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Positive solutions of Schrödinger-Kirchhoff-Poisson system without compact condition
Boundary Value Problems volume 2017, Article number: 156 (2017)
Abstract
Purpose
The existence of positive solutions for a class of nonlinear Schrödinger-Kirchhoff-Poisson systems.
Methods
Variational method.
Results
Some results on the existence of positive solutions.
1 Introduction
In this paper, we are concerned with the existence of positive solutions for the following nonlinear Schrödinger-Kirchhoff-Poisson system:
by using the variational methods, where \(a>0\) and \(b\geqslant0\) are constants, \(1< p<5\), \(q>0\), and \(K, f: \mathbb {R}^{3} \rightarrow \mathbb {R}\) are two nonnegative functions.
The Kirchhoff equation
which is related to the stationary analogue of the equation
arises in many mathematical physics contexts. Equation (1.2) was proposed by Kirchhoff [1] in 1883 as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings. Kirchhoff’s model considers the changes in the length of the string produced by transverse vibrations.
The Schrödinger-Poisson system
describes a charged wave interacting with its own electrostatic field [2]. We also refer the readers to [3, 4] and the references therein for more mathematical and physical background of (1.3). The Schödinger-Kirchhoff-Poisson system \((\mathcal{SK})\) is a more generalized Kirchhoff-type system of (1.3).
In recent years, there have been enormous results on the existence and multiplicity of solutions of problem (1.3) for \(q>0\) (see e.g. [5–8]). To the best of our knowledge, there are a few articles on the existence of solutions to problem (1.3) for \(q<0\). Recently, in [9], the author proved that problem (1.3) has a positive ground state solution for \(q<0\) and \(f(x,u)=a(x)\vert u\vert ^{p-1}u\), \(3< p<5\). In [10], the author proved that problem (1.3) has a positive solution for \(q<0\). When \(q\equiv1\), the problem \((\mathcal{S}\mathcal{K})\) can be reduced to
Li and Ye [11] proved that problem (1.4) has a a positive ground state solution for \(K(x)\equiv0\), \(f(x)\equiv1\), \(2< p<5\).
Motivated by the works mentioned, we consider system \((\mathcal{SK})\) with the Kirchhoff term \((\int_{\mathbb {R}^{3}}\vert \nabla u\vert ^{2})\triangle u\), which implies that the equation in \((\mathcal{SK})\) is no longer a pointwise identity and is different from the classical Schrödinger-Poisson system. We must emphasize that the combined effects of the nonlocal term it contains and the negative coefficient at ϕu make problem (1.4) an interesting problem.
To obtain our main results, we first require some conditions on f and K:
- (\(f_{1}\)):
-
\(f(x)\geqslant f_{\infty} =\lim_{\vert x\vert \rightarrow+\infty}f(x)>0\), and \(\alpha(x)=f(x)-f_{\infty} \in L^{\frac {6}{5-p}}(\mathbb {R}^{3})\);
- (\(f_{2}\)):
-
\((\nabla f,x)\in L^{\frac{6}{5-p}}(\mathbb {R}^{3})\setminus\{0\} \), and \((\nabla f,x)\geqslant0\);
- (\(K_{1}\)):
-
\(K\in L^{2}(\mathbb {R}^{3})\);
- (\(K_{2}\)):
-
\((\nabla K,x)\in L^{2}(\mathbb {R}^{3})\setminus\{0\}\);
- (\(K_{3}\)):
-
\(K(x)\geqslant K_{\infty}=\lim_{\vert x\vert \rightarrow +\infty}K(x)\geqslant0\), and \(K-K_{\infty} \in L^{2}(\mathbb {R}^{3})\);
- (\(K_{4}\)):
-
\(K(x)+(\nabla K,x)\geqslant0\), \(x\in \mathbb {R}^{3}\).
2 Conclusion
In this paper, we get the existence of a positive solution for all \(1< p<5\). Our main results are as follows.
Theorem 2.1
Let (\(f_{1}\)), (\(f_{2}\)), (\(K_{1}\)), and (\(K_{2}\)) hold. Then there exists \(q_{0}>0\) such that, for any \(0< q< q_{0}\), problem \((\mathcal{S}\mathcal{K})\) admits a positive solution \((u,\phi_{u}) \in H^{1}(\mathbb {R}^{3})\times D^{1,2}(\mathbb {R}^{3})\) for all \(1< p<5\).
Theorem 2.2
Let (\(f_{1}\)), (\(f_{2}\)), and (\(K_{2}\))-(\(K_{4}\)) hold. Then problem (1.4) admits a positive ground state solution \((u,\phi_{u}) \in H^{1}(\mathbb {R}^{3})\times D^{1,2}(\mathbb {R}^{3})\) for all \(1< p<5\).
As a consequence of Theorem 2.2, we have that the problem
has a positive ground state solution for all \(1< p<5\). This can be viewed as an extension of a recent result of Li and Ye [11] concerning the existence of positive ground state solutions.
The paper is organized as follows. In Section 3, we describe the notation and preliminaries. In Section 4, we give a proof of Theorem 2.1. In Section 5, we give a proof of Theorem 2.2.
3 Notation and preliminaries
-
Let \(a>0\) is fixed, and let \(H^{1}(\mathbb {R}^{3})\) be the usual Sobolev space endowed with the scalar product and norm
$$(u,v)= \int_{\mathbb {R}^{3}}(a\nabla u\nabla v+uv) \qquad \mbox{and}\qquad \Vert u\Vert = \biggl( \int_{\mathbb {R}^{3}}\bigl(a\vert \nabla u\vert ^{2}+u^{2} \bigr) \biggr)^{{\frac{1}{2}}}. $$ -
\(H_{r}^{1}(\mathbb {R}^{3}):= \{u:u\in H^{1}(\mathbb {R}^{3}),u(x)=u(\vert x\vert ) \}\).
-
\(D^{1,2}(\mathbb {R}^{3})\) is the completion of \(C_{0}^{\infty}(\mathbb {R}^{3})\) with respect to the norm
$$\Vert u\Vert _{D^{1,2}}= \biggl( \int_{\mathbb {R}^{3}}\vert \nabla u\vert ^{2} \biggr)^{{\frac{1}{2}}}. $$ -
For any \(u\in H^{1}(\mathbb {R}^{3})\) and \(t>0\), we define \({u}_{t}(x)=t^{\frac{1}{2}}u(\frac{x}{t})\).
-
For any \(z\in{ \mathbb {R}^{3}}\) and \(\rho>0\), \(B_{\rho}(z)\) denotes the ball of radius ρ centered at z.
-
For any \(s\in[1,+\infty)\), \(\vert \cdot \vert _{s}\) denotes the usual norm of the Lebesgue space \(L^{s}({\mathbb {R}^{3}})\).
-
\(S_{1}\) is the best Sobolev constant for the embedding of \(H^{1}(\mathbb {R}^{3})\) in \(L^{6}(\mathbb {R}^{3})\), that is,
$$S_{1}=\operatorname{inf}_{u\in H^{1}(\mathbb {R}^{3})\backslash\{0\}}\frac {\Vert u\Vert }{\vert u\vert _{6}}. $$ -
\(S_{2}\) is the best Sobolev constant for the embedding of \(D^{1,2}(\mathbb {R}^{3})\) in \(L^{6}(\mathbb {R}^{3})\), that is,
$$S_{2}=\operatorname{inf}_{u\in D^{1,2}(\mathbb {R}^{3})\backslash\{0\} }\frac{\Vert u\Vert _{D^{1,2}}}{\vert u\vert _{6}}. $$ -
\(H^{-1}(\mathbb {R}^{3})\) denotes the conjugate Sobolev space of \(H^{1}({\mathbb {R}^{3}})\).
-
C and \(C_{i}\), \(i\in \mathbb {N}\), are various positive constants.
It is well known that problem \((\mathcal{SK})\) can be reduced to a nonlinear Schrödinger-Kirchhoff equation with a nonlocal term. For any \(u \in H^{1}({\mathbb {R}^{3}})\), define the linear functional \(L_{u}\) in \(D^{1,2}(\mathbb {R}^{3})\) by
Then (\(K_{1}\)) or (\(K_{3}\)) and the Hölder and Sobolev inequalities imply
Hence, by the Lax-Milgram theorem there exists a unique \(\phi_{u}\in D^{1,2}(\mathbb {R}^{3})\) such that
that is, \(\phi_{u}\) is a weak solution of \(-\triangle\phi=qK(x)u^{2}\), and
Moreover, \(\Vert \phi_{u}\Vert _{D^{1,2}(\mathbb {R}^{3})}^{2}=q\int_{\mathbb {R}^{3}}K(x)\phi_{u}u^{2}\) and \(\phi_{u}>0\) when \(u\neq0\). Then (3.3) inserted into the first equation of \((\mathcal{SK})\) gives
Problem (3.4) is variational, and its solutions are the critical points of the functional defined in \(H^{1}(\mathbb {R}^{3})\) by
It is clear to see that \(I_{q}\) is well defined on \(H^{1}(\mathbb {R}^{3})\) and is of class \(C^{1}\), and for any \(u,v\in H^{1}(\mathbb {R}^{3})\),
Thus, if \(u\in H^{1}(\mathbb {R}^{3})\) is a critical point of \(I_{q}\), then the pair \((u,\phi_{u})\) with \(\phi_{u}\) as in (3.3) is a solution of the problem \((\mathcal{SK})\).
Define the operator \(\Phi: H^{1}(\mathbb {R}^{3})\rightarrow D^{1,2}(\mathbb {R}^{3})\) as \(\Phi[u]=\phi_{u}\). The following lemma shows that the operator Φ possesses the property.
Lemma 3.1
(See [7])
If K satisfies (\(K_{1}\)) or (\(K_{3}\)), then
-
(i)
Φ is continuous;
-
(ii)
Φ maps bounded sets into bounded sets;
-
(iii)
If \(u_{n}\rightharpoonup u\) in \(H^{1}(\mathbb {R}^{3})\), then \(\Phi [u_{n}]\rightharpoonup\Phi[u]\) in \(D^{1,2}(\mathbb {R}^{3})\).
To obtain the boundedness of (PS) sequences, we recall the indirect approach developed by Jeanjean [12].
Proposition 3.1
Let (\(X, \Vert \cdot \Vert \)) be a real Banach space, and let \(J\subset \mathbb {R}^{+}\) be an interval. Consider the following family of \(C^{1}\) functionals on X:
with \(B(u)\geqslant0\) and either \(A(u)\rightarrow+\infty\) or \(B(u)\rightarrow+\infty\) as \(\Vert u\Vert \rightarrow+\infty\) and such that \(I_{\lambda}(0)=0\). For every \(\lambda\in J\), set
and
Then, for almost every \(\lambda\in J\), there is a bounded \((\mathrm{PS})_{c_{\lambda}}\) sequence \(\{u_{n}\}\subset X\).
Lemma 3.2
(See [13], Lemma 2.3)
Under the assumptions of Proposition 3.1, the map \(\lambda\rightarrow c_{\lambda}\) is nonincreasing and left-continuous.
4 A mountain pass solution
In this section, let (\(f_{1}\)), (\(f_{2}\)), (\(K_{1}\)), and (\(K_{2}\)) hold, and let \(1< p<5\). Using (\(K_{1}\)), (3.2), and the Sobolev inequality, we have
and
To overcome the difficulty of finding bounded (PS) sequences for the associated function \(I_{q}\), we use the cut-off function \(\chi \in C^{\infty}_{0}(\mathbb {R}^{+},\mathbb {R})\) satisfying
and study the following modified functional \(I_{q}^{T}:H^{1}(\mathbb {R}^{3})\rightarrow \mathbb {R}\):
Letting \(J=[ \frac{1}{2},1]\), we consider the following family of functionals on \(X=H^{1}(\mathbb {R}^{3})\):
Then \(I_{q,\lambda}^{T}(u)=A(u)-\lambda B(u)\), where
and
Moreover,
The following lemma implies that \(I_{q,\lambda}^{T}\) satisfies the conditions of Proposition 3.1.
Lemma 4.1
-
(i)
For any \(q_{0}>0\), there exists a constant \(\eta>0\) such that \(c_{\lambda}\geqslant\eta>0\) for all \(\lambda\in J\) and \(q\in (0,q_{0}]\);
-
(ii)
\(\Gamma_{\lambda}\neq\emptyset\) for all \(\lambda\in J\).
Proof
(i) For any \(\lambda\in J\) and \(q_{0}>0\), from (\(f_{1}\)), (\(K_{1}\)), (4.2)-(4.3), and the Hölder and Sobolev inequalities it follows that
Since \(1< p<5\), there exists \(\rho>0\) such that \(I_{q,\lambda}^{T}(u)> 0\) for \(0<\Vert u\Vert \leqslant\rho\). In particular, there exists \(\eta>0\) such that \(I_{q,\lambda }^{T}(u)\geqslant\eta\) for \(\Vert u\Vert =\rho\). Now fix \(\lambda \in J\) and \(\gamma\in\Gamma_{\lambda}\). Since \(I_{q,\lambda}^{T}(\gamma(1))<0\), \(\Vert \gamma(1)\Vert >\rho \). By \(\gamma(0)=0\) and the continuity of γ we can deduce that there exists \(t_{\gamma}\in(0,1)\) such that \(\Vert \gamma(t_{\gamma})\Vert =\rho\). Hence, for any \(\lambda\in J\),
(ii) Set \(w\in H^{1}(\mathbb {R}^{3})\backslash\{0\}\) with \(\vert w\vert _{6}=1\). Define \(\gamma:[0,1]\rightarrow H^{1}(\mathbb {R}^{3})\) as
where \(\overline{w}={w}_{\theta}\) and \(\theta>2T\). It is clear to see that γ is a continuous path from 0 to w̅. Moreover, for all \(\lambda\in J\), it follows from (4.3) that
As \(1< p<5\), we get \(I_{q,\lambda}^{T}(\gamma(1))<0\) for θ large enough. □
Lemma 4.2
For any \(\lambda\in J\), each bounded (PS) sequence of the functional \(I_{q,\lambda}^{T}\) admits a convergent subsequence.
Proof
Let \(\lambda\in J\), and let \(\{u_{n}\}\) be a bounded (PS) sequence for \(I_{q,\lambda}^{T}\), that is,
Up to a subsequence, we may suppose that there exists \(u\in H^{1}(\mathbb {R}^{3})\) such that
Now we prove that
and
First, by \(u_{n}\rightharpoonup u \) in \(H^{1}(\mathbb {R}^{3})\) and Lemma 3.1(iii) we have that
Moreover, in view of the Sobolev embedding theorem and Lemma 3.1(iii), we deduce that
Thus, given \(\varepsilon>0\), it follows from (4.11) that
and for any fixed \(w\in H^{1}(\mathbb {R}^{3})\), we have
Since \(K\in L^{2}(\mathbb {R}^{3})\), there exists \(\rho={\rho}(\varepsilon)>0\) such that
and
for large n. Hence, by (4.12), (4.13), (4.15), and (4.16) we deduce
for large n. Analogously, by (4.12), (4.14), (4.15), and (4.17) we infer
for large n. This completes the proof of (4.9) and (4.10). Similarly, we can get
and
Thus, it follows from (4.18)-(4.20) that
Moreover, by (4.7)-(4.10) and (4.3) we have
By (4.8)(a) and (4.3) we deduce
and
It follows (4.3), (4.9), and (4.10) that
Therefore, it follows from (4.21)-(4.26) that
This implies \(\Vert u_{n}-u\Vert ^{2}=o(1)\). Thus, \(u_{n}\rightarrow u\) in \(H^{1}(\mathbb {R}^{3})\). □
Lemma 4.3
For almost every \(\lambda\in J\), there exists \(u^{\lambda}\in H^{1}(\mathbb {R}^{3})\backslash\{0\}\) such that
Proof
By Lemma 3.1, Lemma 3.2, and Proposition 3.1, for almost every \(\lambda\in J\), there exists a bounded \((\mathrm{PS})_{c_{\lambda }}\) sequence \(\{u_{n}^{\lambda}\}\subset H^{1}(\mathbb {R}^{3})\). Up to a subsequence, by Lemma 4.2 we can suppose that there exists \(u^{\lambda} \in H^{1}(\mathbb {R}^{3})\) such that \(u_{n}^{\lambda} \rightarrow u^{\lambda} \) in \(H^{1}(\mathbb {R}^{3})\). Then we have \(I_{q,\lambda}^{T}(u^{\lambda})=c_{\lambda}\) and \((I_{q,\lambda}^{T})'(u^{\lambda})=0\). □
Lemma 4.4
(Pohoz̆aev identity)
If \(u \in H^{1}(\mathbb {R}^{3})\) is a weak solution of
then we have
where \(\phi_{u}\) is defined by (3.3).
Proof
The proof is standard, and we omit it (see [14]). □
Lemma 4.5
Let \(m_{i}\) (\(i=1,2,3\)) be positive constants, \(p>1\), and \(g(t)=m_{1}t^{2}+m_{2}t^{4}-m_{3}t^{3+\frac{p+1}{2}}\), \(t\geqslant 0\). Then g has a unique positive critical point, which corresponds to its maximum.
Proof
The proof is similar to that of Lemma 3.3 in [6] and is elementary. We omit the proof. □
Lemma 4.6
Let \(\lambda_{n} \in J\), and let \(u_{n}\) be a critical point of \(I_{q,\lambda_{n}}^{T}\) at level \(c_{\lambda_{n}}\) for every \(n\in \mathbb {N}\). Then, for \(T>0\) sufficiently large, there exists \(q_{0}=q_{0}(T)\) such that, for any \(0< q< q_{0}\), up to a subsequence, \(\vert u_{n}\vert _{6}\leqslant T\) for all \(n\in \mathbb {N}\).
Proof
We argue by contradiction. First of all, it follows from \((I_{q,\lambda_{n}}^{T})'(u_{n})=0\) and Lemma 4.4 that \(u_{n}\) satisfies the following Pohoz̆aev identity:
Since \(I_{q,\lambda_{n}}^{T}(u_{n})=c_{\lambda_{n}}\), we have
Hence, by (4.28) and (4.29) we obtain
Moreover, combining (\(f_{2}\)), (\(K_{2}\)), (4.1), (4.2), (4.3), (4.30), Lemma 3.1(ii), and the Sobolev embedding theorem, we deduce
By the definition of \(c_{\lambda_{n}}\) and by (4.6) we have
where
By Lemma 4.5, \(A_{1}\) is a finite number. If \({\theta} ^{6}\geqslant {2T^{6}}\), then \(A_{2}(T)=0\). Otherwise, it follows from (4.2) that
We also have
Then we deduce
By the Sobolev embedding theorem we have
We suppose by contradiction that there exists no subsequence of \(\{ u_{n}\}\) that is uniformly bounded by T. Then we can assume that \(\vert u_{n}\vert _{6}>T\), \(n\in \mathbb {N}\). Therefore by (4.32) we conclude that
which is not true for T large and q small enough. Indeed, we can find \(T_{0}>0\) such that \(T_{0}^{2}>C_{7}+1\) and \(q_{0}=q_{0}(T_{0})\) such that \(C_{8}q^{2}T^{4}<1\) for any \(0< q< q_{0}\). □
Proof of Theorem 2.1
Let T, \(q_{0}\) be as in Lemma 4.6 and fix \(0 < q < q_{0}\). According to Lemma 4.3, there exist sequences \(\{\lambda_{n}\} \subset J\) and \(\{u_{n}\} \subset H^{1}(\mathbb {R}^{3})\) such that
where \(c_{\lambda_{n}}\) is defined by (3.7). We will prove that \(\{u_{n}\}_{n}\) is a bounded (PS) sequence for \(I_{q}=I_{q,1}^{T}\). By Lemma 4.6 we know that
Using the Hölder inequality, we get
Since \(\langle(I_{q,\lambda}^{T})'(u_{n}),u_{n}\rangle=0\), it follows from (\(f_{1}\)), (4.2), (4.3), (4.5), (4.34), and (4.35) that
As \(1< p<5\), \(\int_{\mathbb {R}^{3}} u_{n}^{2}\) is bounded. Combining (4.31), we deduce that \(\{u_{n}\}\) is bounded in \(H^{1}(\mathbb {R}^{3})\).
On the other hand, by (4.34), (4.3), and (4.4) we have
Since \(\lambda_{n}\rightarrow1^{-}\), we can prove that \(\{u_{n}\}\) is a (PS) sequence for \(I_{q}=I_{q,1}^{T}\) by similar arguments as in Theorem 1.1 of [15]. We finish as in Lemma 4.3. □
5 A ground state solution
To apply the global compactness lemma to solve problem (1.4), first of all, we need to consider the existence of ground state solutions of the associated ‘limit problem’ of (1.4), which is given as
and the corresponding least energy of the associated limited functional
Set
where
We remark that
and
Lemma 5.1
\(I_{\infty}\) is not bounded from below.
Proof
For any \(u\in H^{1}(\mathbb {R}^{3})\setminus\{0\}\), we have
As \(1< p<5\), \(I_{\infty}(u_{t})\rightarrow-\infty\) as \(t\rightarrow +\infty\). □
Lemma 5.2
Let \(m_{i}\) (\(i=1,2,3,4\)) be positive constants, \(1< p<5\), and \(g(t)=m_{1}t^{2}+m_{2}t^{4}-m_{3}t^{7}-m_{4}t^{\frac{p+7}{2}}\) for \(t\geqslant0\). Then g has a unique positive critical point, which corresponds to its maximum.
Proof
The proof is similar to Lemma 3.3 of [6], and we omit it. □
Lemma 5.3
For any \(u\in H^{1}(\mathbb {R}^{3})\setminus\{0\}\), there is a unique \({\overline{t}}>0\) such that \(u_{\overline{t}}\in\mathcal{M}_{\infty }\). Moreover, \(I_{\infty}(u_{\overline{t}})=\max _{t>0 } I_{\infty}(u_{t})\).
Proof
For any \(u\in H^{1}(\mathbb {R}^{3})\setminus\{0\}\) and \(t\geqslant0\), we denote
By Lemma 5.2, γ has a unique critical point \(\overline{t}>0\) corresponding to its maximum. Then \(\gamma(\overline{t})=\max _{t>0 }\gamma(t) \) and \(\gamma'(\overline{t})=0\). It follows that \(I_{\infty}(u_{\overline{t}})=\max _{t>0 } I_{\infty}(u_{t})\), \(G_{\infty}(u_{\overline{t}})=\overline{t}\gamma '(\overline{t})=0\) a.e., and \(u_{\overline{t}} \in{\mathcal{M}_{\infty }}\). □
Set
where
The constant \(c_{3}\) turns out to be a nontrivial number, as we will prove in the next lemma, which contains the statement of the main properties of \(\mathcal{M}_{{\infty}}\).
Lemma 5.4
-
(i)
There exists \(\sigma>0\) such that \(\Vert u\Vert \geqslant\sigma \) for all \(u \in\mathcal{M}_{{\infty}}\);
-
(ii)
\(I_{{\infty}}\) is bounded from below on \(\mathcal{M}_{{\infty}}\) by a positive constant, i.e. \(c_{3}>0\);
-
(iii)
u is a free critical point of \(I_{\infty}\) if and only if u is a critical point of \(I_{\infty}\) constrained on \(\mathcal{M}_{\infty}\).
Proof
(i) Let \(u \in\mathcal{M}_{\infty}\). By the Hölder inequality we see that
As \(1< p<5\), there exists \(\sigma>0\) such that
(ii) It follows from (5.4) and (5.5) that
Thus, \(c_{3}>0\).
(iii) The proof consists of two steps.
Step 1. \(G'_{\infty}(u)\neq0\) for any \(u\in\mathcal {M}_{\infty}\), and hence \(\mathcal{M}_{\infty}\) is a \(C^{1}\)-manifold.
We will prove this by contradiction. Suppose that \(G'_{\infty}(u) =0\) for some \(u\in\mathcal{M}_{\infty}\). Denote
In a weak sense, the equation \(G'_{\infty}(u) =0\) can be written as
where \(\phi_{u}(x)= \frac{K_{\infty}}{4\pi}\int_{\mathbb {R}^{3}}\frac {u^{2}(y)}{\vert x-y\vert }\,dy\). Combining \(G_{\infty}(u)=0\), (5.6), and its Pohoz̆ave identity, we have
Thus, we can conclude that
which is impossible since ν and δ are positive. Thus, \(G'_{\infty}(u)\neq0\) for any \(u\in\mathcal{M}_{\infty}\), and \(\mathcal{M}_{\infty}\) is a \(C^{1}\)-manifold by the implicit function theorem.
Step 2. Every critical point of \(I_{\infty}\vert_{\mathcal {M}_{\infty}}\) is the critical point of \(I_{\infty}\) in \(H^{1}(\mathbb {R}^{3})\).
If u is a critical point of \(I_{\infty}\vert_{\mathcal{M}_{\infty }}\), then there exists a Lagrange multiplier \(\lambda\in \mathbb {R}\) such that
We claim that \(\lambda=0\). The equation \(I'_{\infty}(u)-\lambda G'_{\infty}(u)=0\) can be written in the weak sense as
that is, u solves the equations \(-\triangle\phi=K_{\infty}u^{2}\) and
for \(x\in \mathbb {R}^{3}\). Using the notation in Step 1, we have that
The coefficient matrix of (5.7) is
and its determinant is
As \(1< p<5\),
Now we prove that \(\lambda=0\) by excluding the other two possibilities:
-
(1)
If \(\lambda=\frac{1}{4}\), then the last equation of (5.7) is
$$\frac{1}{4}\alpha+ \frac{15}{16}\nu+ \frac{3(p-1)}{8(p+1)}\delta=0. $$This is a contradiction since \(\alpha>0\), \(\delta>0\), \(\nu\geqslant0\), and \(1< p<5\).
-
(2)
If \(\lambda\neq0\), \(\lambda\neq\frac{1}{4}\), then the linear system (5.7) has a unique solution. We obtain
$$\delta=-\frac{(p+1)(32k+15\nu)}{(p+3)(p-1)}. $$As \(1< p<5\), \(k>0\) and \(\nu\geqslant0\), we have \(\delta<0\). This is impossible since \(\delta>0\). □
Lemma 5.5
\(c\triangleq c_{1} =c_{2}=c_{3}=c_{4}\).
Proof
(i) \(c_{1} =c_{2}=c_{3}\). The proof is similar to the argument of Nehari manifold method [16].
(ii) \(c_{2}=c_{4}\). It is clear that \(c_{2}=c_{3}\leqslant c_{4}\). We next prove that \(c_{2}\geqslant c_{4}\). For any \(u \in H^{1}(\mathbb {R}^{3})\setminus\{0\}\), it follows from (5.3) that there exists a unique \({\overline{t}}>0\) such that \(u_{\overline {t}}\in\mathcal{M}_{\infty}\) and \(I_{\infty}(u_{\overline {t}})=\max _{t>0 } I_{\infty}(u_{t})\). Similarly, for \(u^{*}\), the Schwarz symmetric arrangement of u, there exists a unique \(t^{*}>0\) such that \(u^{*}_{t^{*}} \in\mathcal{M}_{\infty}\), that is, \(u^{*}_{t^{*}} \in\mathcal{M}_{\infty} \cap H^{1}_{r}(\mathbb {R}^{3})\). Note that
and (see [17])
Then we have that
By the preceding we deduce that
□
Lemma 5.6
(See [6])
Let \(\{u_{n}\}\) be a sequence satisfying \(u_{n}\rightharpoonup u\) in \(H_{r}^{1}(\mathbb {R}^{3})\). Then
Theorem 5.1
Problem (5.1) admits a positive ground state solution \((u,\phi_{u}) \in H^{1}(\mathbb {R}^{3})\times D^{1,2}(\mathbb {R}^{3})\).
Proof
By Lemmas 5.4-5.5 we only need to prove that c is attained at some \(u\in H^{1}_{r}(\mathbb {R}^{3})\cap\mathcal{M}_{\infty}\). Letting \(\{u_{n}\}\subset H^{1}_{r}(\mathbb {R}^{3})\cap\mathcal{M}_{\infty }\) be a minimizing sequence for \(I_{\infty}\), it follows from (5.4) that
for n large enough. Therefore, \(\{u_{n}\}\) is bounded in \(H_{r}^{1}(\mathbb {R}^{3})\). Passing, if necessary, to a subsequence, we may assume that there exists a function \(\tilde{u}\in H_{r}^{1}(\mathbb {R}^{3})\) such that
Now we show that \(u_{n} \rightarrow\tilde{u}\) in \(H^{1}(\mathbb {R}^{3})\). Indeed, it follows from (5.8), Fatou’s lemma, and Lemma 5.6 that
and
Clearly, we have \(G_{\infty}(\tilde{u})\leqslant0\). Therefore, by Lemma 5.3 there exists a unique \(t_{0}\in(0,1]\) such that \(\tilde {u}_{t_{0}}\in\mathcal{M}_{\infty}\). If \(t_{0}\in(0,1)\), then it follows from (5.3) that
which is impossible. Thus \(t_{0}=1\), and c is attained at \(\tilde{u} \in\mathcal{M}_{\infty}\).
By the standard regularity arguments as in the proof of Theorem 1.4 of [11] we see that ũ is a positive ground state solution for problem (5.1). □
Assume that (\(f_{1}\))-(\(f_{2}\)) and (\(K_{2}\))-(\(K_{4}\)) hold. We apply Proposition 3.1 to prove Theorem 2.2. Set \(J=[\frac{1}{2},1]\). We consider a family of functionals on \(H^{1}(\mathbb {R}^{3})\),
Then \(I_{\lambda}(u)=A(u)-\lambda B(u)\), where
and
Lemma 5.7
Assume that (\(f_{1}\))-(\(f_{2}\)) and (\(K_{3}\)) hold. Then
-
(i)
there exists \(v\in H^{1}(\mathbb {R}^{3})\setminus\{0\}\) such that \(I_{\lambda}(v)< 0\) for all \(\lambda\in J\);
-
(ii)
\(c_{\lambda}:=\operatorname{inf}_{\gamma\in\Gamma_{\lambda}}\max _{t\in[0,1]}I_{\lambda}(\gamma(t))>0\) for all \(\lambda\in J\), where
$$\Gamma_{\lambda}:= \bigl\{ \gamma\in C\bigl([0,1],H^{1}\bigl( \mathbb {R}^{3}\bigr)\bigr)\mid\gamma (0)=0,\gamma(1)=v \bigr\} . $$
Proof
The proof is similar to those of Lemma 3.1 and Lemma 3.2, and we omit it. □
By Proposition 3.1 we see that, for any \(\lambda\in[ \frac{1}{2}, 1]\), the associated limit problem
has a positive ground state solution \((u_{\lambda},\phi_{u_{\lambda}}) \in H^{1}(\mathbb {R}^{3})\times D^{1,2}(\mathbb {R}^{3})\), that is, for any \(\lambda \in[\frac{1}{2}, 1]\),
is achieved at \(u_{\lambda}\in\mathcal{M}_{\lambda}^{\infty} \triangleq\{u\in H^{1}(\mathbb {R}^{3})\backslash\{0\}: G_{\lambda}^{\infty }(u)=0\}\), where
Lemma 5.8
Assume that (\(f_{1}\))-(\(f_{2}\)) and (\(K_{2}\))-(\(K_{3}\)) hold and that \(K\not\equiv K_{\infty}\) or \(f\not \equiv f_{\infty}\). Then \(c_{\lambda}< m_{\lambda}^{\infty}\) for any \(\lambda\in J\).
Proof
Let \(u_{\lambda}\) be the minimizer of \(m_{\lambda }^{\infty}\). By Lemma 5.3 we see that \(I_{\lambda}^{\infty}(u_{\lambda})= \max _{t>0 } I_{\lambda}^{\infty}(t^{\frac{1}{2}}u_{\lambda}(\frac{x}{t}))\). Then, choosing \(v(x)=(u_{\lambda})_{t_{0}}\) for \(t_{0}\) large enough in Lemma 5.7(i), we have that, for any \(\lambda\in J\), there exists \(\widehat{t}\in(0,t_{0})\) such that
By the assumptions on K and f, since \(u_{\lambda}>0\), we derive
Thus, we conclude that
□
To prove that the functional \(I_{\lambda}\) satisfies \((\mathrm{PS})_{c_{\lambda }}\) condition for a.e. \(\lambda\in J\), we need the following global compactness lemma, which is suitable for Kirchhoff equations.
Lemma 5.9
Assume that (\(f_{1}\)), (\(f_{2}\)), and (\(K_{3}\)) hold. Then for \(c > 0\), \(\lambda\in J\), and a bounded \((\mathrm{PS})_{c}\) sequence \(\{ u_{n}\}\subset H^{1}(\mathbb {R}^{3})\) of \(I_{\lambda}\), there exist \(u\in H^{1}(\mathbb {R}^{3})\) and \(A \in \mathbb {R}\) such that \(J'_{\lambda}(u)=0\), where
Moreover, there exists a finite (possibly, empty) set \(\{ w^{1},w^{2},\ldots, w^{l}\}\subset H^{1}(\mathbb {R}^{3})\) of nontrivial positive solutions of
and \(\{y_{n}^{1},y_{n}^{2},\ldots,y_{n}^{l}\}\subset \mathbb {R}^{3}\) such that
and
where
Proof
The proof is similar to that of Lemma 3.4 in [11]. Since \(\{u_{n}\}\) is bounded in \(H^{1}( \mathbb {R}^{3})\), there exist \(u\in H^{1}( \mathbb {R}^{3})\) and \(A\in \mathbb {R}\) such that
Moreover, using (\(f_{1}\))-(\(f_{2}\)) and (\(K_{2}\))-(\(K_{3}\)), we have
Therefore, \(I'_{\lambda}(u_{n})\rightarrow0\) implies that
that is, \(J'_{\lambda}(u)=0\). Since
and
we conclude that
We next show that
where either \(l=0\) or \(l>0\).
Step 1: Set \(u_{n}^{1}=u_{n}-u\). By (5.15) and the Bresiz-Lieb lemma we obtain that
-
(A.1)
\(\vert \nabla u_{n}^{1}\vert _{2}^{2}=\vert \nabla u_{n}\vert _{2}^{2}-\vert \nabla u\vert _{2}^{2}+o(1)\),
-
(B.1)
\(\vert u_{n}^{1}\vert _{2}^{2}=\vert u_{n}\vert _{2}^{2}-\vert u\vert _{2}^{2}+o(1)\),
-
(C.1)
\(J_{\lambda}^{\infty}(u_{n}^{1})\rightarrow c+\frac {bA^{4}}{4}-J_{\lambda}(u)\), and
-
(D.1)
\((J_{\lambda}^{\infty})'(u_{n}^{1})\rightarrow0\) in \(H^{-1}( \mathbb {R}^{3})\).
Let
Similarly to (4.10), we have that
Vanishing: If \(\sigma^{1}=0\), then it follows from Lemma 1.21 of [12] that \(u_{n}^{1}\rightarrow0\) in \(L^{s}( \mathbb {R}^{3})\) for \(s\in(2,2^{*})\). Thus, \(\int_{ \mathbb {R}^{3}}\vert u_{n}^{1}\vert ^{p-1}u_{n}^{1}\rightarrow0\) in \(H^{-1}( \mathbb {R}^{3})\). Since \((J_{\lambda}^{\infty})'(u_{n}^{1})\rightarrow0\), we get that \(u_{n}^{1}\rightarrow0\) in \(H^{1}( \mathbb {R}^{3})\), and the proof is completed.
Nonvanishing: If \(\sigma^{1}>0\), then there exists a sequence \(\{ y_{n}^{1}\}\in \mathbb {R}^{3}\) such that
Set \(w_{n}^{1}\triangleq u_{n}^{1}(\cdot+y_{n}^{1})\). Then \(\{ w_{n}^{1}\}\) is bounded, and we may assume that \(w_{n}^{1}\rightharpoonup w^{1}\) in \(H^{1}( \mathbb {R}^{3})\). Hence \((J_{\lambda}^{\infty})'(w^{1})=0\). As
we see that \(w^{1}\neq0\). Moreover, \(u_{n}^{1}\rightharpoonup0\) in \(H^{1}( \mathbb {R}^{3})\) implies that \(\{y_{n}^{1}\}\) is unbounded. Hence, we may assume that \(\vert y_{n}^{1}\vert \rightarrow+\infty\).
Step 2: Setting \(u_{n}^{2}=u_{n}-u-w^{1}(\cdot-y_{n}^{1})\), we can similarly get that
-
(A.2)
\(\vert \nabla u_{n}^{2}\vert _{2}^{2}=\vert \nabla u_{n}\vert _{2}^{2}-\vert \nabla u\vert _{2}^{2}-\vert \nabla w^{1} \vert _{2}^{2}+o(1)\),
-
(B.2)
\(\vert u_{n}^{2}\vert _{2}^{2}=\vert u_{n}\vert _{2}^{2}-\vert u\vert _{2}^{2}-\vert w^{1} \vert _{2}^{2}+o(1)\),
-
(C.2)
\(J_{\lambda}^{\infty}(u_{n}^{2})\rightarrow c+\frac {bA^{4}}{4}-J_{\lambda}(u)-J_{\lambda}^{\infty}(w^{1})\),
-
(D.2)
\((J_{\lambda}^{\infty})'(u_{n}^{2})\rightarrow0\) in \(H^{-1}( \mathbb {R}^{3})\).
Similarly to the arguments in Step 1, let
If vanishing occurs, then \(\Vert u_{n}^{2}\Vert \rightarrow0\), that is, \(\Vert u_{n}-u-w^{1}(\cdot-y_{n}^{1})\Vert \rightarrow0\). Moreover, it follows from (5.16) and from (A.2) and (C.2) that
If nonvanishing occurs, then there exist a sequence \(\{y_{n}^{2}\}\in \mathbb {R}^{3}\) and a nontrivial \(w^{2}\in H^{1}( \mathbb {R}^{3})\) such that \(w_{n}^{2}\triangleq u_{n}^{2}(\cdot+y_{n}^{2}) \rightharpoonup w^{2}\) in \(H^{1}( \mathbb {R}^{3})\). Then by (D.2) we have that \((J_{\lambda }^{\infty})'(w^{2})=0\). Furthermore, \(u_{n}^{2}\rightharpoonup0\) in \(H^{1}( \mathbb {R}^{3})\) implies that \(\vert y_{n}^{2}\vert \rightarrow\infty\) and \(\vert y_{n}^{2}-y_{n}^{1}\vert \rightarrow\infty\).
We next proceed by iteration. Recall that if \(w^{k}\) is a nontrivial solution of \(I_{\lambda}^{\infty}\), then \(I_{\lambda}^{\infty}(w^{k})>0\). So there exists some finite \(l\in \mathbb {N}\) such that only the vanishing case occurs in Step l. Then the lemma is proved. □
Lemma 5.10
Assume that (\(f_{1}\))-(\(f_{2}\)) and (\(K_{2}\))-(\(K_{4}\)) hold. For \(\lambda\in J\), let \(\{u_{n}\}\subset H^{1}(\mathbb {R}^{3})\) be a bounded \((\mathrm{PS})_{c_{\lambda }}\) sequence of \(I_{\lambda}\). Then there exists a nontrivial \(u_{\lambda} \in H^{1}(\mathbb {R}^{3})\) such that
Proof
By Lemma 5.9 we know that, for \(\lambda\in J\), there exist \(u_{\lambda} \in H^{1}(\mathbb {R}^{3})\) and \(A_{\lambda} \in \mathbb {R}\) such that
Now we prove that \(u_{n}\rightarrow u_{\lambda}\) in \(H^{1}(\mathbb {R}^{3})\), that is, \(\{w_{1},w_{2},\ldots,w_{l}\}=\emptyset\). By contradiction we assume that there exist a positive integer l and \(\{y_{n}^{k}\}_{k=1}^{l}\subset \mathbb {R}^{3}\) with \(\vert y_{n}^{k}\vert \rightarrow\infty\) as \(n\rightarrow \infty\) for each \(1\leqslant k \leqslant l\), and nontrivial positive solutions \(w^{1},w^{2},\ldots, w^{l}\) of problem (5.14) such that
and
In the weak sense, the equation \(J'_{\lambda}(u_{\lambda})=0\) can be written as
where \(\phi_{u_{\lambda}}= \frac{1}{4\pi}\int_{\mathbb {R}^{3}}\frac {K(y)u_{\lambda}^{2}(y)}{\vert x-y\vert }\,dy\). Denote
It is clear that α, β, μ are nonnegative. Then (\(f_{1}\))-(\(f_{2}\)) and (\(K_{3}\)) imply that δ, δ̅, ν are nonnegative too. Then the Pohožaev identity of (5.20) and \(\langle J'_{\lambda }(u_{\lambda}),u_{\lambda}\rangle=0\) can be written as follows:
Note that (\(K_{4}\)) implies that \(\nu+\overline{\nu}\) is nonnegative. Then we conclude
Hence, we derive
On the other hand, for each nontrivial positive solution \(w^{k}\) (\(k=1,2,\ldots, l\)) of problem (5.14), we have the following Pohožaev identity:
It follows from (5.19) that
Then there exists \(t_{k}\in(0,1]\) such that \((w^{k})_{t_{k}}\in\mathcal {M}_{\lambda}^{\infty}\), that is, \(G_{\lambda}^{\infty}((w^{k})_{t_{k}})=0\). By direct calculation we obtain
Hence we get that \(c_{\lambda}\geqslant m_{\lambda}^{\infty}\), which contradicts to Lemma 5.8. So \(u_{n}\rightarrow u_{\lambda}\) in \(H^{1}(\mathbb {R}^{3})\). As a consequence, we obtain that \(I_{\lambda }'(u_{\lambda})=0\) and \(I_{\lambda}(u_{\lambda})=c_{\lambda}\). □
To prove that problem (1.4) has a positive ground state solution, we define
Proof of Theorem 2.2
We complete the proof in three steps.
Step 1. \(\mathcal{S}\neq\emptyset\).
By Lemma 5.7, Lemma 5.10, and Proposition 3.1, for a.e. \(\lambda\in J\), there exists \(u_{\lambda}\in H^{1}(\mathbb {R}^{3})\backslash\{0\}\) such that \(I_{\lambda }'(u_{\lambda})=0\) and \(I_{\lambda}(u_{\lambda})=c_{\lambda}\). Choosing a sequence \(\{\lambda_{n}\}\subset J\) satisfying \(\lambda _{n}\rightarrow1^{-}\), we have a sequence \(\{u_{{\lambda}_{n}}\}\subset H^{1}(\mathbb {R}^{3})\backslash\{0\}\) such that \(I_{{\lambda}_{n}}'(u_{{\lambda}_{n}})=0\) and \(I_{{\lambda }_{n}}(u_{{\lambda}_{n}})=c_{{\lambda}_{n}}\). For simplicity, denoting \(\{u_{n}\}\) instead of \(\{u_{{\lambda}_{n}}\} \), we next show that \(\{u_{n}\}\) is bounded in \(H^{1}(\mathbb {R}^{3})\).
Denote
Then we have
Hence, we have
Note that \(\alpha_{n}\), \(\beta_{n}\), \(\mu_{n}\) are nonnegative. Conditions (\(f_{1}\))-(\(f_{2}\)), (\(K_{3}\)), and (\(K_{4}\)) imply that \(\delta_{n}\), \(\overline {\delta}_{n}\), \(\nu_{n}\), \(\nu_{n}+\overline{\nu}_{n}\) are nonnegative too. Then we conclude that \(\delta_{n}\) and \(\nu_{n}\) are bounded. Moreover, by the third equation we have that \(\alpha_{n} +\beta_{n}\) is bounded, that is, \(\{u_{n}\}\) is bounded in \(H^{1}(\mathbb {R}^{3})\). Therefore, we get
and
that is, \(\{u_{n}\}\) is a bounded \((\mathrm{PS})_{c_{1}}\) sequence for \(I_{1}\). By Lemma 5.10 there exists \(u_{0}\in H^{1}(\mathbb {R}^{3})\backslash\{0\}\) such that \(I_{1}'(u_{0})=0\) and \(I_{1}(u_{0})=c_{1}\).
Step 2. \(0< m<\infty\).
It is clear that \(m\leqslant c_{1}<\infty\). Next, we prove \(m\geqslant0\). For all \(u\in\mathcal{S}\), we have \(\langle I_{1}'(u),u\rangle=0\). Then by a standard argument we get that \(\Vert u\Vert \geqslant\delta\) for some \(\delta>0\) (see Lemma 5.4(i)). On the other hand, by the Pohoz̆aev identity of problem (1.4),
where \(\phi_{u}(x)= \frac{1}{4\pi}\int_{\mathbb {R}^{3}}\frac {K(y)u^{2}(y)}{\vert x-y\vert }\,dy\), we have
From (\(f_{1}\)), (\(f_{2}\)), and (\(K_{2}\))-(\(K_{4}\)) we infer
Therefore, we obtain that \(m\geqslant0\). In the following, we will prove that \(m>0\). By contradiction, let \(\{u_{n}\}\) be a \((\mathrm{PS})_{0}\) sequence of \(I_{1}\). Inequality (5.24) implies that \(\lim _{n\rightarrow \infty} \vert \nabla u_{n}\vert ^{2}=0\). This conclusion, combined with \(\langle I'_{1}(u),u \rangle=0\), implies that \(\lim _{n\rightarrow\infty} u_{n}^{2}=0\). Therefore, we obtain \(\lim _{n\rightarrow\infty} \Vert u_{n}\Vert =0\), a contradiction with \(\Vert u_{n}\Vert \geqslant\delta>0\) for all n.
Step 3. m is is attained at some \(u\in\mathcal{S}\).
Let \(\{u_{n}\}\subset\mathcal{S}\) satisfy \(I_{1}(u_{n})\rightarrow m\). Using the same arguments as in Step 1, we can deduce that \(\{u_{n}\}\) is bounded in \(H^{1}(\mathbb {R}^{3})\), that is, \(\{u_{n}\}\) is a bounded \((\mathrm{PS})_{m}\) sequence of \(I_{1}\). Similarly to the arguments in Lemma 5.10, there exists a nontrivial point \(u\in H^{1}(\mathbb {R}^{3})\) such that \(I_{1}(u)=m\) and \(I_{1}'(u)=0\). By the standard regularity arguments as in the proof of Theorem 1.4 of [11] we see that u is a positive ground state solution for problem (1.4). □
References
Kirchhoff, G: Mechanik. Teubner, Leipzig (1883)
Benci, V, Fortunato, D: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 11, 283-293 (1998)
Kirchhoff, G: Vorlesungen Über Mathematische Physik. Teubner, Leipzig (1876)
Benci, V, Fortunato, D: Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations. Rev. Math. Phys. 14, 409-420 (2002)
Azzollini, A, Pomponio, A: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 345, 90-108 (2008)
Ruiz, D: The Schrödinger-Poisson equation under the effect of a nonlinear local term. J. Funct. Anal. 237, 655-674 (2006)
Cerami, G, Vaira, G: Positive solutions of some non-autonomous Schrödinger-Poisson systems. J. Differ. Equ. 248, 521-543 (2010)
Huang, L, Rocha, EM, Chen, J: Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity. J. Differ. Equ. 255, 2463-2483 (2013)
Vaira, G: Ground states for Schrödinger-Poisson type systems. Ric. Mat. 60, 263-297 (2011)
Gao, Y, Yu, S, Tang, C: Positive ground state solutions to Schrödinger-Poisson systems with a negative non-local term. Electron. J. Differ. Equ. 2015, 118 (2015)
Li, G, Ye, H: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \(\mathbb {R}^{3}\). J. Differ. Equ. 257, 566-600 (2014)
Jeanjean, L: Local condition insuring bifurcation from the continuous spectrum. Math. Z. 232, 651-664 (1999)
Jeanjean, L: On the existence of bounded Palais-Smal sequences and application to a Landsman-Lazer-type problem set on \(\mathbb {R}^{N}\). Proc. Edinb. Math. Soc. 129(2), 787-809 (1999)
Willem, M: Minimax Theorems. Springer, Media (1997)
Azzollini, A, Pomponio, A: On the Schrödinger equation in \(\mathbb {R}^{n}\) under the effect of a general nonlinear term. Indiana Univ. Math. J. 58, 1361-1378 (2009)
Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Different Equations. Cbms, vol. 65. AMS, Providence (1986)
Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Cambridge University Press, Cambridge (1952)
Acknowledgements
The authors would like to thank the referee for his/her valuable observations, which greatly improved the paper. SW is supported by the Science Council of Shanxi Province (No. 201601D102001).
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FL participated in the design of the study and drafted the manuscript. SW carried out the theoretical studies and helped to draft the manuscript. Both authors read and approved the final manuscript.
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Liu, F., Wang, S. Positive solutions of Schrödinger-Kirchhoff-Poisson system without compact condition. Bound Value Probl 2017, 156 (2017). https://doi.org/10.1186/s13661-017-0884-8
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DOI: https://doi.org/10.1186/s13661-017-0884-8