A strong maximum principle for mixed local and nonlocal p -Laplace equations

Abstract

We establish a strong maximum principle for weak solutions of the mixed local and nonlocal p-Laplace equation $\begin{array}{l} - Δ_{p} u + {(- Δ)}_{p}^{s} u = c (x) | u |^{p - 2} u in Ω, \end{array}$ where $Ω \subset R^{N}$ is an open set, $p \in (1, \infty)$ , $s \in (0, 1)$ and $c \in C (\overline{Ω})$ .

Keywords

Strong maximum principle weak solutions viscosity solutions

1. Introduction

Let $Ω \subset R^{N}$ be an open set. In this work, we devote to proving a strong maximum principle for the following mixed problem $\begin{array}{l} (1.1) & - Δ_{p} u + {(- Δ)}_{p}^{s} u = c (x) | u |^{p - 2} u in Ω, \end{array}$ where $p \in (1, \infty)$ , $s \in (0, 1)$ and $c \in C (\overline{Ω})$ . As usual, $Δ_{p}$ is the p-Laplace operator defined as $\begin{array}{l} - Δ_{p} u : = - div (| \nabla u |^{p - 2} \nabla u), \end{array}$ while the nonlocal p-Laplace operator is given by $\begin{array}{l} (1.2) & {(- Δ)}_{p}^{s} u (x) : = 2 P . V . \int_{R^{N}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y))}{| x - y |^{N + s p}} d y, \end{array}$ where $P . V .$ stands for the Cauchy principal value. The mixed local and nonlocal problems describing the superposition of classical random walk and Lévy flight have received much attention recently and possess important applications in biology [15] and plasma physics [7].

The strong maximum principle is an essential tool for obtaining information on solutions in partial differential equations. It is also widely applied in the moving plane method to get the symmetry and monotonicity of solutions [4,11,23]. In the following, let us review some relevant results. In the local setting, we refer readers to the monograph [28], in which the strong maximum principle, Hopf lemma, and Harnack inequality were proved in detail. For the nonlocal case, Greco–Servadei [21] showed a strong minimum principle and Hopf lemma for pointwise solutions of fractional Laplacian operator. After that, the results were generalized to the nonlinear operator (1.2) by Del Pezzo–Quaas [14]; see [1] for the fractional $(p, q)$ -Laplacian operator. When considering the fractional elliptic and parabolic equations with mixed boundary conditions, Barrios–Medina [2] established a strong maximum principle and Hopf lemma by presenting some comparison results for solutions. More results can be found in [9,10,24,26,27].

For the mixed local and nonlocal problem $\begin{array}{l} (1.3) & - Δ_{p} u + {(- Δ)}_{p}^{s} u = 0, \end{array}$ some qualitative results about existence, maximum principles for weak and classical solutions, and regularity theory were provided by Biagi–Dipierro–Valdinoci–Vecchi [5] when $p = 2$ . In the case of $p ⩾ 2$ , Del Pezzo–Ferreira–Rossi [13] discussed the Dirichlet eigenvalue problem and proved a strong maximum principle, where they pointed out that the main difficulty in deriving such principle under $1 < p < 2$ is how to define viscosity solutions. In the non-homogeneous analogue of (1.3), Biagi–Mugnai–Vecchi [6] investigated the global boundedness and a strong maximum principle for weak solutions in a connected and bounded domain if the non-homogeneous term satisfies some growth and monotonicity conditions. For more results concerned with the qualitative properties and regularity of solutions to mixed problems, we refer the readers to [6,12,16–19] and references therein.

Motivated by the works mentioned above, our intention in the present paper is to prove a strong maximum principle for weak solutions of mixed local and nonlocal problem (1.1) in the full range $1 < p < \infty$ . Due to the properties of mixed operators, we could remove the assumption that Ω is connected and only assume that $Ω \subset R^{N}$ is an open set. The proof relies on the logarithmic type lemma and the relationship between weak and viscosity solutions.

Before stating our main results, we introduce some definitions of spaces. For simplicity, we denote $Ω^{c} = R^{N} ∖ Ω$ . Define the space $X_{0}^{1, p} (Ω)$ by $\begin{array}{l} X_{0}^{1, p} (Ω) = {u \in W^{1, p} (R^{N}) ∣ u = 0 in Ω^{c}} . \end{array}$ The dual space of $X_{0}^{1, p} (Ω)$ is denoted by $X^{- 1, p^{'}} (Ω)$ . For any bounded $Ω^{'} \subset \subset Ω$ , we set $\begin{array}{l} X^{1, p} (Ω) = {u \in L_{loc}^{p} (R^{N}) | \exists U \supset \supset Ω^{'} s.t. ‖ u ‖_{W^{1, p} (U)} + \int_{R^{N}} \frac{| u (x) |^{p - 1}}{{(1 + | x |)}^{N + s p}} d x < \infty}, \end{array}$ where U is an open set. In fact, when Ω is bounded, we say $u \in X^{1, p} (Ω)$ if and only if for all functions $u \in L_{loc}^{p} (R^{N})$ , there is an open set $U \supset \supset Ω$ such that $\begin{array}{l} ‖ u ‖_{W^{1, p} (U)} + \int_{R^{N}} \frac{| u (x) |^{p - 1}}{{(1 + | x |)}^{N + s p}} d x < \infty . \end{array}$

Next, let us recall the definition of weak solutions.

Definition 1.1.
Let Ω be bounded and $f \in X^{- 1, p^{'}} (Ω)$ . A function $u \in X^{1, p} (Ω)$ is a weak super(sub)-solution of $- Δ_{p} u + {(- Δ)}_{p}^{s} u = f$ in Ω if $\begin{array}{l} \int_{Ω} | \nabla u |^{p - 2} \nabla u \cdot \nabla φ d x + \int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y)) (φ (x) - φ (y))}{| x - y |^{N + s p}} d x d y \\ ⩾ (⩽) ⟨ f, φ ⟩ \end{array}$ for any non-negative function $φ \in X_{0}^{1, p} (Ω)$ .

When Ω is unbounded and $f \in X^{- 1, p^{'}} (Ω)$ , a function $u \in X^{1, p} (Ω)$ is called a weak super(sub)-solution of $- Δ_{p} u + {(- Δ)}_{p}^{s} u = f$ in Ω if for any bounded open set $Ω^{'} \subset Ω$ , u is a super(sub)-solution of $- Δ_{p} u + {(- Δ)}_{p}^{s} u = f$ in $Ω^{'}$ . A function u is a weak solution of $- Δ_{p} u + {(- Δ)}_{p}^{s} u = f$ in Ω if and only if u is both a supersolution and subsolution.

The first result we present is a strong maximum principle as follows.
Theorem 1.2.
Assume that $c \in L_{loc}^{1} (Ω)$ is a non-positive function and $u \in X^{1, p} (Ω)$ is a weak supersolution of ( 1.1 ).
If Ω is bounded, and $u ⩾ 0$ a.e. in $Ω^{c}$ then either $u > 0$ a.e. in Ω or $u = 0$ a.e. in $R^{N}$ ;

If $u ⩾ 0$ a.e. in $R^{N}$ then either $u > 0$ a.e. in Ω or $u = 0$ a.e. in $R^{N}$ .

Remark 1.3.
If $c \in L_{loc}^{1} (Ω)$ , then $c (x) ⩾ c^{-} (x) = min {0, c (x)} \in L_{loc}^{1} (Ω)$ . Thus if u is a weak supersolution of (1.1) and $u ⩾ 0$ a.e. in $R^{N}$ , we know that u is also a weak supersolution of $\begin{array}{l} - Δ_{p} u + {(- Δ)}_{p}^{s} u = c^{-} (x) u^{p - 1} in Ω . \end{array}$ By Theorem 1.2, it follows that $u > 0$ a.e. in Ω or $u = 0$ a.e. in $R^{N}$ . In other words, it is not necessary to assume that $c (x)$ is non-positive when $u ⩾ 0$ a.e. in $R^{N}$ .

In fact, if c and u are continuous functions in Theorem 1.2, we can remove “a.e.” in the above statement according to the conclusion to be proved in Section 2 that continuous weak supersolutions are viscosity supersolutions. Theorem 1.4.
Let $c \in C (\overline{Ω})$ be a non-positive function. Suppose that $u \in X^{1, p} (Ω) \cap C (\overline{Ω})$ is a weak supersolution of ( 1.1 ).
If Ω is bounded, and $u ⩾ 0$ a.e. in $Ω^{c}$ then either $u > 0$ in Ω or $u = 0$ a.e. in $R^{N}$ ;

If $u ⩾ 0$ a.e. in $R^{N}$ then either $u > 0$ in Ω or $u = 0$ a.e. in $R^{N}$ .

Remark 1.5.
If Ω is a bounded domain in $R^{N}$ with $C^{1, 1}$ boundary, the condition $u \in X^{1, p} (Ω) \cap C (\overline{Ω})$ in the above theorem can be weakened as $u \in X^{1, p} (Ω) \cap L^{\infty} (Ω)$ , since the continuity of solutions can be deduced from the boundedness (see [20, Remark 2.4]).

2. Preliminaries

In this section, we will give some notations. Moreover, the definition of viscosity solutions and some auxiliary lemmas will be introduced.

For a function $u : Ω \to R$ , we denote $\begin{array}{l} u_{+} (x) = max {u (x), 0}, u_{-} (x) = max {- u (x), 0} \end{array}$ and $\begin{array}{l} Ω_{+} = {x \in Ω : u (x) > 0}, Ω_{-} = {x \in Ω : u (x) < 0} . \end{array}$

To begin with, we state a comparison principle.

Lemma 2.1 (Lemma 2.1, [13]).

Suppose that Ω is bounded, $u, v \in X^{1, p} (Ω)$ satisfy $u ⩾ v$ in $Ω^{c}$ and $\begin{array}{l} \int_{Ω} | \nabla u |^{p - 2} \nabla u \cdot \nabla φ + \int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y)) (φ (x) - φ (y))}{| x - y |^{N + s p}} d x d y \\ ⩾ \int_{Ω} | \nabla v |^{p - 2} \nabla v \cdot \nabla φ + \int_{R^{N}} \int_{R^{N}} \frac{| v (x) - v (y) |^{p - 2} (v (x) - v (y)) (φ (x) - φ (y))}{| x - y |^{N + s p}} d x d y \end{array}$ for all non-negative functions $φ \in X_{0}^{1, p} (Ω)$ , then we have $u ⩾ v$ a.e. in Ω.

In the rest of this section, we always assume that Ω is a bounded open set with smooth boundary and $c \in C (\overline{Ω})$ .

It is well-known that the p-Laplace operator is singular in the range of $1 < p < 2$ , we must ensure the valid definition in this case when $\nabla u$ vanishes. As for the fractional p-Laplace operator, we face the same problem in the singular range $1 < p < \frac{2}{2 - s}$ . Thus we need to work in the following more restricted class. Let $D \subset Ω$ be an open set, define $\begin{array}{l} C_{β}^{2} (D) : = {u \in C^{2} (Ω) | sup_{x \in D} (\frac{min {dist (x, N_{u}), 1}^{β - 1}}{| \nabla u (x) |} + \frac{| D^{2} u (x) |}{{(dist (x, N_{u}))}^{β - 2}}) < + \infty}, \end{array}$ where $N_{u}$ denotes the set of critical points of the differential function u, that is $\begin{array}{l} N_{u} : = {x \in Ω ∣ \nabla u (x) = 0} . \end{array}$

We also denote $\begin{array}{l} L_{s p}^{p - 1} (R^{N}) = {u \in L_{loc}^{p - 1} (R^{N}) | \int_{R^{N}} \frac{| u (x) |^{p - 1}}{{(1 + | x |)}^{N + s p}} d x < + \infty} . \end{array}$ Influenced by [3,22,25] and considering the properties of mixed local and nonlocal operators, we give the definition of viscosity solutions.

Definition 2.2.
Let Ω be a bounded open set with smooth boundary and $c \in C (\overline{Ω})$ . A function $u : R^{N} \to [- \infty, + \infty]$ is called a viscosity supersolution of (1.1) if
$u < + \infty$ a.e. in $R^{N}$ , $u > - \infty$ a.e. in Ω.

u is lower semicontinuous in Ω.

If $ψ \in C^{2} (B_{r} (x_{0})) \cap L_{s p}^{p - 1} (R^{N})$ for some $B_{r} (x_{0}) \subset Ω$ such that $ψ (x_{0}) = u (x_{0})$ , $ψ ⩽ u$ in $R^{N}$ and one of the following conditions holds:
if $p ⩾ 2$ ;

if $\frac{2}{2 - s} < p < 2$ and $\nabla ψ (x) \neq 0$ for $x \neq x_{0}$ ;

if $1 < p ⩽ \frac{2}{2 - s}$ , $\nabla ψ (x_{0}) = 0$ with $x_{0}$ an isolated critical point of ψ, and $ψ \in C_{β}^{2} (B_{r} (x_{0}))$ for some $β > \frac{s p}{p - 1}$ ,
it holds that $\begin{array}{l} (2.1) & lim_{r \to 0} sup_{x \in B_{r} (x_{0}) ∖ {x_{0}}} - Δ_{p} ψ_{r} (x) + {(- Δ)}_{p}^{s} ψ_{r} (x_{0}) ⩾ c (x_{0}) {| u (x_{0}) |}^{p - 2} u (x_{0}), \end{array}$ where $\begin{array}{l} ψ_{r} (x) = \{\begin{matrix} ψ & if x \in B_{r} (x_{0}), \\ u (x) & otherwise . \end{matrix} \end{array}$

$u_{-} \in L_{s p}^{p - 1} (R^{N})$ .

Remark 2.3.
We remark that when $p ⩾ 2$ we can simplify (2.1) by $\begin{array}{l} - Δ_{p} ψ_{r} (x_{0}) + {(- Δ)}_{p}^{s} ψ_{r} (x_{0}) ⩾ c (x_{0}) {| u (x_{0}) |}^{p - 2} u (x_{0}) . \end{array}$
Lemma 2.4.
Let $u \in X^{1, p} (Ω) \cap C (\overline{Ω})$ be a weak supersolution of ( 1.1 ). If $u ⩾ 0$ in $Ω^{c}$ , then u is a viscosity supersolution of ( 1.1 ).
Proof.
By assumption, it is easy to check that u satisfies (VS1), (VS2) and (VS4). In the following, we devote to verifying (VS3) by contradiction. Suppose that there exist $x_{0} \in Ω$ and $ψ \in C^{2} (B_{r} (x_{0}))$ such that
$ψ (x_{0}) = u (x_{0})$ and $ψ ⩽ u$ in $B_{r} (x_{0}) \subset Ω$ ;

Either (a) or (b) or (c) in (VS3) holds;

$\begin{array}{l} lim_{r \to 0} sup_{x \in B_{r} (x_{0}) ∖ {x_{0}}} - Δ_{p} ψ (x) + {(- Δ)}_{p}^{s} ψ_{r} (x_{0}) \\ < c (x_{0}) {| u (x_{0}) |}^{p - 2} u (x_{0}) = c (x_{0}) {| ψ_{r} (x_{0}) |}^{p - 2} ψ_{r} (x_{0}) . \end{array}$

Due to $c \in C (\overline{Ω})$ , $ψ \in C^{2} (B_{r} (x_{0}))$ and the continuity of ${(- Δ)}_{p}^{s} ψ_{r}$ (Lemma 3.8, [22]), we can deduce that there exist $μ_{0} > 0$ and $0 < r_{1} < r$ such that $\begin{array}{l} - Δ_{p} ψ_{r} (x) + {(- Δ)}_{p}^{s} ψ_{r} (x) < c (x) {| ψ_{r} (x) |}^{p - 2} ψ_{r} (x) - μ_{0} for x \in B_{r_{1}} (x_{0}) ∖ {x_{0}} . \end{array}$ In view of Lemma 3.9 in [22], we have for any $ε > 0$ and $r_{2} > 0$ , there exist $0 < θ_{1} = θ_{1} (ε, r_{1})$ , $0 < r_{2} < \frac{r_{1}}{2}$ and $η \in C_{0}^{2} (B_{\frac{r_{2}}{2}} (x_{0}))$ satisfying $0 ⩽ η ⩽ 1$ with $η (x_{0}) = 1$ such that $\begin{array}{l} sup_{x \in B_{r_{2}} (x_{0})} | {(- Δ)}_{p}^{s} ψ_{r} (x) - {(- Δ)}_{p}^{s} (ψ_{r} + θ η) | < ε \end{array}$ for any $0 < θ < θ_{1}$ . Moreover, by the definition of p-Laplace operator, we can write $\begin{array}{l} {(- Δ)}_{p} (ψ_{r} + θ η) (x) - {(- Δ)}_{p} ψ_{r} (x) \\ = {| \nabla (ψ_{r} + θ η) |}^{p - 2} [Δ (ψ_{r} + θ η) + (p - 2) \nabla^{2} (ψ_{r} + θ η) \frac{\nabla (ψ_{r} + θ η)}{| \nabla (ψ_{r} + θ η) |} \cdot \frac{\nabla (ψ_{r} + θ η)}{| \nabla (ψ_{r} + θ η) |}] \\ - | \nabla ψ_{r} |^{p - 2} [Δ ψ_{r} + (p - 2) \nabla^{2} ψ_{r} \frac{\nabla ψ_{r}}{| \nabla ψ_{r} |} \cdot \frac{\nabla ψ_{r}}{| \nabla ψ_{r} |}] \\ = {| \nabla (ψ_{r} + θ η) |}^{p - 2} θ Δ η + [{| \nabla (ψ_{r} + θ η) |}^{p - 2} - | \nabla ψ_{r} |^{p - 2}] Δ ψ_{r} \\ + (p - 2) {| \nabla (ψ_{r} + θ η) |}^{p - 2} θ \nabla^{2} η \frac{\nabla (ψ_{r} + θ η)}{| \nabla (ψ_{r} + θ η) |} \cdot \frac{\nabla (ψ_{r} + θ η)}{| \nabla (ψ_{r} + θ η) |} \\ + (p - 2) {| \nabla (ψ_{r} + θ η) |}^{p - 2} \nabla^{2} ψ_{r} (\frac{\nabla (ψ_{r} + θ η)}{| \nabla (ψ_{r} + θ η) |} \cdot \frac{\nabla (ψ_{r} + θ η)}{| \nabla (ψ_{r} + θ η) |} - \frac{\nabla ψ_{r}}{| \nabla ψ_{r} |} \cdot \frac{\nabla ψ_{r}}{| \nabla ψ_{r} |}) \\ + (p - 2) [{| \nabla (ψ_{r} + θ η) |}^{p - 2} - | \nabla ψ_{r} |^{p - 2}] \nabla^{2} ψ_{r} \frac{\nabla ψ_{r}}{| \nabla ψ_{r} |} \cdot \frac{\nabla ψ_{r}}{| \nabla ψ_{r} |} . \end{array}$ Since $ψ_{r} \in C^{2} (B_{r} (x_{0}))$ , there exist η as we set above and $0 < θ_{2} = θ_{2} (ψ_{r}, η)$ such that $\begin{array}{l} sup_{x \in B_{r_{2}} (x_{0}) ∖ {x_{0}}} | {(- Δ)}_{p} ψ_{r} (x) - {(- Δ)}_{p} (ψ_{r} + θ η (x)) | < ε \end{array}$ for every $0 < θ < θ_{2}$ . Choosing $0 < θ < min {θ_{1}, θ_{2}}$ small enough, we arrive at $\begin{array}{l} sup_{x \in B_{r_{2}} (x_{0}) ∖ {x_{0}}} | {(- Δ)}_{p} ψ_{r} (x) - {(- Δ)}_{p} (ψ_{r} + θ η (x)) | \\ + sup_{x \in B_{r_{2}} (x_{0})} | {(- Δ)}_{p}^{s} ψ_{r} (x) - {(- Δ)}_{p}^{s} (ψ_{r} + θ η (x)) | < ε . \end{array}$ Let $ε = \frac{μ_{0}}{2}$ , we can obtain $\begin{array}{l} - Δ_{p} (ψ_{r} + θ η) + {(- Δ)}_{p}^{s} (ψ_{r} + θ η) & < - Δ_{p} ψ_{r} + {(- Δ)}_{p}^{s} ψ_{r} + \frac{μ_{0}}{2} \\ ⩽ c (x) {| ψ_{r} (x) |}^{p - 2} ψ_{r} (x) - \frac{μ_{0}}{2} \\ (2.2) & ⩽ c (x) | ψ_{r} + θ η |^{p - 2} (ψ_{r} + θ η) \end{array}$ for $x \in B_{\frac{r_{2}}{2}} (x_{0})$ , where in the last line we let θ small enough and used $ψ_{r} \in C^{2} (B_{r} (x_{0}))$ . By integration by parts, we can verify that (2.2) also holds in the weak sense. Thus $ψ_{r} + θ η$ is a weak subsolution of $\begin{array}{l} (2.3) & - Δ_{p} v + {(- Δ)}_{p}^{s} v = c (x) | v |^{p - 2} v, x \in B_{\frac{r_{2}}{2}} (x_{0}) . \end{array}$ Moreover, we know that u is a weak supersolution of (2.3) and $\begin{array}{l} (ψ_{r} + θ η) (x) = ψ_{r} (x) ⩽ u (x), x \in R^{N} ∖ B_{\frac{r_{2}}{2}} (x_{0}) . \end{array}$ By lemma 2.1, we obtain $u ⩾ ψ_{r} + θ η$ in $B_{\frac{r_{2}}{2}} (x_{0})$ , which implies that $\begin{array}{l} u (x_{0}) ⩾ ψ_{r} (x_{0}) + θ η (x_{0}) > ψ_{r} . \end{array}$ It contradicts the definition of $ψ_{r}$ . Thus we finish the proof. □

3. Strong maximum principle

In this section, we will establish several crucial lemmas and prove our main results. Using the method of proving Lemma 3.4 in [17], we can deduce the logarithmic lemma as follows.

Lemma 3.1.

Let $c \in L_{loc}^{1} (Ω)$ . Assume that $u \in X^{1, p} (Ω)$ is a weak supersolution of ( 1.1 ). If $u ⩾ 0$ a.e. in $B_{R} (x_{0}) \subset \subset Ω$ , then we have $\begin{array}{l} \int_{B_{r}} {| \nabla log (u + h) |}^{p} d x + \int_{B_{r}} \int_{B_{r}} \frac{1}{| x - y |^{N + s p}} {| log (\frac{u (x) + h}{u (y) + h}) |}^{p} d x d y \\ (3.1) & ⩽ C r^{N} (r^{- p} + r^{- s p} + h^{1 - p} r^{- p} \int_{B_{2 r}^{c}} \frac{u_{-} {(y)}^{p - 1}}{| y - x_{0} |^{N + s p}} d y) + ‖ c ‖_{L^{1} (B_{2 r})}, \end{array}$ where $B_{r} = B_{r} (x_{0}) \subset B_{\frac{R}{2}} (x_{0})$ and $h > 0$ .

Proof.

Let $ϕ \in C_{0}^{\infty} (B_{\frac{3 r}{2}} (x_{0}))$ satisfy $\begin{array}{l} 0 ⩽ ϕ ⩽ 1, ϕ \equiv 1 in B_{r} (x_{0}), | \nabla ϕ | ⩽ C r^{- 1} in B_{\frac{3 r}{2}} (x_{0}) . \end{array}$ Choosing $v = {(u + h)}^{1 - p} ϕ^{p} \in X_{0}^{1, p} (Ω)$ as the text function in (1.1), we obtain $\begin{array}{l} \int_{B_{\frac{3 r}{2}}} c (x) \frac{u^{p - 1} (x) ϕ^{p} (x)}{{(u (x) + h)}^{p - 1}} d x \\ ⩽ \int_{B_{\frac{3 r}{2}}} | \nabla u |^{p - 2} \nabla u \cdot \nabla (\frac{ϕ^{p} (x)}{{(u (x) + h)}^{p - 1}}) d x \\ + \int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y))}{| x - y |^{N + s p}} (\frac{ϕ^{p} (x)}{{(u (x) + h)}^{p - 1}} - \frac{ϕ^{p} (y)}{{(u (y) + h)}^{p - 1}}) d x d y \\ ⩽ C r^{N} (r^{- p} + r^{- s p} + h^{1 - p} r^{- p} \int_{B_{2 r}^{c}} \frac{u_{-} {(y)}^{p - 1}}{| y - x_{0} |^{N + s p}} d y) \\ (3.2) & - \int_{B_{r}} {| \nabla log (u + h) |}^{p} d x - \int_{B_{r}} \int_{B_{r}} \frac{1}{| x - y |^{N + s p}} {| log (\frac{u (x) + h}{u (y) + h}) |}^{p} d x d y, \end{array}$ where the last estimate follows the proof of Lemma 3.4 in [17] and C depends on N, p, s. Considering (3.2) and the fact that $0 ⩽ u^{p - 1} {(u + h)}^{1 - p} ϕ^{p} ⩽ 1$ in $B_{\frac{3 r}{2}} (x_{0})$ , we can get the conclusion. □

Lemma 3.2.

Assume that $c \in L_{loc}^{1} (Ω)$ is a non-positive function and u is a weak supersolution of ( 1.1 ). Then,

if $u ⩾ 0$ a.e. in $Ω^{c}$ and $u = 0$ a.e. in Ω then $u = 0$ a.e. in $R^{N}$ ;

if Ω is bounded, and $u ⩾ 0$ a.e. in $Ω^{c}$ then $u ⩾ 0$ a.e. in Ω.

Proof.

We prove (i) first. Let $φ \in C_{0}^{\infty} (Ω)$ be a non-negative function. Since $u = 0$ a.e. in Ω, then $\begin{array}{l} 0 & ⩽ \int_{Ω} | \nabla u |^{p - 2} \nabla u \cdot \nabla φ d x + \int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y)) |^{p - 2} (u (x) - u (y)) (φ (x) - φ (y))}{| x - y |^{N + s p}} d x d y \\ = - 2 \int_{Ω^{c}} \int_{Ω} \frac{u {(y)}^{p - 1} φ (x)}{| x - y |^{N + s p}} d x d y . \end{array}$ By assumption, $u ⩾ 0$ a.e. in $Ω^{c}$ then $u = 0$ a.e. in $Ω^{c}$ . Thus, $u = 0$ a.e. in $R^{N}$ .

Next, we prove (ii). By $u \in X^{1, p} (Ω)$ and $u ⩾ 0$ in $Ω^{c}$ , we can check that $u_{-} \in X_{0}^{1, p} (Ω)$ . Since u is a weak supersolution of (1.1) and $c (x) ⩽ 0$ , we have $\begin{array}{l} 0 & ⩽ - \int_{Ω} c (x) {| u_{-} (x) |}^{p} d x \\ = \int_{Ω} c (x) {| u (x) |}^{p - 2} u (x) u_{-} (x) d x \\ ⩽ \int_{Ω} | \nabla u |^{p - 2} \nabla u \cdot \nabla (u_{-} (x)) d x \\ + \int_{R^{N}} \int_{R^{N}} \frac{| u (x) - u (y) |^{p - 2} (u (x) - u (y)) (u_{-} (x) - u_{-} (y))}{| x - y |^{N + s p}} d x d y . \end{array}$ Notice that $\begin{array}{l} {| u (x) - u (y) |}^{p - 2} (u (x) - u (y)) (u_{-} (x) - u_{-} (y)) \\ ⩽ \{\begin{matrix} - | u_{-} (x) - u_{-} (y) |^{p} & if x, y \in Ω_{-}, \\ - | u_{-} (x) + u_{+} (y) |^{p - 2} (u_{-} (x) + u_{+} (y)) u_{-} (x) & if x \in Ω_{-}, y \in Ω_{-}^{c} . \end{matrix} \end{array}$ Thus, we obtain $\begin{array}{l} 0 & ⩽ \int_{Ω_{-}} | \nabla u |^{p - 2} \nabla u \cdot \nabla u_{-} d x - \int_{Ω_{-}} \int_{Ω_{-}} \frac{| u_{-} (x) - u_{-} (y) |^{p}}{| x - y |^{N + s p}} d x d y \\ - 2 \int_{Ω_{-}^{c}} \int_{Ω_{-}} \frac{| u_{-} (x) + u_{+} (y) |^{p - 2} (u_{-} (x) + u_{+} (y)) u_{-} (x)}{| x - y |^{N + s p}} d x d y \\ ⩽ 0 . \end{array}$ Hence $u_{-} \equiv 0$ a.e. in $R^{N}$ , which implies that $u ⩾ 0$ a.e. in $R^{N}$ . □

In what follows, we prove a strong maximum principle with Ω being connected.

Lemma 3.3.

Assume that $c (x) ⩽ 0$ belongs to $L_{loc}^{1} (Ω)$ and $u \in X^{1, p} (Ω)$ is a weak supersolution of ( 1.1 ).

If Ω is bounded and connected, and $u ⩾ 0$ a.e. in $Ω^{c}$ , then either $u > 0$ a.e. in Ω or $u = 0$ a.e. in Ω;

If Ω is connected, and $u ⩾ 0$ a.e. in $R^{N}$ , then either $u > 0$ a.e. in Ω or $u = 0$ a.e. in Ω.

Proof.

The proof of (i) follows the arguments used in [8, Theorem A.1]. For completeness of the article, we give the details here. In view of Lemma 3.2, $u ⩾ 0$ a.e. in $R^{N}$ . Thus we assume $u \neq 0$ a.e. in Ω and prove $u > 0$ a.e. in Ω. We aim at proving that for any connected compact set $K \subset \subset Ω$ , if $u ≢ 0$ in K then $u > 0$ a.e. in K. Observe that $K \subset {x \in Ω : dist (x, \partial Ω) > 2 r}$ for some $r > 0$ and it can be covered by a finite number of balls $B_{r / 2} (x_{1}), \dots, B_{r / 2} (x_{k})$ such that $x_{i} \in K$ and $\begin{array}{l} (3.3) & | B_{\frac{r}{2}} (x_{i}) \cap B_{\frac{r}{2}} (x_{i + 1}) | > 0, i = 1, \dots, k - 1 . \end{array}$ Since $u ≢ 0$ in K, we can suppose the set $Z : = {x \in B_{\frac{r}{2}} (x_{i}) : u (x) = 0} \subset K$ for some $i \in {1, \dots, k - 1}$ has positive measure. Moreover, by Poincare equality ([8], page 795), it follows that $\begin{array}{l} \int_{B_{\frac{r}{2} (x_{i})}} {| log (1 + \frac{u (x)}{δ}) |}^{p} d x ⩽ \frac{r^{N + s p}}{| Z |} \int_{B_{\frac{r}{2} (x_{i})}} \int_{B_{\frac{r}{2} (x_{i})}} \frac{1}{| x - y |^{N + s p}} {| log (\frac{u (x) + δ}{u (y) + δ}) |}^{p} d x d y \end{array}$ with $δ > 0$ . Recalling Lemma 3.1 and $u_{-} = 0$ , we obtain $\begin{array}{l} \int_{B_{\frac{r}{2} (x_{i})}} {| log (1 + \frac{u (x)}{δ}) |}^{p} d x ⩽ \frac{C r^{N + s p}}{| Z |} (C r^{N - s p} + C r^{N - p} + ‖ c ‖_{L^{1} (B_{r} (x_{i}))}), \end{array}$ where $C > 0$ is independent of δ. Then we let $δ \to 0$ , which leads to $u = 0$ a.e. in $B_{\frac{r}{2}} (x_{i})$ . In view of (3.3), we can follow the same argument to get $u = 0$ a.e. in K which contradicts our assumption. Thus we finish the proof of (i).

In the case of Ω being unbounded and connected, we can find a sequence of bounded connected open sets ${Ω_{n}}_{n \in N}$ that satisfy $Ω_{n} \subset Ω_{n + 1} \subset Ω$ for any $n \in N$ and $Ω = ⋃_{n \in N} Ω_{n}$ . In fact, if $u \neq 0$ a.e. in Ω, there exists $n_{0} \in N$ such that $u > 0$ a.e. in $Ω_{n}$ for all $n ⩾ n_{0}$ . Since $u \neq 0$ a.e. in $Ω_{n}$ and u is a non-negative weak supersolution of ${(- Δ)}_{p} u + {(- Δ)}_{p}^{s} u = c (x) | u |^{p - 2} u$ in $Ω_{n}$ , we can conclude that $u > 0$ a.e. in Ω. □

In addition, since the mixed local and nonlocal operators are considered, the condition that Ω is connected in the above lemma can be removed by using the method of proving Lemma 3.4 in [14].

Lemma 3.4.

Suppose that the non-positive function $c (x) \in L_{loc}^{1} (Ω)$ . Let $u \in X^{1, p} (Ω)$ be a weak supersolution of ( 1.1 ).

If Ω is bounded and $u ⩾ 0$ a.e. in $Ω^{c}$ , then either $u > 0$ a.e. in Ω or $u = 0$ a.e. in Ω;

If $u ⩾ 0$ a.e. in $R^{N}$ , then either $u > 0$ a.e. in Ω or $u = 0$ a.e. in Ω.

Combining Lemma 3.2 and Lemma 3.4 yields Theorem 1.2.

Finally, we discuss the following lemma which plays a crucial role in proving Theorem 1.4.

Lemma 3.5.

Assume that $c \in C (\overline{B_{R} (x_{0})})$ and $u \in X^{1, p} (B_{R} (x_{0})) \cap C (\overline{B_{R} (x_{0})})$ is a weak supersolution of $\begin{array}{l} (3.4) & - Δ_{p} u + {(- Δ)}_{p}^{s} u = c (x) | u |^{p - 2} u in B_{R} (x_{0}) . \end{array}$ If $u ⩾ 0$ in $B_{R} {(x_{0})}^{c}$ then either $u > 0$ in $B_{R} (x_{0})$ or $u = 0$ a.e. in $R^{N}$ .

Proof.

We will deduce that if there is $x_{⋆} \in B_{R} = B_{R} (x_{0})$ such that $u (x_{⋆}) = 0$ and then $u = 0$ a.e. in $R^{N}$ . Using Lemma 3.2, $u ⩾ 0$ a.e. in $B_{R}$ and recalling Theorem 1.2, we get either $u > 0$ a.e. in $B_{R}$ or $u = 0$ a.e. in $R^{N}$ . Moreover, by Lemma 2.4, u is a viscosity supersolution of (3.4). Since $u ⩾ 0$ in $B_{R}$ , we can choose $\begin{array}{l} ψ^{ε} = - ε | x - x_{⋆} |^{β} \end{array}$ as the text function for all $ε > 0$ and $β > max {2, \frac{2}{2 - s}}$ . Thus it follows that $\begin{array}{l} lim_{r \to 0} sup_{x \in B_{r} (x_{⋆}) ∖ {x_{⋆}}} - Δ_{p} ψ_{r}^{ε} (x) + {(- Δ)}_{p}^{s} ψ_{r}^{ε} (x_{⋆}) ⩾ c (x_{⋆}) {| ψ_{r}^{ε} (x_{⋆}) |}^{p - 2} ψ_{r}^{ε} (x_{⋆}) = 0 \end{array}$ for some $r \in (0, R - | x_{⋆} - x_{0} |)$ , where $\begin{array}{l} ψ_{r}^{ε} (x) = \{\begin{matrix} ψ^{ε} & if x \in B_{r} (x_{⋆}), \\ u (x) & otherwise . \end{matrix} \end{array}$ Thus for any $x \in B_{r} (x_{⋆}) ∖ {x_{⋆}}$ , we have $\begin{array}{l} 0 & ⩽ - div ({| \nabla (- ε | x - x_{⋆} |^{β}) |}^{p - 2} \nabla (- ε | x - x_{⋆} |^{β})) \\ + ε^{p - 1} \int_{B_{r} (x_{⋆})} | y - x_{⋆} |^{β (p - 1) - N - s p} d y - \int_{B_{r} {(x_{⋆})}^{c}} \frac{u {(y)}^{p - 1}}{| y - x_{⋆} |^{N + s p}} d y \\ = - ε^{p - 2} ε {| \nabla {(x - x_{⋆})}^{β} |}^{p - 2} \\ \times [Δ | x - x_{⋆} |^{β} + (p - 2) \nabla^{2} (| x - x_{⋆} |^{β}) \frac{\nabla (| x - x_{⋆} |^{β})}{| \nabla (| x - x_{⋆} |^{β}) |} \cdot \frac{\nabla (| x - x_{⋆} |^{β})}{| \nabla (| x - x_{⋆} |^{β}) |}] \\ + ε^{p - 1} \int_{B_{r} (x_{⋆})} | y - x_{⋆} |^{β (p - 1) - N - s p} d y - \int_{B_{r} {(x_{⋆})}^{c}} \frac{u {(y)}^{p - 1}}{| y - x_{⋆} |^{N + s p}} d y \\ = : I_{1} + I_{2} + I_{3} + I_{4} . \end{array}$ Employing $ψ^{ε} \in C_{β}^{2} (B_{r} (x_{⋆}))$ and $β > max {2, \frac{2}{2 - s}}$ , we can deduce that $I_{1}, I_{2}, I_{3} \to 0$ as $ε \to 0$ . Thus we derive $u = 0$ a.e. in $B_{r} {(x_{⋆})}^{c}$ , which implies that $u = 0$ a.e. $R^{N}$ . □

Proof of Theorem 1.4.

Utilizing Theorem 1.2, we know $u > 0$ a.e. in Ω or $u = 0$ a.e. in $R^{N}$ . Let $x_{0} \in Ω$ such that $u (x_{0}) = 0$ . By the assumptions that $c, u \in C (\overline{Ω})$ and Ω is an open set, there exists $R > 0$ such that $B_{R} (x_{0}) \subset Ω$ and $c, u \in C (\overline{B_{R} (x_{0})})$ . In addition, we have u is a weak supersolution of $\begin{array}{l} - Δ_{p} u + {(- Δ)}_{p}^{s} u = c (x) | u |^{p - 2} u in B_{R} (x_{0}) . \end{array}$ Thus, we get $u = 0$ a.e. in $R^{N}$ by Lemma 3.5. We conclude that $u > 0$ in Ω or $u = 0$ in Ω. □

Footnotes

Acknowledgement

This work was supported by the National Natural Science Foundation of China (No. 12071098).

References

Ambrosio, A strong maximum principle for the fractional

(p, q)

-Laplacian operator, Appl. Math. Lett. 126 (2022), Paper No. 107813, 10 pp. doi:10.1016/j.aml.2021.107813.

Barrios and

Medina, Strong maximum principles for fractional elliptic and parabolic problems with mixed boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 150(1) (2020), 475–495. doi:10.1017/prm.2018.77.

Barrios and

Medina, Equivalence of weak and viscosity solutions in fractional non-homogeneous problems, Math. Ann. 381 (2021), 1979–2012. doi:10.1007/s00208-020-02119-w.

Barrios,

Montoro and

Sciunzi, On the moving plane method for nonlocal problems in bounded domains, J. Anal. Math. 135(1) (2018), 37–57. doi:10.1007/s11854-018-0031-1.

Biagi,

Dipierro,

Valdinoci and

Vecchi, Mixed local and nonlocal elliptic operators: Regularity and maximum principles, Comm. Partial Differential Equations 47(3) (2022), 585–629. doi:10.1080/03605302.2021.1998908.

Biagi,

Mugnai and

Vecchi, A Brezis–Oswald approach for mixed local and nonlocal operators, arXiv:2103.11382v3.

Blazevski and

del-Castillo-Negrete, Local and nonlocal anisotropic transport in reversed shear magnetic fields: Shearless Cantori and nondiffusive transport, Phys. Rev. E 87(6) (2013), 1–15.

Brasco and

Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J. 37(3) (2014), 769–799. doi:10.2996/kmj/1414674621.

Cassani and

Tarsia, Maximum principle for higher order operators in general domains, Adv. Nonlinear Anal. 11(1) (2022), 655–671. doi:10.1515/anona-2021-0210.

10.

Chen and

Li, Maximum principles for the fractional-Laplacian and symmetry of solutions, Adv. Math. 335 (2018), 735–758. doi:10.1016/j.aim.2018.07.016.

11.

Cheng,

Huang and

Li, The maximum principles for fractional Laplacian equations and their applications, Commun. Contemp. Math. 19(6) (2017), 1750018, 12 pp. doi:10.1142/S0219199717500183.

12.

De Filippis and

Mingione, Gradient regularity in mixed local and nonlocal problems, arXiv:2204.06590.

13.

Del Pezzo,

Ferreira and

Rossi, Eigenvalues for a combination between local and nonlocal p-Laplacians, Fract. Calc. Appl. Anal. 22(5) (2019), 1414–1436. doi:10.1515/fca-2019-0074.

14.

Del Pezzo and

Quaas, A Hopf’s lemma and a strong minimum principle for the fractional p-Laplacian, J. Differential Equations 263(1) (2017), 765–778. doi:10.1016/j.jde.2017.02.051.

15.

Dipierro and

Valdinoci, Description of an ecological niche for a mixed local/nonlocal dispersal: An evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes, Phys. A 575 (2021), 20 pp. doi:10.1016/j.physa.2021.126052.

16.

Fang,

Shang and

Zhang, Regularity theory for mixed local and nonlocal parabolic p-Laplace equations, J. Geom. Anal. 32(1) (2022), Paper No. 22, 33 pp. doi:10.1007/s12220-021-00768-0.

17.

Garain and

Kinnunen, On the regularity theory for mixed local and nonlocal quasilinear elliptic equations, Trans. Amer. Math. Soc. (2021). doi:10.1090/tran/8621.

18.

Garain and

Lindgren, Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations, arXiv:2204.13196.

19.

Garain and

Ukhlov, Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems, Nonlinear Anal. 223 (2022), Paper No. 113022, 35 pp. doi:10.1016/j.na.2022.113022.

20.

Giacomoni,

Kumar and

Sreenadh, Global regularity results for non-homogeneous growth fractional problems, J. Geom. Anal. 32(1) (2022), Paper No. 36, 41 pp. doi:10.1007/s12220-021-00837-4.

21.

Greco and

Servadei, Hopf’s lemma and constrained radial symmetry for the fractional Laplacian, Math. Res. Lett. 23(3) (2016), 863–885. doi:10.4310/MRL.2016.v23.n3.a14.

22.

Korvenpää,

Kuusi and

Lindgren, Equivalence of solutions to fractional p-Laplace type equations, J. Math. Pures Appl. 132 (2019), 1–26. doi:10.1016/j.matpur.2017.10.004.

23.

Liu, Maximum principles and monotonicity of solutions for fractional p-equations in unbounded domains, J. Differential Equations 270 (2021), 1043–1078. doi:10.1016/j.jde.2020.09.001.

24.

López-Soriano and

Ortega, A strong maximum principle for the fractional Laplace equation with mixed boundary condition, Fract. Calc. Appl. Anal. 24(6) (2021), 1699–1715. doi:10.1515/fca-2021-0073.

25.

Medina and

Ochoa, On viscosity and weak solutions for non-homogeneous p-Laplace equations, Adv. Nonlinear Anal. 8(1) (2019), 468–481. doi:10.1515/anona-2017-0005.

26.

Musina and

Nazarov, Strong maximum principles for fractional Laplacians, Proc. Roy. Soc. Edinburgh Sect. A 149(5) (2019), 1223–1240. doi:10.1017/prm.2018.81.

27.

N.S.

Papageorgiou,

V.D.

Rădulescu and

Repovš, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys. 69(4) (2018), Paper No. 108, 21 pp. doi:10.1007/s00033-018-1001-2.

28.

Pucci and

Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and Their Applications, Vol. 73, Birkhäuser Verlag, Basel, 2007.