In this work, we consider the following degenerate (or singular) elliptic equations of *p*-Laplacian type:

$-\mathrm{div}\left({|\nabla u|}^{p-2}\nabla u\right)=f(x,u,\nabla u),$(1.1)

defined in an open and bounded set $\mathrm{\Omega}\subset {\mathbb{R}}^{n}$ and for $1<p<\mathrm{\infty}$. The modulus of ellipticity of the *p*-Laplace operator is ${|\nabla u|}^{p-2}$. When $p>2$, the modulus vanishes whenever $\nabla u=0$, and the equation is called degenerate at those points where that occurs. On the other hand, for $p<2$, the modulus becomes infinite when $\nabla u=0$, and the equation is called singular at those points. Observe that the case $p=2$ is just the linear case and corresponds to the Laplace operator.

Different notions of solutions have been formulated for equation (1.1). We are interested in the relation between Sobolev weak solutions and viscosity solutions. For the homogeneous *p*-Laplace equation, this relation has already been studied by Juutinen, Lindqvist and Manfredi in [9], via the notion of p-harmonic, p-subharmonic and p-superharmonic functions. Roughly speaking, a p-harmonic function is a continuous function which solves, weakly, the homogeneous *p*-Laplace equation, and a p-superharmonic (p-subharmonic) function is a lower (upper) semicontinuous function that admits comparison with p-harmonic functions from below (above).

In [9], Juutinen, Lindqvist and Manfredi showed that the notion of p-harmonic solution is equivalent to the notion of viscosity solution. Moreover, it was shown in [13] that locally bounded p-harmonic functions are weak solutions. Conversely, every weak solution to the homogeneous *p*-Laplace equation has a representative which is lower semicontinuous and it is p-harmonic. We refer the interested reader to [5] for further details. In this way, there is an equivalence between the notion of weak and viscosity solutions for the homogeneous framework. It is worth to mention that a different and simpler proof of this equivalence was stated by Julin and Juutinen in [6] by using inf and sup convolutions. In turn, this reasoning was extended in [10] to more general second-order differential equations.

For the non-homogeneous case, the notion of p-harmonic functions is lost and we need to study directly the link between viscosity and Sobolev weak solutions. In [6], the authors showed that viscosity solutions of (1.1) are weak solutions in the case where *f* is continuous and depends only on *x*.

Our main goal in the present manuscript is to prove the equivalence of these two notions of solutions for the general structure (1.1). The implication that viscosity solutions are weak solutions is partially based on the work [6], but the non-homogeneous nature of the equation under consideration requires some extra effort to deal with the lower-order term.

On the other hand, the converse statement relies on comparison principles for weak solutions. To the best of our knowledge, the available comparison results for the full case $f=f(x,s,\eta )$ require additional limitations in the degenerate case which do not appear in the singular context (compare Theorem A.1 and Theorem A.2). Moreover, we believe that the assumption that weak subsolutions and weak supersolutions belong to ${\mathcal{\mathcal{C}}}^{1}$ or to the Sobolev space ${W}_{\mathrm{loc}}^{1,\mathrm{\infty}}$ in order to have comparison is not a strong limitation since we are interested in the equivalence of weak and viscosity solutions, and for weak solutions the ${\mathcal{\mathcal{C}}}^{1,\alpha}$-regularity holds (see [4, 16]). Finally, in the quasi-linear case $f=f(x,u)$ there is no need to impose higher regularity than ${W}_{\mathrm{loc}}^{1,p}\cap \mathcal{\mathcal{C}}$ on the solutions. We refer the reader to [15] for a survey of maximum principles and comparison results for general structures in divergence form.

Finally, we stress that the equivalence between weak and viscosity solutions may be used to prove relevant properties on the solutions. As an example, in [7], Juutinen and Lindqvist prove a Radó’s-type theorem for p-harmonic functions. Roughly speaking, they state that if a function *u* solves, weakly, the homogeneous *p*-Laplace equation in the complement of the set where *u* vanishes, then it is a solution in the whole set. It is an open problem to obtain a similar result for equations like (1.1). We shall return to this issue in a subsequent paper.

We recall that the *p*-Laplace operator is defined as

${\mathrm{\Delta}}_{p}u:=\mathrm{div}\left({|\nabla u|}^{p-2}\nabla u\right).$

Let us state the different type of solutions to (1.1) we will manage.

#### Definition 1.1 (Sobolev weak solution).

*A function $u\mathrm{\in}{W}_{\mathrm{loc}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ is a weak supersolution to (1.1) if*

${\int}_{\mathrm{\Omega}}{|\nabla u|}^{p-2}\nabla u\cdot \nabla \psi \ge {\int}_{\mathrm{\Omega}}f(x,u,\nabla u)\psi $

*for all non-negative $\psi \mathrm{\in}{\mathcal{C}}_{\mathrm{0}}^{\mathrm{\infty}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$. On the other hand, **u* is a weak subsolution if $\mathrm{-}u$ is a weak supersolution of the equation $\mathrm{-}{\mathrm{\Delta}}_{p}\mathit{}u\mathrm{=}\mathrm{-}f\mathit{}\mathrm{(}x\mathrm{,}\mathrm{-}u\mathrm{,}\mathrm{-}\mathrm{\nabla}\mathit{}u\mathrm{)}$. We call *u* a weak solution if it is both a weak subsolution and a weak supersolution to (1.1).

Due to the non-homogeneous nature of (1.1), viscosity solutions are stated as in [6], considering semicontinuous envelopes of the *p*-Laplace operator. More precisely, we have the following definition.

#### Definition 1.2.

*A lower semicontinuous function $u\mathrm{:}\mathrm{\Omega}\mathrm{\to}\mathrm{(}\mathrm{-}\mathrm{\infty}\mathrm{,}\mathrm{+}\mathrm{\infty}\mathrm{]}$ is a viscosity supersolution to (1.1) if $u\mathrm{\not\equiv}\mathrm{+}\mathrm{\infty}$ and for every $\varphi \mathrm{\in}{\mathcal{C}}^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ such that $\varphi \mathit{}\mathrm{(}{x}_{\mathrm{0}}\mathrm{)}\mathrm{=}u\mathit{}\mathrm{(}{x}_{\mathrm{0}}\mathrm{)}$, $u\mathit{}\mathrm{(}x\mathrm{)}\mathrm{\ge}\varphi \mathit{}\mathrm{(}x\mathrm{)}$ and $\mathrm{\nabla}\mathit{}\varphi \mathit{}\mathrm{(}x\mathrm{)}\mathrm{\ne}\mathrm{0}$ for all $x\mathrm{\ne}{x}_{\mathrm{0}}$, there holds*

$\underset{r\to 0}{lim}\underset{x\in {B}_{r}({x}_{0})\setminus \{{x}_{0}\}}{sup}(-{\mathrm{\Delta}}_{p}\varphi (x))\ge f({x}_{0},u({x}_{0}),\nabla \varphi ({x}_{0})).$(1.2)

*A function **u* is a viscosity subsolution if $\mathrm{-}u$ is a viscosity supersolution to the equation $\mathrm{-}{\mathrm{\Delta}}_{p}\mathit{}u\mathrm{=}\mathrm{-}f\mathit{}\mathrm{(}x\mathrm{,}\mathrm{-}u\mathrm{,}\mathrm{-}\mathrm{\nabla}\mathit{}u\mathrm{)}$, and it is a viscosity solution if it is both a viscosity sub- and supersolution.

We now list the main contributions of our work. The results are stated for supersolutions, but they hold for subsolutions as well.

#### Theorem 1.4.

*Let $\mathrm{1}\mathrm{<}p\mathrm{<}\mathrm{\infty}$. Assume that $f\mathrm{=}f\mathit{}\mathrm{(}x\mathrm{,}s\mathrm{,}\eta \mathrm{)}$ is uniformly continuous in $\mathrm{\Omega}\mathrm{\times}\mathrm{R}\mathrm{\times}{\mathrm{R}}^{n}$, non-increasing in **s*, and satisfies the growth condition

$|f(x,s,\eta )|\le \gamma (|s|){|\eta |}^{p-1}+\varphi (x),$(1.3)

*where $\gamma \mathrm{\ge}\mathrm{0}$ is continuous, and $\varphi \mathrm{\in}{L}_{\mathrm{loc}}^{\mathrm{\infty}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$. Hence,
if $u\mathrm{\in}{L}_{\mathrm{loc}}^{\mathrm{\infty}}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ is a viscosity supersolution to (1.1), then it is a weak supersolution to (1.1).*

A converse of Theorem 1.4 is given below.

#### Theorem 1.5.

*Assume that $f\mathrm{=}f\mathit{}\mathrm{(}x\mathrm{,}s\mathrm{,}\eta \mathrm{)}$ is continuous in $\mathrm{\Omega}\mathrm{\times}\mathrm{R}\mathrm{\times}{\mathrm{R}}^{n}$, non-increasing in **s*, and locally Lipschitz continuous with respect to η. Hence we have the following:

(i)

*If *
$1<p\le 2$
* and if *
$u\in {W}_{\mathrm{loc}}^{1,\mathrm{\infty}}(\mathrm{\Omega})$
* is a weak supersolution to (*
1.1
*), then it is a viscosity supersolution to (*
1.1
*) in *
Ω.

(ii)

*If *
$p>2$,
$f(x,s,0)=0$
* for *
$x\in \mathrm{\Omega}$
* and *
$s\in \mathbb{R}$
*, and if *
$u\in {\mathcal{\mathcal{C}}}^{1}(\mathrm{\Omega})$
* is a weak supersolution to (*
1.1
*), then it is a viscosity supersolution to (*
1.1
*) in *
Ω.

(iii)

*Finally, if *
$p>2$
* and if *
$u\in {W}_{\mathrm{loc}}^{1,\mathrm{\infty}}(\mathrm{\Omega})$
* is a weak supersolution to (*
1.1
*), with *
$\nabla u\ne 0$
* in *
Ω
*, then it is a viscosity supersolution to (*
1.1
*).*

In view of the available regularity theory for weak solutions of (1.1), we have the following equivalence.

#### Corollary 1.7.

*Let $\mathrm{1}\mathrm{<}p\mathrm{<}\mathrm{\infty}$. Assume that $f\mathrm{=}f\mathit{}\mathrm{(}x\mathrm{,}s\mathrm{,}\eta \mathrm{)}$ is uniformly continuous, locally Lipschitz in η, non-increasing in **s*, and satisfies the growth condition (1.3). Additionally, assume that $f\mathit{}\mathrm{(}x\mathrm{,}s\mathrm{,}\mathrm{0}\mathrm{)}\mathrm{=}\mathrm{0}$ for $x\mathrm{\in}\mathrm{\Omega}$ and $s\mathrm{\in}\mathrm{R}$ when $p\mathrm{>}\mathrm{2}$. Then *u* is a weak solution to (1.1) if and only if it is a viscosity solution to (1.1).

We point out that, in the degenerate case, it is possible to remove the assumption $f(x,s,0)=0$ by imposing the non-vanishing of the gradient of the weak solution in the whole Ω. This is a straightforward consequence of Theorem 1.5 (iii).

In the particular case where *f* does not depend on η, we have the following converse to Theorem 1.4 which does not require the locally Lipschitz regularity of the solutions.

#### Theorem 1.8.

*Let $\mathrm{1}\mathrm{<}p\mathrm{<}\mathrm{\infty}$. Suppose that $f\mathrm{=}f\mathit{}\mathrm{(}x\mathrm{,}s\mathrm{)}$ is continuous in $\mathrm{\Omega}\mathrm{\times}\mathrm{R}$ and non-increasing in **s*. If $u\mathrm{\in}{W}_{\mathrm{loc}}^{\mathrm{1}\mathrm{,}p}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}\mathrm{\cap}\mathcal{C}\mathit{}\mathrm{(}\mathrm{\Omega}\mathrm{)}$ is a weak supersolution to (1.1), then it is a viscosity supersolution to (1.1).

Let us briefly discuss the above hypotheses on *f*. Firstly, assuming that *f* is non-increasing and introducing the operator

$F(x,s,\eta ,\mathcal{\mathcal{X}}):=-{|\eta |}^{p-2}\left(\mathrm{tr}(\mathcal{\mathcal{X}})+\frac{p-2}{{|\eta |}^{2}}\mathcal{\mathcal{X}}\eta \cdot \eta \right)-f(x,s,\eta ),p\ge 2,$

we derive that *F* is proper, that is, *F* is non-increasing in $\mathcal{\mathcal{X}}$ and non-decreasing in *s*, which is a standard and useful assumption in the theory of viscosity solutions [1]. For instance, it allows to get the equivalence between classical solutions (${\mathcal{\mathcal{C}}}^{2}$ functions which satisfy the equations pointwise) and ${\mathcal{\mathcal{C}}}^{2}$ viscosity solutions. On the other hand, the growth property (1.3) implies the ${\mathcal{\mathcal{C}}}^{1,\alpha}$-regularity of weak solutions to (1.1) (see [4, 17, 16]). Moreover, under a regular Dirichlet boundary condition $\phi \in {\mathcal{\mathcal{C}}}^{1,\alpha}$, the ${\mathcal{\mathcal{C}}}^{1,\alpha}$-regularity up to the boundary of weak solutions follows. For further details, see the reference [12]. Finally, the extra assumption $f(x,s,0)=0$ appearing in Theorem 1.5 in the degenerate case is used to remove critical sets of points of the weak solution (see reference [8]). Hence, it allows the application of comparison results without assuming the non-vanishing of the gradients. We point out that other properties of $f=f(x,s,\eta )$, as more regularity on *s* and η and convexity-like conditions, may be employed to ensure comparison for weak solutions. We refer the reader to [11] and the references therein for more details.

It is worth mentioning that many equations appearing in the literature have the structure of (1.1) with the lower-order term satisfying the above assumptions on *f*. We refer the reader to [14, 15, 3, 2] and the references therein for examples of such *f*.

The paper is organized as follows: In Section 2, we provide some preliminary results concerning properties of infimal convolutions (which will be the main tool in the proof of Theorem 1.4) and a convergence result. In addition, we prove a Caccioppoli-type estimate that will provide important uniform bounds, fundamental when using approximation arguments. This result is interesting in itself.

Section 3 contains the proof of the main result of the paper, Theorem 1.4, that states under which conditions on the non-homogeneous function *f* in (1.1) viscosity solutions are actually weak solutions. This proof is divided into two major cases: the singular and the degenerate scenario, thus, although both cases rely on the same idea, different approximations and estimates are needed depending on the range of *p*.

In Section 4, we prove the reverse statement, that is, weak solutions of (1.1) are viscosity solutions. This result is based on comparison arguments, and this will determine the conditions we will need to impose on *f*. Finally, in Appendix A we give, for the sake of completeness, precise references and state the comparison results that we use in Section 4.

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