Double-Phase Variable Exponent Problems

• Double-phase variable exponent problems are elliptic PDE models featuring locally variable p(x) and q(x) growth terms that capture distinct material regimes.
• Their variational formulation leverages Musielak–Orlicz Sobolev spaces and critical point theory to establish existence, multiplicity, and regularity of solutions.
• Recent advances extend these models to fractional, manifold, and singular settings using novel vector inequalities and concentration–compactness techniques.

The double-phase variable exponent problem describes a class of elliptic partial differential equations (PDEs) and variational integrals modeling materials with regions of distinct growth regimes determined by pointwise variable exponents. The fundamental feature is a differential operator superposing a $p(x)$-growth term with a $q(x)$-growth term, each weighted locally, capturing spatially heterogeneous ellipticity and anisotropy. This flexibility—which generalizes uniform $p$-Laplacian and double-phase ($p$-$q$) scenarios—has driven a rich body of research in nonlinear analysis, functional spaces, and critical point theory, particularly in settings involving Kirchhoff-type nonlocality, singular potentials, fractional operators, and variational inequalities.

1. Mathematical Structure of Double-Phase Variable Exponent Operators

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz domain, and consider two continuous exponent functions $p,q\colon \overline{\Omega} \to (1,n)$ with $1 < p(x) < q(x) < n$, and a weight $\mu \in L^\infty(\Omega)$, $\mu(x) \geq 0$ a.e. The prototypical double-phase integrand is

$H(x, t) = t^{p(x)} + \mu(x) t^{q(x)}.$

The associated operator is

$$u \mapsto \mathrm{div} \Big[ |\nabla u|^{p(x)-2}\nabla u + \mu(x)|\nabla u|^{q(x)-2}\nabla u \Big].$$

This operator exhibits simultaneous $p(x)$-type and $q(x)$-type growth locally modulated by $\mu(x)$; regions with $\mu(x) > 0$ are “hard” (higher growth), while $\mu(x) = 0$ reduces to the $p(x)$-Laplace case.

A Kirchhoff-type term introduces nonlocality:

$$M\left(\int_\Omega H(x, |\nabla u|)\,dx\right) \cdot \mathrm{div}\Big[ |\nabla u|^{p(x)-2}\nabla u + \mu(x)|\nabla u|^{q(x)-2}\nabla u \Big] = \lambda f(x, u), \quad u|_{\partial\Omega} = 0,$$

where $M$ is a nondecreasing $C^1$-function satisfying structural bounds $$k_1 t^{\alpha_1-1} \leq M(t) \leq k_2 t^{\alpha_2-1}$$ for $1 < \alpha_1 \leq \alpha_2$ and $k_i > 0$ (Avci, 8 Jul 2025, Ho et al., 2022).

2. Functional Framework: Musielak–Orlicz and Variable Exponent Spaces

Analysis of these problems occurs in Musielak–Orlicz (variable-exponent Orlicz) Sobolev spaces:

• Variable-exponent Lebesgue space: For $h \in C_+(\overline{\Omega})$,

$$L^{h(x)}(\Omega) = \{u: \int_\Omega |u(x)|^{h(x)} dx < \infty\},$$

with Luxemburg norm

$$\|u\|_{h(x)} = \inf \{ \gamma > 0 : \int_\Omega |u/\gamma|^{h(x)} dx \leq 1 \}.$$

• Musielak–Orlicz–Sobolev space: The modular is

$\rho_H(u) = \int_\Omega H(x, |u|) dx,$

and the norm is

$$\|u\|_H = \inf\{ \gamma > 0 : \rho_H(u/\gamma) \leq 1 \}.$$

The Sobolev space $W^{1,H}(\Omega)$ consists of $u \in L^H(\Omega)$ with $\nabla u \in L^H(\Omega)$, and $W_0^{1,H}(\Omega)$ is its closure of $C_0^\infty(\Omega)$.

Under standard exponent bounds, these spaces are separable, reflexive, and uniformly convex. Compact embeddings

$$W^{1,H}_0(\Omega) \hookrightarrow L^{r(\cdot)}(\Omega)$$

hold for $r(x) < p^*(x) = n p(x)/(n-p(x))$ (and analogously for the $q$-phase) (Khamsi et al., 8 Jul 2025, Ho et al., 2022).

3. Variational Formulation and Energy Functionals

The typical energy functional associated to the double-phase variable exponent problem is

$$I_\lambda(u) = \widehat{M}\left( \int_\Omega H(x, |\nabla u|)\,dx \right) - \lambda \int_\Omega F(x, u)\,dx,$$

where $F(x, t) = \int_0^t f(x, s)\,ds$ and $\widehat{M}(s) = \int_0^s M(\tau)\,d\tau$ (Avci, 8 Jul 2025, Avci, 22 Feb 2025).

For nonlinearities depending on $u$ and possibly $\nabla u$ (convection-type), direct monotone operator theory (Browder–Minty) applies:

• The map $T(u)$ defined by the sum of double-phase gradient and reaction terms is shown to be coercive, strictly monotone, and hemicontinuous (Avci, 22 Feb 2025, Crespo-Blanco et al., 2021).
• Weak solutions exist via surjectivity results for monotone operators.

For variational functionals admitting critical point theory, the functional is $C^1$, coercive, and sequentially weakly lower semicontinuous. The Gateaux derivative for the Kirchhoff-type energy incorporates both the nonlocal coefficient and the double-phase structure:

$$\langle I_\lambda'(u), \varphi \rangle = M\Big( \rho_H(u) \Big) \int_\Omega \Big[ |\nabla u|^{p(x)-2} \nabla u + \mu(x) |\nabla u|^{q(x)-2} \nabla u \Big] \cdot \nabla \varphi\,dx - \lambda \int_\Omega f(x, u) \varphi\,dx.$$

4. Existence, Multiplicity, and Regularity Results

A central achievement is the establishment of existence and multiplicity of solutions under Carathéodory growth, superlinear and critical nonlinearities, and Kirchhoff-type nonlocality:

• Three solutions (Kirchhoff–double-phase): By applying Bonanno–Chinn (Bonanno–Marano) three-critical-point theorem to the energy functional $I_\lambda$, existence of at least three distinct weak solutions is shown under explicit growth conditions (on $f$), suitable construction of test functions, and verification of coercivity, monotonicity, and compactness properties. This includes new vector inequalities that enable $C^1$-regularity of nonhomogeneous energy functionals (Avci, 8 Jul 2025).
• Infinitely many small solutions: Kajikiya’s abstract critical point theorem allows construction of a sequence of nontrivial solutions $u_n$ with amplitudes tending to zero, exploiting symmetric properties of the functional, a priori $L^\infty$ bounds, and compact truncations (Ho et al., 2022, Ha et al., 2023).
• Regularity: While higher regularity (e.g., $C^{1,\alpha}$ estimates) is not pursued in some works, $C^1$-regularity of the energy functional is proven using new vector inequalities to manage discontinuity and nonhomogeneity (Avci, 8 Jul 2025). References for further regularity results in double-phase, variable exponent settings are available (Baroni–Colombo–Mingione, Colombo–Mingione).

5. Novel Analytical Tools and Techniques

Distinctive mathematical apparatus is developed and used:

• n-dimensional vector inequalities: New inequalities (e.g., Proposition 2.2 in (Avci, 8 Jul 2025)) bound quotient of weighted gradients and are essential to prove differentiability and continuity of the variational functional’s derivative.
• Concentration–compactness principle: Adapted for double-phase, variable exponent settings, this tool manages loss of compactness due to critical growth and yields appropriate measures for defect mass at concentration points (Ha et al., 2023, Arora et al., 29 Jan 2025).
• Sub-supersolution/trapping region constructions: For systems and problems with constraints, sub-supersolution and monotone operator methods ensure the existence of weak solutions in order intervals, even in the absence of full coercivity (Carl et al., 2022, Guarnotta et al., 2022).
• Mountain-Pass, Nehari manifold, genus theory: Multiplicity is proved using advanced critical point theory (e.g., genus and symmetric mountain-pass arguments), tailored to the nonhomogeneous functional structure (Crespo-Blanco et al., 2022, Ha et al., 2023).

6. Extensions: Fractional, Manifold, and Singular Double-Phase Problems

The double-phase variable exponent paradigm extends to fractional-order and geometric PDE settings:

• Fractional double-phase operators: Nonlocal versions are constructed via variable order Gagliardo seminorms; variational frameworks on fractional Sobolev spaces, concentration–compactness, and Morse theory demonstrate existence and multiplicity without Ambrosetti–Rabinowitz conditions (Biswas et al., 2021, Aberqi et al., 2023, Avci, 4 Jan 2026).
• Manifolds: Double-phase problems are formulated on compact Riemannian manifolds using Sobolev–Orlicz spaces and embedding theorems. Multiplicity and nodal structure of solutions are proved via Nehari manifold methods (Aberqi et al., 2021).
• Singular potentials: Incorporate Hardy-type singular terms and establish positivity results using strong minimum principles (Avci, 4 Jan 2026).

7. Principal Theorems and Model Results

Result Type	Main Hypotheses	Conclusion (Solution Type)
Existence (Bonanno–Marano critical pt.)	$f$ Carathéodory, growth, $M$ bounds	$\geq$3 weak solutions for interval $\lambda\in (1/\sigma^r, 1/\sigma_r)$ (Avci, 8 Jul 2025)
Infinitely many small solutions	$p,q,a$, $M$, $f$ odd, subcritical	Sequence $u_n$ with $\|u_n\|_{1,H}\to 0$, $\|u_n\|_\infty\to 0$ (Ho et al., 2022)
Uniqueness (modular convexity)	$p,q$ continuous, $\mu(x)\geq 0$, dual source	Unique minimizer of energy (and weak solution) (Khamsi et al., 8 Jul 2025)

The field synthesizes variable exponent function space theory, generalized modular analysis, and critical point methods, underpinning the mathematical modeling of phase transitions, heterogeneous media, and nonhomogeneous phenomena in elliptic PDEs.