Abstract
In this work, we study the following Schrödinger-Poisson system
where \(\Delta _{H}\) is the Kohn-Laplacian on the first Heisenberg group \(\mathbb{H}^{1}\), and \(\Omega \subset \mathbb{H}^{1}\) is a smooth bounded domain, \(\mu =\pm 1\), \(0<\gamma <1\), and \(\lambda >0\) are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for \(\mu =1\) and each \(\lambda >0\). Multiple solutions of the system are also considered for \(\mu =-1\) and \(\lambda >0\) small enough using the critical point theory for nonsmooth functional.
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1 Introduction and main results
This paper consider the following singular Schrödinger-Poisson system
where \(\Delta _{H}\) is the Kohn-Laplacian on the first Heisenberg group \(\mathbb{H}^{1}\), Ω is a smooth bounded domain of \(\mathbb{H}^{1}\), \(\mu =\pm 1\), \(0<\gamma <1\), and \(\lambda >0\) are some real parameters.
Over the years, many scholars have been widely studied the Heisenberg group due to its crucial role in several branches of mathematics, such as quantum mechanics, complex variables, and harmonic analysis, so one can refer to [9, 12, 19] and the references therein.
In 2022, Liu et al. [15] investigated the following Schrödinger-Poisson system on the Heisenberg group
where \(\Omega \subset \mathbb{H}^{1}\) is a smooth bounded domain, \(a, b>0\), \(1< q<2\) or \(2< q<4\), \(\lambda >0\), and \(\mu \in \mathbb{R}\) are some real parameters. They obtained the existence and multiplicity of solutions. In particular, when \(a=1\), \(b=0\), An and Liu in [1] established the existence and multiplicity of solutions of problem (1.2). Using the Green representation formula, the concentration compactness, and the critical point theory, they proved that the above system has at least two positive solutions for \(\mu < S\times \operatorname{meas}(\Omega )^{-\frac{1}{2}}\) and λ small enough. In addition, they also established that there is a positive ground-state solution for (1.2).
Lei and Liao [13] considered the following system
where \(0<\gamma <1\), \(0\leq \beta <\frac{5+\gamma}{2}\) and \(\lambda >0\) is parameter, they obtained two positive solutions using the variational method and the Nehari manifold method.
In [20], Pucci and Ye studied the logarithmic and critical nonlinearities for the Kirchhoff-Poisson system
where Ω is a smooth bounded domain of \(\mathbb{H}^{1}\), \(q\in (2\theta ,4)\), \(\mu \in \mathbb{R}\), and \(\lambda >0\) are some real parameters. Under suitable assumptions on the Kirchhoff function M, covering the degenerate case, they proved the existence of nontrivial solutions for the above system when \(\lambda >0\) is sufficiently large. For more on the results of the Heisenberg group, we refer the reader to [2, 8, 10, 14, 16–18, 21] and the references therein.
Furthermore, for the system (1.1) in the Heisenberg group, there is no result that this paper answers positively. Before giving the theorem, we define the solutions of (1.1) if u satisfies
we say that u is a solution of problem (1.1).
Theorem 1.1
Assume that \(0<\gamma <1\), \(\mu =1\) and \(\lambda >0\), then system (1.1) has a unique solution.
Theorem 1.2
Assume that \(0<\gamma <1\) and \(\mu =-1\), then there exists \(\Lambda _{0}>0\) such that for every \(\lambda \in (0,\Lambda _{0})\), system (1.1) has at least two positive solutions.
Remark 1.3
Our approach is novel, unlike the Euclidean case, since the presence of singular terms gives us great difficulties; the critical point theory for nonsmooth functional is used to overcome the difficulties, generalizing the results of literature [22].
2 Some preliminary results
In this section, we review the Heisenberg group. For more results, see [7, 11]. Let \(\mathbb{H}^{1}\) be the Heisenberg group of topological dimension 3, that is, the Lie group where underlying manifold is \(\mathbb{R}^{3}\), endowed with the non-Abelian law
where
for \(\forall\xi , \xi '\in \mathbb{H}^{1}\), with \(\xi =(x, y, t)\) and \(\xi '=(x', y', t')\), satisfy the inverse operation. Consider the family of dilations on \(\mathbb{H}^{1}\) defined by
so \(\delta _{s}(\xi \circ \xi ')=\delta _{s}(\xi )\circ \delta _{s}(\xi ')\) (see [19]). The number \(Q=4\) is the homogeneous dimension of \(\mathbb{H}^{1}\), definition
where \(B_{H}(\xi _{0}, r)\) is the Heisenberg ball of radius r centered at \(\xi _{0}\), i.e.,
\(d_{H}(\xi _{0}, \xi )=|\xi ^{-1}\circ \xi _{0}|_{H}\) and \(\omega _{Q}=|B_{H}(0, 1)|\).
The Kohn-Laplacian \(\Delta _{H}\) on \(\mathbb{H}^{1}\) is defined as
where \(\nabla _{H}u=(Xu, Yu)\). Indeed, the vector fields
are a basis of the Lie algebra of \(\mathbb{H}^{1}\) thus constituting a set of left invariant vector fields on \(\mathbb{H}^{1}\). Widely known that \(\Delta _{H}\) is a degenerate elliptic operator, and the Bony maximum principle is satisfied (see [4]). In the present section, the existence and multiplicity of solutions of system (1.1), when \(\mu =-1\), are studied. We prove that system (1.1) has two positive solutions using the critical point theory for nonsmooth functional and the variational method for \(\lambda >0\) small enough.
Let us review critical points of nonsmooth functions related concepts. Let \((X,d)\) be a complete metric space with metric d and \(f:X\rightarrow \mathbb{R}\) be a continuous functional in X. Denote by \(|df|(u)\) the supremum of δ in \([0,\infty )\) such that there exist \(r>0\) and a continuous map \(\sigma :U\times [0,r]\rightarrow X\), satisfying
The number \(|df|(u)\) is called the weak slope of f at u. Thus, \(u\in X\) is a critical point of f if \(|df|(u)=0\), and \(c\in \mathbb{R}\) is a critical value of f if there exists a critical point \(u\in X\) of f with \(f(u)=c\).
Since we are solving for the positive solution of system (1.1), so consider the functional \(I_{\lambda}\) defined on the closed positive cone \(U^{+}\) of \(S^{1}_{0}(\Omega )\), which is defined as
The Hilbert space \(S^{1}_{0}(\Omega )\) is defined as the closure of \(C^{\infty}_{0}(\Omega )\) under the inner product \(\langle u,v\rangle =\int _{\Omega}\nabla _{H} u\nabla _{H} v\, d\xi \). Accordingly, the norm is denoted by \(\|u\|=\|u\|_{S^{1}_{0}(\Omega )}= (\int _{\Omega}|\nabla _{H} u|^{2}\,d\xi )^{\frac{1}{2}}\). The norm in \(L^{p}(\Omega )\) is denoted by \(\|u\|_{p}=(\int _{\Omega}|u|^{p}\,d\xi )^{\frac{1}{p}}\). The embedding \(S^{1}_{0}(\Omega )\hookrightarrow L^{p}(\Omega )\) is continuous for \(p\in [1, Q^{*}]\), where \(Q^{*}=\frac{2Q}{Q-2}=4\) is the critical exponent in \(\mathbb{H}^{1}\). Let us denote by \(B_{\rho}\) and \(S_{\rho}\) a closed ball and a sphere, respectively, of a center of zero and radius ρ. Let S be the best Sobolev constant, namely
First, using the Lax-Milgram theorem, for each \(u\in S^{1}_{0}(\Omega )\), there exists a unique solution \(\phi _{u} \in S^{1}_{0}(\Omega )\), which satisfies the second equation of system (1.1). Then, system (1.1) is transformed into the following problem
For problem (2.3), we define the functional
We know that the functional \(I_{\lambda}\) is well defined and \(I_{\lambda}\in C^{1}(S^{1}_{0}(\Omega ), \mathbb{R})\). Besides, we say that u is a weak solution of problem (2.3) if u satisfies
By the Hölder inequality and (2.2), we obtain
Lemma 2.1
(See [1])
For all \(u\in S^{1}_{0}(\Omega )\), there exists a unique solution \(\phi _{u} \in S^{1}_{0}(\Omega )\) of
and
-
(1)
\(\phi _{u}\geq 0\) and \(\phi _{tu}=t^{2}\phi _{u}\) for each \(t>0\);
-
(2)
If \(u_{n}\rightharpoonup u\) in \(S^{1}_{0}(\Omega )\), then \(\phi _{u_{n}}\rightarrow \phi _{u}\) in \(S^{1}_{0}(\Omega )\) and
$$ \begin{aligned} \lim_{n\rightarrow \infty} \int _{\Omega}\phi _{u_{n}}u_{n}v\, d\xi = \int _{\Omega}\phi _{u}uv\, d\xi , \quad \forall v\in S^{1}_{0}(\Omega ); \end{aligned} $$ -
(3)
For all \(u\in S^{1}_{0}(\Omega )\), there holds that
$$ \begin{aligned} \int _{\Omega} \vert \nabla _{H}\phi _{u} \vert ^{2}\,d\xi = \int _{\Omega}\phi _{u}u^{2}\,d\xi \leq S^{-1} \Vert u \Vert ^{4}_{8/3}\leq S^{-3} \vert \Omega \vert ^{\frac{1}{2}} \Vert u \Vert ^{4}; \end{aligned} $$ -
(4)
For \(u, v\in S^{1}_{0}(\Omega )\), \(\int _{\Omega}(\phi _{u}u-\phi _{v}v)(u-v)\,d\xi \geq \frac{1}{2}\| \phi _{u}-\phi _{v}\|^{2}\).
Lemma 2.2
Assume that \(u\in U^{+}\) and \(|dI_{\lambda}|(u)<+\infty \). Then, for all \(v\in U^{+}\), one obtains
Proof
Let \(u\neq v\in U^{+}\) and \(\|v-u\|>2\delta \). Define \(\sigma :U\times [0,\delta ]\rightarrow U^{+}\) by
where U is a neighborhood of u, then \(\|\sigma (z,t)-z\|=t\). By (2.1), there exists \((z,t)\in U\times [0,\delta ]\) such that \(I_{\lambda}(\sigma (z,t))>I_{\lambda}(z)-ct\). Hence, we assume that there exist sequences \(\{u_{n}\}\subset U^{+}\) and \(\{t_{n} \} \subset [0,+\infty )\), such that \(u_{n}\rightarrow u\), \(t_{n} \rightarrow 0^{+}\), and
That is say
where \(s_{n}=\frac{t_{n}}{ \Vert v-u_{n} \Vert } \rightarrow 0^{+}\) as \(n\rightarrow \infty \). Dividing (2.8) by \(s_{n}\), we deduce that
Further, we can infer that
By mean value theorem, one has
where \(\zeta _{n}\in (u_{n}-s_{n} u_{n},u_{n}+s_{n} (v-u_{n} ) )\), that is \(\zeta _{n}\rightarrow u (u_{n} \rightarrow u )\) as \(s_{n}\rightarrow 0^{+}\), since \(I_{1n} \geq 0\) for all n. Applying Fatou’s Lemma to \(I_{1n}\), one gets
For \(I_{2n}\), by the dominated convergence theorem, it holds that
For every \(v\in U^{+}\), and the above information, we have
where \(|dI_{\lambda}|(u)< c\) is arbitrary. □
Lemma 2.3
\(I_{\lambda}\) satisfies the \((P.S. )\) condition.
Proof
Let \(\{u_{n}\}\subset U^{+}\) be \((P.S. )\) sequence of \(I_{\lambda}\), that is
By Lemma 2.2, \(\forall v\in U^{+}\), we can infer that
taking \(v=2u_{n}\in U^{+}\) in (2.11), we have that
Since \(I_{\lambda}(u_{n})\rightarrow c\),
From (2.12) and (2.13), we have
which implies that \(\{u_{n}\}\) is bounded in \(S^{1}_{0}(\Omega )\). Thus, there exists a subsequence, still denoted by itself, and a function \(u \in S^{1}_{0}(\Omega )\), such that \(u_{n}\rightharpoonup u\) in \(S^{1}_{0}( \Omega )\), \(u_{n}(x)\rightarrow u(x)\) a.e. in Ω as \(n\rightarrow \infty \). Choosing \(v=u_{m}\) as the test function in (2.11), we have
Exchanging \(u_{m}\) and \(u_{n}\) gives a similar inequality, and adding two inequalities together and Lemma 2.1(4), it holds that
We have \(\lim_{n\rightarrow \infty} \Vert u_{n}-u_{m} \Vert =0\). Therefore, \(u_{n}\rightarrow u\) in \(S^{1}_{0}(\Omega )\) as \(n \rightarrow \infty \). □
Lemma 2.4
Suppose that \(\vert dI_{\lambda} \vert (u )=0\), then u is a weak solution of the problem (2.3). Namely, \(u^{-\gamma}\varphi \in L^{1} (\Omega )\) for all \(\varphi \in S^{1}_{0}(\Omega )\), there holds
Proof
By Lemma 2.2, we deduce that
for every \(v\in U^{+}\). Letting \(s\in \mathbb{R}\), \(\varphi \in S^{1}_{0}( \Omega )\), taking \(v=(u+s\varphi )^{+}\) and \(v\in U^{+}\) as a test function in (2.7), one gets
since \(\nabla _{H} u(x)=0\) for a.e. \(x\in \Omega \) with \(u(x)=0\), and \(\operatorname{Meas}\{x\in \Omega :u(x)+s\varphi (x)<0,u(x)>0\}\rightarrow 0\) as \(s\rightarrow 0\), one obtains
Therefore
as \(s\rightarrow 0\), we obtain that
By the arbitrariness of φ, also holds for −φ
Hence, we can deduce that (2.15) holds. □
Lemma 2.5
Given \(0<\gamma <1\), there exist constants \(r,\rho ,\Lambda _{0}>0\), such that the functional \(I_{\lambda}\) satisfies the following conditions for \(0<\lambda <\Lambda _{0}\):
-
(i)
\(I_{\lambda}(u)|_{u\in S_{\rho}}\geq r> 0\), \(\inf_{u\in B_{\rho}}I_{\lambda}(u)<0\);
-
(ii)
There exists \(e\in S^{1}_{0}(\Omega )\) with \(\|e\|>\rho \) such that \(I_{\lambda}(e)<0\).
Proof
(i) It follows from (2.6) and Lemma 2.1(3) that
which implies that there exist constants \(r, \rho , \Lambda _{0}>0\), such that \(I_{\lambda}| _{u\in S_{\rho}}\geq r>0\) for every \(\lambda \in (0,\Lambda _{0} )\). Moreover, for \(u\in S^{1}_{0}( \Omega )\backslash \{0 \}\), it holds that
So, we obtain that \(I_{\lambda}(tu)<0\) for all \(u\neq 0\) and t small enough. Therefore, for \(\|u\|\) small enough, one has
(ii) For every \(u^{+}\in S^{1}_{0}(\Omega )\), \(u^{+}\neq 0\) and \(t>0\), we get
as \(t\rightarrow +\infty \). Therefore, we can find \(e\in S^{1}_{0}(\Omega )\) such that \(\|e\|>\rho \) and \(I_{\lambda}(e)<0\). □
3 Proof of main results
In this section, we show that for each \(\lambda >0\), the functional \(I_{\lambda}\) attains the global minimizer in \(S^{1}_{0}(\Omega )\), which is the unique solution of system (1.1) for \(\mu =1\) and multiple solutions of the system for \(\mu =-1\), \(\lambda >0\) small enough.
Proof of Theorem 1.1
We prove Theorem 1.1 in three steps.
Step 1. For every \(\lambda >0\) and \(\mu =1\), the functional \(I_{\lambda}\) attains the global minimizer in \(S^{1}_{0}(\Omega )\), in other words, there exists \(u_{*}\in S^{1}_{0}(\Omega )\) such that
In fact, for all \(u\in S^{1}_{0}(\Omega )\), combining with Lemma 2.1(1) and (2.6), we infer that
this implies that \(I_{\lambda}\) is coercive and bounded from below on \(S^{1}_{0}(\Omega )\) for each \(\lambda >0\). Thus, \(m_{\lambda}=\inf_{S^{1}_{0}(\Omega )}I_{\lambda}\). For \(t>0\) and given \(u\in S^{1}_{0}(\Omega )\backslash \{0\}\),
We deduce from that for \(t>0\) small enough, \(I_{\lambda}(tu)<0\). Therefore, \(m_{\lambda}=\inf_{S^{1}_{0}(\Omega )}I_{\lambda}<0\).
From the definition of \(m_{\lambda}\), existence of minimizing sequence \(\{u_{n}\}\subset S^{1}_{0}(\Omega )\) such that \(\lim_{n\rightarrow \infty}I_{\lambda}(u_{n})=m_{\lambda}<0\). Since \(I_{\lambda}(u_{n})=I_{\lambda}(|u_{n}|)\), we can assume that \(u_{n}\geq 0\). By (3.1), we know that \(\{u_{n}\}\) is bounded in \(S^{1}_{0}(\Omega )\). Suppose there exists a subsequence, still denoted by \(\{u_{n}\}\), and \(u_{*}\in S^{1}_{0}(\Omega )\) such that
Then, combining with the weakly lower semi-continuity of the norm and Lemma 2.1 (2), one has
Furthermore, \(I_{\lambda}(u_{*})\geq m_{\lambda}\), thus \(I_{\lambda}(u_{*})= m_{\lambda}<0\).
In addition, we show \(u_{*}>0\) in Ω. From the information above, \(u_{*}\geq 0\) and \(u_{*}\neq 0\). Fix \(\eta \in S^{1}_{0}(\Omega )\), \(\eta >0\) and \(t\geq 0\), we obtain that
that is
Notice that
where \(\zeta (x)\in (0,1)\) and
Since \((u_{*}(x)+t\eta (x)\zeta (x))^{-\gamma}\eta (x)\geq 0\), using Fatou’s Lemma, from (3.2), it holds
Using a similar approach, the above equation also holds for \(0\leq \eta \in S^{1}_{0}(\Omega )\), that is
Thus,
Note that \(\phi _{u_{*}}(\xi )>0\) for any \(\xi \in \Omega \), \(u_{*}\geq 0\) and \(u_{*}\neq 0\). According to the maximum principle (see [3, 4]), \(u_{*}>0\) in Ω.
Step 2. We prove that \(u_{*}\) satisfies (2.5) for \(\mu =1\). Let \(\delta >0\) and define \(h: [-\delta ,\delta ]\rightarrow \mathbb{R}\) by \(h(t)=I_{\lambda}(u_{*}+tu_{*})\), then h attains its minimum at \(t=0\), and it holds that
We take \(\eta \in S^{1}_{0}(\Omega )\backslash \{0\}\), \(\varepsilon >0\) and define \(\Phi =(u_{*}+\varepsilon \eta )^{+}\). Let
Then, \(\Phi |_{\Omega _{1}}=u_{*}+\varepsilon \eta \), \(\Phi |_{\Omega _{2}}=0\). Inserting Φ into (3.3) and using (3.4), we can get
Due to \(u_{*}>0\) and the measure of the domain \(\Omega _{2}=\{x\in \Omega :u_{*}(x)+\varepsilon \eta (x)\leq 0\}\) tends to zero as \(\varepsilon \rightarrow 0\), there holds
Then, dividing by \(\varepsilon >0\) and letting \(\varepsilon \rightarrow 0\) in (3.5), we have
The above inequality also holds for −η, and we can get
Then, \(u_{*}\in S^{1}_{0}(\Omega )\) is a solution of system (1.1) for \(\lambda >0\) and \(\mu =1\).
Step 3. We prove that \(u_{*}\) is the unique solution of (1.1) for \(\mu =1\). We may assume that \(v_{\star}\in S^{1}_{0}(\Omega )\) is also a solution of system (1.1), and from (2.5), we get
and
Combining with (3.6) and (3.7), it holds that
For \(\gamma \in (0, 1)\), \(u_{*}, v_{\star}>0\) in Ω, and
Hence, by (3.8) and Lemma 2.1(4), we get
that is say
that is \(u_{*}=v_{\star}\). Hence, \(u_{*}\in S^{1}_{0}(\Omega )\) is the unique solution of system (1.1). □
Proof of Theorem 1.2
We prove Theorem 1.2 in two steps.
Step 1. Suppose that \(0<\lambda <\Lambda _{0}\), then system (1.1) admits a positive solution \(u_{*}\) such that \(I_{\lambda} (u_{\lambda} )=m<0\).
In fact, we claim that there exists \(u_{\lambda}\in B_{\rho}\), such that \(I_{\lambda}(u_{\lambda})=m<0\). By the definition of m, we know that there exists a minimizing sequence \(\{u_{n}\}\subset B_{\rho}\subset U^{+}\) such that \(\lim_{n\rightarrow \infty}I_{\lambda}(u_{n})=m<0\). Since \(\{u_{n}\}\) is bounded in \(B_{\rho}\), we may assume that, up to a subsequence still denoted by itself, there exists \(u_{\lambda}\in S^{1}_{0}(\Omega )\), such that
as \(n\rightarrow \infty \). Set \(w_{n}=u_{n}-u_{\lambda}\), and using Brézis-Lieb’s Lemma (see [5]), one has
Hence, by Lemma 2.3, we can infer that
which implies that \(m\geq I_{\lambda}(u_{\lambda})\). Since \(B_{\rho}\) is closed and convex, one has that \(u_{\lambda}\in B_{\rho}\). Thus, we obtain \(I_{\lambda}(u_{\lambda})=m<0\) and \(u_{\lambda}\not \equiv 0\) in Ω. From the above arguments, we know that \(u_{\lambda}\) is a local minimizer of \(I_{\lambda}\).
Now, we prove that \(u_{\lambda}\) is a critical point of \(I_{\lambda}\). Note that \(u_{\lambda}\geq 0\) and \(u_{\lambda}\not \equiv 0\). Then, for any \(\psi \in U^{+}\subset S^{1}_{0}(\Omega )\), let \(t>0\) such that \(u_{\lambda}+t\psi \in S^{1}_{0}(\Omega )\), and one has
Actually, from (3.11), we can see that
Dividing by \(t>0\) and passing to the limit as \(t\rightarrow 0^{+}\), it holds that
Notice that
Where \(\zeta \rightarrow 0^{+}\) and \((u_{\lambda}+\zeta t\psi )^{-\gamma}\psi \rightarrow u_{\lambda}^{- \gamma}\psi \) a.e. \(x\in \Omega \) as \(t\rightarrow 0^{+}\), since \((u_{\lambda}+\zeta t\psi )^{-\gamma}\psi \geq 0\). By Fatou’s Lemma, one has
Therefore, we deduce from (3.12) and the above estimate that
Since \(I_{\lambda}(u_{\lambda})<0\), this, together with Lemma 2.5, implies that \(u_{\lambda}\notin S_{\rho}\); therefore, we obtain \(\|u_{\lambda}\|<\rho \). For \(u_{\lambda}\), there is \(\delta _{1}\in (0,1)\) such that \((1+t)u_{\lambda}\in B_{\rho}\) for \(|t|\leq \delta _{1}\). Define \(k:[-\delta _{1},\delta _{1}]\) by \(k(t)=I_{\lambda}((1+t)u_{\lambda})\). Clearly, \(k(t)\) achieves its minimum at \(t=0\), namely
Suppose that for any \(v\in S^{1}_{0}(\Omega )\), \(\epsilon >0\). Define \(\Psi \in U^{+}\) by
Combining with (3.13) and (3.14), we get
Since the measure of the domain of integration \(\{u_{\lambda}+\epsilon v\leq 0\}\rightarrow 0\) as \(\epsilon \rightarrow 0\), it follows that
Therefore, dividing by ϵ and setting \(\epsilon \rightarrow 0\) in (3.15), one gets
By the arbitrariness of v, the inequality also holds for −v
Since \(u_{\lambda}\geq 0\) and \(u_{\lambda}\not \equiv 0\), from (3.17), there holds
Note that \(u_{\lambda}\geq 0\) and \(u_{\lambda}\neq 0\), then by the maximum principle (see [3, 4]), it suggests that \(u_{\lambda}>0\) in Ω. From the above arguments, we obtain that \(u_{\lambda}\) is a positive solution of system (1.1) with \(I_{\lambda}(u_{\lambda})=m<0\).
Step 2. Assume that \(0<\lambda <\Lambda _{0}\), then system (1.1) has a positive solution \(v_{*}\) such that \(I_{\lambda}(v_{*})>0\).
In fact, by Lemma 2.5, \(I_{\lambda}\) satisfies the geometric structure of the mountain pass Lemma. By Lemma 2.3, there exists a sequence \(\{v_{n}\}\) such that
We know that \(\{v_{n}\}\subset S^{1}_{0} (\Omega )\) has a convergent subsequence, still denoted by \(\{v_{n}\}\), we may assume that \(v_{n}\rightarrow v_{*}\) in \(S^{1}_{0}(\Omega )\), and
Applying Theorem 1.3.1 in [6], similar to step 1, \(v_{*}\) satisfies problem (2.3) with \(I_{\lambda}(v_{*})=c>0\). Thus, \(v_{*}\) is the second positive solution of system (1.1). □
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Funding
This work is supported by the Science and Technology Project of Bijie (No. BKH [2023]26) and the Disciplinary Construction Project of Mathematics of Guizhou University of Engineering Science (2022).
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Tian, G., An, Y. & Suo, H. Multiple positive solutions for Schrödinger-Poisson system with singularity on the Heisenberg group. J Inequal Appl 2024, 19 (2024). https://doi.org/10.1186/s13660-024-03096-3
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DOI: https://doi.org/10.1186/s13660-024-03096-3