Nonvariational Elliptic PDEs & Gradient Dependence

• Nonvariational elliptic PDEs with gradient dependence are second-order equations featuring nonlinear terms in both the solution and its gradient without a variational structure.
• They use advanced methods such as truncation and fixed-point schemes to rigorously establish existence and multiplicity of solutions.
• Recent research focuses on optimal regularity, explicit gradient estimates, and transformation techniques to overcome analytical challenges.

Nonvariational elliptic partial differential equations (PDEs) with gradient dependence refer to a broad class of second-order elliptic PDEs in which the nonlinearity involves both the unknown function and its gradient, but the system lacks a variational structure (i.e., does not arise as the Euler-Lagrange equation for any classical energy functional). These problems exhibit distinctive analytical challenges due to the presence of $\nabla u$ in the nonlinear term, which breaks the usual self-adjointness and compactness properties. The field encompasses models of the form $-\Delta u = f(x,u,\nabla u)$ and includes fully nonlinear analogs and systems, with significant recent developments on existence, multiplicity, regularity, and qualitative properties of solutions.

1. Model Problems and Structural Hypotheses

The prototypical nonvariational elliptic PDE with gradient dependence on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$ ($n\ge3$) is: $$-\Delta u(x) = f(x, u(x), \nabla u(x)),\quad x\in\Omega;\qquad u=0,\ x\in\partial\Omega$$ where $u\in H_0^1(\Omega)$ is sought as a weak solution. Under minimal conditions, the nonlinearity $f$ must satisfy Carathéodory regularity, controlled growth (e.g., $|f(x,t,\xi)| \le c_1[1+|t|^{s-1}+|\xi|]$ for $1\le s<2^* = 2n/(n-2)$), and uniform Lipschitz continuity in $t$ and $\xi$ with a spectral gap constraint linking the $L_2$-Lipschitz constant to the first Dirichlet eigenvalue of $-\Delta$.

Fully nonlinear and quasilinear generalizations cover equations of the form

$F(x, u, \nabla u, D^2u) = 0$

with $F$ nonvariational (i.e., not expressible as divergence of a function of $u$ and $\nabla u$ alone), encompassing source terms $g(\nabla u, S(u))$ potentially with nonlocal or higher-order dependence (Cavaterra et al., 2019).

In the system case (e.g., for $m$ coupled equations), nonvariationality persists when each component's equation depends on both the unknown functions and their gradients, as well as on the cross-gradients of the system (Dellouche et al., 2021, Biagi et al., 2019, Cianciaruso et al., 2017).

2. Solution Concepts and Frameworks

Weak solution concepts originate in Sobolev spaces, where $u\in H_0^1(\Omega)$ (or $W_0^{1,p}$ for quasilinear problems) solves

$$\int_\Omega \nabla u \cdot \nabla v\,dx = \int_\Omega f(x, u, \nabla u)\,v\,dx\quad\forall\,v \in H_0^1(\Omega).$$

For fully nonlinear equations, viscosity solutions—$C^0$ sub- and super-solutions defined using second-order test functions—are essential, especially when standard regularity fails or for equations in nondivergence form (Silva et al., 2020, Koike et al., 2010, Oliveira, 2 Nov 2025). In the multivalued or inclusion setting, solutions are defined as triples $(u,\xi,f)$, with $\xi$ (for example) in the subdifferential of a convex function and $f\in F(x,u,\nabla u)$ a.e. (Papageorgiou et al., 2018).

For systems, existence and uniqueness depend on monotonicity, positivity, and suitable cone-theoretic or topological indices, with solutions often constructed in cones of $C^1$ nonnegative functions with explicit gradient bounds (Dellouche et al., 2021, Biagi et al., 2019, Cianciaruso et al., 2017).

3. Existence, Multiplicity, and Truncation Methods

Recent work has resolved the existence of infinitely many solutions to nonvariational elliptic PDEs with gradient dependence, previously a longstanding open problem (Bahrouni, 24 Dec 2025). The methodology is based on refined truncation—constructing "local" nonlinearities $f_n^{\pm}$ supported in disjoint intervals of $u$—combined with a fixed-point/iteration scheme:

• For fixed $n$, solve the truncated problem using a minimization at frozen gradient, yielding variational minimizers confined to a-priori bounds.
• Use the contraction mapping principle (under spectral/Lipschitz hypotheses) to iterate solutions to a genuine fixed point, thereby generating sequences of positive and negative solutions corresponding to distinct intervals in $u$.
• Disjointness of intervals ensures multiplicity and sequence separation: distinct solutions are localized in nonoverlapping regions of phase space.

Analogous strategies have been adapted for singular, quasilinear systems, but often require additional a priori bounds or barrier constructions to accommodate singularities or lack of uniform ellipticity (Dellouche et al., 2021). Topological fixed-point index and degree arguments have also been used extensively in systems with gradient-dependent BCs or constraints (Biagi et al., 2019, Cianciaruso et al., 2017, Papageorgiou et al., 2018). For certain multi-equation setups, barrier functions and cone invariance under Green's integral operators are essential for trapping solutions within desired ranges and achieving nontriviality.

4. Regularity and Gradient Estimates

Sharp regularity properties are available even under broad forms of gradient dependence:

• For equations of the form $-\Delta u = f(u) + g(\nabla u, S(u))$ with general structure, pointwise global gradient estimates can be obtained via the $P$-function method, which controls $|\nabla u|$ by functional expressions involving $F(u) = \int f$ and the structure of the elliptic operator (Cavaterra et al., 2019). Specifically, for $p$-Laplacian-type models, these yield explicit bounds of Modica type—$|\nabla u(x)| \le [C(F(u)-C_u)]^{1/p}$—uniformly in $\mathbb{R}^n$.
• In fully nonlinear and unbounded coefficient scenarios, regularity theories yield optimal $C^{0,\alpha}$, $C^{1,\alpha}$, $C^{1,\log\text{-}Lip}$, and $C^{2,\alpha}$ estimates, with the precise exponent dictated by scaling, integrability of the drift (gradient) term, and the regime of gradient growth ($m$-homogeneity, $0<m\le2$) (Silva et al., 2020).
• For singular quasilinear systems, a combination of Hardy-Sobolev inequalities, global $L^\infty$-gradient bounds, and Schauder-type estimates ensure existence in $C^{1,\alpha}$ with explicit control over both solution and gradient norms (Dellouche et al., 2021).
• Special cases with weight degeneracy and arbitrary power-law gradient dependence are handled by Bernstein-type methods, establishing explicit local and global gradient bounds and extending Liouville-type rigidity (Ching et al., 2018).

5. Comparison Principles, Uniqueness, and Nonexistence

The comparison principle for viscosity solutions is foundational for uniqueness and qualitative analysis in nonvariational, possibly degenerate, fully nonlinear equations with superlinear gradient terms. When $H(x,\nabla u)$ is convex and superlinear and the remaining terms satisfy proper ellipticity and regularity, one can prove that classical or viscosity subsolutions never exceed supersolutions, provided all functions are of at most polynomial growth (Koike et al., 2010). This framework remains robust even in presence of nonconvex Hamiltonians (under geometric degeneracy), and extends to monotone systems of PDEs.

Nonexistence results generally rely on energy estimates, test function arguments, and violation of growth or sign conditions. For certain large in-ball radii, explicit integral constraints, or excessive parameter regimes, nontrivial solutions are ruled out (Oliveira, 2 Nov 2025, Biagi et al., 2019, Cianciaruso et al., 2017).

6. Transformations, Change of Variables, and Viscosity Solution Theory

A remarkable structural phenomenon is the Kazdan–Kramer type change of variable for second-order PDEs with "natural" gradient terms: $M(x, Du, D^2u) + g(u)N(x, Du, D^2u) + f(x, u) = 0$ where $N$ is a quadratic term encoding interaction between $M$ and $Du$. Via the monotonic diffeomorphism $v = \Phi_g(u)$ with $\Phi_g'(u) = e^{G(u)}$, $G(u) = \int g$, one can exactly "gauge away" the gradient term, reducing to a PDE for $v$ without $N$ but with transformed zeroth-order term (Oliveira, 2 Nov 2025). This encompasses Laplacian, $m$-Laplacian, $k$-Hessian, and even the infinity Laplacian, providing explicit means to transfer existence, uniqueness, and nonexistence results from the transformed problem to the original.

For equations such as the infinity Laplacian with quartic gradient dependence, this transformation reduces the existence and uniqueness of viscosity solutions with Dirichlet data to the classic, well-developed theory for the homogeneous inhomogeneous infinity Laplace equation (Oliveira, 2 Nov 2025).

7. Systems, Multivalued Inclusions, and Advanced Operator Structures

The extension to systems and inclusions significantly broadens the landscape of nonvariational gradient-dependent elliptic problems. Elliptic systems with both $u$ and $\nabla u$ appearing nonlinearly and cross-componentally arise, for instance, in coupled $p$-Laplacian and reaction-diffusion frameworks (Dellouche et al., 2021, Biagi et al., 2019, Cianciaruso et al., 2017). Existence often leverages monotonicity, cone invariance, and coupled barrier constructions.

For inclusions and equations with unilateral constraints, e.g., nonlinear Neumann inclusions with sources in convex subdifferentials and multivalued terms depending on $\nabla u$, the solution theory exploits pseudo-monotonicity, Moreau–Yosida approximation, and topological (Bader's) alternatives, with key a priori $C^{1,\alpha}$ estimates providing compactness for passage to the limit (Papageorgiou et al., 2018).

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The modern theory of nonvariational elliptic PDEs with gradient dependence therefore centers on the existence of rich multiplicity structures via truncation and fixed-point methods, optimal regularity theory accommodating superlinear or singular gradient growth, comparison results and transformations to eliminate or transfer gradient dependence, and robust frameworks for the analysis of systems, inclusions, and various boundary condition contexts. The area remains active with open questions on finer regularity (e.g., $W^{1,p}$ theory in superlinear cases), extensions to parabolic and obstacle problems, and further weakening of structural conditions, as well as the development of unified approaches dovetailing existence, uniqueness, and regularity for both variational and nonvariational regimes (Bahrouni, 24 Dec 2025, Silva et al., 2020, Oliveira, 2 Nov 2025).