Abstract
In this paper, by using a change of variable and the mountain-pass theorem, we show that the following quasilinear Schrödinger systems
have a nontrivial solution \((u, v)\) for all \(\lambda >\lambda _{1}(\kappa )\), where \(N\geq 3, V_{1}(x), V_{2}(x)\) are positive continuous functions, κ, λ are positive parameters, and nonlinear terms \(f, h\in C(\mathbb{R}^{N}\times \mathbb{R}^{2}, \mathbb{R})\). Our main contribution is that we can deal with the case when \(\kappa >0\) is large for the above systems.
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1 Introduction
In this paper, we consider the following quasilinear Schrödinger systems of the form
where \(N\geq 3, V_{1}(x), V_{2}(x)\) are positive continuous functions, κ, λ are positive parameters and the nonlinear term \(f, h\in C(\mathbb{R}^{N}\times \mathbb{R}^{2}, \mathbb{R})\).
Quasilinear Schrödinger systems like (1.1) are in part motivated by the following quasilinear Schrödinger equations
where \(z: \mathbb{R}\times \mathbb{R}^{N}\rightarrow \mathbb{C}\), \(W: \mathbb{R}^{N}\rightarrow \mathbb{R}\) is a given potential, κ is a positive parameter and \(l: \mathbb{R}\rightarrow \mathbb{R}\), \(k: \mathbb{R}^{N}\times \mathbb{R}\rightarrow \mathbb{R}\) are continuous functions. The quasilinear Schrödinger equations (1.2) describe several physical phenomena with different l and k, see [3, 4, 10, 12, 14] and the references therein. In this paper, we are interested in the existence of standing-wave solutions for (1.2), that is, solutions of the form \(z(t, x)=\exp (-iEt)u(x)\), where \(E\in \mathbb{R}\) and u is a real function. Submitting \(z(t,x)\) into (1.2) and denoting \(k(x, z)=\zeta (z^{2})z\), we obtain
If we take \(l(t)=t\), \(\zeta (u^{2})u=\omega (u)\), then (1.3) can be reduced to the following equations:
Letting \(\kappa \neq 0\), equations (1.4) are quasilinear Schrödinger equations and \([\triangle |u|^{2}]u\) is a quasilinear term. Compared to the semilinear case, quasilinear equations become more complicated due to the quasilinear and nonconvex term \([\triangle |u|^{2}]u\). One of the main difficulties of (1.4) is that the functional \(\int _{\mathbb{R}^{N}}u^{2}|\nabla u|^{2}\,dx\) of the quasilinear term \([\triangle |u|^{2}]u\) is not smooth in usual Sobolev space \(H^{1}(\mathbb{R}^{N})\). By using the change of variable (dual approach) \(s=G^{-1}(t)\) for \(t\in [0, +\infty )\), where
and \(G^{-1}(t)=-G^{-1}(-t)\) for \(t\in (-\infty , 0)\), quasilinear equations (1.4) can be reduced to the semilinear one
Then, an Orlicz-space framework can be used to prove the existence of nontrivial solutions via minimax methods. Since \(1-\kappa t^{2}\) may be negative with \(\kappa >0\), a change of variable (1.5) is not adequate to study the existence of nontrivial solutions for these quasilinear equations. Letting \(1-\kappa t^{2}>0\), integral (1.5) makes sense and the inverse function \(G^{-1}(t)\) exists. When \(\kappa >0\) is small enough, many papers have studied the existence results of nontrivial solutions or multiple solutions for (1.4) via dual-approach techniques and variational methods, see [1, 2, 6, 7, 16, 18–20].
Moreover, when \(\kappa >0\) is large, letting \(1-\kappa t^{2}>0\), a change of variable (1.5) is also used to study the existence of solutions for quasilinear equations, but now the nonlinearity \(\omega (u)\) needs to be multiplied by a large constant λ. For example, Huang and Jia [11] (\(\kappa =2\)), Li and Huang [13], and Liang, Gao and Li [15] obtained the existence of nontrivial solutions for the following equations
For quasilinear Schrödinger systems like (1.1), when \(\kappa >0\) is small enough, Chen and Zhang proved the existence of a positive ground-state solution [8] and a nonradially symmetrical nodal solution [9] for the following quasilinear Schrödinger systems, respectively,
and
where \(\alpha , \beta >1\), \(2<\alpha +\beta <2^{*}\), potential function \(A(x)\) is radially symmetric and \(B>0\). However, these papers did not consider the existence of the nontrivial solutions for systems (1.1) when \(\kappa >0\) is large. To the best of our knowledge, there are no results for this problem in the literature. In this paper, we will be concerned with this problem.
Throughout this paper, we assume that \(V_{1}(x), V_{2}(x)\in C(\mathbb{R}^{N}, \mathbb{R})\) and satisfy the following conditions
- (\(\mathcal{V}_{1}\)):
-
\(0< V_{0}:=\min \{\inf_{x\in \mathbb{R}^{N}}V_{1}(x), \inf_{x\in \mathbb{R}^{N}}V_{2}(x) \}\);
- (\(\mathcal{V}_{2}\)):
-
there exists \(M_{0}>0\) such that for all \(M\geq M_{0}\)
$$ \mu \bigl(\bigl\{ x\in \mathbb{R}^{N}: V_{i}(x)\leq M \bigr\} \bigr)< +\infty ,\quad i=1, 2, $$
where μ denotes the Lebesgue measure in \(\mathbb{R}^{N}\). Moreover, suppose the nonlinearities f, h satisfy the following conditions
-
(h1)
\(\lim_{(s,t)\to (0, 0)}\frac{f(x, s, t)}{|(s, t)|}=\lim_{(s,t)\to (0, 0)}\frac{h(x, s, t)}{|(s, t)|}=0\);
-
(h2)
There is a constant \(C>0\), such that \(\langle \nabla \eta (x, s, t), (s, t)\rangle \leq C(|(s, t)|+|(s, t)|^{q})\), \(\forall t\in \mathbb{R}\), \(2< q<\frac{2^{*}+2}{2}\);
-
(h3)
There is \(\theta \in (2, 2^{*})\) such that \(0<\theta \eta (x, s, t)\leq (s, t)\nabla \eta (x, s, t)\) and \(uf(x, u, v)\geq 0\), \(vh(x, u, v)\geq 0\), where \(\nabla \eta (x, s, t)=(f(x, s, t), h(x, s, t))\).
Our main results in this paper are as follows.
Theorem 1.1
Assume that (\(\mathcal{V}_{1}\)), (\(\mathcal{V}_{2}\)), and (h1)–(h3) hold. Then, for given \(\kappa >0\), there exists \(\lambda _{1}(\kappa )>0\) such that for all \(\lambda >\lambda _{1}(\kappa )\), systems (1.1) have a nontrivial solution \((u, v)\in \mathcal{H}\) satisfying \(\max_{x\in \mathbb{R}^{N}}|(u(x), v(x)|\leq \sqrt{ \frac{1}{2\kappa}}\).
Remark 1.1
In [17], Sever and Silva obtained the existence of nontrivial solutions for (1.1) with \(\kappa =-2\), \(\lambda =1\) under conditions (h1)–(h3). \(f(x, u, v)=\frac{2\alpha}{\alpha +\beta}|u|^{\alpha -2}u|v|^{\beta}\), \(h(x, u, v)=\frac{2\beta}{\alpha +\beta}|u|^{\alpha}|v|^{\beta -2}v\) satisfy (h1)–(h3), where \(\alpha , \beta >1\).
The remainder of this paper is organized as follows. In Sect. 2, we give some preliminaries. In Sect. 3, we show the existence of a nontrivial solution \((z_{\kappa}, w_{\kappa})\) for the modified problem via the mountain-pass theorem. In Sect. 4, we use the Morse iteration technique to obtain \(L^{\infty}\)-estimate for \((z_{\kappa}, w_{\kappa})\) and finally we obtain the solutions for the original systems (1.1).
Throughout this paper, we use the standard notations. We use \(\|\cdot \|_{q}\) (\(1< q\leq \infty \)) that is a standard norm in the usual Lebesgue space \(L^{q}({\mathbb{R}^{N}})\). \(o_{n}(1)\) will always denote the quantities tending to 0 as \(n\to \infty \). ⇀ and → denote weak and strong convergence. \(B_{R}(0)\) denotes a ball centered at the origin with radius \(R>0\). \(C, C_{0}, C_{1}, \ldots \) denote positive constants.
2 Preliminaries
The energy functional associated with (1.1) is
where \(\nabla \eta (x, s, t)=(f(x, s, t), h(x, s, t))\). Since the terms \(\int _{\mathbb{R}^{N}}u^{2}|\nabla u|^{2}\,dx\) and \(\int _{\mathbb{R}^{N}}v^{2}|\nabla v|^{2}\,dx\) are not well defined in the usual Sobolev spaces, the functional \(I_{\kappa}\) may not be smooth. Hence, we cannot directly apply variational methods to obtain the critical points of \(I_{\kappa}\).
We define \(\mathcal{H}=\mathcal{H}_{1}\times \mathcal{H}_{2}\) with the norm
where \(\mathcal{H}_{i}, i=1, 2\) are Banach spaces and
endowed with the norm
\(H^{1}(\mathbb{R}^{N})\) is the usual Sobolev space.
We say \((u, v): \mathbb{R}^{N}\times \mathbb{R}^{N}\to \mathbb{R}\times \mathbb{R}\) is a (weak) solution of (1.1) if \((u, v)\in \mathcal{H}\) and it holds that
for all \(\varphi , \psi \in C^{\infty}_{0}(\mathbb{R}^{N})\).
In order to obtain nontrivial (weak) solutions of (1.1), we assume that \(1-\kappa u^{2}>0\). Moreover, we define \(g: \mathbb{R}\to \mathbb{R}^{+}\) as follows
Then, \(g\in C^{1} (\mathbb{R}, (\frac{\sqrt{2}}{4}, 1 ] )\), g is even, increasing in \((-\infty , 0)\) and decreasing in \([0, +\infty )\).
Motivated by [2], we consider the existence of nontrival solutions for the following modified quasilinear Schrödinger systems:
where \(g(t)\) is defined in (2.1). Also, we say \((u, v)\) is a (weak) solution of (2.2) if \((u, v)\in \mathcal{H}\) and
for \(\varphi , \psi \in C^{\infty}_{0}(\mathbb{R}^{N})\).
Clearly, if we obtain a solution u of (2.2) that satisfies \(\|(u, v)\|_{\infty}<1/\sqrt{2\kappa}\), then \((u, v)\) is a solution of (1.1). By using the following change of variable
then we see that the problem (2.2) can be reduced to the following semilinear Schrödinger systems:
where \(G^{-1}(z)\), \(G^{-1}(w)\) are the inverse of \(G(u)\), \(G(v)\). The energy functional associated with (2.4) is
It is easy to prove that \(J_{\kappa}\) is well defined in \(\mathcal{H}\) and \(J_{\kappa}\in C^{1}(\mathcal{H}, \mathbb{R})\) under our assumptions and the following lemma (cf. [19, Lemma 2.1]).
Lemma 2.1
The functions \(g(t)\), \(G(t)\) enjoy the following properties:
-
(i)
G is inverse, \(G(t)\) and the inverse \(G^{-1}(t)\) are odd;
-
(ii)
\(-1\leq \frac{t}{g(t)}g'(t)\leq 0\) for all \(t\in \mathbb{R}\);
-
(iii)
\(|t|\leq |G^{-1}(t)|\leq 2\sqrt{2}|t|\) for all \(t\in \mathbb{R}\);
-
(iv)
\(\lim_{t\rightarrow 0}\frac{G^{-1}(t)}{t}=1\), \(\lim_{t \rightarrow \infty}\frac{G^{-1}(t)}{t}=2\sqrt{2}\);
-
(v)
\(g(G^{-1}(t))\leq \frac{t}{G^{-1}(t)}\) for all \(t\in \mathbb{R}\).
The following lemma shows that any critical point \((z, w)\in \mathcal{H}\) of \(J_{\kappa}\) is a (weak) solution of (2.2).
Lemma 2.2
Assume that \((\mathcal{V}_{1})\), \((\mathcal{V}_{2})\), and (h1)–(h3) hold. If \((z, w)\in \mathcal{H}\) is a critical point of \(J_{\kappa}\), then \((u, v)=(G^{-1}(z), G^{-1}(w))\) is a weak solution of (2.2).
Proof
Since \((z, w)\in \mathcal{H}\) is a critical point of \(J_{\kappa}\), we have
for all \((\varphi , \psi )\in \mathcal{H}\). It also implies from \((\mathcal{V}_{1})\) and Lemma 2.1(iii) that \((u, v):=(G^{-1}(z), G^{-1}(w))\in \mathcal{H}\). For each \(\varphi _{1}, \psi _{1}\in C^{\infty}_{0}(\mathbb{R}^{N})\) and taking \((\varphi , \psi ):=(g(u)\varphi _{1}, g(v)\psi _{1})\in \mathcal{H}\) in (2.6), we obtain that
Note that \(z=G(u)\), \(w=G(v)\) and \(\nabla z=g(u)\nabla u\), \(\nabla w=g(v)\nabla v\), then we obtain (2.3). Therefore, \((u, v)\) is a weak solution of (2.2). □
Denote \([L^{r}(\mathbb{R}^{N}) ]^{2}=L^{r}(\mathbb{R}^{N}) \times L^{r}(\mathbb{R}^{N})\) with the norm
where \(L^{r}(\mathbb{R}^{N})\) is the Lebesgue function space with the norm
Now, we state the Sobolev embedding Lemma.
Lemma 2.3
Assume that \((\mathcal{V}_{1})\) and \((\mathcal{V}_{2})\) hold. Let \(\{z_{n}\}\) and \(\{w_{n}\}\) be bounded in \(\mathcal{H}\). Then, there exist \(z, w\in \mathcal{H}\cap L^{r}(\mathbb{R}^{N})\) such that up to a subsequence, \(z_{n}\to z\), \(w_{n}\to w\) in \(L^{r}(\mathbb{R}^{N}), r\in [2, 2^{*})\).
Proof
It is analogous to the proof of [5]. □
3 The modified systems
In this section, we shall prove the existence of nontrivial solutions for the modified systems (2.4) via the mountain-pass theorem.
Lemma 3.1
Assume that (h1)–(h3) hold, then
-
(i)
there exist \(\rho , \alpha >0\) such that \(J_{\kappa}(z, w)\geq \alpha \) for all \((z, w)\) satisfying \(\|(z, w)\|=\rho \);
-
(ii)
there is \((z, w)\in \mathcal{H}\setminus \{(0, 0)\}\) such that \(J_{\kappa}(z, w)\leq 0\).
Proof
(i) By (h1) and (h2), for each \(\varepsilon >0\) there exists a constant \(C=C(\varepsilon )>0\) such that
where \(2< q<2^{*}\). Then,
Letting \(\varepsilon =\min \{\frac{V_{1}(x)}{16\lambda}, \frac{V_{2}(x)}{16\lambda} \}\), it follows from Lemma 2.1 (iii), (3.2), and Lemma 2.3 that there exists a constant \(C_{r}>0\) such that
hence, we may choose \(\|(z, w)\|=\rho \) so small that
(ii) Choose \((\tau _{1}, \tau _{2})\in \mathcal{H}\) with \(\tau _{1}, \tau _{2}>0\). Then, by Lemma 2.1 (iii) we have \(|\tau _{i}|^{2}\leq \frac{|G^{-1}(t\tau _{i})|^{2}}{t^{2}}\leq 8| \tau _{i}|^{2}\), \(i=1, 2\). It follows from (h3) that \(\lim_{|(\nu _{1}, \nu _{2})|\to +\infty} \frac{\eta (x, \nu _{1}, \nu _{2})}{|(\nu _{1}, \nu _{2})|^{2}}=+ \infty \). Hence, we have
therefore, there is a sufficiently large \(t_{0}>0\); let \((z, w)=(t_{0}\tau _{1}, t_{0}\tau _{2})\) with \(\|(z, w)\|>\rho \) such that \(J_{\kappa}(z, w)\leq 0\). □
It follows from Lemma 3.1 that \(J_{\kappa}\) has a (PS)c sequence \(\{(z_{n}, w_{n})\}\subset \mathcal{H}\) such that
where
Lemma 3.2
Assume that (h3) holds, then any (PS)c sequence \(\{(z_{n}, w_{n})\}\subset \mathcal{H}\) obtained in (3.3) is bounded.
Proof
By (3.3), Lemma 2.1(ii), (iii), and (h3), one has
which implies that \(\{(z_{n}, w_{n})\}\subset \mathcal{H}\) is bounded. □
Since the sequence \(\{(z_{n}, w_{n})\}\) given by Lemma 3.2 is bounded in \(\mathcal{H}\), there exists \((z, w)\in \mathcal{H}\) and a subsequence of \(\{(z_{n}, w_{n})\}\), still denoted by \(\{(z_{n}, w_{n})\}\), such that
Now, we denote the functional \(J_{\kappa}\) given in (2.5) by
where
and
Lemma 3.3
Assume that (h1), (h2), \((\mathcal{V}_{1})\), \(\textit{and} (\mathcal{V}_{2})\) hold, \(\{(z_{n}, w_{n})\}\subset \mathcal{H}\) is a (PS)c sequence such that \((z_{n}, w_{n})\rightharpoonup (z, w)\) in \(\mathcal{H}, n\to \infty \), then
Proof
By Lemma 2.3, since \(z_{n}\to z\), \(w_{n}\to w\) in \(L^{r}(\mathbb{R}^{N}), r\in [2, 2^{*})\), for every small \(\varepsilon _{1}>0\), there exists \(R_{1}>0\) such that
Then,
We derive from (3.5) that
By Lemma 2.1(iii), since \(\vert \frac{G^{-1}(t)}{g(G^{-1}(t))}t \vert \leq 8|t|^{2}\), it follows from (3.11) that
Together with Lemma 2.1 (iii), (3.1), and the Hölder inequality, we obtain
By (3.8), we have
and
Hence,
It follows from (3.5) that
Combining (3.13) and (3.14), we have
Then, we conclude (3.6) from (3.11), (3.12), and (3.15). (3.7) can be proved similarly. □
Lemma 3.4
Suppose that \((\mathcal{V}_{1})\), \((\mathcal{V}_{2})\), (h1) and (h2) hold, then any (PS)c sequence \(\{(z_{n}, w_{n})\}\subset \mathcal{H}\) obtained in (3.3) has a strong convergence subsequence.
Proof
By Lemma 3.2, \(\{(z_{n}, w_{n})\}\) is bounded in \(\mathcal{H}\), up to a subsequence, we may assume that \((z_{n}, w_{n})\rightharpoonup (z, w)\in \mathcal{H}\) as \(n\to \infty \), and the fact that \(\langle J'_{\kappa}(z_{n}, w_{n}), (z_{n}, w_{n})\rangle =o_{n}(1)\) and Lemma 3.3 imply
On the other hand, it follows from \(\langle J'_{\kappa}(z_{n}, w_{n}), (z, w)\rangle =o_{n}(1)\) that
Then,
that is,
Hence, \((z_{n}, w_{n})\to (z, w)\) in \(\mathcal{H}\). □
By Lemmas 3.1–3.4, we have the following result.
Theorem 3.5
Suppose that \((\mathcal{V}_{1})\), \((\mathcal{V}_{2})\), and (h1)–(h3) hold, then the problem (2.4) has a nontrivial solution.
Proof
By Lemmas 3.1 and 3.2, \(J_{\kappa}\) has a bounded (PS)c sequence \(\{(z_{n}, w_{n})\}\subset \mathcal{H}\). It follows from Lemma 3.4 that there is a sequence of \(\{(z_{n}, w_{n})\}\), up to a subsequence, such that \((z_{n}, w_{n})\to (z_{\kappa}, w_{\kappa})\) in \(\mathcal{H}\) as \(n\to \infty \) satisfying \(J_{\kappa}(z_{\kappa}, w_{\kappa})=c\geq \rho >0\), which means that \((z_{\kappa}, w_{\kappa})\) is a nontrivial solution of (2.4). □
4 Proof of the main results
In this section, we will prove the main results. We note that the solution \((u_{\kappa}, v_{\kappa})=(G^{-1}(z_{\kappa}), G^{-1}(w_{\kappa}))\) may not be a solution of the systems (1.1). In order to find a solution of the original systems (1.1), we establish a result of \(L^{\infty}\)-estimate for \((z_{\kappa}, w_{\kappa})\).
Lemma 4.1
Assume that \((z_{\kappa}, w_{\kappa})\) (denoted by \((z, w)\)) is a nontrivial critical point of \(J_{\kappa}\) and \(J_{\kappa}(z, w)=c\), then there exists a positive constant C independent of λ such that
Proof
By (h3), Lemma 2.1 (ii) and (iii), and (3.4), we obtain
Hence,
This completes the proof. □
Lemma 4.2
Suppose that \((z, w)\) is a positive solution of (2.4), then there exists a constant \(C_{1}>0\) that is independent of λ such that
Proof
For each \(m\in \mathbb{N}\), let \(\beta >1\) be a constant to be determined, we set
and
Clearly, \((u_{m}, v_{m}), (z_{m}, w_{m})\in \mathcal{H}\). Since \((z, w)\) is a nontrivial solution of Eq. (2.4), we obtain
Furthermore, we have
It follows from (4.2) and (4.3) that
By (4.4), (4.5), and the fact of \(\beta >1\), we deduce that
for \(0<\varepsilon <\min \{\frac{V_{1}(x)}{\lambda}, \frac{V_{2}(x)}{\lambda} \}\). It follows from (iii) of Lemma 2.1, \(1<\frac{1}{g(t)}<2\sqrt{2}\), and \(zu_{m}=z^{2}_{m}\), \(wv_{m}=w^{2}_{m}\) that
If \(\mu (a)+\mu (b)\leq \nu (a)+\nu (b)\), we have \(\mu (a)\leq \nu (a)\), \(\mu (b)\leq \nu (b)\). By (4.7), we can obtain
By the Sobolev inequality, there is \(S>0\) such that
combining (4.6) and the Hölder inequality, we know that
where \(\frac{1}{q_{1}}+\frac{q-2}{2^{*}}=1\). Note that \(|z_{m}|=|z|^{\beta}\) in \(A_{m}\) and \(|z_{m}|\leq |z|^{\beta}\) in \(\mathbb{R}^{N}\), hence
Letting \(m\rightarrow \infty \) in the above inequality, we have
Denote \(\sigma =\frac{2^{*}}{2q_{1}}\). Now, taking \(\beta =\sigma \) in (4.8), we see that
Taking \(\beta =\sigma ^{2}\) in (4.8), we obtain that
It follows from (4.9) and (4.10) that
Continuing in this way by taking \(\beta =\sigma ^{i}\) (\(i=1, 2, \ldots \)) in (4.8), we obtain
It follows from the Sobolev inequality and letting \(j\rightarrow +\infty \), we obtain
Similarly, we may obtain
Therefore, by (4.11) and (4.12), we obtain
where \(C_{1}, C_{2}>0\) are independent of λ. □
Proof of Theorem 1.1
Let \(\delta >0\) be such that the set
is nonempty. By (h3), for \(x\in T\), there exists \(C_{1}>0\) such that
By Theorem 3.5, let \((z, w)\) be a critical point of \(J_{\kappa}\) and \(J_{\kappa}(z, w)=c\), together with Lemma 3.1 (3) and (4.14), one has
By (4.1), (4.13), and the continuous embedding \(\mathcal{H}\hookrightarrow L^{r}_{K}{(\mathbb{R}^{N})}\), \(r\in [2, 2^{*}]\), we have
Since \(2< q<(2^{*}+2)/2\), for fixing \(\kappa >0\), there is \(\lambda _{1}(\kappa )= (16C^{2}_{3}\kappa )^{ \frac{(2^{*}-q)(q-2)}{2(2^{*}-2q+2)}}\) such that for any \(\lambda >\lambda _{1}(\kappa )\), it holds that
thus, \((u, v)=(G^{-1}(z), G^{-1}(w))\) is a nontrivial solution of systems (1.1). □
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Funding
This work was supported by the National Natural Science Foundation of China (Grant No. 11901345), Yunnan Local Colleges Applied Basic Research Projects (Grant No. 202001BA070001-032), the Technology Innovation Team of University in Yunnan Province (Grant No. 2020CXTD25), and Yunnan Fundamental Research Projects (Grant No. 202101AT070057).
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Li, G. On the existence of nontrivial solutions for quasilinear Schrödinger systems. Bound Value Probl 2022, 40 (2022). https://doi.org/10.1186/s13661-022-01623-z
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DOI: https://doi.org/10.1186/s13661-022-01623-z