Monotone operator methods for a class of nonlocal multi-phase variable exponent problems

In this article, we study the following nonlocal multi-phase variable exponent problem

with

where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ $(N\geq 2)$ with Lipschitz boundary; $f\in W_{0}^{1,\mathcal{T}}(\Omega)^{*}$; $\mathcal{M}$ is a $C^{1}$-continuous nondecreasing function; $p,q,r\in C_{+}(\overline{\Omega})$ with $1<p(x)<q(x)<r(x)$; and $0\leq\mu_{1}(\cdot),\mu_{2}(\cdot)\in L^{\infty}(\Omega)$.

The operator

governs anisotropic and heterogeneous diffusion and is associated with the energy functional

This operator is referred to as a ”multi-phase” operator because it encapsulates three distinct types of elliptic behavior within a unified framework. Such a structure allows the model to describe phenomena where materials or processes exhibit varying properties in different regions—for instance, materials that are harder in some areas and softer in others.

The energy functional given in (2) was first introduced in [1] for constant exponents, where the authors established regularity results for multi-phase variational problems. Later, [2] investigated Dirichlet problems driven by multi-phase operators with variable exponents, providing a priori upper bounds for weak solutions. More recently, [3] examined multi-phase operators with variable exponents, analyzing the associated Musielak-Orlicz Sobolev spaces, extending Sobolev embedding results, and establishing essential regularity properties. Additionally, they demonstrated existence and uniqueness results for Dirichlet problems involving gradient-dependent nonlinearity and derived local regularity estimates.

To provide historical context, we also discuss the development of double-phase operators associated with the energy functional

This type of functional was introduced in [4], and since then, numerous studies have explored its properties and applications (see, e.g., [5, 6, 7, 8, 9, 10]). The significance of this model extends across multiple disciplines, underscoring its broad applicability.

While preparing this article, we could only find the paper [11] where a Kirchhoff-type (i.e. nonlocal) problem involving a multi-phase operator with variable exponents is studied. In this paper, the author investigate a Kirchhoff-type problem involving a multi-phase operator with three variable exponents. The problem features a right-hand side comprising a Carathéodory perturbation, which is defined locally, along with a Kirchhoff term. By employing a generalized version of the symmetric mountain pass theorem and leveraging recent a priori upper bounds for multi-phase problems, the author establishes the existence of sequence of nontrivial solutions which converges to zero in the corresponding Musielak-Orlicz Sobolev space as well as in $L^{\infty}(\Omega)$.

The paper is organised as follows. In Section 2, we first provide some background for the theory of variable Sobolev spaces $W_{0}^{1,p(x)}(\Omega)$ and the Musielak-Orlicz Sobolev space $W_{0}^{1,\mathcal{H}}(\Omega)$, and then obtain a crucial auxiliary result. In Section 3, we set up the first problem where we work with a general nonlinearity $f\in W_{0}^{1,\mathcal{T}}(\Omega)^{*}$, and obtain the existence and uniqueness result for ($\mathcal{P}$). In Section 4, we study the second problem where we specify the nonlinearity $f$ as $f=f(x,u,\nabla u)$, and obtain an existence result for ($\mathcal{P}$).

We start with some basic concepts of variable Lebesgue-Sobolev spaces. For more details, and the proof of the following propositions, we refer the reader to [12, 13, 14, 15, 16].

For $h\in C_{+}(\overline{\Omega})$ denote

For any $h\in C_{+}\left(\overline{\Omega}\right)$, we define the variable exponent Lebesgue space by

Then, $L^{h(x)}(\Omega)$ endowed with the norm

becomes a Banach space. The convex functional $\rho:L^{h(x)}(\Omega)\rightarrow\mathbb{R}$ defined by

is called modular on $L^{h(x)}(\Omega)$.

The variable exponent Sobolev space $W^{1,h(x)}(\Omega)$ is defined by

with the norm

for all $u\in W^{1,h(x)}(\Omega)$.

The space $W_{0}^{1,h(x)}(\Omega)$ is defined as $\overline{C_{0}^{\infty}(\Omega)}^{\|\cdot\|_{1,h(x)}}=W_{0}^{1,h(x)}(\Omega)$, and hence, it is the smallest closed set that contains $C_{0}^{\infty}(\Omega)$. Therefore, $W_{0}^{1,h(x)}(\Omega)$ is also a separable and reflexive Banach space due to the inclusion $W_{0}^{1,h(x)}(\Omega)\subset W^{1,h(x)}(\Omega)$.
Note that as a consequence of Poincaré inequality, $\|u\|_{1,h(x)}$ and $|\nabla u|_{h(x)}$ are equivalent norms on $W_{0}^{1,h(x)}(\Omega)$. Therefore, for any $u\in W_{0}^{1,h(x)}(\Omega)$ we can define an equivalent norm $\|u\|$ such that

In the sequel, we introduce the multi-phase operator, the Musielak–Orlicz space, and the Musielak–Orlicz Sobolev space, respectively.
We make the following assumptions.

• $(H_{1})$

  $p,q,r,s\in C_{+}(\overline{\Omega})$ with $p(x)<N$; $1<p^{-}\leq p(x)<q(x)<r(x)<s(x)<p^{*}(x)$; $s^{+}<p^{*}(x)$.

• $(H_{2})$

  $\mu_{1}(\cdot),\mu_{2}(\cdot)\in L^{\infty}(\Omega)$ such that $\mu_{1}(x)\geq 0$ and $\mu_{2}(x)\geq 0$ for all $x\in\overline{\Omega}$.

Under the assumptions $(H_{1})$ and $(H_{2})$, we define the nonlinear function $\mathcal{T}:\Omega\times[0,\infty]\to[0,\infty]$, i.e. the multi-phase operator, by

Then the corresponding modular $\rho_{\mathcal{T}}(\cdot)$ is given by

The Musielak-Orlicz space $L^{\mathcal{T}}(\Omega)$, is defined by

endowed with the Luxemburg norm

Analogous to Proposition 2.1, there are similar relationship between the modular $\rho_{\mathcal{T}}(\cdot)$ and the norm $\|\cdot\|_{\mathcal{T}}$, see [3, Proposition 3.2] for a detailed proof.

The Musielak-Orlicz Sobolev space $W^{1,\mathcal{T}}(\Omega)$ is defined by

and equipped with the norm

where $\|\nabla u\|_{\mathcal{T}}=\|\,|\nabla u|\,\|_{\mathcal{T}}$.
The space $W_{0}^{1,\mathcal{T}}(\Omega)$ is defined as $\overline{C_{0}^{\infty}(\Omega)}^{\|\cdot\|_{1,\mathcal{T}}}=W_{0}^{1,\mathcal{T}}(\Omega)$. Notice that $L^{\mathcal{T}}(\Omega),W^{1,\mathcal{T}}(\Omega)$ and $W_{0}^{1,\mathcal{T}}(\Omega)$ are uniformly convex and reflexive Banach spaces, and the following embeddings hold [3, Propositions 3.1, 3.3].

As a conclusion of Proposition 2.6:
We have the continuous embedding $W_{0}^{1,\mathcal{T}}(\Omega)\hookrightarrow L^{h(\cdot)}(\Omega)$, and there is a constant $c_{\mathcal{T}}$ such that

As well, $W_{0}^{1,\mathcal{H}}(\Omega)$ is compactly embedded in $L^{\mathcal{H}}(\Omega)$.
Thus, $W_{0}^{1,\mathcal{T}}(\Omega)$ can be equipped with the equivalent norm

We lastly introduce the seminormed spaces

and

which are endow with the seminorms

and

respectively. We have $L^{\mathcal{T}}(\Omega)\hookrightarrow L^{q(\cdot)}_{\mu_{1}}(\Omega)$ and $L^{\mathcal{T}}(\Omega)\hookrightarrow L^{r(\cdot)}_{\mu_{2}}(\Omega)$ continuously [3, Proposition 3.3].

The following result is obtained by the author in his recently submitted paper, which is still under review. However, since it plays a crucial part to obtain the main regularity results of the present paper, we provide its proof for the convenience of the reader.

Let us define the functional $\mathcal{G}:W_{0}^{1,\mathcal{H}}(\Omega)\rightarrow\mathbb{R}$ as

Therefore, we can define the operator $\mathcal{H}:W_{0}^{1,\mathcal{T}}(\Omega)\rightarrow W_{0}^{1,\mathcal{T}}(\Omega)^{*}$ by

where $\mathcal{H}(u)=\mathcal{G}^{\prime}(u)$. As it is well-know from the theory of monotone operators [17], due to (16) and (18), one way to show that $u\in W_{0}^{1,\mathcal{T}}(\Omega)$ is a solution to problem ($\mathcal{P}$) for all test functions $\varphi\in W_{0}^{1,\mathcal{T}}(\Omega)$ is to solve the operator equation

We employ the following well-known result from nonlinear monotone operator theory (see, e.g., [17] for further details).

The following is the first main result.