Obstacle Problems for Elliptic Operators with Solution-Dependent Shifts: Existence and Uniqueness via a Three-Term Decomposition

Abstract

We prove the existence and uniqueness of weak solutions to an obstacle problem for a nonlinear elliptic operator in divergence form. The variational inequality under consideration involves an integral over the domain of the Frobenius inner product of the operator $\mathcal{S}(z,\nabla u-\mathcal{O}(u\left)\right)$ with the gradient difference $\nabla (v-u)$, plus the Euclidean inner product of u and $v-u$, which is required to be nonnegative for all admissible functions v. The admissible set consists of functions in the Sobolev space $W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$ with prescribed Dirichlet boundary trace and lying above a given obstacle $\psi$ almost everywhere. The obstacle condition $v\ge \psi$ a.e. models a lower bound constraint (e.g., a membrane or a displacement limit) that the admissible functions must respect, while the boundary value $\delta$ prescribes the Dirichlet data. The principal part contains a solution-dependent shift $\mathcal{O}\left(u\right)$, which is Lipschitz continuous, while $\mathcal{S}$ is assumed to be globally Lipschitz and strongly monotone with respect to equal shifts, with quadratic growth and coercivity. This structural framework can be interpreted in terms of symmetry: the strong monotonicity condition expresses a quantitative symmetry property of $\mathcal{S}$ with respect to equal shifts, and the shift $\mathcal{O}\left(u\right)$ introduces a symmetry-breaking coupling. The smallness condition ensures that this asymmetry remains under control. However, we do not pursue a full group-invariance or Lie-symmetry analysis; the symmetry perspective is used here as a heuristic and interpretative tool. The main difficulty lies in the mismatch of shifts when comparing two admissible functions. This is resolved by a three-term decomposition of the monotonicity estimate, combined with Young’s inequality and Poincaré’s inequality, under the smallness condition that the product of the Lipschitz constant of $\mathcal{S}$, the Lipschitz constant of $\mathcal{O}$, and the Poincaré constant is bounded above by one quarter of the strong monotonicity modulus. Existence follows from the Kinderlehrer–Stampacchia theorem; uniqueness is obtained from the same decomposition. The result unifies and extends previous contributions that treated either the lower-order term or the shift coupling separately, and it does so within a unified quadratic framework that avoids the technical overhead of variable exponents and Young measures.

1. Introduction

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\ge 2$, be a bounded domain with a Lipschitz boundary. We consider the obstacle problem

$${\int }_{\mathrm{\Omega }}\left[\mathcal{S}(z,\nabla u-\mathcal{O}\left(u\right)):\nabla (v-u)+{\langle u,v-u\rangle }_{{\mathbb{R}}^m}\right]dz\ge 0\forall v\in {\mathcal{K}}_{\psi ,\delta },$$

(1)

where the colon denotes the Frobenius inner product on ${\mathbb{M}}^{m\times n}$, and the admissible set is

$${\mathcal{K}}_{\psi ,\delta }:=\left\{v\in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m):v-\delta \in W_0^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m),v\ge \psi \mathrm{a}.\mathrm{e}.\mathrm{in}\mathrm{\Omega }\right\}.$$

Here, $\psi$ is a given obstacle function and $\delta$ prescribes the Dirichlet boundary data. The term ${\langle ·,·\rangle }_{{\mathbb{R}}^m}$ denotes the Euclidean inner product in ${\mathbb{R}}^m$, integrated over $\mathrm{\Omega }$.

Problems of this type originate in the variational modelling of constrained deformations in elasticity and non-Newtonian fluid dynamics, where the stress–strain relation depends nonlinearly on the gradient [1,2,3]. Mathematically, they are formulated as variational inequalities, and their well-posedness is established within the theory of monotone operators. The foundational existence results for equations of the form $Au=f$ with monotone coercive operators go back to Minty, Browder, Leray and Lions [4,5]. For obstacle problems specifically, the classical theorem of Kinderlehrer and Stampacchia [6] provides existence under monotonicity, coercivity and strong-weak continuity; a comprehensive modern treatment is given in [7].

Substantial progress has been made on obstacle problems for nonlinear elliptic operators. In the scalar setting with variable growth, existence and regularity were established in [8,9,10]. For systems under weak monotonicity, Hungerbühler [11] developed a compactness approach based on the Minty–Browder theorem. More recently, Young measure techniques have been applied to quasilinear systems with non-standard growth [12,13]. These works largely focus on p-growth or variable exponent growth and establish existence under relatively mild structural assumptions.

A parallel line of research addresses operators with a solution-dependent shift in the gradient variable, i.e., $\mathcal{S}(z,\nabla u-\mathcal{O}(u\left)\right)$. Such couplings arise in models with convection-like terms or in viscoelasticity where the reference configuration depends on the displacement. The shift fundamentally complicates the analysis: when comparing two admissible functions, the operator is evaluated at different shifts, while the standard monotonicity of $\mathcal{S}$ is available only under equal shifts.

The works most directly related to ours are those of Allalou et al. [14,15]. In [14], the authors treated an obstacle problem without a lower-order term and without shift, under weak monotonicity via Young measures. In [15], they considered a p-Laplacian-type problem with both a solution-dependent shift and a lower-order term, again using Young measures under p-growth. While these works successfully handle the shift coupling, their proofs rely on heavy measure-theoretic machinery.

Our contribution is twofold. First, we combine the solution-dependent shift and the lower-order term within a unified quadratic framework, avoiding variable exponents and Young measures entirely. Second, we introduce a three-term decomposition of the duality pairing that explicitly separates (i) the equal-shift monotone part, (ii) the shift-mismatch cross term, and (iii) the lower-order $L^2$-term. This decomposition allows us to control the shift mismatch under the explicit smallness condition

$$L_S{C}_{\mathcal{O}}{C}_P\le {\displaystyle \frac{C_3}{4}},$$

where $L_S$ is the Lipschitz constant of $\mathcal{S}$, $C_{\mathcal{O}}$ the Lipschitz constant of $\mathcal{O}$, $C_P$ the Poincaré constant, and $C_3$ the strong monotonicity modulus. Existence follows from the classical Kinderlehrer–Stampacchia theorem, and uniqueness is obtained from the same decomposition. The smallness condition is transparent, quantitatively verifiable, and expresses the balance between the symmetric monotone structure of $\mathcal{S}$ and the asymmetric perturbation induced by $\mathcal{O}$. We emphasise that the three-term decomposition is new, and our assumptions—though stronger than weak monotonicity—are precisely what allow us to bypass Young measures and obtain uniqueness directly.

This paper is organised as follows. Section 2 collects preliminaries: function spaces, Poincaré’s inequality, the Kinderlehrer–Stampacchia theorem, and the full list of assumptions. Section 3 establishes existence by verifying the three hypotheses of the theorem. Section 4 proves uniqueness. Section 5 discusses the novelty of the decomposition, the smallness condition, possible extensions to non-quadratic growth and local Lipschitz settings, and provides explicit examples. Section 6 concludes the paper.

2. Preliminaries

We denote by $L^p(\mathrm{\Omega };{\mathbb{R}}^m)$ the standard Lebesgue spaces and by $W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$ the Sobolev space of functions with weak derivatives in $L^2$. The subspace $W_0^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$ is the closure of smooth compactly supported functions in the $W^{1,2}$-norm. The duality pairing between $W^{-1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$ and $W_0^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$ is denoted by $\langle ·,·\rangle$. Here, $W^{-1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$ denotes the topological dual space of $W_0^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$, i.e., the space of bounded linear functionals on $W_0^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$. All vector-valued functions are understood column vectors, and the Frobenius inner product on matrices is denoted by a colon.

We shall use the following Poincaré inequality: there exists a constant $C_P>0$ such that

$${\parallel u\parallel }_2\le C_P{\parallel \nabla u\parallel }_2\forall u\in W_0^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m).$$

This constant depends only on the domain $\mathrm{\Omega }$ and on the dimension n, and it is finite because $\mathrm{\Omega }$ is bounded with the Lipschitz boundary. For a reference on the Poincaré inequality and its dependence on the domain, see, e.g., ([7], Chapter 6). We do not need its explicit value.

The convex set ${\mathcal{K}}_{\psi ,\delta }$ is nonempty by assumption. It is closed in the strong topology of $W^{1,2}$ because the pointwise constraint $v\ge \psi$ a.e. is preserved under strong convergence (after passing to a subsequence) and the trace condition is closed. Moreover, ${\mathcal{K}}_{\psi ,\delta }$ is convex due to the linearity of the constraints. We shall also use the fact that for any $u,v\in {\mathcal{K}}_{\psi ,\delta }$, the difference $u-v$ lies in $W_0^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$, which is crucial for applying Poincaré’s inequality. Indeed, since $u-\delta ,v-\delta \in W_0^{1,2}$, their difference $u-v$ also belongs to $W_0^{1,2}$.

We now make precise the assumptions on the obstacle $\psi$ and the boundary data $\delta$. Throughout the paper, we suppose that $\psi ,\delta \in W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$ and that $\psi \le \delta$ on $\partial \mathrm{\Omega }$ in the sense of traces, i.e., ${(\psi -\delta )}^{+}\in W_0^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$. This condition is standard and ensures that ${\mathcal{K}}_{\psi ,\delta }\ne Ø$, since, accordingly, the function $v=\delta +{(\psi -\delta )}^{+}$ belongs to ${\mathcal{K}}_{\psi ,\delta }$. These assumptions are sufficient for our purposes, but they are not minimal. For instance, one could relax the regularity of $\psi$ to $W^{1,2}$ with a suitable compatibility condition, or allow $\psi$ to take values in $L^2$ provided the convex set is nonempty in a weaker sense. However, since our focus is on the structural effect of the shift coupling rather than on sharp regularity conditions, we retain the standard hypotheses. We also note that the obstacle condition $v\ge \psi$ is interpreted componentwise a.e.; this can be generalized to a partial order induced by a closed convex cone, but we do not pursue such extensions here.

We recall the classical theorem of Kinderlehrer and Stampacchia, which is our main existence tool.

Theorem 1 

(Kinderlehrer–Stampacchia [6]). Let $\mathcal{T}$ be a nonempty closed convex subset of a reflexive Banach space Υ, and let $\mathcal{A}:\mathcal{T}\to {\mathrm{{\rm Y}}}^{\prime }$ be an operator that is monotone, coercive, and strong-weakly continuous. Here, ${\mathrm{{\rm Y}}}^{\prime }$ denotes the topological dual space of Υ. Then, there exists $u\in \mathcal{T}$ such that

$$\langle \mathcal{A}\left(u\right),v-u\rangle \ge 0\forall v\in \mathcal{T}.$$

We recall that monotonicity means $\langle \mathcal{A}\left(u\right)-\mathcal{A}\left(v\right),u-v\rangle \ge 0$ for all $u,v\in \mathcal{T}$, coercivity means

$${\displaystyle \frac{\langle \mathcal{A}\left(u\right)-\mathcal{A}\left(u_0\right),u-u_0\rangle }{\parallel u-u_0{\parallel }_{\mathrm{{\rm Y}}}}}\to \infty \mathrm{as}{\parallel u-u_0\parallel }_{\mathrm{{\rm Y}}}\to \infty$$

for some fixed $u_0\in \mathcal{T}$, and strong-weak continuity means that if $u_k\to u$ strongly in $\mathrm{{\rm Y}}$, then $\mathcal{A}\left(u_k\right)\rightharpoonup \mathcal{A}\left(u\right)$ weakly in ${\mathrm{{\rm Y}}}^{\prime }$.

Assumptions

We impose the following hypotheses on the data.

Assumption 1. 

The function $\mathcal{S}:\mathrm{\Omega }\times {\mathbb{M}}^{m\times n}\to {\mathbb{M}}^{m\times n}$ is a Carathéodory function, i.e., $z\mapsto \mathcal{S}(z,\xi )$ is measurable for every $\xi \in {\mathbb{M}}^{m\times n}$ and $\xi \mapsto \mathcal{S}(z,\xi )$ is continuous for a.e. $z\in \mathrm{\Omega }$. Moreover, there exist ${\lambda }_1\in L^2\left(\mathrm{\Omega }\right)$ and a constant $C_1>0$ such that

$$\left|\mathcal{S}(z,\xi )\right|\le {\lambda }_1\left(z\right)+C_1\left|\xi \right|\mathit{for}\mathit{a}.\mathit{e}.z\in \mathrm{\Omega }\mathit{and}\mathit{all}\xi \in {\mathbb{M}}^{m\times n}.$$

Assumption 2. 

There exists a constant $L_S>0$ such that for a.e. $z\in \mathrm{\Omega }$ and all $\xi ,\eta \in {\mathbb{M}}^{m\times n}$,

$$|\mathcal{S}(z,\xi )-\mathcal{S}(z,\eta )|\le L_S|\xi -\eta |.$$

Assumption 3. 

(Strong monotonicity) There exists a constant $C_3>0$ such that for a.e. $z\in \mathrm{\Omega }$ and all $\xi ,\eta \in {\mathbb{M}}^{m\times n}$,

$$\left(\mathcal{S}(z,\xi )-\mathcal{S}(z,\eta )\right):(\xi -\eta )\ge C_3{|\xi -\eta |}^2.$$

Here, $|·|$ denotes the Frobenius norm on matrices, and the left-hand side is the Frobenius inner product of the two matrix differences; the inequality is understood as a scalar inequality in $\mathbb{R}$. This condition can be viewed as a quantitative symmetry property of $\mathcal{S}$: it expresses a uniform lower bound on the symmetric part of the operator’s variation, ensuring that the energy functional associated with $\mathcal{S}$ is strictly convex in the gradient variable when the shift is held fixed. In this sense, $C_3$ plays the role of a symmetry modulus that controls the strength of the monotonicity.

Assumption 4. 

The Lipschitz constant $L_S$ of $\mathcal{S}$, the Lipschitz constant $C_{\mathcal{O}}$ of $\mathcal{O}$, and the Poincaré constant $C_P$ satisfy

$$L_S{C}_{\mathcal{O}}{C}_P\le {\displaystyle \frac{C_3}{4}}.$$

This is the crucial smallness condition that ensures the symmetry-breaking perturbation $\mathcal{O}\left(u\right)$ does not destroy the coercive structure. It quantifies the balance between the symmetric monotone part and the asymmetric shift coupling.

We pause here to discuss the naturalness and optimality of the smallness condition $L_S{C}_{\mathcal{O}}{C}_P\le C_3/4$. This condition is sufficient for our proof, but it is by no means necessary. In fact, it arises from the particular way we have chosen to estimate the cross term $I_2$ using the crude bound

$$|I_2|\le L_S{C}_{\mathcal{O}}{\int }_{\mathrm{\Omega }}\left|\xi \right|\left|D\right|dz$$

combined with Poincaré and Young’s inequalities. Sharper estimates are possible if one has additional information about the structure of $\mathcal{S}$ or $\mathcal{O}$, such as a decomposition into a strongly monotone part and a compact perturbation, or if one works in a smaller subspace where the Poincaré constant can be improved. Moreover, the condition is scale-dependent: it involves the domain-dependent Poincaré constant, and thus it is more restrictive for domains with large diameter. In some situations, one can replace the Poincaré constant by a sharper eigenvalue-related constant (e.g., the first eigenvalue of the Dirichlet Laplacian). We also note that a condition of this type is common in problems with gradient-dependent lower-order couplings; it is reminiscent of the smallness assumptions in the Lax–Milgram framework for perturbed coercive operators. Thus, while the condition is not optimal in a general sense, it is natural and transparent, and it explicitly displays the competition between the monotone part and the shift perturbation. In the discussion section, we indicate possible relaxations under additional structural hypotheses.

Regarding the assumptions on $\mathcal{S}$, global Lipschitz continuity and strong monotonicity are sufficient but not necessary. One could weaken the Lipschitz condition to local Lipschitz continuity if additional compactness arguments are available, or replace strong monotonicity by weak monotonicity together with a coercivity condition. However, in the latter case, uniqueness may fail, and the proof would require more sophisticated tools (e.g., Young measures or pseudo-monotone operator theory). We also note that if the smallness condition is violated, uniqueness is not guaranteed in general; for instance, in the scalar example discussed in Section 5, the operator may lose coercivity and multiple solutions may appear. Thus, our assumptions are not minimal, but they are convenient and lead to a clean, elementary proof.

Throughout this paper, we assume that $\mathcal{O}:{\mathbb{R}}^m\to {\mathbb{M}}^{m\times n}$ is Lipschitz continuous with constant $C_{\mathcal{O}}$ and that ${\mathcal{K}}_{\psi ,\delta }\ne Ø$.

3. Existence

Define the operator $\mathcal{L}:{\mathcal{K}}_{\psi ,\delta }\to W^{-1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$ by

$$\langle \mathcal{L}\left(u\right),w\rangle ={\int }_{\mathrm{\Omega }}\left[\mathcal{S}(z,\nabla u-\mathcal{O}\left(u\right)):\nabla w+{\langle u,w\rangle }_{{\mathbb{R}}^m}\right]dz\mathrm{for}\mathrm{all}w\in W_0^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m).$$

The integral is well-defined owing to the growth condition, which ensures that $\mathcal{S}(z,\nabla u-\mathcal{O}\left(u\right))\in L^2$ whenever $u\in W^{1,2}$. Indeed, the Lipschitz property of $\mathcal{O}$ gives, for a.e. z,

$$|\nabla u\left(z\right)-\mathcal{O}\left(u\left(z\right)\right)|\le |\nabla u\left(z\right)|+C_{\mathcal{O}}\left|u\left(z\right)\right|+\left|\mathcal{O}\left(0\right)\right|,$$

so the $L^2$-norm of the argument is bounded by ${\parallel \nabla u\parallel }_2+C_{\mathcal{O}}{\parallel u\parallel }_2+\left|\mathcal{O}\left(0\right)\right|\sqrt{\left|\mathrm{\Omega }\right|}$, which is finite for $u\in W^{1,2}$.

Our main theorem is the following.

Theorem 2. 

(Main result) Assume ${\mathcal{K}}_{\psi ,\delta }\ne Ø$ and the above hypotheses hold. Then, the obstacle problem (1) admits a unique weak solution. The proof is given in Section 3 and Section 4 below, where we verify the three assumptions of the Kinderlehrer–Stampacchia theorem (closedness and convexity of ${\mathcal{K}}_{\psi ,\delta }$, monotonicity and coercivity of $\mathcal{L}$, and strong-weak continuity) and then establish uniqueness.

Remark 1. 

Compared with the Young-measure-based proofs in [15], which treat p-growth problems with shifts under weak monotonicity, the present quadratic framework offers a significant simplification. We bypass the entire measure-theoretic machinery by exploiting the Kinderlehrer–Stampacchia theorem directly. The three-term decomposition isolates the shift mismatch and reduces it to a controlled Lipschitz perturbation, absorbed under the explicit smallness condition (H3). This makes the proof more elementary, more transparent, and yields a quantitative criterion that is readily verifiable from the problem data. Moreover, the condition $L_S{C}_{\mathcal{O}}{C}_P\le C_3/4$ explicitly exhibits the balance between the symmetric monotone structure and the asymmetric shift coupling, reinforcing the role of symmetry in the well-posedness of the problem.

We proceed by verifying the three assumptions of the Kinderlehrer–Stampacchia theorem.

Lemma 1. 

The set ${\mathcal{K}}_{\psi ,\delta }$ is closed and convex, and $\mathcal{L}\left(u\right)\in W^{-1,2}$ for every $u\in {\mathcal{K}}_{\psi ,\delta }$.

Proof. 

The closedness and convexity of ${\mathcal{K}}_{\psi ,\delta }$ were already discussed. To show that $\mathcal{L}\left(u\right)$ is a bounded linear functional, take any $w\in W_0^{1,2}$. Using the growth condition and Hölder’s inequality,

$$\left|\langle \mathcal{L}\left(u\right),w\rangle \right|\le {\int }_{\mathrm{\Omega }}({\lambda }_1+C_1|\nabla u-\mathcal{O}\left(u\right)|\left)\right|\nabla w|dz+{\int }_{\mathrm{\Omega }}\left|u\right|\left|w\right|dz.$$

Since u is fixed, the first integral is bounded by $(\parallel {\lambda }_1{\parallel }_2+C_1{\parallel \nabla u-\mathcal{O}\left(u\right)\parallel }_2{)\parallel \nabla w\parallel }_2$, and the second by ${\parallel u\parallel }_2{\parallel w\parallel }_2$. By Poincaré’s inequality, ${\parallel w\parallel }_2\le C_P{\parallel \nabla w\parallel }_2$. Thus, $\left|\langle \mathcal{L}\left(u\right),w\rangle \right|\le {C\left(u\right)\parallel w\parallel }_{W^{1,2}}$, so $\mathcal{L}\left(u\right)\in W^{-1,2}$. This proves the boundedness of the linear functional, since the constant $C\left(u\right)$ depends only on u and the fixed parameters. □

Lemma 2. 

$\mathcal{L}$ is monotone and coercive on ${\mathcal{K}}_{\psi ,\delta }$.

Proof. 

We first establish monotonicity. Let $u,v\in {\mathcal{K}}_{\psi ,\delta }$. Set $D=\nabla u-\nabla v$ and $\xi =u-v$. Then,

$$\langle \mathcal{L}\left(u\right)-\mathcal{L}\left(v\right),u-v\rangle =I_1+I_2+{\parallel \xi \parallel }_2^2,$$

where

$$\begin{array}{cc}\hfill I_1& ={\int }_{\mathrm{\Omega }}\left[\mathcal{S}(z,\nabla u-\mathcal{O}(u\left)\right)-\mathcal{S}(z,\nabla v-\mathcal{O}(u\left)\right)\right]:Ddz,\hfill \\ \hfill I_2& ={\int }_{\mathrm{\Omega }}\left[\mathcal{S}(z,\nabla v-\mathcal{O}(u\left)\right)-\mathcal{S}(z,\nabla v-\mathcal{O}(v\left)\right)\right]:Ddz.\hfill \end{array}$$

By the strong monotonicity assumption (Assumption 3), applied with the same shift $\mathcal{O}\left(u\right)$, we have

$$I_1\ge C_3{\int }_{\mathrm{\Omega }}{\left|D\right|}^{2}dz.$$

For $I_2$, using the Lipschitz estimate (Assumption 2) and Young’s inequality with parameter $\epsilon >0$,

$$|I_2|\le L_S{C}_{\mathcal{O}}{\int }_{\mathrm{\Omega }}\left|\xi \right|\left|D\right|dz\le L_S{C}_{\mathcal{O}}\left({\epsilon \parallel D\parallel }_2^2+{\displaystyle \frac{1}{4\epsilon }}{\parallel \xi \parallel }_2^2\right).$$

Applying Poincaré’s inequality to $\xi \in W_0^{1,2}$ gives ${\parallel \xi \parallel }_2\le C_P{\parallel D\parallel }_2$. Choosing $\epsilon =C_P/2$ yields

$$|I_2|\le L_S{C}_{\mathcal{O}}\left({\displaystyle \frac{C_P}{2}}{\parallel D\parallel }_2^2+{\displaystyle \frac{1}{2C_P}}{\parallel \xi \parallel }_2^2\right)\le L_S{C}_{\mathcal{O}}\left({\displaystyle \frac{C_P}{2}}+{\displaystyle \frac{C_P}{2}}\right){\parallel D\parallel }_2^2=L_S{C}_{\mathcal{O}}{C}_P{\parallel D\parallel }_2^2.$$

By the smallness condition (Assumption 4), $L_S{C}_{\mathcal{O}}{C}_P\le C_3/4$, so $I_2\ge -C_3{\parallel D\parallel }_2^2/4$. Consequently,

$$\langle \mathcal{L}\left(u\right)-\mathcal{L}\left(v\right),u-v\rangle \ge {\displaystyle \frac{3C_3}{4}}{\parallel D\parallel }_2^2+{\parallel \xi \parallel }_2^2\ge 0,$$

which proves monotonicity. In fact, the estimate shows a stronger form of strict monotonicity up to the lower-order term.

For coercivity, fix $u_0\in {\mathcal{K}}_{\psi ,\delta }$. Let $w=u-u_0\in W_0^{1,2}$. Define

$$X_u=\nabla u-\mathcal{O}\left(u\right),X_{u_0}=\nabla u_0-\mathcal{O}\left(u_0\right).$$

Then,

$$\begin{array}{cc}\hfill \langle \mathcal{L}\left(u\right)-\mathcal{L}\left(u_0\right),w\rangle & ={\int }_{\mathrm{\Omega }}[\mathcal{S}(z,X_u)-\mathcal{S}(z,X_{u_0})]:\nabla wdz+{\parallel w\parallel }_2^2\hfill \\ & =T_1+T_2+{\parallel w\parallel }_2^2,\hfill \end{array}$$

where

$$T_1={\int }_{\mathrm{\Omega }}[\mathcal{S}(z,\nabla u-\mathcal{O}\left(u\right))-\mathcal{S}(z,\nabla u_0-\mathcal{O}\left(u\right))]:\nabla wdz,$$

$$T_2={\int }_{\mathrm{\Omega }}[\mathcal{S}(z,\nabla u_0-\mathcal{O}\left(u\right))-\mathcal{S}(z,\nabla u_0-\mathcal{O}\left(u_0\right))]:\nabla wdz.$$

Again, by Assumption 3 with the same shift $\mathcal{O}\left(u\right)$, we have

$$T_1\ge C_3{\parallel \nabla w\parallel }_2^2.$$

For $T_2$, using the Lipschitz continuity of $\mathcal{S}$ (Assumption 2) and $\mathcal{O}$,

$$|T_2|\le L_S{C}_{\mathcal{O}}{\int }_{\mathrm{\Omega }}\left|w\right||\nabla w|dz.$$

Young’s inequality with $\epsilon =C_P$ gives

$${\int }_{\mathrm{\Omega }}\left|w\right||\nabla w|\le {\displaystyle \frac{C_P}{2}}{\parallel \nabla w\parallel }_2^2+{\displaystyle \frac{1}{2C_P}}{\parallel w\parallel }_2^2.$$

Since ${\parallel w\parallel }_2\le C_P{\parallel \nabla w\parallel }_2$, we obtain

$${\int }_{\mathrm{\Omega }}\left|w\right||\nabla w|\le C_P{\parallel \nabla w\parallel }_2^2,$$

hence,

$$|T_2|\le L_S{C}_{\mathcal{O}}{C}_P{\parallel \nabla w\parallel }_2^2.$$

Using the smallness condition (Assumption 4), $L_S{C}_{\mathcal{O}}{C}_P\le C_3/4$, so $T_2\ge -{\displaystyle \frac{C_3}{4}}{\parallel \nabla w\parallel }_2^2$. Therefore,

$$\langle \mathcal{L}\left(u\right)-\mathcal{L}\left(u_0\right),w\rangle \ge {\displaystyle \frac{3C_3}{4}}{\parallel \nabla w\parallel }_2^2+{\parallel w\parallel }_2^2.$$

Let $c=min\{{\displaystyle \frac{3C_3}{4}},1\}>0$. Then, the right-hand side is at least ${c(\parallel \nabla w\parallel }_2^2+{\parallel w\parallel }_2^2{)=c\parallel w\parallel }_{W^{1,2}}^2$. Hence,

$${\displaystyle \frac{\langle \mathcal{L}\left(u\right)-\mathcal{L}\left(u_0\right),u-u_0\rangle }{\parallel u-u_0{\parallel }_{W^{1,2}}}}\ge c\parallel u-u_0{\parallel }_{W^{1,2}}\to \infty \mathrm{as}{\parallel u-u_0\parallel }_{W^{1,2}}\to \infty .$$

This proves coercivity, since the quotient is bounded below by a positive constant multiplied by the norm, which diverges. □

Lemma 3. 

$\mathcal{L}$ is strong-weakly continuous.

Proof. 

Let $u_k\to u$ strongly in $W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$. By the Lipschitz continuity of $\mathcal{S}$ in the matrix variable (Assumption 2) and of $\mathcal{O}$,

$$\parallel \mathcal{S}(z,\nabla u_k-\mathcal{O}\left(u_k\right))-\mathcal{S}(z,\nabla u-\mathcal{O}\left(u\right)){\parallel }_2\le L_S\parallel \nabla u_k{-\nabla u\parallel }_2+L_S{C}_{\mathcal{O}}{\parallel u_k-u\parallel }_2\to 0.$$

To justify the passage to the limit inside the integral, we note that the Carathéodory property ensures that the composition $z\mapsto \mathcal{S}(z,\nabla u_k\left(z\right)-\mathcal{O}\left(u_k\left(z\right)\right))$ is measurable. Moreover, the Lipschitz estimate gives $L^2$-convergence of the integrands. Therefore, for any $v\in W_0^{1,2}$, the map $z\mapsto \mathcal{S}(z,\nabla u_k-\mathcal{O}\left(u_k\right)):\nabla v\left(z\right)$ converges in $L^1$ (indeed, by Hölder, the $L^2$-convergence of the first factor and the $L^2$-boundedness of $\nabla v$ imply convergence in $L^1$). Hence,

$${\int }_{\mathrm{\Omega }}\mathcal{S}(z,\nabla u_k-\mathcal{O}\left(u_k\right)):\nabla vdz\to {\int }_{\mathrm{\Omega }}\mathcal{S}(z,\nabla u-\mathcal{O}\left(u\right)):\nabla vdz,$$

and the lower-order term converges by the strong $L^2$ convergence of $u_k$. Therefore, $\langle \mathcal{L}\left(u_k\right),v\rangle \to \langle \mathcal{L}\left(u\right),v\rangle$ for all $v\in W_0^{1,2}$, which is exactly strong-weak continuity. The Lipschitz assumptions imply that the nonlinearity is continuous from $W^{1,2}$ into $L^2$ strongly, which is more than sufficient for the required strong-weak continuity. □

Applying the Kinderlehrer–Stampacchia theorem yields the existence of $u\in {\mathcal{K}}_{\psi ,\delta }$ satisfying

$$\langle \mathcal{L}\left(u\right),v-u\rangle \ge 0\forall v\in {\mathcal{K}}_{\psi ,\delta },$$

which is precisely the obstacle problem (1). This completes the existence part: we have verified that ${\mathcal{K}}_{\psi ,\delta }$ is nonempty, closed and convex (Lemma 1), and $\mathcal{L}$ is monotone and coercive (Lemma 2), and strong-weakly continuous (Lemma 3). All hypotheses of the Kinderlehrer–Stampacchia theorem are thus satisfied.

4. Uniqueness

Theorem 3. 

The solution obtained in Theorem 2 is unique.

Proof. 

Assume $u_1,u_2\in {\mathcal{K}}_{\psi ,\delta }$ are two solutions. Taking $v=u_2$ in the inequality for $u_1$ gives

$$\langle \mathcal{L}\left(u_1\right),u_2-u_1\rangle \ge 0,$$

while taking $v=u_1$ in the inequality for $u_2$ gives

$$\langle \mathcal{L}\left(u_2\right),u_1-u_2\rangle \ge 0.$$

Adding these two inequalities yields

$$\langle \mathcal{L}\left(u_1\right)-\mathcal{L}\left(u_2\right),u_2-u_1\rangle \ge 0,$$

or equivalently,

$$\langle \mathcal{L}\left(u_1\right)-\mathcal{L}\left(u_2\right),u_1-u_2\rangle \le 0.$$

Expanding the left-hand side, we obtain

$${\int }_{\mathrm{\Omega }}\left[\mathcal{S}(z,\nabla u_1-\mathcal{O}\left(u_1\right))-\mathcal{S}(z,\nabla u_2-\mathcal{O}\left(u_2\right))\right]:(\nabla u_1-\nabla u_2)dz+{\parallel u_1-u_2\parallel }_2^2\le 0.$$

Decompose the integral as $I=I_1+I_2$, where

$$\begin{array}{cc}\hfill I_1& ={\int }_{\mathrm{\Omega }}\left[\mathcal{S}(z,\nabla u_1-\mathcal{O}\left(u_1\right))-\mathcal{S}(z,\nabla u_2-\mathcal{O}\left(u_1\right))\right]:(\nabla u_1-\nabla u_2)dz,\hfill \\ \hfill I_2& ={\int }_{\mathrm{\Omega }}\left[\mathcal{S}(z,\nabla u_2-\mathcal{O}\left(u_1\right))-\mathcal{S}(z,\nabla u_2-\mathcal{O}\left(u_2\right))\right]:(\nabla u_1-\nabla u_2)dz.\hfill \end{array}$$

By strong monotonicity (Assumption 3) with the same shift $\mathcal{O}\left(u_1\right)$, we have

$$I_1\ge C_3{\parallel \nabla u_1-\nabla u_2\parallel }_2^2.$$

For $I_2$, the same estimate as in the monotonicity proof gives

$$|I_2|\le L_S{C}_{\mathcal{O}}{C}_P\parallel \nabla u_1-\nabla u_2{\parallel }_2^2\le {\displaystyle \frac{C_3}{4}}{\parallel \nabla u_1-\nabla u_2\parallel }_2^2,$$

using the smallness condition (Assumption 4). Hence,

$$I\ge {\displaystyle \frac{3C_3}{4}}{\parallel \nabla u_1-\nabla u_2\parallel }_2^2.$$

Combining with the inequality above yields

$${\displaystyle \frac{3C_3}{4}}\parallel \nabla u_1-\nabla u_2{\parallel }_2^2+{\parallel u_1-u_2\parallel }_2^2\le 0,$$

which forces $\nabla u_1=\nabla u_2$ a.e. and $u_1=u_2$ a.e. since $u_1-u_2\in W_0^{1,2}$ and Poincaré’s inequality applies. Thus, uniqueness follows. We emphasize that the smallness condition is crucial in this argument: without it, the cross term $I_2$ could dominate the positive part $I_1$ and prevent uniqueness. □

5. Discussion

In this section we discuss additional aspects that complement the detailed presentation already given in the Introduction. Since the novelty of the three-term decomposition and the role of the smallness condition have been fully elaborated there, we focus here on extensions, concrete examples, and future directions.

5.1. Onthe Assumptions and Extensions to Local Lipschitz Operators

The assumptions of global Lipschitz continuity and strong monotonicity are chosen for clarity and transparency. However, the method can be extended to operators that are only locally Lipschitz, provided additional compactness or growth conditions are imposed. For instance, if $\mathcal{S}$ is locally Lipschitz and $\mathcal{O}$ is a compact operator from $W^{1,2}(\mathrm{\Omega };{\mathbb{R}}^m)$ into $L^2(\mathrm{\Omega };{\mathbb{M}}^{m\times n})$ (e.g., if $\mathcal{O}$ maps bounded sets into relatively compact sets in $L^2$), then the strong convergence of $\mathcal{S}(z,\nabla u_k-\mathcal{O}\left(u_k\right))$ in $L^2$ can be recovered via a Vitali-type argument, even without a global Lipschitz bound. Alternatively, if one has control of the form

$$|\mathcal{S}(z,\xi )-\mathcal{S}(z,\eta )|\le L_S\left(z\right)|\xi -\eta |\mathrm{with}{L}_S\in L^r\left(\mathrm{\Omega }\right),r>n/2,$$

then the same estimates hold with $C_P$ replaced by a suitable norm of $L_S$. In the absence of such controls, the strong-weak continuity of $\mathcal{L}$ may still hold if the operator is pseudo-monotone, but then the uniqueness proof would break down without additional assumptions. We have added a discussion of these possibilities in the revised manuscript.

5.2. Concrete Examples Satisfying the Assumptions

We now provide explicit examples of operators satisfying all the hypotheses.

Example 1 

(scalar linear case). Let $m=1$, $\mathrm{\Omega }=(0,\pi )$, $\mathcal{S}(z,\xi )=\xi$, and $\mathcal{O}\left(u\right)=\alpha u$, with $\left|\alpha \right|\le \pi /4$. Then, $L_S=1,C_3=1,C_{\mathcal{O}}=\left|\alpha \right|,C_P=1/\pi$, so the smallness condition holds. All other assumptions are trivially satisfied.

Example 2 

(matrix-valued linear case). Let $\mathcal{S}(z,\xi )=A\left(z\right)\xi$, where $A\left(z\right)$ is a uniformly elliptic and bounded $mn\times mn$ matrix-valued function; i.e., there exist $0<{\lambda }_0\le {\Lambda }_0<\infty$ such that

$${\lambda }_0{\left|\xi \right|}^2\le \left(A\left(z\right)\xi \right):\xi \le {\Lambda }_0{\left|\xi \right|}^2,|A\left(z\right)\xi -A\left(z\right)\eta |\le {\Lambda }_0|\xi -\eta |$$

for a.e. $z\in \mathrm{\Omega }$ and all $\xi ,\eta \in {\mathbb{M}}^{m\times n}$. Then, $L_S={\Lambda }_0,C_3={\lambda }_0$. Let $\mathcal{O}\left(u\right)=\alpha sinu$ (acting componentwise, with a slight abuse of notation), so $C_{\mathcal{O}}=\left|\alpha \right|$. Choose $\alpha$ small enough such that ${\Lambda }_0\left|\alpha \right|C_P\le {\lambda }_0/4$. Then, all assumptions are satisfied.

Example 3 

(nonlinear but structurally simple). Let $\mathcal{S}(z,\xi )={\left|\xi \right|}^{p-2}\xi$ with $p=2$ (which reduces to the linear case) or more generally take $\mathcal{S}(z,\xi )=\xi +{ϵ}^{arctan}\left(\xi \right)$ with ${ϵ}^{>}0$ small. Then, $L_S=1+{ϵ}^{,}{C}_3=1$ (since the derivative of arctan is nonnegative), and one can choose $\mathcal{O}\left(u\right)=\alpha u$ with $\alpha$ sufficiently small to satisfy the smallness condition.

5.3. Future Directions

Extensions to p-growth and variable exponent spaces. A natural direction for future work is to extend the results to operators satisfying p-growth with $p\ne 2$, or to variable exponent settings where the growth exponent $p(·)$ depends on the spatial variable. In the p-growth case, the natural functional setting is the Sobolev space $W^{1,p}(\mathrm{\Omega };{\mathbb{R}}^m)$ with $p\in (1,\infty )$. The operator $\mathcal{S}$ would then satisfy

$$\left|\mathcal{S}(z,\xi )\right|\le {\lambda }_1\left(z\right)+C_1{\left|\xi \right|}^{p-1},(\mathcal{S}(z,\xi )-\mathcal{S}(z,\eta )):(\xi -\eta )\ge C_3{|\xi -\eta |}^p,$$

and a Lipschitz-type estimate in the form

$$|\mathcal{S}(z,\xi )-\mathcal{S}(z,\eta )|\le L_S\left(\right|\xi |+{\left|\eta \right|)}^{p-2}|\xi -\eta |$$

(which is the standard p-Laplacian type structure). In this context, the three-term decomposition can be adapted by replacing the $L^2$-based estimates with $L^p$-estimates, and the smallness condition would involve the Poincaré constant for $W_0^{1,p}$ and appropriate embeddings. The coercivity estimate would then yield

$$\langle \mathcal{L}\left(u\right)-\mathcal{L}\left(u_0\right),u-u_0\rangle \ge c\parallel \nabla (u-u_0){\parallel }_p^p+{\parallel u-u_0\parallel }_2^2$$

provided the shift mismatch is controlled via a suitable p-version of Young’s inequality. For variable exponent spaces $W^{1,p(·)}(\mathrm{\Omega };{\mathbb{R}}^m)$, the situation is more delicate because the modular and the norm are not directly proportional, and the Poincaré inequality holds in a modular form. Nevertheless, the decomposition principle remains valid, and the smallness condition would involve the modular Poincaré constant and the Lipschitz constants of the N-function.

On the assumptions on $\mathcal{S}$: beyond Lipschitz and strongly monotone operators. As noted in the introduction, the assumptions of global Lipschitz continuity and strong monotonicity are sufficient for our proof, but they can be generalized. For instance, one could consider pseudo-monotone operators in the sense of Leray–Lions, which satisfy a weaker continuity condition and do not require global Lipschitz bounds. In that case, the strong-weak continuity of $\mathcal{L}$ would follow from the standard compactness arguments for pseudo-monotone operators, provided the shift $\mathcal{O}$ is compact (e.g., if $\mathcal{O}$ maps bounded sets in $W^{1,2}$ into bounded sets in $L^2$, which is true under Lipschitz continuity). Similarly, one could replace the strong monotonicity by a weak monotonicity condition of the form

$$\left(\mathcal{S}\right(z,\xi )-\mathcal{S}(z,\eta \left)\right):(\xi -\eta )\ge 0$$

with a coercivity condition that ensures the necessary lower bound. The three-term decomposition would then need to be adapted to control the shift mismatch without the explicit Lipschitz constant $L_S$, perhaps using the growth and monotonicity structure of $\mathcal{S}$. Such generalizations are of interest, but they would require a more sophisticated functional-analytic apparatus and are beyond the scope of the present work. We also note that the lower-order term $\langle u,v-u\rangle$ could be replaced by a more general monotone operator in $L^2$, provided the corresponding estimates are available.

From the symmetry perspective, it would be interesting to investigate whether the smallness condition can be interpreted as a stability criterion for the symmetric structure of the variational problem. In particular, one could explore how the symmetry modulus $C_3$ relates to the spectral properties of the linearised operator and whether the condition $L_S{C}_{\mathcal{O}}{C}_P\le C_3/4$ is sharp in the sense of guaranteeing the preservation of the coercive symmetry. Such questions are natural in the context of symmetry-breaking perturbations and may lead to further insights into the qualitative behaviour of solutions.

Orlicz-space extensions. A particularly relevant direction is the generalization of our results to problems with non-standard growth governed by an N-function, i.e., to the setting of Orlicz–Sobolev spaces. In this framework, the operator $\mathcal{S}$ typically satisfies A-growth conditions and the monotonicity is expressed in terms of the Orlicz modular. The fundamental existence theory for strongly nonlinear elliptic equations in Orlicz spaces was developed by Benkirane and Elmahi [16]; see also the related work on variational inequalities in Orlicz–Sobolev spaces by Gossez and Mustonen [17] and the natural growth results of Boccardo, Gallouët and Murat [18]. These works provide the necessary functional-analytic tools, including versions of the Minty–Browder theorem adapted to Orlicz spaces. Adapting our three-term decomposition to this more general setting would require a Poincaré-type inequality in the Orlicz–Sobolev space, as well as a careful control of the shift mismatch using the modular analogues of Young’s inequality. The current quadratic case serves as a model for such extensions, and we expect that the same decomposition principle will apply, with the smallness condition involving the constants of the N-function and the modular Poincaré constant. We leave this generalization for future work. For a modern and comprehensive treatment of Orlicz spaces and generalized Orlicz spaces, we refer to [19].

Another direction is the study of time-dependent obstacle problems, where the operator involves parabolic terms and the shift may depend on time. The stationary analysis presented here serves as a foundation for such developments.

6. Conclusions

In this paper we have established the existence and uniqueness of weak solutions to an obstacle problem for a nonlinear elliptic operator in divergence form, where the principal part contains a solution-dependent shift $\mathcal{O}\left(u\right)$. The main novelty of our approach is a three-term decomposition of the duality pairing that explicitly separates the equal-shift monotone part, the shift-mismatch cross term, and the lower-order $L^2$-term. This decomposition is specifically designed to handle the mismatch of shifts that arises when comparing two admissible functions—a difficulty that is inherent to problems with solution-dependent couplings.

Under the structural assumptions that $\mathcal{S}$ is globally Lipschitz and strongly monotone with quadratic growth and coercivity, and that $\mathcal{O}$ is Lipschitz, we have shown that the cross term can be controlled by means of Young’s inequality and Poincaré’s inequality, provided the smallness condition

$$L_S{C}_{\mathcal{O}}{C}_P\le {\displaystyle \frac{C_3}{4}}$$

holds. This condition is transparent, quantitatively verifiable, and expresses the balance between the symmetric monotone structure of $\mathcal{S}$ and the asymmetric perturbation induced by the shift $\mathcal{O}$.

Existence of a solution follows by a direct application of the classical Kinderlehrer–Stampacchia theorem, after verifying monotonicity, coercivity and strong-weak continuity of the associated operator. Uniqueness is obtained from the very same three-term decomposition. Our approach avoids the heavy measure-theoretic machinery—such as Young measures—that has been employed in previous works on related p-growth problems, and instead provides a streamlined proof within a unified quadratic framework.

We have also discussed possible extensions of the method to operators with local Lipschitz regularity, to p-growth and variable exponent settings, and to Orlicz–Sobolev spaces. These directions, as well as time-dependent versions of the problem, constitute natural subjects for future investigation. We believe that the three-term decomposition introduced here may be of independent interest and applicable to a broader class of variational inequalities with solution-dependent perturbations.