Existence and Uniqueness of Positive Solutions for Singular Asymmetric Multi-Phase Equations

Abstract

In this work, we establish the existence of positive solutions for a problem driven by a multi-phase operator composed of two distinct exponent Laplacian-type operators and a generalised lower-order term, which ensures asymmetric behaviour across three subregions of the domain under consideration. The reaction term involves a mild singularity at zero and includes a possibly sign-changing perturbation function. Under additional restrictive conditions, we also obtain a uniqueness result for the problem. Our existence result is based on pseudomonotone operator theory. Moreover, a detailed analysis, combined with a Díaz–Saá-type argument, allows us to also establish a uniqueness theorem. To the best of our knowledge, this is the first work addressing such a generalisation of the multi-phase operator. These novel results can serve as a foundation for more general physical and engineering models.

1. Introduction

In recent works, De Filippis [1] and De Filippis-Oh [2] introduced the so called multi-phase operator

$$\mathcal{G}\left(u\right)=div({a\left(x\right)|\nabla u|}^{p-2}\nabla u+{b\left(x\right)|\nabla u|}^{q-2}\nabla u+{|\nabla u|}^{r-2}\nabla u)\hspace{1em}\forall u\in W_0^{1,\mathcal{H}}(\Omega ),$$

where $a,b\ge 0$, $p\ge q>r>1$. In particular, if $b\left(x\right)\equiv 0$, the operator reduces to the classical double phase operator (see Zhikov [3,4]). Moreover, Dai-Vetro [5], Vetro [6,7], and Vetro-Efendiev [8] have extended the study of the multi-phase operator to the variable exponent setting.

We point out that double phase-type operators have a large range of applications in physical and engineering models, for instance, see Bahrouni-Rădulescu-Repovš [9] for transonic flows models, Benci-D’Avenia-Fortunato-Pisani [10] for models in quantum physics, Bonheure-D’Avenia-Pomponio [11] for Born-Infeld equation in electromagnetism, Charkaoui-Ben Loghfyry-Zeng [12,13] and Harjulehto-Hästö [14] in image restoration, Cherfils-Il’yasov [15] for model on reaction-diffusion systems, and Zhikov [3,4] for elasticity theory. See also the work of He-Anjum-He-Alsolami [16] for a possible generalisation via Fourier transform techniques.

From a mathematical perspective, double phase-type operators present deep analytical challenges. In particular, the Lavrentiev phenomenon has been a central topic in Borowski-Chlebicka-De Filippis-Miasojedow [17], De Filippis-Mingione [18], and Zhikov [19]. We also refer the interested reader to a number of seminal contributions on existence and multiplicity results for problems involving the double phase operator: Bai-Papageorgiou-Zeng [20], D’Aguì-Sciammetta-Tornatore-Winkert [21], Gambera-Guarnotta-Papageorgiou [22], Gasiński-Papageorgiou [23,24], Gasiński-Winkert [25,26,27], Guarnotta-Livrea-Winkert [28], Marino-Winkert [29], Papageorgiou-Vetro-Vetro [30], Sciammetta-Tornatore-Winkert [31]. Regarding regularity theory, we mention: Baroni-Colombo-Mingione [32], Colombo-Mingione [33], Ho-Winkert [34], and Ragusa-Tachikawa [35], as well as the references therein. For a comprehensive overview of unbalanced growth operators, see the surveys by Mingione-Rădulescu [36] and Papageorgiou [37].

Our aim in this work is to establish the existence and uniqueness of solutions to a singular problem driven by a generalised multi-phase operator, at least in the constant exponent setting. We generalise the lower-order term by introducing an elliptic operator of the following form:

$$u\mapsto div\left(\alpha \right(\nabla u\left)\right),$$

where the function $\alpha \left(y\right)={\alpha }_0\left(\right|y\left|\right)y$ has a Uhlenbeck structure and satisfies some general assumptions. This formulation allows us to encompass several classical cases. For instance, we can consider:

• ${\alpha }_0\left(t\right)=1$ for $t>0$, the classical Laplacian operator $\Delta u$;
• ${\alpha }_0\left(t\right)={\left|t\right|}^{r-2}$ for $t>0$, the r-Laplacian ${\Delta }_{r}u$;
• ${\alpha }_0\left(t\right)={\left|t\right|}^{r-2}+{\left|t\right|}^{s-2}$ with $r<s$, the $(r,s)$-Laplacian, ${\Delta }_{r}u+{\Delta }_{s}u$;
• ${\alpha }_0\left(t\right)={(1+t^2)}^{{\displaystyle \frac{r-2}{2}}}$, the r-mean curvature operator;
• ${\alpha }_0\left(t\right)={(1-t^2)}^{-{\displaystyle \frac{1}{2}}}$, the Minkowski-curvature operator;

from the nonlinear elasticity problem, we can consider

• ${\alpha }_0\left(t\right)=2\gamma {(1+t^2)}^{\gamma -1}$, for $t>0$, $\gamma >1$;
• ${\alpha }_0\left(t\right)=\gamma {\displaystyle \frac{{(\sqrt{(1+t^2)}-1)}^{\gamma -1}}{\sqrt{1+t^2}}}$, for $t>0$ and $\gamma \ge 1$;

and from plasticity theory, we have

• ${\alpha }_0\left(t\right)={\displaystyle \frac{t^{r-2}(1+t)ln(1+t)+t^{r-1}}{1+t}}$, for $t>0$ and ${\displaystyle \frac{-1+\sqrt{1+4N}}{2}}>1$;
• ${\alpha }_0\left(t\right)={\left|t\right|}^{r-2}+{\displaystyle \frac{{\left|t\right|}^{r-2}}{1+{\left|t\right|}^2}}$, for all $t>0$.

For recent developments on problems involving this operator, we refer the interested reader to the works of Candito-Gasiński-Livrea [38], Carvalho-Goncalves-Silva-Santos [39], Fragnelli-Mugnai-Papageorgiou [40], and Gambera-Guarnotta [41]. See also Guarnotta-Marano-Motreanu [42], Öztürk-Papageorgiou [43], and Tan-Fang [44], as well as the references therein for additional results and extensions.

Motivated by the recent studies of Bai-Gasiński-Papageorgiou [45] and Failla-Gasiński-Papageorgiou-Skupień [46], we investigate a singular problem perturbed by a sign-changing reaction term. Singularities pose substantial mathematical challenges but are central in many models across biology and engineering. We refer the interested reader to Anjum-He [47] for the microelectromechanical system (MEMS) oscillator equation of the type $u^{″}+u=k/(1-u)$, Callegari-Nachman [48] for the boundary layer problem, Callegari-Nachman [49] for models of pseudo-plastic fluids, Carleman [50] for applications in kinetic gas theory, Cohen-Keller [51] for models of heat generation in electrical conductors, El Dib [52] and He-Liu [53] for singular wave phenomena, Nowosad [54] for communication models, and Gierer-Meinhardt [55] and Turing [56] for pattern formation and morphogenesis in biological systems. From a mathematical point of view, for mildly singular problems ($0<\eta <1$), we refer to Bai-Papageorgiou-Zeng [20], Candito-Failla-Livrea [57], and Papageorgiou-Rădulescu-Repovš [58]; while for the study of strongly singular problems ($\eta >1$), we refer to Papageorgiou-Rădulescu-Yuan [59], and Papageorgiou-Rădulescu-Zhang [60]. Finally, for a comprehensive overview of singular problems, we refer to the survey by Guarnotta-Livrea-Marano [61].

In this work, we prove the existence and uniqueness results for the following problem

$$\{\begin{array}{cc}-{\Delta }_p^{a}u-{\Delta }_q^{b}u-div\left(\alpha \right(\nabla u\left)\right)=\beta \left(x\right)u^{-\eta }+f(x,u)\hfill & \mathrm{in}\Omega ,\hfill \\ u>0\hfill & \mathrm{in}\Omega ,\hfill \\ u=0\hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}$$

(P)

with $\Omega \subset {\mathbb{R}}^N$$(N\ge 2)$ being a bounded domain with $C^2$-boundary. We indicate by

$${\Delta }_s^{\zeta }u=div({\zeta \left(x\right)|\nabla u|}^{s-2}\nabla u)\hspace{1em}\forall u\in W_0^{1,\mathcal{H}}(\Omega ),$$

the weighted s-Laplacian with the weight $\zeta$. Clearly, if $\zeta \equiv 1$ we come back to the classical s-Laplacian operator. Then, our problem is driven by the following operator

$$u\mapsto -div({a\left(x\right)|\nabla u|}^{p-2}\nabla u+b\left(x\right){|\nabla u|}^{q-2}\nabla u+\alpha (\nabla u\left)\right)\hspace{1em}\forall u\in W_0^{1,\mathcal{H}}(\Omega ),$$

where the growth is unbalanced in a “multi-phase” way (so, in fact, we have an asymmetric behaviour caused by the operator on three subregions of the domain $\Omega$).

We assume that

$\left(H_1\right)$

$0<\eta <1<r<q<p<min\{N,r^{*}\}$, $a(·),b(·)\in C^{0,1}\left(\overline{\Omega }\right)$, such that $a,b\ge 0$.

$\left(H_2\right)$

$\beta \in W_0^{1,\infty }(\Omega )$, $\beta \left(x\right)>0$ for all $x\in \Omega$.

These assumptions allow us to obtain a suitable structure for the multi-phase operator. Indeed, the relations between p and q and the regularity on the functions $a,b$ guarantees the embeddings of Musielak–Orlicz–Sobolev spaces, see Section 2 for details.

Moreover, we assume the following Uhlenbeck structure on the function $\alpha$, i.e.,

$$\alpha \left(y\right)={\alpha }_0\left(\right|y\left|\right)y\hspace{1em}\forall y\in {\mathbb{R}}^N$$

where ${\alpha }_0:(0,+\infty )\to (0,+\infty )$ is a suitable $C^1$-function. Furthermore, we assume that there exists $\gamma \in C^1(0,+\infty )$, such that $\gamma \left(t\right)>0$ for all $t>0$, and there exist positive constants $c_1,c_2,c_3,c_4$, such that, for $1<s<r<N$, we have

$$0<c_1\le {\displaystyle \frac{{\gamma }^{\prime }\left(t\right)t}{\gamma \left(t\right)}}\le c_2\mathrm{and}{c}_3{t}^{r-1}\le \gamma \left(t\right)\le c_4(t^{s-1}+t^{r-1}).$$

(1)

Further hypotheses needed in the sequel are as follows.

${\left(H_{\alpha }\right)}_1$

the map $t\mapsto t{\alpha }_0\left(t\right)$ is strictly increasing on $(0,+\infty )$ and

$$\underset{t\to 0^{+}}{lim}{\alpha }_0\left(t\right)t=0,\underset{t\to 0^{+}}{lim}{\displaystyle \frac{{\alpha }_0^{\prime }\left(t\right)t}{{\alpha }_0\left(t\right)}}>-1;$$

${\left(H_{\alpha }\right)}_2$

there exists a constant $M>0$, such that

$$|\nabla \alpha \left(y\right)|\le M{\displaystyle \frac{\gamma \left(\right|y\left|\right)}{\left|y\right|}}\hspace{1em}\forall y\in {\mathbb{R}}^N\setminus \left\{0\right\};$$

${\left(H_{\alpha }\right)}_3$

for all $y\in {\mathbb{R}}^N\setminus \left\{0\right\}$ and for all $\xi \in {\mathbb{R}}^N$, we have

$${\displaystyle \frac{\gamma \left(\right|y\left|\right)}{\left|y\right|}}{\left|\xi \right|}^2\le {\langle \nabla \alpha \left(y\right)\xi ,\xi \rangle }_{{\mathbb{R}}^N}.$$

Note that the structural assumptions imposed on the function $\alpha$ ensure the ellipticity of the corresponding lower-order operator. In particular, these assumptions allow us to encompass several classical models arising in applications; we refer to the Introduction for a comprehensive list. Specifically, the condition $\left(H_{\alpha }\right)$ enables us to address two key properties of the operator $div\left(\alpha \right(\nabla u\left)\right)$, namely:

1.

to obtain solutions with global regularity, see Lieberman [62];

2.

to apply maximum principle, see Pucci-Serrin [63].

Remark 1.

Arguing as in Corollary 2.3 of Fragnelli-Mugnai-Papageorgiou [40], setting

$${\displaystyle G_0\left(t\right)={\int }_0^{t}s{\alpha }_0\left(s\right)ds,}$$

we can show that the function

$$G\left(y\right)=G_0\left(\right|y\left|\right)\hspace{1em}\forall y\in {\mathbb{R}}^N,$$

has a balance growth, namely

$$c_5{\left|y\right|}^r\le G\left(y\right)\le c_6{(1+|y|}^r)\hspace{1em}\forall y\in {\mathbb{R}}^N$$

with some constants $c_5,c_6>0$.

Using the same reasoning as in Gasiński-Winkert [25] and Marino-Winkert [29], Section 3, and using the Moser iteration scheme, as in Guedda-Véron [64], Proposition 3.1, we can show that problem

$$\{\begin{array}{cc}-{\Delta }_p^{a}u-{\Delta }_q^{b}u-div\left(\alpha \right(\nabla u\left)\right)=g\left(x\right)\hfill & \mathrm{in}\Omega ,\hfill \\ u>0\hfill & \mathrm{in}\Omega ,\hfill \\ u=0\hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}$$

(2)

where $g\in L^{\infty }(\Omega )$ possesses a unique positive solution u, such that

$${\parallel u\parallel }_{L^{\infty }(\Omega )}\le m_0{\parallel g\parallel }_{L^{\infty }(\Omega )}^{{\displaystyle \frac{1}{r-1}}},$$

(3)

for some $m_0>0$ (independent on $g\in L^{\infty }(\Omega )$).

Finally, we state the following hypotheses on the reaction term f:

${\left(H_f\right)}_1$

$f:\Omega \times (0,+\infty )\to \mathbb{R}$ is a Carathéodory function and there exist $m_1>0$ and $1<\tau \le r$, such that $f(x,t)\le m_1{t}^{\tau -1}$ for a.a. $x\in \Omega$ and all $t\ge 0$, where in addition $m_1{m}_0<1$ in case if $\tau =r$ ($m_0$ being given by (3)). Moreover, for any $\rho >0$ there exists ${\xi }_{\rho }>0$, such that

$$\left|f(x,t)\right|\le {\xi }_{\rho }\mathrm{for} \mathrm{a}.\mathrm{a}.x\in \Omega ,\mathrm{for} \mathrm{all}0\le t\le \rho .$$

${\left(H_f\right)}_2$

there exists $\delta >0$ such that $f(x,0)=0\le f(x,t)$ for a.a. $x\in \Omega$, for all $0\le t\le \delta$.

As we seek strictly positive solutions, the assumptions on the function f are asymmetric on the real line; in fact, we may assume that $f(x,t)$ vanishes whenever t is nonpositive.

We emphasise that our assumptions on the reaction term allow for sign-changing perturbations in f, representing a novel approach recently introduced in the works of Bai-Gasiński-Papageorgiou [45] and Failla-Gasiński-Papageorgiou-Skupień [46]. Moreover, the framework adopted here encompasses several classical structures commonly found in applied models—see the Introduction for a detailed discussion.

The rest of the paper is organised as follows: in Section 2, we introduce the functional setting and some auxiliary tools. Section 3 presents key results related to the pure singular problem, which are essential to handle the singularities. The main existence result is established in Section 4. Finally, in Section 5, we prove a uniqueness result under additional restrictive assumptions.

2. Preliminaries

2.1. Spaces Setting

We introduce the natural settings for our problem; see Adams-Fournier [65] and Brézis [66] for a complete overview on this topic. Let us begin by considering the Lebesgue space. For $1\le r<+\infty$, we define

$$L^r(\Omega )=\{u:\Omega \to \mathbb{R}\mathrm{measurable}:{\int }_{\Omega }{\left|u\right|}^{r}dx<\infty \},$$

with the classical r-norm

$${\parallel u\parallel }_r={\left({\int }_{\Omega }{\left|u\right|}^{r}dx\right)}^{{\displaystyle \frac{1}{r}}}.$$

For $r=+\infty$, we consider

$$L^{\infty }(\Omega )=\{u:\Omega \to \mathbb{R}\mathrm{measurable}:\underset{x\in \Omega }{ess\; sup}\left|u\left(x\right)\right|<+\infty \},$$

with the sup-norm

$${\parallel u\parallel }_{L^{\infty }(\Omega )}=\underset{x\in \Omega }{ess\; sup}\left|u\left(x\right)\right|.$$

Here, we can consider the ordered cone

$$L_{+}^{\infty }(\Omega )=\{u\in L^{\infty }(\Omega ):u\left(x\right)\ge 0\mathrm{for}\mathrm{a}.\mathrm{a}.x\in \Omega \},$$

with the nonempty interior given by

$$\mathrm{int}{L}_{+}^{\infty }(\Omega )=\{u\in L_{+}^{\infty }(\Omega ):\underset{x\in \Omega }{ess\; inf}u\left(x\right)>0\}.$$

Moreover, for $1\le r<+\infty$, we use the Sobolev space

$$W^{1,r}(\Omega )=\{u\in L^r(\Omega ):\nabla u\in L^r(\Omega )),$$

with the norm

$${\parallel u\parallel }_{1,r}={\parallel u\parallel }_r+{\parallel \nabla u\parallel }_r.$$

In particular, we introduce the Sobolev space

$$W_0^{1,r}(\Omega )={\overline{C_0^{\infty }(\Omega )}}^{{\parallel ·\parallel }_{1,r}},$$

(4)

endowed with the equivalent norm

$$\parallel u\parallel ={\parallel \nabla u\parallel }_r.$$

For our purpose, we introduce the notion of Musielak–Orlicz space and Musielak-Orlicz-Sobolev space; see Harjulehto-Hästö [67] for details. Let us define the nonlinear function $\mathcal{H}:\Omega \times [0,+\infty )\to [0,+\infty )$, by

$$\mathcal{H}(x,t)=a\left(x\right)t^p+b\left(x\right)t^q+t^r,$$

which in fact has an unbalanced growth, and let us consider the related modular function

$${\rho }_{\mathcal{H}}\left(u\right)={\int }_{\Omega }\mathcal{H}(x,|u\left|\right)dx.$$

(5)

Then, we introduce the so-called Musielak–Orlicz space, as follows

$$L^{\mathcal{H}}(\Omega )=\{u:\Omega \to \mathbb{R}\mathrm{measurable}:{\rho }_{\mathcal{H}}\left(u\right)<+\infty \},$$

endowed with the Luxemburg norm

$${\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}=inf\{\lambda >0:{\rho }_{\mathcal{H}}({\displaystyle \frac{u}{\lambda }})\le 1\}.$$

Analogously, we introduce the Musielak–Orlicz-Sobolev space as

$$W^{1,\mathcal{H}}(\Omega )=\{u\in L^{\mathcal{H}}(\Omega ):|\nabla u|\in L^{\mathcal{H}}(\Omega )\},$$

with the norm

$${\parallel u\parallel }_{W^{1,\mathcal{H}}(\Omega )}={\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}+{\parallel \nabla u\parallel }_{L^{\mathcal{H}}(\Omega )}.$$

Moreover, using the Musielak–Orlicz version of the Poincaré inequality (see, e.g., Harjulehto-Hästö [67], Theorem 6.2.8, p. 130), we can consider the space

$$W_0^{1,\mathcal{H}}(\Omega )={\overline{C_0^{\infty }(\Omega )}}^{{\parallel ·\parallel }_{W^{1,\mathcal{H}}(\Omega )}},$$

equipped with the equivalent norm

$$\parallel u\parallel ={\parallel \nabla u\parallel }_{L^{\mathcal{H}}(\Omega )}.$$

Furthermore, using Remark 1 and Crespo Blanco-Gasiński-Harjuletho-Winkert [68], Proposition 2.13 (see also Vetro [6], Proposition 2.4), we have the following result.

Lemma 1.

Assume that $\left(H_{\alpha }\right)$ and $\left(H_1\right)$ hold. Let $u\in L^{\mathcal{H}}(\Omega )$ and let ${\rho }_{\mathcal{H}}$ be defined by (5). Then, the following hold

1. 

if $u\ne 0$ and $\lambda \in \mathbb{R}$, then, ${\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}=\lambda \iff {\rho }_{\mathcal{H}}\left({\displaystyle \frac{u}{\lambda }}\right)=1$;

2. 

${\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}<1(=1;>1)\iff {\rho }_{\mathcal{H}}\left(u\right)<1(=1;>1)$;

3. 

if ${\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}<1$, then ${\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}^p\le {\rho }_{\mathcal{H}}\left(u\right)\le {\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}^r$;

4. 

if ${\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}>1$, then ${\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}^r\le {\rho }_{\mathcal{H}}\left(u\right)\le {\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}^p$;

5. 

${\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}\to 0$ if and only if ${\rho }_{\mathcal{H}}\left(u\right)\to 0$;

6. 

${\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}\to +\infty$ if and only if ${\rho }_{\mathcal{H}}\left(u\right)\to +\infty$;

7. 

${\parallel u\parallel }_{L^{\mathcal{H}}(\Omega )}\to 1$ if and only if ${\rho }_{\mathcal{H}}\left(u\right)\to 1$.

2.2. Some Useful Tools

First, we introduce the operator $V:W_0^{1,\mathcal{H}}(\Omega )\to {\left(W_0^{1,\mathcal{H}}(\Omega )\right)}^{*}$, by

$$\langle V\left(u\right),v\rangle ={\int }_{\Omega }({a\left(x\right)|\nabla u|}^{p-2}\nabla u+b\left(x\right){|\nabla u|}^{q-2}\nabla u+\alpha (\nabla u))\nabla vdx\hspace{1em}\forall v\in W_0^{1.\mathcal{H}}(\Omega ).$$

Analogously, as in Hu-Papageorgiou [69], Proposition 5.47, we can obtain the following properties.

Lemma 2.

Assume that $\left(H_{\alpha }\right)$ and $\left(H_1\right)$ hold. Then, the operator V is bounded, continuous, strictly monotone, and of the $\left(S_{+}\right)$-type.

We recall the definition of the pseudomonotone operator (see Papageorgiou-Winkert [70], Definition 6.1.50).

Definition 1.

Let $K:X\to X^{*}$ be an operator, with $X^{*}$ being the topological dual of X. We say that K is pseudomonotone if the following holds: if $u_n\to u$ weakly in X, $K\left(u_n\right)\to u^{*}$ in $X^{*}$ and $\underset{n\to +\infty }{lim\; sup}\langle K\left(u_n\right),u_n-u\rangle \le 0$, then, $u^{*}=K\left(u\right)$ and ${\langle K\left(u_n\right),u_n\rangle }_X\to {\langle K\left(u\right),u\rangle }_X$.

For the reader’s convenience, we recall the following Hardy’s inequality, a tool to manage singularity (see Papageorgiou-Winkert [70], Theorem 6.8.33).

Theorem 1.

If $r>1$, then there exist two constants $c=c(r,N)>0$ and ${\kappa }_0={\kappa }_0(r,N)>0$, such that

$${\parallel {\displaystyle \frac{u}{d^{1-\kappa }}}\parallel }_r\le c\parallel |\nabla u|d^{\kappa }{\parallel }_r\hspace{1em}\forall u\in W_0^{1,r}(\Omega ),$$

for all $0\le \kappa <{\kappa }_0$, where d is the distance function, defined by

$$d\left(x\right)=d(x,\partial \Omega )=\underset{y\in \partial \Omega }{min}|x-y|\hspace{1em}\forall x\in \Omega .$$

Throughout the paper, we use the following notion.

Definition 2.

Let $u:\Omega \to \mathbb{R}$ be a measurable function, we write $0\prec u$ if for all $D\subset \Omega$ compact, there exists a positive constant $c_D>0$, such that

$$0<c_D\le u\left(x\right)\mathrm{for}\mathrm{a}.\mathrm{a}.x\in \Omega .$$

Note that if $0\prec u$, then, in particular, $0<u\left(x\right)$ for a.a. $x\in \Omega$.

3. Purely Singular Problem

In this section, we prove the existence and uniqueness of solutions for a parametric pure singular problem. This result will be helpful in the next section to control the singular term. Let us consider the problem

$$\{\begin{array}{cc}-{\Delta }_p^{a}u-{\Delta }_q^{b}u-div\left(\alpha \right(\nabla u\left)\right)=\lambda \beta \left(x\right)u^{-\eta }\hfill & \mathrm{in}\Omega ,\hfill \\ u>0\hfill & \mathrm{in}\Omega ,\hfill \\ u=0\hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}$$

(S)

where $\lambda >0$.

Proposition 1.

Assume that $\left(H_{\alpha }\right)$, $\left(H_1\right)$, and $\left(H_2\right)$ hold. Then, for each $\lambda >0$, the problem (S) admits a unique solution ${\overline{u}}_{\lambda }\in W_0^{1,\mathcal{H}}(\Omega )\cap L^{\infty }(\Omega )$, such that $0\prec {\overline{u}}_{\lambda }$. Moreover, ${\overline{u}}_{\lambda }\to 0$ as $\lambda \to 0^{+}$.

Proof. 

Arguing as in Proposition 3.1 of Bai-Papageorgiou-Zeng [20], there exists a unique solution ${\overline{u}}_{\lambda }\in W_0^{1,\mathcal{H}}(\Omega )\cap L^{\infty }(\Omega )$, such that $0\prec {\overline{u}}_{\lambda }$. Now, we prove that

$${\overline{u}}_{\lambda }\to 0\mathrm{in}{L}^{\infty }(\Omega )\mathrm{as}\lambda \to 0^{+}.$$

(6)

First, note that $\beta \left(x\right){\overline{u}}_{\lambda }^{-\eta }\in L^{\infty }(\Omega )$. Indeed, following Papageorgiou-Peng [71], Proposition 5.2, we have that there exists $0<c_7\le \lambda$, such that

$$c_{7}d\left(x\right)\le {\overline{u}}_{\lambda }\left(x\right)\hspace{1em}\mathrm{for} \mathrm{a}.\mathrm{a}.x\in \Omega .$$

Let $1\le s<+\infty$. From Hardy’s inequality, we obtain

$${\int }_{\Omega }{\left({\displaystyle \frac{\beta \left(x\right)}{{\overline{u}}_{\lambda }^{\eta }}}\right)}^{s}dx={\int }_{\Omega }{\left({\overline{u}}_{\lambda }^{1-\eta }{\displaystyle \frac{\beta \left(x\right)}{{\overline{u}}_{\lambda }}}\right)}^{s}dx\le c_8{\int }_{\Omega }{\left({\displaystyle \frac{\beta \left(x\right)}{d\left(x\right)}}\right)}^{s}dx\le c_9{\parallel \nabla \beta \parallel }_s^s$$

for some constants $c_8,c_9>0$. Passing to the limit as $s\to +\infty$ (see Gasiński-Papageorgiou [72], Problem 1.9), we obtain

$$\parallel \beta {\overline{u}}_{\lambda }^{-\eta }{\parallel }_{L^{\infty }(\Omega )}\le c_{10}{\parallel \nabla \beta \parallel }_{L^{\infty }(\Omega )}$$

(7)

for some constants $c_{10}>0$. So, indeed $\beta \left(x\right){\overline{u}}_{\lambda }^{-\eta }\in L^{\infty }(\Omega )$.

Then, by (3),

$$\parallel {\overline{u}}_{\lambda }{\parallel }_{L^{\infty }(\Omega )}\le m_0{\left(\lambda \parallel \beta {\overline{u}}_{\lambda }^{-\eta }{\parallel }_{L^{\infty }(\Omega )}\right)}^{{\displaystyle \frac{1}{r-1}}},$$

that is

$$\parallel {\overline{u}}_{\lambda }{\parallel }_{L^{\infty }(\Omega )}^{1+{\displaystyle \frac{\eta }{r-1}}}\le m_0{\left({\lambda \parallel \beta \parallel }_{L^{\infty }(\Omega )}\right)}^{{\displaystyle \frac{1}{r-1}}}$$

and so (6) holds.

Finally, through the variational formulation of a weak solution for (S) and choosing $h={\overline{u}}_{\lambda }\in W_0^{1,\mathcal{H}}(\Omega )$ as a test function, we have

$${\rho }_{\mathcal{H}}(\nabla {\overline{u}}_{\lambda })=\lambda {\int }_{\Omega }\beta \left(x\right){\overline{u}}_{\lambda }^{1-\eta }dx\le \lambda c_{11}\parallel {\overline{u}}_{\lambda }\parallel ,$$

for some constants $c_{11}>0$. This implies that

$${\overline{u}}_{\lambda }\to 0\mathrm{in}{W}_0^{1,\mathcal{H}}(\Omega )\mathrm{as}\lambda \to 0^{+}$$

(see Hewitt-Stromberg [73], Theorem 13.17 and Lemma 1). □

Using Proposition 1, we infer that we can choose $\lambda \in (0,1)$ to be small enough, such that

$${\parallel {\overline{u}}_{\lambda }\parallel }_{L^{\infty }(\Omega )}\le \delta \mathrm{and}{\parallel {\displaystyle \frac{\beta }{{\overline{u}}_{\lambda }^{\eta }}}\parallel }_{L^{\infty }(\Omega )}>1,$$

(8)

where $\delta >0$ is as in hypothesis ${\left(H_f\right)}_2$. Fix such a $\lambda$ and set $\overline{u}={\overline{u}}_{\lambda }$. Clearly, as $\lambda \in (0,1)$, and from assumption ${\left(H_f\right)}_2$, we have

$$-{\Delta }_p^a\overline{u}-{\Delta }_q^b\overline{u}-div\left(\alpha \right(\nabla \overline{u}\left)\right)=\lambda \beta {\overline{u}}^{-\eta }\le \beta {\overline{u}}^{-\eta }+f(x,\overline{u}).$$

(9)

The next proposition permits building a super-solution for our problem.

Proposition 2.

Assume that $\left(H_{\alpha }\right)$, $\left(H_1\right)$, $\left(H_2\right)$ and $\left(H_f\right)$ hold. Then, for any $\mu >1$, the problem

$$\{\begin{array}{cc}-{\Delta }_p^{a}u-{\Delta }_q^{b}u-div(\alpha (\nabla u))=\mu {\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}\hfill & in\Omega ,\hfill \\ u>0\hfill & in\Omega ,\hfill \\ u=0\hfill & on\partial \Omega ,\hfill \end{array}$$

(10)

admits a unique solution $\widehat{u}\in W_0^{1,\mathcal{H}}(\Omega )\cap L^{\infty }(\Omega )$ such that $0\prec \widehat{u}$ and $\overline{u}\le \widehat{u}$.

Proof. 

We rewrite problem (10) as

$$V\left(u\right)=\mu {\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}.$$

As V is coercive, continuous, and strictly monotone (see Lemma 2), there exists a unique (by strict monotonicity) solution $\widehat{u}\in W_0^{1,\mathcal{H}}(\Omega )$. Clearly,

$$\widehat{u}\ge 0 \mathrm{and} \widehat{u}\ne 0.$$

Let $k>1$ and set

$$L_k=\{x\in \Omega :\widehat{u}\left(x\right)>k\}.$$

Choose k to be large enough, such that

$$\parallel {(\widehat{u}-k)}^{+}\parallel \le 1.$$

From Lemma 1, we have that

$${\parallel {(\widehat{u}-k)}^{+}\parallel }^p\le {\rho }_{\mathcal{H}}(\nabla {(\widehat{u}-k)}^{+})={\int }_{\Omega }\mu {\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}{(\widehat{u}-k)}^{+}dx.$$

(11)

As $r<q<p<r_{*}$, we can choose $\sigma \in (p,r^{*})$. Remember that the characteristic function ${\chi }_{L_k}\in L^{{\sigma }^{\prime }}(\Omega )$, where ${\sigma }^{\prime }$ is the Hölder conjugate exponent of $\sigma$. Moreover,

$${(\widehat{u}-k)}^{+}\in W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{\sigma }(\Omega ).$$

Thus, Hölder’s inequality guarantees that

$${\parallel {(\widehat{u}-k)}^{+}\parallel }^p\le \mu {\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}|L_k{|}^{{\displaystyle \frac{1}{{\sigma }^{\prime }}}}\parallel {(\widehat{u}-k)}^{+}{\parallel }_{L^{\sigma }(\Omega )}\le c_{12}\mu {\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}|L_k{|}^{{\displaystyle \frac{1}{{\sigma }^{\prime }}}}\parallel {(\widehat{u}-k)}^{+}\parallel ,$$

for some constant $c_{12}>0$, i.e.,

$${\parallel {(\widehat{u}-k)}^{+}\parallel }^{p-1}\le c_{12}\mu {\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}{\left|L_k\right|}^{{\displaystyle \frac{1}{{\sigma }^{\prime }}}}.$$

(12)

Next, let $j>k$. As $L_j\subseteq L_k$, we have

$$(j-k)|L_j{|}^{{\displaystyle \frac{1}{\sigma }}}\le {\left({\int }_{L_j}{(\widehat{u}-k)}^{\sigma }dx\right)}^{{\displaystyle \frac{1}{\sigma }}}\le {\left({\int }_{L_k}{(\widehat{u}-k)}^{\sigma }dx\right)}^{{\displaystyle \frac{1}{\sigma }}}\le c_{13}\parallel {(\widehat{u}-k)}^{+}\parallel ,$$

(13)

for some constant $c_{13}>0$.

Combining (12) and (13), we obtain

$$\left|L_j\right|\le {\displaystyle \frac{c_{14}}{{(j-k)}^{\sigma }}}{\left(\mu {\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}\right)}^{{\displaystyle \frac{\sigma }{p-1}}}{\left|L_k\right|}^{{\displaystyle \frac{\sigma p^{\prime }}{{\sigma }^{\prime }p}}},$$

for some constant $c_{14}>0$. Notice that ${\displaystyle \frac{\sigma p^{\prime }}{{\sigma }^{\prime }p}}>1$. Thus, Lemma B.1 of Kinderlehrer-Stampacchia [74], permits obtaining $R>1$, such that $|L_R|=0$. So,

$$\widehat{u}\in W_0^{1,\mathcal{H}}(\Omega )\cap L^{\infty }(\Omega ).$$

Moreover, arguing as in Papageorgiou-Vetro-Vetro [30], Proposition 2.4, we infer that

$$0\prec \widehat{u}.$$

Finally, as $0<\lambda <1<\mu$, and choosing ${(\overline{u}-\widehat{u})}^{+}\in W_0^{1,\mathcal{H}}(\Omega )$ as the test function, we have

$$\langle V\left(\overline{u}\right),{(\overline{u}-\widehat{u})}^{+}\rangle \le \langle V\left(\widehat{u}\right),{(\overline{u}-\widehat{u})}^{+}\rangle .$$

From the monotonicity of V (see Lemma 2), we obtain that $\overline{u}\le \widehat{u}$. □

4. Existence of Positive Solutions

In this section, we prove our main existence result. The proof is based on the pseudomonotone operator approach. Let us start with $\overline{u}\le \widehat{u}$ as in (8) and Proposition 2. Note that, by (3) and problem (10), we have

$${\parallel \widehat{u}\parallel }_{L^{\infty }(\Omega )}\le m_0{\left(\mu {\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}\right)}^{{\displaystyle \frac{1}{r-1}}}.$$

(14)

Moreover, as $\overline{u}\le \widehat{u}$ (see Proposition 2) and from assumption ${\left(H_f\right)}_1$, we have

$$\begin{array}{ccc}& & -{\Delta }_p^a\widehat{u}-{\Delta }_q^b\widehat{u}-div\left(\alpha \right(\nabla \widehat{u}\left)\right)-\beta \left(x\right){\widehat{u}}^{-\eta }-f(x,\widehat{u})\hfill \\ & \ge & \mu {\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}-{\parallel {\displaystyle \frac{\beta }{{\widehat{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}-m_1{\parallel \widehat{u}\parallel }^{\tau -1}\hfill \\ & \ge & (\mu -1){\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}-m_1{m}_0{\left(\mu {\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}\right)}^{{\displaystyle \frac{\tau -1}{r-1}}}.\hfill \end{array}$$

Then, in the case $1<\tau <r$, for $\mu >1$ that is large enough, we have

$$-{\Delta }_p^a\widehat{u}-{\Delta }_q^b\widehat{u}-div\left(\alpha \right(\nabla \widehat{u}\left)\right)\ge \beta \left(x\right){\widehat{u}}^{-\eta }+f(x,\widehat{u})\mathrm{in}\Omega .$$

(15)

Furthermore, by ${\left(H_f\right)}_1$ and assumption $m_0{m}_1<1$, the inequality also holds true for $\tau =r$ and $\mu >{\displaystyle \frac{1}{1-m_0{m}_1}}$ large.

Next, consider the truncation $\ell :W_0^{1,\mathcal{H}}(\Omega )\to W_0^{1,\mathcal{H}}(\Omega )$, defined by

$$\ell \left(u\right)\left(x\right)=\{\begin{array}{cc}\overline{u}\left(x\right)\hfill & \mathrm{if}u\left(x\right)<\overline{u}\left(x\right);\hfill \\ u\left(x\right)\hfill & \mathrm{if}\overline{u}\le u\left(x\right)\le \widehat{u}\left(x\right);\hfill \\ \widehat{u}\left(x\right)\hfill & \mathrm{if}u\left(x\right)>\widehat{u}\left(x\right),\hfill \end{array}$$

(16)

and, from Papageorgiou-Winkert [70], Corollary 4.5.19, we have

$$\nabla \ell \left(u\right)\left(x\right)=\{\begin{array}{cc}\nabla \overline{u}\left(x\right)\hfill & \mathrm{if}u\left(x\right)<\overline{u}\left(x\right);\hfill \\ \nabla u\left(x\right)\hfill & \mathrm{if}\overline{u}\le u\left(x\right)\le \widehat{u}\left(x\right);\hfill \\ \nabla \widehat{u}\left(x\right)\hfill & \mathrm{if}u\left(x\right)\ge \widehat{u}\left(x\right).\hfill \end{array}$$

(17)

Finally, $\ell :W_0^{1,\mathcal{H}}(\Omega )\to W_0^{1,\mathcal{H}}(\Omega )$ is continuous (see Papageorgiou-Peng [71], Proposition 5.1).

Next, we consider the operator $N:W_0^{1,\mathcal{H}}(\Omega )\to L^{{\tau }^{\prime }}(\Omega )$, defined by

$$N\left(u\right)=\beta \left(u\right)\ell \left(u\right){\left(x\right)}^{-\eta }+f(x,\ell \left(u\right)\left(x\right)),$$

where ${\tau }^{\prime }$ is the Hölder conjugate exponent of $\tau$. Note that, from Proposition 1 and $\left(H_f\right)$, the operator N is well defined. Furthermore, as the embedding $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{\tau }(\Omega )$ is compact, so is $L^{{\tau }^{\prime }}(\Omega )\hookrightarrow {\left(W_0^{1,\mathcal{H}}(\Omega )\right)}^{*}$ (see, e.g., Gasiński-Papageorgiou [75], Lemma 2.2.27). Thus, $N:W_0^{1,\mathcal{H}}(\Omega )\to {\left(W_0^{1,\mathcal{H}}(\Omega )\right)}^{*}$.

Our goal is to study the following bounded map

$$\begin{array}{c}\hfill K=V-N:W_0^{1,\mathcal{H}}(\Omega )\to {\left(W_0^{1,\mathcal{H}}(\Omega )\right)}^{*}.\end{array}$$

Proposition 3.

Assume that $\left(H_{\alpha }\right)$, $\left(H_1\right)$, $\left(H_2\right)$, and $\left(H_f\right)$ hold. Then, K is a pseudomonotone operator.

Proof. 

Let $\left\{u_n\right\}\subseteq W_0^{1,\mathcal{H}}(\Omega )$ be a sequence, such that $u_n\to u$ weakly in $W_0^{1,\mathcal{H}}(\Omega )$, $K\left(u_n\right)\to u_{*}$ in ${\left(W_0^{1,\mathcal{H}}(\Omega )\right)}^{*}$ and $\underset{n\to +\infty }{lim\; sup}\langle K\left(u_n\right),u_n-u\rangle$. From the compactness of the embedding $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{\tau }(\Omega )$, we have that $u_n\to u$ in $L^{\tau }(\Omega )$. Then, clearly, by $\left(H_f\right)$ and (16), we obtain

$${\int }_{\Omega }f(x,\ell \left(u_n\right)\left(x\right))(u_n-u)dx\to 0\mathrm{as}n\to +\infty .$$

Moreover, we infer that

$$\begin{array}{ccc}& & |{\int }_{\Omega }\beta \left(x\right)\ell \left(u_n\right){\left(x\right)}^{-\eta }(u_n-u)dx|\hfill \\ & \le & \underset{\{u_n<\overline{u}\}}{\int }\beta \left(x\right){\overline{u}}^{-\eta }(u_n-u)dx+\underset{\{\overline{u}\le u_n\le \widehat{u}\}}{\int }\beta \left(x\right)u_n^{-\eta }(u_n-u)dx+\underset{\{\widehat{u}<u_n\}}{\int }\beta \left(x\right){\widehat{u}}^{-\eta }(u_n-u)dx\hfill \\ & \le & 3{\int }_{\Omega }\beta \left(x\right){\overline{u}}^{-\eta }(u_n-u)dx\hfill \\ & \stackrel{H\ddot{o}lder}{\le }& 3{\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}{\parallel u_n-u\parallel }_{L^1(\Omega )}\to 0\mathrm{as}n\to +\infty .\hfill \end{array}$$

That is, we obtain that

$$\underset{n\to +\infty }{lim}\langle V\left(u_n\right),u_n-u\rangle \le 0.$$

(18)

So, the $\left(S_{+}\right)$-property of V (see Lemma 2) ensures that

$$u_n\to u\hspace{1em}\mathrm{in}{W}_0^{1,\mathcal{H}}(\Omega ).$$

From the continuity of V (see Lemma 2), we obtain

$$V\left(u_n\right)\to V\left(u\right)\hspace{1em}\mathrm{in}{\left(W_0^{1,\mathcal{H}}(\Omega )\right)}^{*}.$$

Analogously, from $\left(H_f\right)$ and (16), it follows that

$$N\left(u_n\right)\to N\left(u\right)\hspace{1em}\mathrm{in}{\left(W_0^{1,\mathcal{H}}(\Omega )\right)}^{*}.$$

Then,

$$K\left(u_n\right)\to K\left(u\right)=V\left(u\right)-N\left(u\right)\mathrm{as}n\to +\infty \mathrm{in}{W}_0^{1,\mathcal{H}}(\Omega ).$$

(19)

Thus, we deduce that $u_{*}=V\left(u\right)-N\left(u\right)=K\left(u\right)$ and $\langle K\left(u_n\right),u_n\rangle \to \langle K\left(u\right),u\rangle$ as $n\to +\infty$. □

Proposition 4.

Assume that $\left(H_{\alpha }\right)$, $\left(H_1\right)$, $\left(H_2\right)$ and $\left(H_f\right)$ hold. Then, K is strongly coercive.

Proof. 

Note that, from hypothesis $\left(H_f\right)$, Formula (16), and the compactness of the embedding $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{\tau }(\Omega )$, we have

$${\int }_{\Omega }f(x,\ell \left(u\right)\left(x\right))udx\le \underset{\{u<\overline{u}\}}{\int }f(x,\overline{u})udx+\underset{\{\overline{u}\le u\le \widehat{u}\}}{\int }f(x,u)udx+\underset{\{\widehat{u}<u\}}{\int }f(x,\widehat{u})udx\le c_{15}\parallel u\parallel$$

(20)

for some constant $c_{15}>0$. Let $u\in W_0^{1.\mathcal{H}}(\Omega )$ be such that $\parallel u\parallel \ge 1$. Recalling that

$${\rho }_H(\nabla u)\ge {\parallel u\parallel }^r$$

(see Lemma 1), we infer that

$$\begin{array}{ccc}\hfill \langle K\left(u\right),u\rangle & =& \langle V\left(u\right),u\rangle -\langle N\left(u\right),u\rangle \hfill \\ & =& {\rho }_{\mathcal{H}}(\nabla u)-{\int }_{\Omega }(\beta \left(x\right)\ell \left(u\right){\left(x\right)}^{-\eta }+f(x,\ell \left(u\right)\left(x\right)))udx\hfill \\ & \ge & {\parallel u\parallel }^r-3{\parallel {\displaystyle \frac{\beta }{{\overline{u}}^{\eta }}}\parallel }_{L^{\infty }(\Omega )}{\parallel u\parallel }_{L^1(\Omega )}-c_{15}\parallel u\parallel \hfill \\ & \ge & {\parallel u\parallel }^r-c_{16}\parallel u\parallel \hfill \end{array}$$

for some constants $c_{16}>0$. As $r>1$, it follows that

$${\displaystyle \frac{\langle K\left(u\right),u\rangle }{\parallel u\parallel }}\to +\infty \mathrm{as}\parallel u\parallel \to +\infty ,$$

that is, K is strongly coercive. □

Now, we are ready to prove our main existence result for the problem (P).

Theorem 2.

Assume that $\left(H_{\alpha }\right)$, $\left(H_1\right)$, $\left(H_2\right)$, and $\left(H_f\right)$ hold. Then, problem (P) admits at least one solution $u\in W_0^{1,\mathcal{H}}(\Omega )\cap L^{\infty }(\Omega )$, such that $0\prec u$.

Proof. 

First, note that using Propositions 3 and 4, K is pseudomonotone and coercive (and bounded), then, it is also surjective (see Motreanu-Motreanu-Papageorgiou [76], Theorem 2.63). Then, there exists $u_0\in W_0^{1,\mathcal{H}}(\Omega )$, such that $K\left(u_0\right)=0$, i.e.,

$$V\left(u_0\right)=N\left(u_0\right)\mathrm{in}{\left(W_0^{1,\mathcal{H}}(\Omega )\right)}^{*}.$$

(21)

Choosing one time $v={(\overline{u}-u_0)}^{+}$ and the other time $v={(u_0-\widehat{u})}^{+}$ in $W_0^{1,\mathcal{H}}(\Omega )$ as test functions in (21), and using (9) and (15), we conclude that

$$\langle V\left(u_0\right),{(\overline{u}-u_0)}^{+}\rangle ={\int }_{\Omega }(\beta \left(x\right){\overline{u}}^{-\eta }+f(x,\overline{u})){(\overline{u}-u_0)}^{+}dx\ge \langle V\left(\overline{u}\right),{(\overline{u}-u_0)}^{+}\rangle ,$$

(22)

and

$$\langle V\left(u_0\right),{(u_0-\widehat{u})}^{+}\rangle ={\int }_{\Omega }(\beta \left(x\right){\widehat{u}}^{-\eta }+f(x,\widehat{u})){(u_0-\widehat{u})}^{+}dx\le \langle V\left(\widehat{u}\right),{(u_0-\widehat{u})}^{+}\rangle .$$

(23)

Using the monotonicity of V (see Lemma 2) applied to (22) and (23), we infer that $\overline{u}\le u_0\le \widehat{u}$. Finally, using (16), we come back to the solution of (P), such that $u_0\in W_0^{1,\mathcal{H}}(\Omega )\cap L^{\infty }(\Omega )$ and $0\prec u_0$. □

Example 1.

A possible perturbation function that satisfies assumption $\left(H_f\right)$ is the following (for simplicity we drop the x-dependence):

$$f\left(t\right)=\{\begin{array}{ccc}0\hfill & \mathit{if}\hfill & t\le 0,\hfill \\ t^{\tau -1}(1+sin\left(5t\right))\hfill & \mathit{if}\hfill & t>0.\hfill \end{array}$$

(24)

5. Uniqueness

In this last section, some more restrictive assumptions on the perturbation f in the reaction term and on the operator G will allow us to ensure the uniqueness of solutions for problem (P). The main idea is to obtain a Díaz–Saá-type result (see Díaz-Saá [77]) for our generalised multi-phase operator. In particular, we follow some ideas introduced in Fragnelli-Mugnai-Papageorgiou [40] to deal with our general operator.

In addition to the previous hypotheses, we assume that

${\left(H_u\right)}_1$

the map $(0,+\infty )\ni t\mapsto {\displaystyle \frac{f(x,t)}{t^{\tau -1}}}$ is nonincreasing for a.a. $x\in \Omega$;

${\left(H_u\right)}_2$

the map $(0,+\infty )\ni t\mapsto G_0(t^{{\displaystyle \frac{1}{\tau }}})$ is convex.

Our main result regarding uniqueness is as follows.

Theorem 3.

Assume that $\left(H_{\alpha }\right)$, $\left(H_1\right)$, $\left(H_2\right)$, $\left(H_f\right)$, and $\left(H_u\right)$ hold. Then, the problem (P) admits a unique solution $u\in W_0^{1,\mathcal{H}}(\Omega )\cap L^{\infty }(\Omega )$, with $0\prec u$.

Proof. 

Clearly, using Theorem 2, there exists at least one solution u for problem (P), such that $u\in W_0^{1,\mathcal{H}}(\Omega )\cap L^{\infty }(\Omega )$ and $0\prec u$. Now, we prove the uniqueness. Let us start by considering the following functional $j:L^1(\Omega )\to \mathbb{R}$,

$$j\left(u\right)=\{\begin{array}{cc}{\displaystyle {\int }_{\Omega }(\frac{a\left(x\right)}{p}{|\nabla u|}^{\frac{p}{\tau }}+\frac{b\left(x\right)}{q}{|\nabla u|}^{\frac{q}{\tau }}+G_0({|\nabla u|}^{\frac{1}{\tau }}\left)\right)dx}\hfill & \mathrm{if}u\ge 0\mathrm{and}{u}^{{\displaystyle \frac{1}{\tau }}}\in W_0^{1,\mathcal{H}}(\Omega ),\hfill \\ +\infty \hfill & \mathrm{otherwise}.\hfill \end{array}$$

(25)

Clearly, $j\not\equiv +\infty$. Let us set

$$\mathrm{dom}\left(j\right)=\{u\in L^1(\Omega ):j\left(u\right)<+\infty \},$$

as the effective domain of j. Note that, by Díaz-Saá [77], Lemma 1 and ${\left(H_u\right)}_2$, we have that j is convex. Moreover, Fatou’s Lemma (see, e.g., Gasiński-Papageorgiou [75], Theorem A.2.1) implies that j is lower semicontinuous and it is clear that j is Gâteaux differentiable and the convexity of j implies the monotonicity of $j^{\prime }$. Suppose that $v\in W_0^{1,\mathcal{H}}(\Omega )\cap L^{\infty }(\Omega )$, $0\prec v$ is another solution to (P). Let $\epsilon \in (0,1)$ and set $u_{\epsilon }=u+\epsilon$ and $v_{\epsilon }=v+\epsilon$. Note that $u_{\epsilon },v_{\epsilon }\in \mathrm{int}{L}_{+}^{\infty }(\Omega )$ and by Hu-Papageorgiou [69], Proposition 2.86, we infer

$${\displaystyle \frac{u_{\epsilon }}{v_{\epsilon }}}\in L^{\infty }(\Omega ) \mathrm{and} {\displaystyle \frac{v_{\epsilon }}{u_{\epsilon }}}\in L^{\infty }(\Omega ).$$

(26)

Let $h=u_{\epsilon }^{\tau }-v_{\epsilon }^{\tau }\in W_0^{1,\mathcal{H}}(\Omega )\cap L^{\infty }(\Omega )$. Moreover, for $t\in (0,1)$, using (26), we have that $u_{\epsilon }^{\tau }+th,v_{\epsilon }^{\tau }+th\in \mathrm{dom}\left(j\right)$. Then, Green’s identity (see Hu-Papageorgiou [78], Theorem 4.106) and Fragnelli-Mugnai-Papageorgiou [40], Theorem 3.5) imply that

$$\begin{array}{c}\hfill j^{\prime }\left(u_{\epsilon }\right)\left(h\right)={\displaystyle \frac{1}{\tau }}{\int }_{\Omega }{\displaystyle \frac{-{\Delta }_p^{a}u-{\Delta }_q^{b}u-div\left(\alpha \right(\nabla u\left)\right)}{u_{\epsilon }^{\tau -1}}}hdx\end{array}$$

(27)

and

$$\begin{array}{c}\hfill j^{\prime }\left(v_{\epsilon }\right)\left(h\right)={\displaystyle \frac{1}{\tau }}{\int }_{\Omega }{\displaystyle \frac{-{\Delta }_p^{a}v-{\Delta }_q^{b}v-div\left(\alpha \right(\nabla v\left)\right)}{v_{\epsilon }^{\tau -1}}}hdx.\end{array}$$

(28)

Then, adding the two above derivatives and exploiting the monotonicity of $j^{\prime }$, we infer

$$\begin{array}{ccc}\hfill 0& \le & {\int }_{\Omega }{\displaystyle \frac{-{\Delta }_p^{a}u-{\Delta }_q^{b}u-div\left(\alpha \right(\nabla u\left)\right)}{u_{\epsilon }^{\tau -1}}}hdx-{\int }_{\Omega }{\displaystyle \frac{-{\Delta }_p^{a}v-{\Delta }_q^{b}v-div\left(\alpha \right(\nabla v\left)\right)}{v_{\epsilon }^{\tau -1}}}hdx\hfill \\ & =& {\int }_{\Omega }\beta \left(x\right)({\displaystyle \frac{1}{u^{\eta }{u}_{\epsilon }^{\tau -1}}}-{\displaystyle \frac{1}{v^{\eta }{v}_{\epsilon }^{\tau -1}}})(u_{\epsilon }^{\tau }-v_{\epsilon }^{\tau })dx+{\int }_{\Omega }({\displaystyle \frac{f(x,u)}{u_{\epsilon }^{\tau -1}}}-{\displaystyle \frac{f(x,v)}{v_{\epsilon }^{\tau -1}}})(u_{\epsilon }^{\tau }-v_{\epsilon }^{\tau })dx.\hfill \end{array}$$

Finally, passing the limits as $\epsilon \to 0$, we obtain

$$0\le {\int }_{\Omega }\beta \left(x\right)({\displaystyle \frac{1}{u^{\tau +\eta -1}}}-{\displaystyle \frac{1}{v^{\tau +\eta -1}}})(u^{\tau }-v^{\tau })dx+{\int }_{\Omega }({\displaystyle \frac{f(x,u)}{u^{\tau -1}}}-{\displaystyle \frac{f(x,v)}{v^{\tau -1}}})(u^{\tau }-v^{\tau })dx.$$

(29)

Note that the map $t\mapsto {\displaystyle \frac{1}{t^{\tau +\eta -1}}}$ is strictly decreasing in $(0,+\infty )$. From ${\left(H_u\right)}_1$, we deduce that $u\equiv v$. □

We conclude this section with a possible example of a function f that satisfies the uniqueness assumptions ${\left(H_u\right)}_1$. Notice that the function in Example 1 does not satisfy the monotonicity hypothesis ${\left(H_u\right)}_1$.

Example 2.

Consider the function $f:\mathbb{R}\to \mathbb{R}$ defined by (for simplicity, we drop the x-dependence):

$$f\left(t\right)=\{\begin{array}{ccc}0\hfill & \mathit{if}\hfill & t\le 0,\hfill \\ {\widehat{k}}_1{t}^{\tau -1}\hfill & \mathit{if}\hfill & 0<t<1,\hfill \\ {\widehat{k}}_2{t}^{\tau -2}\hfill & \mathit{if}\hfill & t\ge 1,\hfill \end{array}$$

(30)

where ${\widehat{k}}_1\ge {\widehat{k}}_2>0$ are some constants. Clearly, the function f satisfies $\left(H_f\right)$ and ${\left(H_u\right)}_1$.

6. Conclusions

In this paper, we provide a meaningful generalization of singular problems driven by unbalanced growth operators. In particular, the inclusion of a general lower-order term enables us to obtain results applicable to a broad class of singular problems, with potential applications in physics and engineering. For an overview of related models in physical, biological, and engineering contexts, we refer to the Introduction.

Finally, we highlight several promising directions for future research, including:

1.

Extension to the variable exponent setting;

2.

Analysis in the fractional (nonlocal) framework;

3.

Treatment of the strongly singular case in the reaction term, i.e., $\eta >1$;

4.

Further generalisations using new analytical techniques, as in He-Anjum-He-Alsolami [16] and related works.