1. Introduction

In this note we analyse some quantitative and qualitative results for the following partial differential inequalities (PDI in the sequel)

(1.1)\begin{align} & -\mathrm{div}(\mathcal{A}(x,u,\nabla u))+ \mathcal{B}_1(x,u,\nabla u)\leq0\text{ in }\Omega, \end{align}

(1.2)\begin{align} & -\mathrm{div}(\mathcal{A}(x,u,\nabla u))\geq \mathcal{B}_2(x,u,\nabla u)\text{ in }\Omega, \end{align}

where $\Omega$ will be either a bounded open set in $\mathbb {R}^N$ or the whole space itself, $N> 1$, $\mathcal {A}:\Omega \times \mathbb {R}\times \mathbb {R}^N\to \mathbb {R}^N$ is a Carathéodory function (namely measurable in $x$ and continuous in the $u,\,\nabla u$ entries) such that

(A)\begin{equation*} |\mathcal{A}(x,s,\xi)|\leq \nu|\xi|^{p-1},\quad\nu>0, \end{equation*}

for every $(s,\,\xi )\in \mathbb {R}\times \mathbb {R}^N$, a.e. $x\in \Omega$, and

(B1)\begin{gather} \mathcal{B}_1(x,s,\xi)\geq c_H|\xi|^\gamma+\lambda s-f(x), \end{gather}

(B2)\begin{gather} \mathcal{B}_2(x,s,\xi)\geq c_H|\xi|^\gamma. \end{gather}

Here $\mathcal {B}_1:\Omega \times \mathbb {R}\times \mathbb {R}^N\to \mathbb {R}$, $\mathcal {B}_2:\Omega \times \mathbb {R}\times \mathbb {R}^N\to \mathbb {R}$ , $\lambda \geq 0$, $c_H>0$ and $f$ is a measurable source term belonging to $\mathcal {L}^{1,\frac {N}{q}}(\Omega )$, the Morrey space of parameters $(1,\,N/q)$, $q\geq 1$, $\gamma >p-1$, $p>1$. A prototype equation satisfying the previous assumptions is the quasilinear Hamilton–Jacobi/Riccati equation driven by the $p$-Laplacian [Reference Bidaut-Véron, Garcia-Huidobro and Véron7, Reference Cirant and Goffi14, Reference Cirant and Goffi16, Reference Leonori and Porretta30, Reference Lions31]

\[ -\mathrm{div}(|\nabla u|^{p-2}\nabla u)+\lambda u+|\nabla u|^\gamma=g(x)\ . \]

Our main results are the following. The first is a quantitative result for semi-solutions to PDIs, that is summarized in the next

Theorem 1.1 Let $\Omega$ be an open bounded and connected subset of $\mathbb {R}^N$ having Lipschitz boundary and satisfying the uniform interior sphere condition. Assume (A) and (B1), $\gamma >p,$ $\lambda \geq 0$ and $f\in \mathcal {L}^{1,\frac {N}{q}}(\Omega )$ for some $q>\frac {N}{\gamma }$. Let $u\in W^{1,\gamma }_{\mathrm {loc}}(\Omega )$ be such that $\lambda u^-\in \mathcal {L}^{1,\frac {N}{q}}(\Omega )$ satisfying, in the sense of distributions, the inequality

\[ -\mathrm{div}(\mathcal{A}(x,u,\nabla u))+\mathcal{B}_1(x,u,\nabla u)\leq0\text{ in }\Omega. \]

Then $u$ is Hölder continuous up to the boundary (namely in the whole $\overline {\Omega }$) and satisfies

\[ |u(x)-u(y)|\leq K|x-y|^\alpha\ ,\forall x,y\in\overline{\Omega}, \]

where

\[ \alpha=\min\left\{1-\frac{N}{q\gamma},\frac{\gamma-p}{\gamma-(p-1)}\right\}, \]

and $K$ depends on $c_H,\,p,\,q,\,\gamma,\,N,\,\nu,\,\Omega,\, \|f+\lambda u^-\|_{\mathcal {L}^{1,\frac {N}{q}}(\Omega )}$. In particular, the estimate is satisfied when $f,\,\lambda u^-\in L^q(\Omega )$, $q>\frac {N}{\gamma }$.

The second is a Liouville/Bernstein-type result and states the nonexistence of nontrivial solutions to (1.2) for the corresponding homogeneous PDE/PDI without zero-th order terms on noncompact complete Riemannian manifolds $M$ in terms of the growth rate of the area of the geodesic spheres.

Theorem 1.2 Let $(M,\,g)$ be a noncompact complete Riemannian manifold. If $p>1,$ $\gamma >p-1$ and

(1.3)\begin{equation} \int^{+\infty}\frac{1}{(\mathrm{area}(\partial B_t(o)))^{\frac{\gamma-(p-1)}{p-1}}}\,{\rm d}t={+}\infty \end{equation}

for some origin $o\in M$, then any distributional solution of

(1.4)\begin{equation} -\mathrm{div} (\mathcal{A}(x,u,\nabla u))\geq \mathcal{B}_2(x,u,\nabla u)\text{ in }M, \end{equation}

where $\mathcal {A}$ satisfies (A), must be a.e. constant on $M$. In particular, if (1.3) holds, any distributional solution to

(1.5)\begin{equation} -{\rm div} ({\mathcal{A}}(x,u,\nabla u))+ c_{H}|\nabla u|^{\gamma}=0\text{ in }M\end{equation}

with $\gamma >p-1$ must be a.e. constant on $M$.

The previous result leads to the following property when specialized to the Euclidean space $M=\mathbb {R}^N$.

Corollary 1.3 Assume (A) and (B2). Any distributional solution to the inequality

(1.6)\begin{equation} -\mathrm{div}(\mathcal{A}(x,u,\nabla u))\geq \mathcal{B}_2(x,u,\nabla u) \text{ in }\mathbb{R}^N \end{equation}

must be a.e. constant in $\mathbb {R}^N$ provided that

(1.7)\begin{equation} p-1<\gamma\leq \frac{N(p-1)}{N-1}\ ,1< p< N\ . \end{equation}

Similarly, any distributional solution to

(1.8)\begin{equation} -\mathrm{div} (\mathcal{A}(x,u,\nabla u))+c_H|\nabla u|^\gamma=0\text{ in }\mathbb{R}^N \end{equation}

must be a.e. constant when $\gamma,\,p$ vary in the range (1.7).

These results rely on interior integral gradient estimates of Morrey-type of the form

\[ \int_{B_r}|\nabla u|^\gamma\,{\rm d}x\leq CR^\alpha \]

for some $\alpha \in \mathbb {R}$, see lemma 2.3, remark 2.5 and lemma 4.4. Such bounds are obtained either by simple integral arguments that exploit test function methods or combine a weak version of the divergence theorem and the co-area formula.

Related properties to those in theorem 1.1 for problems with supernatural gradient growth, i.e. $\gamma >p$, have been already analysed in the literature. The paper by I. Capuzzo-Dolcetta, F. Leoni and A. Porretta [Reference Capuzzo Dolcetta, Leoni and Porretta11] treated a large class of diffusion operators in non-divergence form (that also includes fully nonlinear uniformly elliptic operators and the $p$-Laplacian) with bounded right-hand side $f$ (with estimates depending on $\|u\|_\infty$) in the realm of viscosity solutions. This was revisited with a shorter proof by G. Barles in [Reference Barles3], again in the framework of viscosity (semi-)solutions. This regularity problem has been much less investigated for unbounded data, where mostly optimal regularity at the level of Lebesgue spaces has been studied, see e.g. [Reference Grenon, Murat and Porretta24]. An analogue to our theorem 1.1 with unbounded source terms $f\in L^q$ was proved by A. Dall'Aglio and A. Porretta in the case $p=2$, and it has been the starting point of our work. At this stage, we stress that distributional solutions for these problems are not unique, cf § 3.2, and hence the quantitative result in theorem 1.1 takes on a stronger meaning.

Recent advances on the Hölder regularity for problems driven by the Laplacian with coercive gradient terms and space-time $L^q$ right-hand side have been obtained for parabolic equations in [Reference Cirant and Goffi14] through quite different integral/duality methods, which, unfortunately, do not seem to extend to quasi-linear operators satisfying (A). We further mention that the Hölder regularity for solutions to these equations when $p=2$ has been also the focus of the recent work [Reference Cirant and Verzini17], but for solutions solving the PDE in strong sense, looking thus beyond the range of the Hölder bound stated in theorem 1.1 for source terms with better summability. Indeed, we also show that, as far as the Hölder regularity is concerned, the range found in theorem 1.1 is sharp for our class of functions, cf § 3.2. It is well-known that these regularity properties are the cornerstone for the analysis of many different problems ranging from those arising in ergodic control and homogenization to problems with state constraints, at least when $p=2$, see [Reference Barles3, Reference Birindelli, Demengel and Leoni9]. Nonetheless, the case of quasilinear equations with terms having supernatural growth appears, to our knowledge, still an open line of research, cf [Reference Birindelli, Demengel and Leoni9, Reference Leonori and Porretta30, Reference Nguyen and Phuc34] for some recent contributions.

Instead, statements like theorem 1.2, and its byproduct corollary 1.3, have been studied (almost) thoroughly and for many years through various different approaches, see e.g. Remark 3.7, § 3.3.1 and the general reference [Reference Bianchini, Mari, Pucci and Rigoli6]. Although the latter is a widely studied problem, we decided to propose a new unifying proof of the result, even in a more general context, due to its shortness, along with the treatment of some new endpoint cases (e.g. the borderline case $\gamma =\frac {N(p-1)}{N-1}$ in corollary 1.3). Here, our approach to tackle the Liouville property has been inspired by the works [Reference Lions31, Reference Pigola, Rigoli and Setti36, Reference Rigoli and Setti40]. At this point it is remarkable to emphasize that when $p=2$ the threshold $\gamma =\frac {N}{N-1}$ is critical for the solvability and the regularity of solutions of the so-called viscous Hamilton–Jacobi equation, see e.g. [Reference Cirant and Goffi16, Reference Hansson, Maz'ya and Verbitsky27].

Notably, the results in theorems 1.1, 1.2 and corollary 1.3 highlight a striking effect of the superlinear gradient term. Indeed, a simple zoom of the equation suggests that such a term is the dominating one at small scales in the supernatural regime $\gamma >p$ (i.e. in the regime of theorem 1.1), and it turns out to be the sole responsible of all the properties analysed throughout this manuscript, as one can realise by a careful inspection of the proofs of theorems 1.1 and 1.2. The link between the Liouville and the local regularity properties of nonlinear elliptic problems has been sometimes pointed out throughout some works, see e.g. [Reference Beck, Bulíček and Frehse4, Reference Meier33] among others, and our analysis underlines once more the bridge between nonexistence results and the Hölder regularity for semi-solutions of PDEs through the Morrey-type bounds in lemma 2.3 and remark 2.5.

Some comments on the aforementioned results are now in order. On one hand, theorem 1.1 holds for merely distributional semi-solutions to PDEs with supernatural gradient growth and unbounded right-hand side. This is unusual for diffusive problems (e.g. of second order) with first-order terms above the natural growth imposed by the diffusion $\gamma =p$, since such estimates hold, in general, for solutions of uniformly elliptic problems with subnatural gradient growth $\gamma < p$ and, usually, bounded data [Reference Bensoussan and Frehse5, Reference Ladyzhenskaya and Ural'tseva29]. Moreover, the estimate in theorem 1.1 holds up to the boundary of the domain and it is a universal estimate for positive solutions.

On the other hand, the approach leading to theorem 1.2 does not require any one-side bound on the solution, being thus consistent with the companion results for solutions to the same kind of equations obtained through the Bernstein gradient estimates by L. Peletier-J. Serrin [Reference Peletier and Serrin35] and P.-L. Lions [Reference Lions31], and later in [Reference Bidaut-Véron, Garcia-Huidobro and Véron7] (see also [Reference Birindelli, Demengel and Leoni9, theorem 2.4] and [Reference Filippucci, Pucci and Souplet22, theorem 1.1] for recent refinements). Notably, in the context of Riemannian manifolds theorem 1.2 does not require curvature conditions on the background geometry. As a matter of fact, we emphasize that curvature bounds have been frequently imposed to derive the Liouville property for solutions to $-\Delta _pu+|\nabla u|^\gamma =0$ when the first-order term has ‘superlinear’ growth with respect to the diffusion (i.e. $\gamma >p-1$) and the equation is posed on a noncompact manifold, see e.g. [Reference Bidaut-Véron, Garcia-Huidobro and Véron7, corollary 4.3]. This is a consequence of the use of Bernstein-type methods through the Böchner's identity. In this paper we adopt a different viewpoint, as in [Reference Pigola, Rigoli and Setti37, Reference Rigoli and Setti40]. Indeed, theorem 1.2 shows, in the case of distributional supersolutions to these homogeneous quasi-linear equations, that the Liouville property can be obtained under some area-growth conditions, at least for slowly increasing gradient terms, see § 4.2. It seems an open problem whether the Liouville result in [Reference Bidaut-Véron, Garcia-Huidobro and Véron7] holds for less regular solutions and with less restrictive bounds on the geometry in the general superlinear range $\gamma >p-1$. We further remark that the assumptions we impose in theorem 1.2 are closer to the ones used in [Reference Rigoli and Setti40, Reference Sun, Xiao and Xu43] rather than in [Reference Bidaut-Véron, Garcia-Huidobro and Véron7].

Our results apply to general diffusion operators in divergence form. For instance $-\mathrm {div}(\mathcal {A}(x,\,u,\,\nabla u))$ can be one of the following:

• • the (negative) Laplacian $-\Delta u=-\mathrm {div}(\nabla u)$ or the mean-curvature operator $-\mathrm {div}(\frac {\nabla u}{\sqrt {1+|\nabla u|^2}})$ when $p=2$;

• • the $p$-Laplacian $-\Delta _pu=-\mathrm {div}(|\nabla u|^{p-2}\nabla u)$ for $p>1$;

• • the generalized mean-curvature operator $-\mathrm {div}(|\nabla u|^{k-2}\frac {\nabla u}{\sqrt {1+|\nabla u|^k}})$, $k\geq 2$, for $p=\frac {k}{2}$.

Moreover, the properties in theorem 1.1 also extend to the subelliptic framework when the standard derivatives in the aforementioned operators are replaced with derivatives along vector fields satisfying the Hörmander's rank condition, i.e. for the subelliptic inequality

\[ -\mathrm{div}(\mathcal{A}(x,u,\nabla_\mathcal{X} u))+ |\nabla_\mathcal{X} u|^\gamma\leq f(x), \]

where $\nabla _\mathcal {X}$ stands for the horizontal gradient built over the frame $\mathcal {X}$, see e.g. theorem 4.2 in § 4.1 for more details. This is the case, for instance, of problems structured over the fields generating a Carnot group of step 2, the main prototype being the Heisenberg group. The interior Hölder regularity for such subelliptic problems with power-like first-order terms on the horizontal gradient seems new to our knowledge.

Moreover, the results in theorems 1.1 and 1.3 can be stated analogously for distributional supersolutions and subsolutions respectively by reversing the signs in the assumptions (B1) and (B2). For instance, the conclusions of theorem 1.1 hold for distributional solutions to the model PDI

(1.9)\begin{equation} -\mathrm{div}(\mathcal{A}(x,u,\nabla u))+\lambda u-|\nabla u|^\gamma\geq f(x)\text{ in }\Omega\ , \end{equation}

under essentially the same regularity assumptions on the data, while the Liouville property in theorem 1.3 holds for the weak PDI

\[ -\mathrm{div}(\mathcal{A}(x,u,\nabla u))+c_H|\nabla u|^\gamma\leq0\text{ in }\mathbb{R}^N \]

under the hypothesis (A). Indeed, the diffusion plays no role here, as in [Reference Capuzzo Dolcetta, Leoni and Porretta11, Reference Dall'Aglio and Porretta18], since the properties can be deduced from the sign of the gradient term in the equation, see remark 2.5. To convince the reader that such properties can be appropriately stated both for sub- and supersolutions to elliptic inequalities with power-growth gradient terms and compare the results with most of the literature on the subject, we decided to state theorem 1.1 for subsolutions and theorem 1.2 for supersolutions to elliptic inequalities.

We conclude by saying that our underlying aim here is also to lay the groundwork to investigate the problem of the optimal gradient regularity in Lebesgue spaces for equations with diffusion in divergence form as those modelled over the $p$-Laplacian, as initiated in [Reference Cirant and Goffi14, Reference Cirant and Goffi16] for the viscous Hamilton–Jacobi equation, through blow-up arguments, see [Reference Beck, Bulíček and Frehse4, Reference Cirant and Verzini17].

Plan of the paper. Section 2 introduces some preliminary definitions and contains the main integral bounds needed for the following sections. Section 3.1 studies the interior and global Hölder regularity for semi-solutions, while § 3.3 the Liouville property for homogeneous PDIs in the Euclidean space, that is corollary 1.3. Section 4.1 and 4.2 conclude the paper with the local Hölder regularity for solutions to quasi-linear subelliptic inequalities and the Liouville properties for inequalities posed on Riemannian manifolds.