Abstract
We consider a Dirichlet problem driven by a (p(z), q(z))-Laplacian and a reaction involving the sum of a parametric singular term plus a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter \(\lambda >0\) varies. Also we show that for every admissible parameter the problem has a smallest positive solution and obtain the monotonicity and continuity properties of the minimal solution map.
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1 Introduction
In this paper, we study the following anisotropic singular Dirichlet problem
In this problem, \(\Omega \subseteq \mathbb {R}^N\) is a bounded domain with a \(C^2\)-boundary \(\partial \Omega \). If \(\displaystyle E_1=\{r\in C(\overline{\Omega }): 1<\min _{\overline{\Omega }}r\}\), then for \(r\in E_1\), by \(\Delta _{r(z)}\) we denote the anisotropic r-Laplace differential operator defined by
In contrast to the isotropic r-Laplacian (that is, \(r(\cdot )\) is constant), the anisotropic one is nonhomogeneous. In (\(\mathrm {P}_\lambda \)) we have the sum of two such operators. So, even in the isotropic case, the differential operator of (\(\mathrm {P}_\lambda \)) is not homogeneous and this makes the study of (\(\mathrm {P}_\lambda \)) more difficult. In the reaction [right-hand side of (\(\mathrm {P}_\lambda \))], we have the combined effects of two nonlinear terms of different nature. One is the parametric singular term \(u\mapsto \lambda u^{-\eta (z)}\) with \(\lambda >0\) being the parameter, with \(\eta \in C(\overline{\Omega })\) and \(\displaystyle 0<\eta _-=\min _{\overline{\Omega }}\eta \le \max _{\overline{\Omega }}\eta =\eta _+<1\). The other is a Carathéodory perturbation f(z, x) (that is, for all \(x\in \mathbb {R}\), \(z\mapsto f(z,x)\) is measurable and for a.a. \(z\in \Omega \), \(x\mapsto f(z,x)\) is continuous). We assume that for a.a. \(z\in \Omega \), \(f(z,\cdot )\) is \((p_+-1)\)-superlinear (\(\displaystyle p_+=\max _{\overline{\Omega }}p\)), but need not satisfy the usual in such cases Ambrosetti–Rabinowitz condition (the AR-condition for short). We search for the existence of positive solutions, and our goal is to produce a precise description of the changes in the set of positive solutions as the parameter \(\lambda \) varies in the open semiaxis \(\mathring{\mathbb {R}}_+=(0,+\infty )\).
Anisotropic boundary value problems have been studied extensively in the last decade. We refer to the books of Diening–Harjulehto–Hästo–Růžička [4] and Rădulescu–Repovš [19] and the references therein. The study of singular anisotropic problems is lagging behind. Very few works on the subject can be found in the literature. We mention two recent ones by Byun–Ko [2] and Saoudi–Ghanmi [22] which are closely related to our work here. Both papers deal with equations driven by the Dirichlet anisotropic p-Laplacian and their hypotheses on the data are more restrictive (see hypothesis (\(p_M\)) in Byun–Ko [2] and hypotheses \((\text {H}_1)\)–\((\text {H}_4)\) in Saoudi–Ghanmi [22]). In addition, our approach is different.
A main difficulty that we encounter when we deal with singular problems is that the corresponding energy (Euler) functional is not \(C^1\), and so we cannot use the results of critical point theory directly on it. Therefore, we need to find a way to isolate the singularity and deal with \(C^1\) functionals. For this reason, first we study an auxiliary purely singular problem for which we prove an existence and uniqueness result. Using this solution we are able to bypass the singularity and then, using variational tools from the critical point theory together with truncation and comparison techniques, we prove a bifurcation-type theorem describing the changes in the set of positive solutions of (\(\mathrm {P}_\lambda \)). According to our theorem, there exists a critical parameter \(\lambda ^*>0\) such that
-
for all \(\lambda \in (0,\lambda ^*)\) problem (\(\mathrm {P}_\lambda \)) has at least two positive smooth solutions;
-
for \(\lambda =\lambda ^*\) problem (\(\mathrm {P}_\lambda \)) has at least one positive smooth solution;
-
for all \(\lambda >\lambda ^*\) problem (\(\mathrm {P}_\lambda \)) has no positive solutions.
Moreover, we show that for every admissible parameter \(\lambda \in (0,\lambda ^*]\), problem (\(\mathrm {P}_\lambda \)) has a smallest positive solution \(u_\lambda ^*\) (minimal or barrier solution) and we establish the monotonicity and continuity properties of the map \(\lambda \mapsto u_\lambda ^*\).
For further details on the study of singular equations, we refer the reader to the papers [20, 21, 23].
2 Mathematical Background: Hypotheses
The study of problem (\(\mathrm {P}_\lambda \)) requires the use of Lebesgue and Sobolev spaces with variable exponents. A comprehensive presentation of the theory of these spaces can be found in the book of Diening–Harjuletho–Hästö–Růžička [4].
Let \(M(\Omega )\) be the vector space of all measurable functions from \(\Omega \) into \(\mathbb {R}\). We identify two such functions which differ only on a Lebesgue-null set. Given \(r\in E_1\), the anisotropic Lebesgue space \(L^{r(z)}(\Omega )\) is defined by
We equip this space with the so-called Luxemburg norm defined by
Closely related to this norm is the modular function \(\rho _r(\cdot )\) defined by
Evidently \(\Vert \cdot \Vert _{r(z)}\) is the Minkowski functional of the set
Also, we have the following proposition which illustrates the close relation between \(\Vert \cdot \Vert _{r(z)}\) and \(\rho _{r}(\cdot )\). Recall that, if \(r\in E_1\), then \(\displaystyle r_-=\min _{\overline{\Omega }}r\), \(\displaystyle r_+=\max _{\overline{\Omega }}r\).
Proposition 2.1
If \(r\in E_1\) and \(\{u_n,u\}_{n\in \mathbb {N}}\subseteq L^{r(z)}(\Omega )\), then
-
(a)
\(\Vert u\Vert _{r(z)}=\vartheta \Leftrightarrow \rho _r\left( \frac{u}{\vartheta }\right) =1\);
-
(b)
\(\Vert u\Vert _{r(z)}<1\) (resp. \(=1\), \(>1\)) \(\Leftrightarrow \; \rho _r(u)<1\) (resp. \(=1\), \(>1\));
-
(c)
\(\Vert u\Vert _{r(z)}\le 1\Rightarrow \Vert u\Vert _{r(z)}^{r_+}\le \rho _r(u)\le \Vert u\Vert _{r(z)}^{r_-}\); \(\Vert u\Vert _{r(z)}\ge 1\Rightarrow \Vert u\Vert _{r(z)}^{r_-}\le \rho _r(u)\le \Vert u\Vert _{r(z)}^{r_+}\);
-
(d)
\(\Vert u_n\Vert _{r(z)}\rightarrow 0\) (resp. \(\rightarrow \infty \)) \(\Leftrightarrow \rho _r (u_n)\rightarrow 0\) (resp. \(\rightarrow \infty \));
-
(e)
\(\Vert u_n-u\Vert _{r(z)}\rightarrow 0 \Leftrightarrow \rho _r (u_n-u)\rightarrow 0\).
The space \(L^{r(z)}(\Omega )\) (\(r\in E_1\)) is a Banach space which is separable, reflexive (in fact uniformly convex). Also, if \(r'\in E_1\) is given by \(r'(z)=\frac{r(z)}{r(z)-1}\) for all \(z\in \overline{\Omega }\) (that is, \(\frac{1}{r(z)}+\frac{1}{r'(z)}=1\) for all \(z\in \overline{\Omega }\)), then
Moreover, we have the following version of Hölder’s inequality
We know that, if \(q,r\in E_1\) and \(q(z)\le r(z)\) for all \(z\in \overline{\Omega }\), then
Using the anisotropic Lebesgue spaces, we can define also anisotropic Sobolev spaces.
Given \(r\in E_1\) we define
We equip this space with the following norm:
Here \(\Vert Du\Vert _{r(z)}=\Vert \,|Du|\,\Vert _{r(z)}\). Suppose that \(r\in E_1 \cap C^{0,1}(\overline{\Omega })\) (that is, assume that the variable exponent is Lipschitz continuous). We define
Both spaces \(W^{1,r(z)}(\Omega )\) and \(W_0^{1,r(z)}(\Omega )\) are Banach spaces which are separable and reflexive (in fact uniformly convex), Moreover, for the space \(W_0^{1,r(z)}(\Omega )\) the Poincaré inequality holds, namely
If \(r\in E_1 \cap C^{0,1}(\overline{\Omega })\), then we define
This is the variable critical Sobolev exponent corresponding to \(r(\cdot )\). Suppose \(q\in C(\overline{\Omega })\) and assume that
Let \(X=W^{1,r(z)}(\Omega )\) or \(X=W_0^{1,r(z)}(\Omega )\). Then, we have that \(X\hookrightarrow L^{q(z)}(\Omega )\) continuously (resp. \(X\hookrightarrow L^{q(z)}(\Omega )\) compactly). This is the so-called anisotropic Sobolev embedding theorem. For \(r\in E_1 \cap C^{0,1}(\overline{\Omega })\), we have
Consider the nonlinear operator \(A_{r(z)}:W_0^{1,r(z)}(\Omega )\rightarrow W^{-1,r'(z)}(\Omega )\) defined by
This operator has the following properties (see Gasiński–Papageorgiou [8], Proposition 2.5 and Rădulescu–Repovš [19], p. 40).
Proposition 2.2
The operator \(A_{r(z)}(\cdot )\) is bounded (maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone too) and of type \((S)_+\), that is, it has the following property
"If \(u_n \xrightarrow {w} u\) in \(W_0^{1,r(z)}(\Omega )\, \) and \(\,\displaystyle \limsup _{n\rightarrow \infty }\langle A_{r(z)}(u_n), u_n-u\rangle \le 0\),
then \(u_n\rightarrow u\) in \(W_0^{1,r(z)}(\Omega )\) as \(n\rightarrow \infty \)".
Another space that we will use (as a result of the anisotropic regularity theory) is the space \(C_0^1(\overline{\Omega })=\{u\in C^1(\overline{\Omega }): u|_{\partial \Omega }=0\}\). This is an ordered Banach space with positive (order) cone \(C_+=\{u\in C_0^1(\overline{\Omega }): u(z)\ge 0\; \text {for all }z\in \overline{\Omega }\}\). This cone has a nonempty interior given by
with \(n(\cdot )\) being the outward unit normal on \(\partial \Omega \).
Our hypotheses on the exponents involved in problem (\(\mathrm {P}_\lambda \)) are the following:
- \(\mathrm{H}_0\)::
-
\(p,q\in C^{0,1}(\overline{\Omega })\), \(\eta \in C(\overline{\Omega })\), \(q_-\le q_+<p_-\le p_+\) and \(0<\eta (z)<1\) for all \(z\in \overline{\Omega }\).
Consider \(h_1,h_2\in M(\Omega )\). We write \(h_1 \preceq h_2\) if and only if for every \(K\subseteq \Omega \) compact we have \(0<c_K\le h_2(z)-h_1(z)\) for a.a. \(z\in K\). Note that if \(h_1,h_2 \in C(\Omega )\) and \(h_1(z)<h_2(z)\) for all \(z\in \Omega \), then \(h_1\preceq h_2\). Using this ordering notion and following the argument in the proof of Proposition 2.4 of Papageorgiou–Rădulescu–Repovš [17], we obtain the following strong comparison theorem for singular problems.
Proposition 2.3
If hypotheses \(\mathrm {H}_0\) hold, \(\widehat{\xi }\in L^\infty (\Omega )\), \(\widehat{\xi }(z)\ge 0\) for a.a. \(z\in \Omega \), \(h_1,h_2\in L^\infty (\Omega )\), \(h_1\preceq h_2\), \(u\in W_0^{1,p(z)}(\Omega )\), \(u\ge 0\), \(u\ne 0\) and \(v\in \mathrm {int}\, C_+\) satisfy
then \(v-u\in \mathrm {int}\, C_+\).
For every \(u\in M(\Omega )\), we set \(u^\pm =\max \{\pm u,0\}\). If \(u\in W_0^{1,p(z)}(\Omega )\), then \(u^\pm \in W_0^{1,r(z)}(\Omega )\), \(u=u^+-u^-\), \(|u|=u^+ +u^-\). Also, if \(u,v\in M(\Omega )\) with \(u(z)\le v(z)\) for a.a. \(z\in \Omega \), then we introduce the following order intervals in \(W_0^{1,p(z)}(\Omega )\):
Throughout this work, by \(\Vert \cdot \Vert \) we denote the norm \(W_0^{1,p(z)}(\Omega )\). On account of the Poincaré inequality, we have
Suppose X is a Banach space and \(\varphi \in C^1(X,\mathbb {R})\). We set
We say that \(\varphi (\cdot )\) satisfies the C-condition, if the following property holds:
"If \(\{u_n\}_{n\in \mathbb {N}}\subseteq X\) is such that \(\{\varphi (u_n)\}_{n\in \mathbb {N}}\subseteq \mathbb {R}\) is bounded and \((1+\Vert u_n\Vert _X)\varphi '(u_n)\rightarrow 0\) in \(X^*\) as \(n\rightarrow \infty \), then \(\{u_n\}_{n\in \mathbb {N}}\) has a strongly convergent subsequence".
Now we will introduce our hypotheses on the perturbation f(z, x).
- \(\mathrm{H}_1\)::
-
\(f:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) is a Carathéodory function such that \(f(z,0)=0\) for a.a. \(z\in \Omega \) and
- (i):
-
\(f(z,x)\le a(z)[1+|x|^{r(z)-1}]\) for a.a. \(z\in \Omega \), all \(x\ge 0\), with \(a\in L^\infty (\Omega )\), \(a(z)\ge 0\) for a.a. \(z\in \Omega \), \(r\in C(\overline{\Omega })\), \(p(z)<r(z)<p^*(z)\) for all \(z\in \overline{\Omega }\);
- (ii):
-
if \(F(z,x)=\int _0^x f(z,s)\,\mathrm {d}s\), then \(\displaystyle \lim _{x\rightarrow +\infty }\frac{F(z,x)}{x^{p_+}}=+\infty \) uniformly for a.a. \(z\in \Omega \) and there exists \(\tau \in C(\overline{\Omega })\) such that
$$\begin{aligned}&\tau (z) \in \left( (r_+-p_-)\max \left\{ \frac{N}{p_-},1\right\} , p^*(z)\right) \quad \text {for all } z\in \overline{\Omega }\\&0<\widehat{\delta }\le \liminf _{x\rightarrow +\infty }\frac{f(z,x)x-p_+F(z,x)}{x^{\tau (z)}}\; \text {uniformly for a.a. } z\in \Omega ; \end{aligned}$$ - (iii):
-
\(\displaystyle \lim _{x\rightarrow 0^+}\frac{f(z,x)}{x^{q(z)-1}}=0\) uniformly for a.a. \(z\in \Omega \) and we have
$$\begin{aligned} 0\le f(z,x)\quad \text {for a.a. } z\in \Omega ,\text { all } x\ge 0; \end{aligned}$$ - (iv):
-
for every \(\rho >0\), we can find \(\widehat{\xi }_\rho >0\) such that for a.a. \(z\in \Omega \), the function
$$\begin{aligned}x\mapsto f(z,x)+\widehat{\xi }_\rho x^{p(z)-1}\end{aligned}$$is nondecreasing on \([0,\rho ]\).
Remarks
Since our goal is to find positive solutions and all the above hypotheses concern the positive semiaxis \(\mathbb {R}_+=[0,+\infty )\), without any loss of generality, we may assume that \(f(z,x)=0\) for a.a. \(z\in \Omega \), all \(x\le 0\). Hypothesis \(\mathrm {H}_1\)(ii) says that for a.a. \(z\in \Omega \), \(f(z,\cdot )\) is \((p_+-1)\)-superlinear. We do not use the AR-condition, which is common in the literature when dealing with superlinear problems (see, for example, Chang [3], p. 147). Hypothesis \(\mathrm {H}_1\)(ii) is less restrictive and incorporates in our framework also superlinear perturbations with "slower" growth as \(x\rightarrow +\infty \). For example, consider the following function
where \(\tau ,s\in C(\overline{\Omega })\) and \(q(z)<\tau (z)\le s(z)\) for all \(z\in \overline{\Omega }\). This function satisfies hypotheses \(\mathrm {H}_1\) but fails to satisfy the AR-condition. Our hypotheses also incorporate the nonlinearity \(f(z,x)=(x^+)^{r(z)-1}\) with \(r\in C(\overline{\Omega })\) and \(p(z)<r(z)\) for all \(z\in \overline{\Omega }\). This is the perturbation used by Byun–Ko [2], where the equation is driven only by the p(z)-Laplacian with \(p(\cdot )\) satisfying stronger conditions (see hypothesis (\(p_M\)) in [2]). Similarly, our hypotheses are more general than those used by Saoudi–Ghanmi [22] (see hypotheses (\(\mathrm {H}_3\)), (\(\mathrm {H}_4\)) in [22]), who also deal with equations driven by the p(z)-Laplacian only. The function f(z, x) given above lies outside the framework provided by the hypotheses in [22].
3 A Purely Singular Problem
As we already indicated in the Introduction, in order to handle the singular term, we will first consider a purely singular problem. The solution of this problem will allow us to bypass the singularity and deal with \(C^1\)-functionals.
So, in this section, we study the following purely singular problem
For this problem, we have the following existence and uniqueness result.
Proposition 3.1
If hypotheses \(\mathrm {H}_0\) hold, then for every \(\lambda >0\) problem (\(\mathrm {Au}_\lambda \)) has a unique positive solution \(\overline{u}_\lambda \in \mathrm {int}\, C_+\) and the map \(\mathring{\mathbb {R}}_+\ni \lambda \mapsto \overline{u}_\lambda \) is nondecreasing and \(\Vert \overline{u}_\lambda \Vert _{C_0^1(\overline{\Omega })}\rightarrow 0\) as \(\lambda \rightarrow 0^+\).
Proof
Given \(\delta \in (0,1)\), first we solve the following approximation to problem (\(\mathrm {Au}_\lambda \)):
To solve this problem, we employ a fixed point argument. So, let \(g\in C_0^1(\overline{\Omega })\) and consider the following Dirichlet problem
The operator \(V=A_{p(z)}+A_{q(z)}: W_0^{1,p(z)}(\Omega )\rightarrow W^{-1,p'(z)}(\Omega )\) is maximal monotone (see Proposition 2.2) and coercive. So \(V(\cdot )\) is surjective (see [15], p. 137). Since \(\frac{\lambda }{[|g|+\delta ]^{\eta (\cdot )}}\in L^\infty (\Omega )\), we can find \(v_\delta ^\lambda \in W_0^{1,p(z)}(\Omega )\) such that
On account of the strict monotonicity of \(V(\cdot )\) (see Proposition 2.2), this solution \(v_\delta ^\lambda \) is unique. We have
From Theorem 4.1 of Fan–Zhao [6], we have that \(v_\delta ^\lambda \in L^\infty (\Omega )\). Then Theorem 1.3 of Fan [5] implies that \(v_\delta ^\lambda \in C_+\setminus \{0\}\). Finally, Proposition 4 of Papageorgiou–Qin–Rădulescu [14] says that \(v_\delta ^\lambda \in \mathrm {int}\, C_+\).
We can define the solution map \(\sigma _\delta ^\lambda : C_0^1(\overline{\Omega })\rightarrow C_0^1(\overline{\Omega })\) by setting
We will show that this map is continuous. To this end, let \(g_n\rightarrow g\) in \(C_0^1(\overline{\Omega })\) and set \((v_\delta ^\lambda )_n=\sigma _\delta ^\lambda (g_n)\), for all \(n\in \mathbb {N}\). We have
In (3.2) we use the test function \((v_\delta ^\lambda )_n\in W_0^{1,p(z)}(\Omega )\). We obtain
This implies that we can find \(c_2>0\) such that
Then Lemma 3.3 of Fukagai–Narukawa [7] says that there exist \(\alpha \in (0,1)\) and \(c_3>0\) such that
Recall that \(C_0^{1,\alpha }(\overline{\Omega })\hookrightarrow C_0^1(\overline{\Omega })\) compactly. So, by passing to a suitable subsequence if necessary, we may assume that
Then, if we pass to the limit as \(n\rightarrow \infty \) in (3.2), we obtain
Also for every \(g\in C_0^1(\overline{\Omega })\) and with \(v_\delta ^\lambda =\sigma _\delta ^\lambda (g)\), we have
As above, using (3.4) and the anisotropic regularity theory, we infer that
Then, the Schauder–Tychonov fixed point theorem (see [15], p. 298) implies that there exists \(\overline{u}_\delta ^\lambda \in C_0^1(\overline{\Omega })\) such that
This \(\overline{u}_\delta ^\lambda \in \mathrm {int}\, C_+\) is a solution of the approximate problem (\(\mathrm {Au}_\lambda ^\delta \)). We show that this solution is unique. To this end, let \(\overline{y}_\delta ^\lambda \) be another solution of (\(\mathrm {Au}_\lambda ^\delta \)). We have
This proves the uniqueness of the solution \(\overline{u}_\delta ^\lambda \in \mathrm {int}\, C_+\) of problem (\(\mathrm {Au}_\lambda ^\delta \)).
Claim: \(0<\delta '<\delta \quad \Rightarrow \quad \overline{u}_\delta ^\lambda \le \overline{u}_{\delta '}^\lambda \).
We have
We introduce the Carathéodory function \(e_\delta ^\lambda : \Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by
We set \(E_\delta ^\lambda (z,x)=\int _0^x e_\delta ^\lambda (z,s)\,\mathrm {d}s\) and consider the \(C^1\)-functional \(\gamma _\lambda : W_0^{1,p(z)}(\Omega )\rightarrow \mathbb {R}\) defined by
We have
Also the anisotropic Sobolev embedding theorem implies that \(\gamma _\lambda (\cdot )\) is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find \(\widetilde{u}_\delta ^\lambda \in W_0^{1,p(z)}(\Omega )\) such that
In (3.7) we choose the test function \(h=-(\widetilde{u}_\delta ^\lambda )^-\in W_0^{1,p(z)}(\Omega )\). We obtain
Next in (3.7) we choose \(h=[\widetilde{u}_\delta ^\lambda -\overline{u}_{\delta '}^\lambda ]^+\in W_0^{1,p(z)}(\Omega )\). We have
So, we have proved that
This proves the claim.
Next we let \(\delta \rightarrow 0^+\) to produce a solution of the auxiliary problem (\(\mathrm {Au}_\lambda \)). Let \(\delta _n \rightarrow 0^+\) and set \(\overline{u}_n^\lambda =\overline{u}_{\delta _n}^\lambda \in \mathrm {int}\, C_+\), for all \(n\in \mathbb {N}\). We have
In (3.10) we use the test function \(\overline{u}_n^\lambda \in W_0^{1,p(z)}(\Omega )\). We obtain
On account of (3.11), we may assume that
From (3.9) we have
Then, for every \(h\in C_c^\infty (\Omega )\), we have that \(\overline{u}_1^\lambda (z)\ge c_h>0\) for all \(z\in \mathrm {supp}\, h\) (recall that \(\overline{u}_1^\lambda \in \mathrm {int}\, C_+\)). Hence, for \(h\in C_c^\infty (\Omega )\), we have
So, if in (3.10) we pass to the limit as \(n\rightarrow \infty \) and use the dominated convergence theorem [see (3.12)], we obtain
The density of \(C_c^\infty (\Omega )\) in \(W_0^{1,p(z)}(\Omega )\) implies that (3.13) holds for all \(h\in W_0^{1,p(z)}(\Omega )\) and so \(-\mathrm {div}\,\zeta =\frac{1}{\overline{u}_\lambda ^{\eta (\cdot )}}\).
As above, we have
Exploiting the density of \(C_c^{\infty }(\Omega )\) in \(W_0^{1,p(z)}(\Omega )\), we infer that (3.14) holds for all \(h\in W_0^{1,p(z)}(\Omega )\). It follows that
From (3.11), (3.15) and Theorem 2.1 of Boccardo–Murat [1], we have that \(D\overline{u}_n^\lambda (z)\rightarrow D\overline{u}_\lambda (z)\) for a.a. \(z\in \Omega \). It follows that
and then
Since \(\overline{u}_1^\lambda \le \overline{u}_\lambda \) (see (3.9), (3.12)), we infer that \(\overline{u}_\lambda \) is a positive solution of (\(\mathrm {Au}_\lambda \)). Then, from Theorem B.1 of Saoudi–Ghanmi [22] (see also Giacomoni–Schindler–Takáč [9], Theorem B.1), we have that \(\overline{u}_\lambda \in \mathrm {int}\, C_+\). As before, we check that \(\overline{u}_\lambda \in \mathrm {int}\, C_+\) is the unique positive solution of (\(\mathrm {Au}_\lambda \)).
Next we check that \(\lambda \mapsto \overline{u}_\lambda \) is nondecreasing from \(\mathring{\mathbb {R}}_+=(0,\infty )\) into \(\mathrm {int}\, C_+\). So, let \(0<\lambda <\lambda '\). We have
We consider the Carathéodory function \(\widehat{g}_\lambda : \Omega \times \mathring{\mathbb {R}}_+\rightarrow \mathring{\mathbb {R}}_+\) defined by
Note that from the proof of the Lemma of Lazer–McKenna [13], we have \(\overline{u}_{\lambda '}(\cdot )^{-\eta (\cdot )}\in L^1(\Omega )\) (recall that \(\overline{u}_{\lambda '}\in \mathrm {int}\,C_+\)). We consider the following Dirichlet problem
Reasoning as in the first part of the proof (using approximations of this problem), we show that it has a unique solution \(\widetilde{u}_\lambda \in \mathrm {int}\, C_+\) and \(\widetilde{u}_\lambda \le \overline{u}_{\lambda '}\). We conclude that \(\widetilde{u}_\lambda = \overline{u}_\lambda \) [see (3.16)] and so \(\overline{u}_\lambda \le \overline{u}_{\lambda '}\).
Finally let \(\lambda _n \rightarrow 0^+\) and let \(\overline{u}_n=\overline{u}_{\lambda _n}\in \mathrm {int}\, C_+\), \(n\in \mathbb {N}\), be the unique solution of \((\mathrm {P}_{\lambda _n})\) produced earlier. We have
Using the test function \(h=\overline{u}_n\in W_0^{1,p(z)}(\Omega )\) we obtain
Then, the anisotropic regularity theory (see [22]), implies that \(\{\overline{u}_n\}_{n\in \mathbb {N}}\subseteq C_0^1(\overline{\Omega })\) is relatively compact. Therefore \(\overline{u}_n\rightarrow 0\) in \(C_0^1(\overline{\Omega })\) [see (3.17)]. \(\square \)
4 Positive Solutions
We introduce the following two sets
Proposition 4.1
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold, then \(\mathscr {L}\ne \varnothing \) and for every \(\lambda >0\), \(S_\lambda \subseteq \mathrm {int}\, C_+\).
Proof
Hypotheses \(\mathrm {H}_1\)(i), (iii) imply that given \(\epsilon >0\), we can find \(c_8=c_8(\epsilon )>0\) such that for \(\widehat{r}\in C(\overline{\Omega })\) with \(\widehat{r}>r\), \(\widehat{r}_->p_+\), we have
Let \(\overline{u}_\lambda \in \mathrm {int}\, C_+\) be the unique solution of the auxiliary problem (\(\mathrm {Au}_\lambda \)). Recall that \(\overline{u}_\lambda (\cdot )^{-\eta (\cdot )}\in L^1(\Omega )\). We introduce the Carathéodory function \(k_\lambda : \Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by
We set \(K_\lambda (z,x)=\int _0^x k_\lambda (z,s)\,\mathrm {d}s\) and consider the \(C^1\)-functional \(\psi _\lambda : W_0^{1,p(z)}(\Omega )\rightarrow \mathbb {R}\) defined by
Using (4.1) and (4.2), we have
Let \(\widehat{d}(\cdot )=d(\cdot , \partial \Omega )\). From Lemma 14.16, p. 355, of Gilbarg–Trudinger [10], we know that we can find \(\delta _0>0\) such that \(\widehat{d}\in C^2(\Omega _{\delta _0})\) where \(\Omega _{\delta _0} = \{z\in \overline{\Omega }: \widehat{d}(z)<\delta _0\}\). It follows that \(\widehat{d}\in C_+\setminus \{0\}\). Since \(\overline{u}_\lambda \in \mathrm {int}\, C_+\), on account of Proposition 4.1.22, p. 274, of Papageorgiou– Rădulescu–Repovš [15], we can find \(c_9>0\) such that
Then we have
where in the last inequality, we have used the anisotropic Hardy inequality due to Harjuletho–Hästö–Koskenoja [11]. Returning to (4.3) and using (4.5), for all \(u\in W_0^{1,p(z)}(\Omega )\) with \(\Vert u\Vert =\rho <1\) we have
Since \(p_+<\widehat{r}_-\), choosing \(\rho \in (0,1)\) small, we have
Then choosing \(\lambda >0\) and \(\epsilon >0\) small, from (4.6) we see that
Let \(v\in \mathrm {int}\,C_+\). Since \(\overline{u}_\lambda \in \mathrm {int}\, C_+\), we can find \(t\in (0,1)\) small such that \(tv\le \overline{u}_\lambda \) (see [15], p. 274). We have
We have seen above that \(\frac{v}{\overline{u}_\lambda ^{\eta (\cdot )}}\in L^1(\Omega )\). Hence
Since \(q_->1\), choosing \(t\in (0,1)\) even smaller if necessary, we have
Consider the closed ball
The reflexivity of \(W_0^{1,p(z)}(\Omega )\) and the Eberlein–Smulian theorem imply that \(\overline{B}_\rho \) is sequentially weakly compact. Also, using the anisotropic Sobolev embedding theorem, we check that \(\psi _\lambda (\cdot )\) is sequentially weakly compact. So, we can find \(u_\lambda \in W_0^{1,p(z)}(\Omega )\) such that
From (4.7), we see that
Then, (4.9) and (4.10) imply that
In (4.11) we use the test function \(h=(\overline{u}_\lambda - u_\lambda )^+\in W_0^{1,p(z)}(\Omega )\). Then
Then (4.12), (4.2) and (4.11) imply that
Moreover, the anisotropic regularity theory (see Theorem B.1 of [22]) and the anisotropic maximum principle (see Proposition 4 of [14]) imply that \(S_\lambda \subseteq \mathrm {int}\, C_+\) for all \(\lambda \in \mathscr {L}\). \(\square \)
The next proposition shows that \(\mathscr {L}\) is connected.
Proposition 4.2
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold, \(\lambda \in \mathscr {L}\) and \(\mu \in (0,\lambda )\), then \(\mu \in \mathscr {L}\).
Proof
Since \(\lambda \in \mathscr {L}\), we can find \(u_\lambda \in S_\lambda \subseteq \mathrm {int}\, C_+\) (see Proposition 4.1). Recall that \(u_\lambda ^{-\eta (\cdot )}\in L^1(\Omega )\). Also we have \(\overline{u}_\mu \le \overline{u}_\lambda \le u_\lambda \) (see Proposition 3.1 and use the fact that \(f\ge 0\)). We introduce the Carathéodory function \(\vartheta _\mu : \Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by
We set \(\Theta _\mu (z,x)=\int _0^x \vartheta _\mu (z,s)\,\mathrm {d}s\) and consider the \(C^1\)-functional \(w_\mu : W_0^{1,p(z)}(\Omega )\rightarrow \mathbb {R}\) defined by
From (4.13), it is clear that \(w_\mu (\cdot )\) is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find \(u_\mu \in W_0^{1,p(z)}(\Omega )\) such that
From (4.14) we have
In (4.15) first we use the test function \(h=(\overline{u}_\mu - u_\mu )^+\in W_0^{1,p(z)}(\Omega )\). Then
Next in (4.15) we choose \(h=(u_\mu -u_\lambda )^+\in W_0^{1,p(z)}(\Omega )\). We have
So, we have proved that
From (4.16), (4.13) and (4.15), we conclude that
\(\square \)
A by-product of the above proof is the following weak monotonicity property for the solution multifunction \(\lambda \mapsto S_\lambda \).
Corollary 4.3
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold, \(\lambda \in \mathscr {L}\), \(u_\lambda \in S_\lambda \subseteq \mathrm {int}\, C_+\) and \(\mu \in (0,\lambda )\), then \(\mu \in \mathscr {L}\) and we can find \(u_\mu \in S_\mu \subseteq \mathrm {int}\, C_+\) such that \(u_\mu \le u_\lambda \).
Using Proposition 2.3, we can improve the assertion of this corollary.
Proposition 4.4
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold, \(\lambda \in \mathscr {L}\), \(u_\lambda \in S_\lambda \subseteq \mathrm {int}\, C_+\) and \(\mu \in (0,\lambda )\), then \(\mu \in \mathscr {L}\) and we can find \(u_\mu \in S_\mu \subseteq \mathrm {int}\, C_+\) such that
Proof
From Corollary 4.3, we already know that \(\mu \in \mathscr {L}\) and there exists \(u_\mu \in S_\mu \subseteq \mathrm {int}\, C_+\) such that
Let \(\rho = \Vert u_\lambda \Vert _\infty \) and let \(\widehat{\xi }_\rho >0\) be as postulated by hypothesis \(\mathrm {H}_1\)(iv). We have
Since \(u_\mu \in \mathrm {int}\, C_+\), we have
So, from (4.18) and Proposition 2.3, it follows that
\(\square \)
Let \(\lambda ^*= \sup \mathscr {L}\).
Proposition 4.5
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold, then \(\lambda ^*<\infty \).
Proof
On account of hypotheses \(\mathrm {H}_1\)(i), (ii), (iii), we can find \(\lambda _0>0\) such that
Let \(\lambda >\lambda _0\) and suppose that \(\lambda \in \mathscr {L}\). Then we can find \(u_\lambda \in S_\lambda \subseteq \mathrm {int}\, C_+\). Consider an open set \(\Omega _0\subseteq \Omega \) with \(C^2\)-boundary such that \(\overline{\Omega }_0\subseteq \Omega \). We set \(\displaystyle m_0=\min _{\overline{\Omega }_0}u_\lambda >0\) (since \(u_\lambda \in \mathrm {int}\, C_+\)). For \(\delta >0\), set \(m_0^\delta =m_0+\delta \), let \(\rho = \Vert u_\lambda \Vert _\infty \) and let \(\widehat{\xi }_\rho >0\) be as postulated by hypothesis \(\mathrm {H}_1\)(iv). We have
Invoking Proposition 6 of Papageorgiou–Rădulescu–Repovš [18], we infer that
a contradiction to the definition of \(m_0\).
Therefore \(\lambda \notin \mathscr {L}\) and so \(\lambda ^*\le \lambda _0<\infty \). \(\square \)
Next we show that \(\lambda \in (0,\lambda ^*)\) we have multiplicity of positive solutions for problem (\(\mathrm {P}_\lambda \)).
Proposition 4.6
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold and \(\lambda \in (0,\lambda ^*)\), then problem (\(\mathrm {P}_\lambda \)) has at least two positive solutions
Proof
Let \(0<\mu<\lambda<\delta <\lambda ^*\). We know that \(\mu ,\delta \in \mathscr {L}\) and on account of Proposition 4.4, we can find \(u_\delta \in S_\delta \subseteq \mathrm {int}\, C_+\), \(u_0 \in S_\lambda \subseteq \mathrm {int}\, C_+\) and \(u_\mu \in S_\mu \subseteq \mathrm {int}\, C_+\) such that
Recall that \(u_{\mu }(\cdot )^{-\eta (\cdot )}, u_\delta (\cdot )^{-\eta (\cdot )}\in L^1(\Omega )\). We introduce the Carathéodory function \(\widehat{\beta }_\lambda :\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by
We set \(\widehat{B}_\lambda (z,x)=\int _0^x \widehat{\beta }_\lambda (z,s)\,\mathrm {d}s\) and consider the \(C^1\)-functional \(\widehat{\zeta }_\lambda : W_0^{1,p(z)}(\Omega )\rightarrow \mathbb {R}\) defined by
We have
Also \(\widehat{\zeta }_\lambda (\cdot )\) is sequentially weakly lower semicontinuous. Therefore, we can find \(\widehat{u}_0\in W_0^{1,p(z)}(\Omega )\) such that
From (4.21) and the anisotropic regularity theory, we have
From (4.22), (4.21) and (4.20), we see that
If \(u_0\ne \widehat{u}_0\), then we already have two positive smooth solutions of (\(\mathrm {P}_\lambda \)) and so we are done. Thus we may assume that
We consider the Carathéodory function \(\beta _\lambda :\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by
We set \(B_\lambda (z,x)=\int _0^x \beta _\lambda (z,s)\,\mathrm {d}s\) and consider the \(C^1\)-functional \(\zeta _\lambda : W_0^{1,p(z)}(\Omega )\rightarrow \mathbb {R}\) defined by
From (4.21) and (4.25), we see that
Combining (4.24), (4.22), (4.20) and (4.26), we see that
From (4.25), it follows that \(K_{\zeta _\lambda }\subseteq [u_\mu )\cap \mathrm {int}\, C_+\). So, we may assume that \(K_{\zeta _\lambda }\) is finite or otherwise we already have an infinity of positive smooth solutions of (\(\mathrm {P}_\lambda \)) and so we are done. Then, from (4.27) and using Theorem 5.7.6, p. 449, of [15], we can find \(\rho \in (0,1)\) small such that
If \(u\in \mathrm {int}\, C_+\), then on account of hypothesis \(\mathrm {H}_1\)(ii), we have
Moreover, Proposition 4.1, of Gasiński–Papageorgiou [8], implies that
Then (4.28), (4.29) and (4.30) permit the use of the mountain pass theorem. So, we can find \(\widehat{u}\in W_0^{1,p(z)}(\Omega )\) such that
\(\square \)
We produce a lower bound for the elements of \(S_\lambda \). This will be helpful in proving the admissibility of the critical parameter \(\lambda ^*\) and also in producing the minimal positive solution for every \(\lambda \in \mathscr {L}\) (see Sect. 5).
Proposition 4.7
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold and \(\lambda \in \mathscr {L}\), then \(\overline{u}_\lambda \le u\) for all \(u\in S_\lambda \).
Proof
Let \(u\in S_\lambda \subseteq \mathrm {int}\, C_+\) and consider the Carathéodory function \(\sigma : \Omega \times \mathring{\mathbb {R}}_+\rightarrow \mathring{\mathbb {R}}_+\) defined by
We consider the following Dirichlet problem
As in the proof of Proposition 3.1, via approximations and fixed point theory, we show that problem (\(\mathrm {Au}_\lambda '\)) as a unique positive solution
From Proposition 3.1, it follows that
\(\square \)
Using this bound, we can prove the admissibility of \(\lambda ^*\).
Proposition 4.8
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold, then \(\lambda ^*\in \mathscr {L}\).
Proof
Let \(\{\lambda _n\}_{n\in \mathbb {N}}\subseteq \mathscr {L}\) such that \(\lambda _n \uparrow \lambda ^*\). From Proposition 3.1, we know that \(\overline{u}_{\lambda _1}\le \overline{u}_{\lambda _n}\) for all \(n\in \mathbb {N}\). Also let \(\widetilde{u}_{n+1}\in S_{\lambda _{n+1}}\subseteq \mathrm {int}\, C_+\). On account of Proposition 4.7, we have \(\overline{u}_{\lambda _1}\le \widetilde{u}_{n+1}\). Then we can introduce the Carathéodory function \(\widehat{l}_n: \Omega \times \mathbb {R}\rightarrow \mathbb {R}\) defined by
We set \(\widehat{L}_n(z,x)=\int _0^x \widehat{l}_n(z,s)\,\mathrm {d}s\) and consider the \(C^1\)-functional \(\widehat{\chi }_n: W_0^{1,p(z)}(\Omega )\!\rightarrow \mathbb {R}\) defined by
From (4.32), we see that \(\widehat{\chi }_n(\cdot )\) is coercive. Also it is sequentially weakly lower semicontinuous. Hence we can find \(u_n\in W_0^{1,p(z)}(\Omega )\) such that
If in (4.33) we use first the test function \(h=(\overline{u}_{\lambda _1}-u_n)^+\in W_0^{1,p(z)}(\Omega )\) and then the test function \(h=(u_n-\widetilde{u}_{n+1})^+\in W_0^{1,p(z)}(\Omega )\), then using (4.32) we obtain
Let \(l_n:\Omega \times \mathbb {R}\rightarrow \mathbb {R}\) be the Carathéodory function defined by
We set \(L_n(z,x)=\int _0^x l_n(z,s)\,\mathrm {d}s\) and consider the \(C^1\)-functional \(\chi _n: W_0^{1,p(z)}(\Omega )\!\rightarrow \mathbb {R}\) defined by
From (4.32) and (4.34), we see that
Also we have
Then, from (4.35) we have
As in Proposition 4.1 of Gasiński–Papageorgiou [8], from (4.36) we deduce that
and
Therefore \(u_*\in S_{\lambda ^*}\subseteq \mathrm {int}\, C_+\) and so \(\lambda ^*\in \mathscr {L}\). \(\square \)
So, we have proved that
We can state the following bifurcation-type theorem about the positive solutions of problem (\(\mathrm {P}_\lambda \)).
Theorem 4.9
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold, then there exists \(\lambda ^*>0\) such that
-
(a)
for all \(\lambda \in (0,\lambda ^*)\) problem (\(\mathrm {P}_\lambda \)) has at least two positive solutions \(u_0,\widehat{u}\in \mathrm {int}\, C_+\), \(u_0\ne \widehat{u}\);
-
(b)
for \(\lambda =\lambda ^*\) problem (\(\mathrm {P}_\lambda \)) has at least one positive solution \(u_*\in \mathrm {int}\, C_+\);
-
(c)
for all \(\lambda >\lambda ^*\) problem (\(\mathrm {P}_\lambda \)) has no positive solutions.
5 Minimal Positive Solution
In this section, we show that for every \(\lambda \in \mathscr {L}=(0,\lambda ^*]\) problem (\(\mathrm {P}_\lambda \)) has a smallest positive solution \(u_\lambda ^*\in \mathrm {int}\, C_+\) (minimal - or barrier - positive solution) and we prove the monotonicity and continuity properties of the map \(\lambda \mapsto u_\lambda ^*\).
Proposition 5.1
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold and \(\lambda \in \mathscr {L}=(0,\lambda ^*]\), then problem (\(\mathrm {P}_\lambda \)) has a smallest positive solution \(u_\lambda ^*\in \mathrm {int}\, C_+\) (that is, \(u_\lambda ^*\le u\) for all \(u\in S_\lambda \)).
Proof
From Papageorgiou–Rădulescu–Repovš [16] (see the proof of Proposition 7) we have that \(S_\lambda \) is downward directed (that is, if \(u_1,u_2\in S_\lambda \), then we can find \(u\in S_\lambda \) such that \(u\le u_1\), \(u\le u_2\)). Invoking Lemma 3.10, p. 178, of Hu–Papageorgiou [12], we can find a decreasing sequence \(\{u_n\}_{n\in \mathbb {N}}\subseteq S_\lambda \subseteq \mathrm {int}\, C_+\) such that
We have
From (5.1) and (5.2), it follows that
From (5.3) and Proposition 2.2 (in particular the \((S)_+\)-property), we obtain
and
Therefore we conclude that
\(\square \)
We consider the map \(\lambda \mapsto u_\lambda ^*\) from \(\mathscr {L}=(0,\lambda ^*]\) into \(\mathrm {int}\, C_+\). We will say that this map is strictly increasing if \(0<\lambda <\lambda '\le \lambda ^*\), then \(u_{\lambda '}^*-u_\lambda ^*\in \mathrm {int}\, C_+\).
Proposition 5.2
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold, then minimal solution map \(\lambda \mapsto u_\lambda ^*\) is
-
(a)
strictly increasing;
-
(b)
left continuous.
Proof
-
(a)
Let \(0<\lambda <\lambda '\le \lambda ^*\). From Proposition 4.4, we know that we can find \(u_\lambda \in S_\lambda \subseteq \mathrm {int}\, C_+\) such that \(u_{\lambda '}^*-u_\lambda \in \mathrm {int}\, C_+\). Since \(u_\lambda ^*\le u_\lambda \), it follows that \(u_{\lambda '}^*-u_\lambda ^*\in \mathrm {int}\, C_+\) and this proves that the minimal solution map is strictly increasing.
-
(b)
Let \(\{\lambda _n\}_{n\in \mathbb {N}}\subseteq \mathscr {L}=(0,\lambda ^*]\) such that \(\lambda _n\rightarrow \lambda ^-\). We have
$$\begin{aligned}&u_{\lambda _1}^* \le u_{\lambda _n}^*\le u_\lambda ^*\quad \text {for all } n\in \mathbb {N}\text { (see (a))}, \end{aligned}$$(5.4)$$\begin{aligned}&\Rightarrow \quad \{u_{\lambda _n}^*\}_{n\in \mathbb {N}}\subseteq W_0^{1,p(z)}(\Omega ). \end{aligned}$$(5.5)From (5.4) and the anisotropic regularity theory (see [22]), we have that \(\{u_{\lambda _n}^*\}_{n\in \mathbb {N}}\subseteq C_0^{1,\alpha }(\overline{\Omega })\) is bounded with \(\alpha \in (0,1)\) and \(C_0^{1,\alpha }(\overline{\Omega })=C^{1,\alpha }(\overline{\Omega })\cap C_0^1(\overline{\Omega })\). From the compact embedding of \(C_0^{1,\alpha }(\overline{\Omega })\) into \(C_0^1(\overline{\Omega })\), it follows that we may assume that at least for a subsequence we have
$$\begin{aligned} u_{\lambda _n}^*\rightarrow \widetilde{u}_\lambda \text { in } C_0^1(\overline{\Omega }). \end{aligned}$$Suppose \(\widetilde{u}_\lambda \ne u_\lambda ^*\). Then we can find \(z_0\in \Omega \) such that
$$\begin{aligned}&u_\lambda ^*(z_0)<\widetilde{u}_\lambda (z_0),\\&\qquad \Rightarrow \quad u_\lambda ^*(z_0)<u_{\lambda _n}^*(z_0)\quad \text {for all } n\ge n_0, \end{aligned}$$which contradicts (a). So, \(\widetilde{u}_\lambda = u_\lambda ^*\) and by Urysohn’s criterion for the convergence of sequences, we infer that for the initial sequence we have
$$\begin{aligned}&u_{\lambda _n}^*\rightarrow u_\lambda ^*\text { in } C_0^1(\overline{\Omega }),\\&\qquad \Rightarrow \quad \lambda \rightarrow u_\lambda ^*\text { is left continuous.} \end{aligned}$$
\(\square \)
So, we can state the following theorem for the minimal positive solutions of problem (\(\mathrm {P}_\lambda \)).
Theorem 5.3
If hypotheses \(\mathrm {H}_0\), \(\mathrm {H}_1\) hold, then for every \(\lambda \in \mathscr {L}=(0,\lambda ^*]\) problem (\(\mathrm {P}_\lambda \)) has a smallest positive solution (minimal positive solution) \(u_\lambda ^*\in \mathrm {int}\, C_+\) and the minimal solution map \(\lambda \mapsto u_\lambda ^*\) from \(\mathscr {L}\) into \(\mathrm {int}\, C_+\) is strictly increasing and left continuous.
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Communicated by Rosihan M. Ali.
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Papageorgiou, N.S., Scapellato, A. Positive Solutions for Anisotropic Singular Dirichlet Problems. Bull. Malays. Math. Sci. Soc. 45, 1141–1168 (2022). https://doi.org/10.1007/s40840-022-01249-5
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DOI: https://doi.org/10.1007/s40840-022-01249-5