb{H}^{N}\). Since \(\{u_{n}\}\) is bounded in E, we have
Substituting \(h=\varphi e^{-M(u_{n}-R)^{-}}\) into (2.1), where \(\varphi\in E\cap L^{\infty}(\mathbb{H}^{N})\), \(\varphi\geq0\), and \(M=\frac{\beta }{\alpha}\), we have
i.e.,
When \(u_{n}\geq R\), by (J3), we have
When \(u_{n}\leq R\), by (J2) and Remark 1.2, we have
By the convergency of \(\{u_{n}\}\), Remark 1.2, \(\omega_{n}\rightarrow0\) in \(E^{*}\) and (f2), we get
as \(n\rightarrow\infty\). We apply Fatou’s lemma to get
Next, let \(\varphi=\psi e^{M(u-R)^{-}}H(\frac{u}{k})\), where \(\psi \geq0\), \(\psi\in E\cap L^{\infty}(\mathbb{H}^{N})\), \(k\in\mathbb{N}\) and \(H(\eta)\in C_{0}^{\infty}(\mathbb{H}^{N})\) with \(H(\eta)=1\) as \(|\eta |_{\mathbb{H}^{N}}\leq1\), \(H(\eta)=0\) as \(|\eta|_{\mathbb{H}^{N}}\geq2\).
Then we have
Putting \(k\rightarrow\infty\), we get
By taking \(h=\varphi e^{-M(u_{n}+R)^{+}}\) and a similar argument we can get the opposite inequality. So we have
Hence,
The proof has been completed. □
In (2.2) we can only select test functions in \(E\cap L^{\infty}(\mathbb {H}^{N})\). In the following lemma, we will enlarge the class of test functions.
Lemma 2.4
Suppose that \(u\in E\) satisfies \(\langle I^{\prime}(u),h\rangle=\langle\omega,h\rangle\), \(h\in E\cap L^{\infty}(\mathbb {H}^{N})\), where \(\omega\in E^{*}\). For \(v\in E\), there exists \(W(\eta )\in L^{1}(\mathbb{H}^{N})\) with
then \(\langle I^{\prime}(u),v\rangle=\langle\omega,v\rangle\).
Proof
Let
Then \(T_{k}(v)\in E\cap L^{\infty}(\mathbb{H}^{N})\) for every \(v\in E\). By Lemma 2.3, we have
i.e.,
Since
by Remark 1.2, we have \(J_{\xi}(\eta,u,\nabla_{\mathbb{H}}u) \nabla_{\mathbb{H}}v\in L^{1}(\mathbb{H}^{N})\). From Lebesgue’s dominated convergence theorem, as \(k\rightarrow\infty\), we have
Then it is easy to see that
from (2.3). By taking the inferior limit in (2.4) and applying Fatou’s lemma, we obtain
Then \(J_{s}(\eta,u,\nabla_{\mathbb{H}^{N}}u)v\in L^{1}(\mathbb{H}^{N})\). Using Lebesgue’s dominated convergence theorem in (2.4) again, we obtain
The lemma has been proved. □
Lemma 2.5
Let \(c\in\mathbb{R}\) and \(\{u_{n}\}\) be a sequence satisfying (2.1) and
Then \(\{u_{n}\}\) is bounded in E.
Proof
By (J3), we have
Further, we get
by Lemma 2.4. From the assumptions we have
From (J3) and (f1), it follows that
There exist \(M>0\) and \(c(M,\lambda)>0\) such that
By (2.7), we obtain
It follows from (f1) and (2.8) that
Therefore,
This implies that \(\{u_{n}\}\) is bounded in E. □
Lemma 2.6
Let \(\{u_{n}\}\) be a subsequence as in Lemma 2.5. Then \(\{u_{n}\}\), possessing a subsequence, converges strongly in E.
Proof
Consider the cut-off function
where \(M=\frac{\beta}{\alpha}\). It is easy to prove \(\{u_{n} e^{\zeta (u_{n})}\}\) is bounded in E, up to a subsequence, having
By Lemma 2.3 we know that \(ue^{\zeta(u)}\) is a critical point of the functional I. Let \(h=u_{n}e^{\zeta(u_{n})}\) in (2.1). It follows from Lemma 2.4 that
We claim
In fact, when \(u_{n}\geq R\), we have
When \(0\leq u_{n}\leq R\), we have
In the case \(u_{n}\leq0\), the proof is similar.
By Lemma 2.3, \(\nabla_{\mathbb{H}}u_{n}\rightarrow\nabla_{\mathbb{H}}u\) a.e. in \(\mathbb{H}^{N}\). By virtue of Fatou’s lemma, we have
Moreover, it is easy to prove \(f(\eta,\cdot): E\rightarrow E^{*}\) is a compact operator, and
By Lemma 2.4, let \(h=ue^{\zeta(u)}\) in (2.2), and then
Combining (2.9)-(2.13), we have
By Fatou’s lemma, we have
Hence, we get
From Remark 1.2,
It follows that
By Lebesgue’s dominated convergence theorem and the weak convergence of \(\{u_{n}\}\) to u in E, we get
By (2.15), (2.16) and (2.17), we have
That is to say, \(\{u_{n}\}\) converges strongly to u in E. □
Remark 2.1
By Lemmas 2.5 and 2.6, for any c, the functional I satisfies the \((\mathit{CPS})_{c}\) condition.
3 Proof of Theorem 1.1
Proof of Theorem 1.1
The functional I is continuous and even. Moreover, by Lemma 2.2 we know that I satisfies \((\mathit{PS})_{c}\) for any \(c\in\mathbb{R}\).
First we verify the condition (2) in Lemma 2.1. Let W be a finite-dimensional subspace of E. For any \(u\in W\), by (f1), we have
Since \((\int_{\mathbb{H}^{N}}|u|^{\rho})^{\frac{1}{\rho }}\) is a norm on W, there exists \(c_{\rho}>0\) such that
Considering \(\theta>2\), there exists \(R>1\) such that \(I(u)<0\) when \(\Vert u \Vert =R\).
Next we consider the condition (1) in Lemma 2.1. By (J2) and (f2), for any \(u\in E\) we have
There exist \(a_{1}(\eta)\in C_{0}^{\infty}\) and \(a_{2}(\eta)\in L^{\frac {2Q}{Q+2}}(\mathbb{H}^{N})\) such that
Let \(V_{k}=\operatorname{span}\{v_{1},v_{2},\ldots,v_{k}\}^{\bot}\) and \(\{v_{j}\}_{j\geq1}\) be an orthonormal basis of eigenvectors of the operator \(\mathfrak{L}\). Then for any \(u\in V_{k}\) we have
When \(\Vert u \Vert =1\), we can choose k large enough and ε small enough such that
Hence the condition (1) of Lemma 2.1 holds with \(V=V_{k}\). □
4 Boundedness of critical points
In this section, we will prove the critical point \(u\in L^{\infty}(\mathbb{H}^{N})\). We make the following hypotheses:
- (\(\mathrm{J}^{*}_{3}\)):
-
there exist \(R>0\), \(\theta>2\), \(1<\gamma<\frac {\theta}{2}\), and \(\alpha_{1}>0\) such that
$$J_{s}(x,s,\xi)s\geq0; $$ - (\(\mathrm{f}^{*}_{2}\)):
-
\(|f(\eta,s)|\leq c|s|^{p}\) a.e. \(\eta\in\mathbb{H}^{N}\) and \(\forall s\in\mathbb{R}\), where \(p<2^{*}-1\) and c is a positive constant.
Theorem 4.1
Suppose that (J3) and (f2) are replaced with (\(\mathrm{J}^{*}_{3}\)) and (\(\mathrm{f}^{*}_{2}\)). If \(u\in E\) is a critical point of I, then \(u\in L^{\infty}(\mathbb{H}^{N})\).
Proof
For \(k>1\) and \(M>0\), define
Let \(\Phi_{M}(s)=\min(G_{k}(s),M)\) and \(\Psi_{M}(s)=\max(G_{k}(s),-M)\). Denote \(s^{+}=\max(s,0)\) and \(s^{-}=\min(s,0)\). Considering that u is a critical point of I, since \(\Phi_{M}(u^{+})\in E\cap L^{\infty }(\mathbb{H}^{N})\), we can take \(\Phi_{M}(u^{+})\) as a test function in \(\langle I^{\prime}(u),h\rangle=0\). Thus
Now, observing that \(u^{+}\Phi_{M}(u^{+})\geq0\) and by (J3), \(A_{s}(\eta,u)\Phi_{M}(u^{+})\geq0\), we get
From (f2) and the fact that \(|\Phi_{M}(u^{+})|\leq|u^{+}|\), we deduce
Taking M to +∞ and taking into account that, as \(M\rightarrow +\infty\), \(\Phi_{M}(u^{+})\rightarrow G_{k}(u^{+})\) a.e. in \(\mathbb {H}^{N}\) and \(\Phi_{M}(u^{+})\rightharpoonup G_{k}(u^{+})\) in E, it follows that
Denote \(\Omega_{k}^{+}= \{\eta\in\mathbb{H}^{N}, u^{+}>k\}\). By Remark 1.2, we obtain
Since \(u\in E\), it implies that \((\int_{\Omega _{k}^{+}}(u^{+}-k)^{p+1})^{1-\frac{2}{p+1}}\leq c\), or
On the other hand, we have
which implies that
Using (4.1), (4.2), and (4.3), the following inequality holds:
From Theorem 5.2, Chapter II of [22], we deduce that \(u^{+}\in L^{\infty}(\mathbb{H}^{N})\). Replacing \(\Phi_{M}(u^{+})\) by \(\Psi _{M}(u^{-})\) and by the same steps we can easily prove that \(u^{-}\in L^{\infty}(\mathbb{H}^{N})\), which yields the conclusion. □
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This work was supported by the National Natural Science Foundation of China (11171220) and the Hujiang Foundation of China (B14005).
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Jia, G., Zhang, Lj. & Chen, J. Multiplicity and boundedness of solutions for quasilinear elliptic equations on Heisenberg group. Bound Value Probl 2015, 131 (2015). https://doi.org/10.1186/s13661-015-0392-7
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DOI: https://doi.org/10.1186/s13661-015-0392-7