Fractional Dirichlet problems with singular and non-locally convective reaction

Laura Gambera; Salvatore A. Marano

doi:10.1515/anona-2025-0082

Article Open Access

Fractional Dirichlet problems with singular and non-locally convective reaction

Laura Gambera and Salvatore A. Marano

Published/Copyright: November 11, 2025

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From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

In this article, the existence of positive weak solutions to a Dirichlet problem driven by the fractional ( p , q ) -Laplacian and with reaction both weakly singular and non-locally convective (i.e., depending on the distributional Riesz gradient of solutions) is established. Due to the nature of the right-hand side, we address the problem via sub-super solution methods, combined with variational techniques, truncation arguments, as well as fixed point results.

Keywords: fractional Dirichlet problem; weakly singular and non-locally convective reaction; fractional (p, q)-Laplacian; distributional Riesz fractional gradient; positive weak solution

MSC 2020: 35J60; 35J75; 35D30

1 Introduction

Let Ω ⊆ R N , N ≥ 2 , be a bounded domain with a C 2 boundary ∂ Ω , let 0 < s 2 ≤ s ≤ s 1 ≤ 1 , and let 2 < q < p < N s 1 with s 1 p > 1 . Consider the problem

(P) ( − Δ ) p s 1 u + ( − Δ ) q s 2 u = f ( x , u ) + g ( x , D s u ) in Ω , u > 0 in Ω , u = 0 in R N \ Ω ,

where, given r > 1 and 0 < t < 1 , ( − Δ ) r t denotes the (negative) fractional r -Laplacian, formally defined by

( − Δ ) r t u ( x ) ≔ 2 lim ε → 0 + ∫ R N \ B ε ( x ) ∣ u ( x ) − u ( y ) ∣ r − 2 ( u ( x ) − u ( y ) ) ∣ x − y ∣ N + t r d y , x ∈ R N .

The symbol D s u indicates the distributional Riesz fractional gradient of u according to [26,27]. If u is sufficiently smooth and appropriately decays at infinity, then

D s u ( x ) ≔ c N , s lim ε → 0 + ∫ R N \ B ε ( x ) u ( x ) − u ( y ) ∣ x − y ∣ N + s x − y ∣ x − y ∣ d y , x ∈ R N ,

with c N , s < 0 ; cf. [27, p. 289]. Moreover, f : Ω × R + → R 0 + and g : Ω × R N → R 0 + stand for Carathéodory’s functions such that

liminf t → 0 + f ( x , t ) ≕ L > 0 uniformly in x ∈ Ω , f ( x , t ) ≤ c 1 t − γ + c 2 t r ∀ ( x , t ) ∈ Ω × R + , ( H f1 )

t ↦ f ( ⋅ , t ) t q − 1 is strictly decreasing on R + , ( H f2 )

g ( x , ξ ) = c 3 ( 1 + ∣ ξ ∣ ζ ) , ( x , ξ ) ∈ Ω × R N , ( H g )

for suitable γ ∈ ( 0 , 1 ) , c i > 0 , i = 1 , 2, 3, r , ζ ∈ ( 1 , p − 1 ) . A simple example of f fulfilling ( H f 1 )–( H f 2 ) is f ( x , t ) ≔ t − γ + t r , ( x , t ) ∈ Ω × R + , where r < q − 1 . Since s 2 ≤ s 1 , we are naturally led to solve problem (P) in the fractional Sobolev space W 0 s 1 , p ( Ω ) . Precisely,

Definition 1.1

A function u ∈ W 0 s 1 , p ( Ω ) is called a weak solution of (P) when u > 0 a.e. in Ω and

∫ R 2 N ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + s 1 p d x d y + ∫ R 2 N ∣ u ( x ) − u ( y ) ∣ q − 2 ( u ( x ) − u ( y ) ) ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + s 2 q d x d y = ∫ Ω f ( ⋅ , u ) φ d x + ∫ Ω g ( ⋅ , D s u ) φ d x ∀ φ ∈ W 0 s 1 , p ( Ω ) .

Let us next point out some hopefully newsworthy aspects, namely,

the driving differential operator is neither local nor homogeneous and no parameters appear on the right-hand side,
f ( x , ⋅ ) can be singular at zero, which means lim t → 0 + f ( x , t ) = + ∞ , and
the reaction is also non-locally convective, because g depends on the distributional fractional gradient of solutions.

In recent years, the study of non-local differential equations has seen significant growth, mainly due to their many applications in real-world problems, such as game theory, finance, image processing, and materials science. A wealth of existence and uniqueness or multiplicity theorems are already available; e.g., [2,5,7,11,13,20,22]. Several regularity results have also been published; let us mention [18,19] for the fractional p -Laplacian and [15] as regards the fractional ( p , q ) -Laplacian. Finally, survey [10] provides an exhaustive account on the corresponding functional framework, i.e., fractional Sobolev spaces.

Dirichlet problems driven by non-local operators and with singular reactions were well investigated in [6,15]. The work [6] treats existence and uniqueness of solutions to the equation

( − Δ ) p s u = a ( x ) u σ in Ω ,

where σ > 0 and a : Ω → R + fulfills appropriate conditions, while [15] studies the more general situation

( − Δ ) p s 1 u + ( − Δ ) q s 2 u = a δ ( x ) u σ in Ω ,

being a δ ∈ L loc ∞ ( Ω ) a function that behaves like d ( x ) − δ , with d ( x ) ≔ dist ( x , ∂ Ω ) and δ ∈ [ 0 , s 1 p ) . It should be noted that, contrary to [6,15], here the reaction term f ( ⋅ , u ) is not necessarily a perturbation of u − σ .

Finally, distributional fractional gradients were first introduced by Horváth [17], but gained traction and saw wider application especially after the two seminal papers of Shieh and Spector [26,27]. Among recent contributions on this subject, we mention [1,5,8,9,25], as well as references therein. The main reasons that contributed to spreading the use of Riesz gradients probably are: (1) Non-local versions of many classical results on Sobolev spaces can be obtained through them. (2) D s u formally tends to ∇ u as s → 1 − . (3) Significant geometrical and physical properties (invariance under translations or rotations, homogeneity of order s , etc.) remain true in this new context; cf. [28].

Although weakly singular (namely, γ < 1 ) problems are normally investigated via variational methods, here the presence of D s u inside the reaction prevents this approach. That is why the existence of at least one positive solution to (P) is established by means of sub-super solution arguments, variational and truncation techniques, besides Schauder’s fixed point theorem.

The article is organized as follows. Preliminary facts are collected in Section 2. Section 3 shows that a suitable auxiliary problem, obtained by freezing the convection term, admits a unique solution, while (P) is solved in Section 4.

2 Preliminaries

Let X be a real Banach space with topological dual X * and duality brackets ⟨ ⋅ , ⋅ ⟩ . A function A : X → X * is called:

monotone when ⟨ A ( x ) − A ( z ) , x − z ⟩ ≥ 0 for all x , z ∈ X .
of type ( S ) + provided
x n ⇀ x in X , limsup n → + ∞ ⟨ A ( x n ) , x n − x ⟩ ≤ 0 ⇒ x n → x in X .

The next elementary result will ensure that condition ( S ) + holds true for the fractional ( p , q ) -Laplacian.

Proposition 2.1

Let A : X → X * be of type ( S ) + and let B : X → X * be monotone. Then, A + B satisfies condition ( S ) + .

Proof

Suppose x n ⇀ x in X and

(2.1) limsup n → + ∞ ⟨ A ( x n ) + B ( x n ) , x n − x ⟩ ≤ 0 .

The monotonicity of B entails

⟨ A ( x n ) , x n − x ⟩ = ⟨ A ( x n ) + B ( x n ) , x n − x ⟩ − ⟨ B ( x n ) , x n − x ⟩ = ⟨ A ( x n ) + B ( x n ) , x n − x ⟩ − ⟨ B ( x n ) − B ( x ) , x n − x ⟩ − ⟨ B ( x ) , x n − x ⟩ ≤ ⟨ A ( x n ) + B ( x n ) , x n − x ⟩ − ⟨ B ( x ) , x n − x ⟩ ∀ n ∈ N .

Using (2.1) we thus obtain

limsup n → + ∞ ⟨ A ( x n ) , x n − x ⟩ ≤ 0 ,

whence x n → x because A is of type ( S ) + .□

Finally, if X and Y are two topological spaces, then X ↪ Y means that X continuously embeds in Y .

Hereafter, Ω is a bounded domain of the real Euclidean N -space ( R N , ∣ ⋅ ∣ ) , N ≥ 2 , with a C 2 -boundary ∂ Ω , ∣ E ∣ indicates the N -dimensional Lebesgue measure of E ⊆ R N ,

t ± ≔ max { ± t , 0 } , t ∈ R ,

while C , C 1 , etc. are positive constants, which may change value from line to line, whose dependencies will be specified when necessary. Denote by d : Ω ¯ → R 0 + the distance function of Ω , i.e.,

d ( x ) ≔ dist ( x , ∂ Ω ) ∀ x ∈ Ω ¯ .

It enjoys a useful summability property (see [16, Proposition 2.1]), namely

Proposition 2.2

If 0 < σ < 1 < q < 1 σ , then d − σ ∈ L q ( Ω ) .

Let X ( Ω ) be a real-valued function space on Ω and let u , v ∈ X ( Ω ) . We simply write u ≤ v when u ( x ) ≤ v ( x ) a.e. in Ω . Analogously for u < v , etc. To shorten notation, define

Ω ( u ≤ v ) ≔ { x ∈ Ω : u ( x ) ≤ v ( x ) } , X ( Ω ) + ≔ { w ∈ X ( Ω ) : w > 0 } .

Henceforth, p ′ indicates the conjugate exponent of p ≥ 1 , the Sobolev space W 0 1 , p ( Ω ) is equipped with Poincaré’s norm

‖ u ‖ 1 , p ≔ ‖ ∣ ∇ u ∣ ‖ p , u ∈ W 0 1 , p ( Ω ) ,

where, as usual,

‖ v ‖ q ≔ ∫ Ω ∣ v ( x ) ∣ q d x 1 ∕ q if 1 ≤ q < + ∞ , ess sup x ∈ Ω ∣ v ( x ) ∣ when q = + ∞ ,

and, given any u ∈ W 0 1 , p ( Ω ) , we set u ≔ 0 a.e. in R N \ Ω ; cf. [10, Section 5]. Moreover, W − 1 , p ′ ( Ω ) ≔ ( W 0 1 , p ( Ω ) ) * while p * is the Sobolev critical exponent for the embedding W 0 1 , p ( Ω ) ↪ L q ( Ω ) . It is known that p * = N p N − p once p < N .

Fix s ∈ ( 0 , 1 ) . The Gagliardo semi-norm of a measurable function u : R N → R is

[ u ] s , p ≔ ∫ R N × R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + p s d x d y 1 ∕ p .

W s , p ( R N ) denotes the fractional Sobolev space

W s , p ( R N ) ≔ { u ∈ L p ( R N ) : [ u ] s , p < + ∞ }

endowed with the norm

‖ u ‖ W s , p ( R N ) ≔ ( ‖ u ‖ L p ( R N ) p + [ u ] s , p p ) 1 ∕ p .

On the space

W 0 s , p ( Ω ) ≔ { u ∈ W s , p ( R N ) : u = 0 a.e. in R N \ Ω } ,

we will consider the equivalent norm

‖ u ‖ s , p ≔ [ u ] s , p , u ∈ W 0 s , p ( Ω ) .

As before, W − s , p ′ ( Ω ) ≔ ( W 0 s , p ( Ω ) ) * and p s * indicates the fractional Sobolev critical exponent, i.e., p s * = N p N − s p when s p < N , p s * = + ∞ otherwise. Thanks to Propositions 2.1 and 2.2, Theorem 6.7, and Corollary 7.2 of [10], one has

Proposition 2.3

If 1 ≤ p < + ∞ , then

0 < s ′ ≤ s ″ ≤ 1 ⇒ W 0 s ″ , p ( Ω ) ↪ W 0 s ′ , p ( Ω ) .
W 0 s , p ( Ω ) ↪ L q ( Ω ) for all q ∈ [ 1 , p s * ] .
The embedding in (b) is compact once q < p s * < + ∞ .

However, contrary to the non-fractional case,

1 ≤ q < p ≤ + ∞ ⇏ W 0 s , p ( Ω ) ⊆ W 0 s , q ( Ω ) ;

cf. [23]. Define, for every u , v ∈ W 0 s , p ( Ω ) ,

⟨ ( − Δ ) p s u , v ⟩ ≔ ∫ R N × R N ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ( v ( x ) − v ( y ) ) ∣ x − y ∣ N + s p d x d y .

The operator ( − Δ ) p s is called (negative) s -fractional p -Laplacian. It possesses the following properties.

( − Δ ) p s : W 0 s , p ( Ω ) → W − s , p ′ ( Ω ) is monotone, continuous, and of type ( S ) + ; vide [13, Lemma 2.1].
( − Δ ) p s maps bounded sets into bounded sets. In fact,
‖ ( − Δ ) p s u ‖ W − s , p ′ ( Ω ) ≤ ‖ u ‖ s , p p − 1 ∀ u ∈ W 0 s , p ( Ω ) .

To deal with distributional fractional gradients, we first introduce the Bessel potential spaces L α , p ( R N ) , where α > 0 . Set, for every x ∈ R N ,

g α ( x ) ≔ 1 ( 4 π ) α ∕ 2 Γ ( α ∕ 2 ) ∫ 0 + ∞ e − π ∣ x ∣ 2 δ e − δ 4 π δ α − N 2 d δ δ .

On account of [24, Section 7.1], one can assert that

g α ∈ L 1 ( R N ) and ‖ g α ‖ L 1 ( R N ) = 1 .
g α enjoys the semigroup property, i.e., g α ∗ g β = g α + β for any α , β > 0 .

Now, put

L α , p ( R N ) ≔ { u : u = g α ∗ u ˜ for some u ˜ ∈ L p ( R N ) }

as well as

‖ u ‖ L α , p ( R N ) = ‖ u ˜ ‖ L p ( R N ) whenever u = g α ∗ u ˜ .

Using (1) and (2) easily yields

0 < α < β ⇒ L β , p ( R N ) ⊆ L α , p ( R N ) ⊆ L p ( R N ) .

Moreover (refer [26, Theorem 2.2]),

Theorem 2.4

If 1 < p < + ∞ and 0 < ε < α , then

L α + ε , p ( R N ) ↪ W α , p ( R N ) ↪ L α − ε , p ( R N ) .

Finally, define

L 0 s , p ( Ω ) ≔ { u ∈ L s , p ( R N ) : u = 0 in R N \ Ω } .

Thanks to Theorem 2.4, we clearly have

(2.2) L 0 s + ε , p ( Ω ) ↪ W 0 s , p ( Ω ) ↪ L 0 s − ε , p ( Ω ) ∀ ε ∈ ( 0 , s ) .

The next basic notion is taken from [26]. For 0 < α < N , let

γ ( N , α ) ≔ Γ ( ( N − α ) ∕ 2 ) π N ∕ 2 2 α Γ ( α ∕ 2 ) , I α ( x ) ≔ γ ( N , α ) ∣ x ∣ N − α , x ∈ R N \ { 0 } .

If u ∈ L p ( R N ) and I 1 − s ∗ u makes sense, then the vector

D s u ≔ ∂ ∂ x 1 ( I 1 − s ∗ u ) , … , ∂ ∂ x N ( I 1 − s ∗ u ) ,

where partial derivatives are understood in a distributional sense, called distributional Riesz s -fractional gradient of u . Theorem 1.2 in [26] ensures that

D s u = I 1 − s ∗ D u ∀ u ∈ C c ∞ ( R N ) .

Further, D s u looks like the natural extension of ∇ u to the fractional framework; cf., e.g., [14] for details. According to [26, Definition 1.5], X s , p ( R N ) denotes the completion of C c ∞ ( R N ) with respect to the norm

‖ u ‖ X s , p ( R N ) ≔ ( ‖ u ‖ L p ( R N ) p + ‖ D s u ‖ L p ( R N ) p ) 1 ∕ p .

Since, by [26, Theorem 1.7], X s , p ( R N ) = L s , p ( R N ) , we can deduce many facts about X s , p ( R N ) from the existing literature on L s , p ( R N ) . In particular, if

X 0 s , p ( Ω ) ≔ { u ∈ X s , p ( R N ) : u = 0 in R N \ Ω } ,

then X 0 s , p ( Ω ) = L 0 s , p ( Ω ) .

3 Freezing the convection term

To address the two troubles (singularity and convection) separately, here we will study an auxiliary equation patterned after that of (P), but with D s u replaced by D s v for fixed v ∈ W 0 s 1 , p ( Ω ) .

Lemma 3.1

Under hypothesis ( H f 1 ), the problem

( − Δ ) p s 1 u + ( − Δ ) q s 2 u = f ( x , u ) i n Ω , u = 0 i n R N \ Ω , P f

possesses a positive sub-solution u ̲ ∈ W 0 s 1 , p ( Ω ) ∩ C 0 , τ ( Ω ¯ ) , where τ ∈ ( 0 , s 1 ) .

Proof

Thanks to ( H f 1 ), for every ε ∈ ( 0 , L ) , there exists δ ∈ ( 0 , 1 ) such that

f ( x , t ) > ε ∀ ( x , t ) ∈ Ω × ( 0 , δ ) .

Let σ > 0 . Theorem 3.15 of [15] provides a positive solution u σ ∈ W 0 s 1 , p ( Ω ) ∩ C 0 , τ ( Ω ¯ ) , τ ∈ ( 0 , s 1 ) , of the torsion problem

( − Δ ) p s 1 u + ( − Δ ) q s 2 u = σ in Ω , u = 0 in R N \ Ω .

Moreover, u σ → 0 in C 0 , τ ( Ω ¯ ) as σ → 0 + . Thus, for any σ sufficiently small one has both σ < ε and ‖ u σ ‖ ∞ < δ . This evidently implies

( − Δ ) p s 1 u σ + ( − Δ ) q s 2 u σ = σ < ε < f ( ⋅ , u σ ) ,

i.e., u ̲ ≔ u σ is a positive sub-solution of ( P f ).□

Remark 3.2

If s 1 ≠ q ′ s 2 , then Hopf’s theorem [15, Proposition 2.12] ensures that

(3.1) η d ( x ) s 1 ≤ u ̲ ( x ) ∀ x ∈ Ω ,

with suitable η > 0 . Otherwise, η d α ≤ u ̲ , being α > s 1 , α ≠ q ′ s 2 , and α ≠ p ′ s 1 ; cf. [15, Remark 2.14].

Now, fixed any v ∈ W 0 s 1 , p ( Ω ) , consider the following problem, where the convective term has been frozen:

( − Δ ) p s 1 u + ( − Δ ) q s 2 u = f ( x , u ) + g ( x , D s v ) in Ω , u = 0 on R N \ Ω . ( P v )

Theorem 3.3

Let ( H f 1 ) and ( H g ) be satisfied. If v ∈ W 0 s 1 , p ( Ω ) , then ( P v ) admits a weak solution u v ∈ W 0 s 1 , p ( Ω ) ∩ C 0 , τ ( Ω ¯ ) , where τ ∈ ( 0 , s 1 ) . Moreover, u v ≥ u ̲ .

Proof

Recalling Lemma 3.1, define

(3.2) f ˜ ( x , t ) ≔ f ( x , max { u ̲ ( x ) , t } ) , ( x , t ) ∈ Ω × R .

Without loss of generality we can suppose s 1 ≠ q ′ s 2 . In fact, by Remark 3.2, the case s 1 = q ′ s 2 is entirely analogous. Thanks to ( H f 1 ) and (3.1) one has

(3.3) f ˜ ( ⋅ , t ) ≤ c 1 ( max { u ̲ , t } ) − γ + c 2 ( max { u ̲ , t } ) r ≤ c 1 u ̲ − γ + c 2 ( u ̲ r + t r ) ≤ c 1 ( η d s 1 ) − γ + c 2 ( max Ω ¯ u ̲ ) r + c 2 t r ≤ C 1 ( d − γ s 1 + t r + 1 ) , t ∈ R + .

The energy functional J ˜ : W 0 s 1 , p ( Ω ) → R associated with the problem

(3.4) ( − Δ ) p s 1 u + ( − Δ ) q s 2 u = f ˜ ( x , u ) + g ( x , D s v ) in Ω , u = 0 in R N \ Ω ,

is written as

J ˜ ( u ) ≔ 1 p ∫ R N × R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + s 1 p d x d y + 1 q ∫ R N × R N ∣ u ( x ) − u ( y ) ∣ q ∣ x − y ∣ N + s 2 q d x d y − ∫ Ω F ˜ ( ⋅ , u ) d x − ∫ Ω G ( ⋅ , D s v ) d x , u ∈ W 0 s 1 , p ( Ω ) ,

where

F ˜ ( x , τ ) ≔ ∫ 0 τ f ˜ ( x , t ) d t , G ( x , ξ ) ≔ ∫ 0 τ g ( x , ξ ) d t = τ g ( x , ξ ) .

Obviously, J ˜ turns out well defined and of class C 1 . Moreover, (3.3) easily entails

(3.5) F ˜ ( x , τ ) ≤ ∫ 0 ∣ τ ∣ f ˜ ( x , t ) d t ≤ C 1 ( d − γ s 1 + 1 ) ∣ τ ∣ + ∣ τ ∣ r r + 1 ∀ τ ∈ R ,

because f ˜ ( x , t ) ≥ 0 . From (3.5), ( H g ), Hölder’s inequality, and fractional Hardy’s inequality [12, Theorem 1.1] (recall that s 1 p > 1 ), it follows that

J ˜ ( u ) ≥ 1 p ∫ R N × R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + s 1 p d x d y − C 1 ∫ Ω d − γ s 1 ∣ u ∣ d x − C 1 r + 1 ∫ Ω ∣ u ∣ r + 1 d x − c 3 ∫ Ω ∣ D s v ∣ ζ ∣ u ∣ d x − ( C 1 + c 3 ) ∫ Ω ∣ u ∣ d x ≥ 1 p ∫ R N × R N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + s 1 p d x d y − C 1 ∫ Ω d ( p − γ ) s 1 ∣ u ∣ d s 1 p d x − C 1 r + 1 ∫ Ω ∣ u ∣ r + 1 d x − c 3 ∫ Ω ∣ D s v ∣ ζ ∣ u ∣ d x − C 2 ‖ u ‖ p ≥ 1 p ‖ u ‖ s 1 , p p − C 3 ( ‖ u ‖ s 1 , p + ‖ u ‖ p r + 1 + ‖ D s v ‖ p ζ ‖ u ‖ p p − ζ + ‖ u ‖ p ) , u ∈ W 0 s 1 , p ( Ω ) .

Since r , ζ ∈ ( 1 , p − 1 ) , through Proposition 2.3 (b) we see that J ˜ is coercive. Thus, by Weierstrass-Tonelli’s theorem, there exists u v ∈ W 0 s 1 , p ( Ω ) fulfilling

J ˜ ( u v ) = inf u ∈ W 0 s 1 , p ( Ω ) J ˜ ( u ) ,

whence u v turns out a weak solution to (3.4). As in the proof of [18, Proposition 2.10] one has ( u ̲ − u v ) + ∈ W 0 s 1 , p ( Ω ) . Testing (3.4) with φ ≔ ( u ̲ − u v ) + yields

⟨ ( − Δ ) p s 1 u v + ( − Δ ) q s 2 u v , φ ⟩ = ∫ Ω f ˜ ( ⋅ , u v ) φ d x + ∫ Ω g ( ⋅ , D s v ) φ d x .

Lemma 3.1 and the inequality g ( x , ξ ) ≥ 0 produce

⟨ ( − Δ ) p s 1 u ̲ + ( − Δ ) q s 2 u ̲ , φ ⟩ ≤ ∫ Ω f ( ⋅ , u ̲ ) φ d x ≤ ∫ Ω f ( ⋅ , u ̲ ) φ d x + ∫ Ω g ( ⋅ , D s v ) φ d x .

Therefore, by ( p 1 ) and (3.2),

⟨ ( − Δ ) p s 1 u ̲ − ( − Δ ) p s 1 u v , φ ⟩ ≤ ⟨ ( − Δ ) p s 1 u ̲ − ( − Δ ) p s 1 u v , φ ⟩ + ⟨ ( − Δ ) q s 2 u ̲ − ( − Δ ) q s 2 u v , φ ⟩ ≤ ∫ Ω ( u v < u ̲ ) ( f ( ⋅ , u ̲ ) − f ˜ ( ⋅ , u v ) ) ( u ̲ − u v ) d x = ∫ Ω ( u v < u ̲ ) ( f ( ⋅ , u ̲ ) − f ( ⋅ , u ̲ ) ) ( u ̲ − u v ) d x = 0 .

Now, Lemma 9 in [21] forces

0 < u ̲ ≤ u v .

Consequently, u v ∈ W 0 s 1 , p ( Ω ) + weakly solves ( P v ). Corollary 2.10 of [15] then ensures that u v ∈ C 0 , τ ( Ω ¯ ) for all τ ∈ ( 0 , s 1 ) .□

Lemma 3.4

If 0 < s < 1 while Φ : W 0 s , p ( Ω ) → R 0 + is defined by

Φ ( u ) ≔ 1 p ∫ R 2 N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + s p d x d y ∀ u ∈ W 0 s , p ( Ω ) ,

then the operator

Φ ˆ ( w ) ≔ Φ ( w 1 ∕ q ) when w ≥ 0 a n d w 1 ∕ q ∈ W 0 s , p ( Ω ) , + ∞ otherwise,

has a nonempty domain and is convex.

Proof

Pick l > q and a non-negative u ∈ W 0 1 , p ( Ω ) ∩ L ∞ ( Ω ) . One has u l ∕ q ∈ W 0 1 , p ( Ω ) , because

∫ Ω ∣ ∇ ( u l ∕ q ) ∣ p d x = ∫ Ω l q u l ∕ q − 1 ∣ ∇ u ∣ p d x ≤ C ∫ Ω ∣ ∇ u ∣ p d x < + ∞ .

So, u l ∕ q ∈ W 0 s , p ( Ω ) by Proposition 2.3. Here, as usual, u ≡ 0 on R N \ Ω . We claim that Φ ˆ ( u l ) < + ∞ . In fact,

Φ ˆ ( u l ) = Φ ( u l ∕ q ) = 1 p ∫ R 2 N ∣ u ( x ) l ∕ q − u ( y ) l ∕ q ∣ p ∣ x − y ∣ N + s p d x d y = 1 p ∫ R N d x ∫ B 1 ( x ) ∣ u ( x ) l ∕ q − u ( y ) l ∕ q ∣ p ∣ x − y ∣ N + s p d y + 1 p ∫ R N d x ∫ R N \ B 1 ( x ) ∣ u ( x ) l ∕ q − u ( y ) l ∕ q ∣ p ∣ x − y ∣ N + s p d y .

Moreover,

∫ R N d x ∫ B 1 ( x ) ∣ u ( x ) l ∕ q − u ( y ) l ∕ q ∣ p ∣ x − y ∣ N + s p d y = ∫ R N d x ∫ B 1 ( 0 ) ∣ u ( x ) l ∕ q − u ( x + y ) l ∕ q ∣ p ∣ y ∣ N + s p d y = ∫ R N d x ∫ B 1 ( 0 ) ∣ u ( x ) l ∕ q − u ( x + y ) l ∕ q ∣ p ∣ y ∣ p 1 ∣ y ∣ N + ( s − 1 ) p d y ≤ ∫ R N d x ∫ B 1 ( 0 ) ∫ 0 1 ∣ ∇ u ( x + τ y ) l ∕ q ∣ d τ p 1 ∣ y ∣ N + ( s − 1 ) p d y ≤ C ∫ R N ∫ B 1 ( 0 ) ∫ 0 1 ∣ ∇ u ( x + τ y ) l ∕ q ∣ p ∣ y ∣ N + ( s − 1 ) p d x d y d τ ≤ C ∫ B 1 ( 0 ) ∫ 0 1 ‖ ∇ u l ∕ q ‖ L p ( R N ) p ∣ z ∣ N + ( s − 1 ) p d y d τ ≤ C 1 ‖ ∇ u l ∕ q ‖ L p ( R N ) p < + ∞ .

Likewise,

∫ R N d x ∫ R N \ B 1 ( x ) ∣ u ( x ) l ∕ q − u ( y ) l ∕ q ∣ p ∣ x − y ∣ N + s p d y ≤ 2 p − 1 ∫ R N d x ∫ R N \ B 1 ( x ) ∣ u ( x ) l ∕ q ∣ p + ∣ u ( y ) l ∕ q ∣ p ∣ x − y ∣ N + s p d y = 2 p − 1 ∫ R N d x ∫ R N \ B 1 ( 0 ) ∣ u ( x ) l ∕ q ∣ p + ∣ u ( x + y ) l ∕ q ∣ p ∣ y ∣ N + s p d y ≤ C 3 ∫ R N ∣ u ( x ) l ∕ q ∣ p d x = C 3 ‖ u l ∕ q ‖ p p < + ∞

because u l ∕ q ∈ W 0 1 , p ( Ω ) . Hence, u l ∈ dom Φ ˆ , and the first conclusion follows. Next, let u 1 , u 2 ∈ dom Φ ˆ and let t ∈ ( 0 , 1 ) . If v i ≔ u i 1 ∕ q , i = 1 , 2, and

v 3 ≔ ( ( 1 − t ) u 1 + t u 2 ) 1 ∕ q ,

then, thanks to discrete hidden convexity [3, Proposition 4.1],

∣ v 3 ( x ) − v 3 ( y ) ∣ p ≤ ( 1 − t ) ∣ v 1 ( x ) − v 1 ( y ) ∣ p + t ∣ v 2 ( x ) − v 2 ( y ) ∣ p ∀ x , y ∈ R N .

This entails

Φ ˆ ( ( 1 − t ) u 1 + t u 2 ) = 1 p ∫ R 2 N ∣ v 3 ( x ) − v 3 ( y ) ∣ p ∣ x − y ∣ N + s p d x d y ≤ 1 − t p ∫ R 2 N ∣ v 1 ( x ) − v 1 ( y ) ∣ p ∣ x − y ∣ N + s p d x d y + t p ∫ R 2 N ∣ v 2 ( x ) − v 2 ( y ) ∣ r ∣ x − y ∣ N + s p d x d y = ( 1 − t ) Φ ˆ ( u 1 ) + t Φ ˆ ( u 2 ) ,

thus completing the proof.□

Remark 3.5

The above result holds true even when q = p , with the same proof.

Theorem 3.6

Under ( H f 1 )–( H f 2 ) and ( H g ), for every fixed v ∈ W 0 s 1 , p ( Ω ) , the solution u v ∈ W 0 s 1 , p ( Ω ) ∩ C 0 , τ ( Ω ¯ ) to problem ( P v ) given by Theorem 3.3 is unique.

Proof

Suppose u v , w v ∈ W 0 s 1 , p ( Ω ) ∩ C 0 , τ ( Ω ¯ ) solve ( P v ), namely,

(3.6) ⟨ ( − Δ ) p s 1 u v + ( − Δ ) q s 2 u v , φ ⟩ = ∫ Ω f ( ⋅ , u v ) φ d x + ∫ Ω g ( ⋅ , D s v ) φ d x ,

(3.7) ⟨ ( − Δ ) p s 1 w v + ( − Δ ) q s 2 w v , ψ ⟩ = ∫ Ω f ( ⋅ , w v ) ψ d x + ∫ Ω g ( ⋅ , D s v ) ψ d x

for all φ , ψ ∈ W 0 s 1 , p ( Ω ) . The functions

φ ≔ u v q − w v q u v q − 1 and ψ ≔ u v q − w v q w v q − 1

lie in W 0 s 1 , p ( Ω ) , because u v , w v ∈ C 0 , τ ( Ω ¯ ) + . Hence, via (3.6)–(3.7), we achieve

(3.8) ⟨ ( − Δ ) p s 1 u v , φ ⟩ − ⟨ ( − Δ ) p s 1 w v , ψ ⟩ + ⟨ ( − Δ ) q s 2 u v , φ ⟩ − ⟨ ( − Δ ) q s 2 w v , ψ ⟩ = ∫ Ω f ( ⋅ , u v ) u v q − 1 − f ( ⋅ , w v ) w v q − 1 ( u v q − w v q ) d x + ∫ Ω g ( ⋅ , D s v ) ( φ − ψ ) d x .

Lemma 3.4 ensures that the functional J ˆ associated with

J ( u ) ≔ 1 p ∫ R 2 N ∣ u ( x ) − u ( y ) ∣ p ∣ x − y ∣ N + s 1 p d x d y , u ∈ W 0 s 1 , p ( Ω ) ,

turns out convex. Therefore, after a standard computation,

(3.9) 0 ≤ q ⟨ J ˆ ′ ( u v q ) − J ˆ ′ ( w v q ) , u v q − w v q ⟩ = ⟨ ( − Δ ) p s 1 u v , φ ⟩ − ⟨ ( − Δ ) p s 1 w v , ψ ⟩ .

An analogous argument produces

(3.10) ⟨ ( − Δ ) q s 2 u v , φ ⟩ − ⟨ ( − Δ ) q s 2 w v , ψ ⟩ ≥ 0 .

Now, gathering (3.8)–(3.10) together yields

(3.11) ∫ Ω f ( ⋅ , u v ) u v q − 1 − f ( ⋅ , w v ) w v q − 1 ( u v q − w v q ) d x + ∫ Ω g ( ⋅ , D s v ) ( φ − ψ ) d x ≥ 0 .

By ( H f 2 ), the function t ↦ f ( ⋅ , t ) t q − 1 is decreasing on R + . This implies

(3.12) ∫ Ω f ( ⋅ , u v ) u v q − 1 − f ( ⋅ , w v ) w v q − 1 ( u v q − w v q ) d x ≤ 0 .

Moreover,

(3.13) ∫ Ω g ( ⋅ , D s v ) ( φ − ψ ) d x ≤ ∫ Ω ( u v ≥ w v ) g ( ⋅ , D s v ) u v q − w v q w v q − 1 − u v q − w v q w v q − 1 d x + ∫ Ω ( u v < w v ) g ( ⋅ , D s v ) u v q − w v q u v q − 1 − u v q − w v q w v q − 1 d x ≤ ∫ Ω ( u v < w v ) g ( ⋅ , D s v ) − w v q − u v q u v q − 1 + w v q − u v q u v q − 1 d x = 0 .

From (3.11)–(3.13), it finally follows that

∫ Ω f ( ⋅ , u v ) u v q − 1 − f ( ⋅ , w v ) w v q − 1 ( u v q − w v q ) d x = 0 ,

whence, due to ( H f 2 ) again, u v ≡ w v , as desired.□

4 Main result

Define, for every v ∈ W 0 s 1 , p ( Ω ) ,

(4.1) T ( v ) ≔ u v ,

u v ∈ W 0 s 1 , p ( Ω ) + being the unique solution of ( P v ) found in Theorem 3.3.

Lemma 4.1

Let ( H f 1 ), ( H f 2 ), ( H g ) be satisfied and let q ′ s 2 ≠ s 1 < 1 p ′ γ . Then, T possesses a fixed point u ∈ W 0 s 1 , p ( Ω ) .

Proof

Given any v ∈ W 0 s 1 , p ( Ω ) , test problem ( P v ) with its solution u v . Through ( H f 1 ) and ( H g ), we thus arrive at

‖ u v ‖ s 1 , p p ≤ ∫ R 2 N ∣ u v ( x ) − u v ( y ) ∣ p ∣ x − y ∣ N + s 1 p d x d y + ∫ R 2 N ∣ u v ( x ) − u v ( y ) ∣ q ∣ x − y ∣ N + s 2 q d x d y ≤ c 1 ∫ Ω u v 1 − γ d x + c 2 ∫ Ω u v r + 1 d x + c 3 ∫ Ω ( u v + ∣ D s v ∣ ζ u v ) d x .

Thanks to Young’s inequality with ε > 0 , each term of the right-hand side is estimated as follows (recall that γ < 1 while r , ζ < p − 1 ):

∫ Ω u v α d x ≤ α p ε ‖ u v ‖ p p + p − α p C ε ( α ) ∣ Ω ∣ , α ∈ { 1 − γ , r + 1,1 } ; ∫ Ω ∣ D s v ∣ ζ u v d x ≤ 1 p ε ‖ u v ‖ p p + 1 p ′ C ε ∫ Ω ∣ D s v ∣ ζ p ′ d x .

Consequently, by Proposition 2.3 (b) and (2.2),

‖ u v ‖ s 1 , p p ≤ c ε ‖ u v ‖ p p + C ε ( 1 + ‖ D s v ‖ ζ p ′ ζ p ′ ) ≤ c ′ ε ‖ u v ‖ s 1 , p p + C ′ ε ( 1 + ‖ D s v ‖ p ζ p ′ ) ≤ c ′ ε ‖ u v ‖ s 1 , p p + C ′ ε ( 1 + ‖ v ‖ X 0 s , p ( R N ) ζ p ′ ) ≤ c ′ ε ‖ u v ‖ s 1 , p p + C ε * ( 1 + ‖ v ‖ s 1 , p ζ p ′ ) .

This entails

( 1 − c ′ ε ) ‖ u v ‖ s 1 , p p ≤ C ε * ( 1 + ‖ v ‖ s 1 , p ζ p ′ ) ,

i.e., after choosing ε < 1 c ′ ,

(4.2) ‖ T ( v ) ‖ s 1 , p p = ‖ u v ‖ s 1 , p p ≤ C ˆ ( 1 + ‖ v ‖ s 1 , p ζ p ′ ) ,

with C ˆ ≔ C ε * 1 − c ′ ε . Since ζ p ′ < p , there exists ρ > 0 such that C ˆ ( 1 + ρ ζ p ′ ) ≤ ρ p . Thus, due to (4.2), ‖ v ‖ s 1 , p ≤ ρ implies ‖ T ( v ) ‖ s 1 , p ≤ ρ , which clearly means T ( K ) ⊆ K , provided

K ≔ { u ∈ W 0 s 1 , p ( Ω ) : ‖ u ‖ s 1 , p ≤ ρ } .

Claim 1: The operator T ⌊ K is compact.

Let { v n } ⊆ K and let u n ≔ T ( v n ) , n ∈ N . The reflexivity of W 0 s 1 , p ( Ω ) yields v n ⇀ v in W 0 s 1 , p ( Ω ) while Proposition 2.3 (c) ensures that

∀ r ∈ [ 1 , p s 1 * ) one has v n → v in L r ( Ω ) ,

where a sub-sequence is considered when necessary. Likewise, from { u n } ⊆ K , it follows u n ⇀ u in W 0 s 1 , p ( Ω ) and, as before,

(4.3) ∀ r ∈ [ 1 , p s 1 * ) one has u n → u in L r ( Ω ) .

Now, testing ( P v ) with φ n ≔ u n − u and using ( H f 1 ), Theorem 3.3, and (3.1), we obtain

⟨ ( − Δ ) p s 1 u n + ( − Δ ) q s 2 u n , φ n ⟩ ≤ ∫ Ω f ( ⋅ , u n ) ∣ φ n ∣ d x + ∫ Ω g ( ⋅ , D s v n ) ∣ φ n ∣ d x ≤ c 1 ∫ Ω u n − γ ∣ φ n ∣ d x + c 2 ∫ Ω u n r ∣ φ n ∣ d x + c 3 ∫ Ω ( 1 + ∣ D s v n ∣ ζ ) ∣ φ n ∣ d x ≤ ∫ Ω [ c 1 ( η d s 1 ) − γ + c 3 ] ∣ φ n ∣ d x + c 2 ∫ Ω u n r ∣ φ n ∣ d x + c 3 ∫ Ω ∣ D s v n ∣ ζ ∣ φ n ∣ d x .

Hence, due to Hölder’s inequality, Proposition 2.2 (recall that s 1 γ < 1 p ′ ), Proposition 2.3 (b), besides (2.2),

(4.4) ⟨ ( − Δ ) p s 1 u n + ( − Δ ) q s 2 u n , φ n ⟩ ≤ C 1 ‖ φ n ‖ p + c 2 ‖ u n ‖ p r ‖ φ n ‖ p p − r + c 3 ‖ D s v n ‖ p ζ ‖ φ n ‖ p p − ζ ≤ C 1 ‖ φ n ‖ p + C 2 ρ r ‖ φ n ‖ p p − r + C 3 ρ ζ ‖ φ n ‖ p p − ζ

for all n ∈ N , where

C 1 ≔ c 1 η − γ ‖ d − s 1 γ ‖ p ′ + c 3 ∣ Ω ∣ 1 ∕ p ′ .

On account of (4.3)–(4.4) one arrives at

limsup n → + ∞ ⟨ ( − Δ ) p s 1 u n + ( − Δ ) q s 2 u n , u n − u ⟩ ≤ 0 ,

whence u n → u in W 0 s 1 , p ( Ω ) because, by Proposition 2.1 and ( p 4 ) , the fractional ( p , q ) -Laplacian

u ↦ ( − Δ ) p s 1 u + ( − Δ ) q s 2 u , u ∈ W 0 s 1 , p ( Ω ) ,

is type ( S ) + .

Claim 2: The operator T ⌊ K turns out continuous.

Let { v n } ⊆ K satisfy v n → v in W 0 s 1 , p ( Ω ) and let u n ≔ T ( v n ) , n ∈ N . Since T ⌊ K is compact, along a sub-sequence if necessary, we have u n → u in W 0 s 1 , p ( Ω ) . Moreover, (4.3) holds. Our claim thus becomes u = T ( v ) . Pick any φ ∈ W 0 s 1 , p ( Ω ) . From (4.1), it follows that

(4.5) ∫ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p − 2 ( u n ( x ) − u n ( y ) ) ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + s 1 p d x d y + ∫ R 2 N ∣ u n ( x ) − u n ( y ) ∣ q − 2 ( u n ( x ) − u n ( y ) ) ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + s 2 q d x d y = ∫ Ω f ( ⋅ , u n ) φ d x + ∫ Ω g ( ⋅ , D s v n ) φ d x ∀ n ∈ N .

Observe that

{ u n } ⊆ K ⇒ ∣ u n ( x ) − u n ( y ) ∣ p − 2 ( u n ( x ) − u n ( y ) ) ∣ x − y ∣ N + s 1 p p ′ bounded in L p ′ ( R 2 N )

and that, by (4.3),

lim n → + ∞ ∣ u n ( x ) − u n ( y ) ∣ p − 2 ( u n ( x ) − u n ( y ) ) ∣ x − y ∣ N + s 1 p p ′ = ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ∣ x − y ∣ N + s 1 p p ′

for almost every ( x , y ) ∈ R 2 N . So, up to sub-sequences,

∣ u n ( x ) − u n ( y ) ∣ p − 2 ( u n ( x ) − u n ( y ) ) ∣ x − y ∣ ∣ x − y ∣ N + s 1 p p ′ ⇀ ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ∣ x − y ∣ N + s 1 p p ′ in L p ′ ( R 2 N ) .

This implies

(4.6) lim n → + ∞ ∫ R 2 N ∣ u n ( x ) − u n ( y ) ∣ p − 2 ( u n ( x ) − u n ( y ) ) ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + s 1 p d x d y = ∫ R 2 N ∣ u ( x ) − u ( y ) ∣ p − 2 ( u ( x ) − u ( y ) ) ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + s 1 p d x d y ,

because

φ ( x ) − φ ( y ) ∣ x − y ∣ N + s 1 p p ∈ L p ( R 2 N ) .

An analogous argument, which employs the continuous embedding W 0 s 1 , p ( Ω ) ↪ W 0 s 2 , q ( Ω ) (cf. Proposition 2.3 (a)), produces

(4.7) lim n → + ∞ ∫ R 2 N ∣ u n ( x ) − u n ( y ) ∣ q − 2 ( u n ( x ) − u n ( y ) ) ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + s 2 q d x d y = ∫ R 2 N ∣ u ( x ) − u ( y ) ∣ q − 2 ( u ( x ) − u ( y ) ) ( φ ( x ) − φ ( y ) ) ∣ x − y ∣ N + s 2 q d x d y .

Let us next focus on the right-hand side of (4.5). Exploiting ( H f 1 ), Theorem 3.3, (3.1), (4.3), and [4, Theorem 4.9], we achieve

∣ f ( ⋅ , u n ) φ ∣ ≤ [ c 1 u n − γ + c 2 u n r ] ∣ φ ∣ ≤ [ c 1 u ̲ − γ + c 2 u n r ] ∣ φ ∣ ≤ [ c 1 ( η d s 1 ) − γ + c 2 ψ r ] ∣ φ ∣ , n ∈ N ,

for some ψ ∈ L r ( Ω ) . Now, by (4.3) and [4, Theorem 4.2], one has

(4.8) lim n → + ∞ ∫ Ω f ( ⋅ , u n ) φ d x = ∫ Ω f ( ⋅ , u ) φ d x .

Finally, observe that, thanks to (2.2),

v n → v in W 0 s 1 , p ( Ω ) ⇒ v n → v in L 0 s , p ( Ω ) ,

as well as

v n → v in L 0 s , p ( Ω ) ⇒ D s v n → D s v in L p ( Ω ) ⇒ ( D s v n ) ζ → ( D s v ) ζ in L p ζ ( Ω ) ,

where a sub-sequence is considered if necessary. Since ( H g ) holds while φ ∈ L p p − ζ ( Ω ) because ζ ∈ ( 1 , p − 1 ) , this forces

(4.9) lim n → + ∞ ∫ Ω g ( ⋅ , D s v n ) φ d x = ∫ Ω g ( ⋅ , D s v ) φ d x .

Letting n → + ∞ in (4.5) and using (4.6)–(4.9), we arrive at u = T ( v ) .

Now, Schauder’s fixed point theorem can be applied to T ⌊ K , which completes the proof.□

Our main result is as follows.

Theorem 4.2

Under hypotheses ( H f 1 ), ( H f 2 ), ( H g ), and the conditions q ′ s 2 ≠ s 1 < 1 p ′ γ , problem (P) admits a weak solution u ∈ W 0 s 1 , p ( Ω ) .

Proof

Simply use Lemma 4.1 and note that fixed points of T weakly solve (P).□

Funding information: This study was partly funded by Research project of MIUR (Italian Ministry of Education, University and Research) Prin 2022 Nonlinear differential problems with applications to real phenomena (Grant No. 2022ZXZTN2). The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission. The authors have contributed equally to this article, from the methodology to the writing, revision, and editing.
Conflict of interest: On behalf of all authors, the corresponding author states that there is no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Received: 2024-11-22

Revised: 2025-03-05

Accepted: 2025-03-24

Published Online: 2025-11-11

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/anona-2025-0082

Keywords for this article

fractional Dirichlet problem; weakly singular and non-locally convective reaction; fractional (p, q)-Laplacian; distributional Riesz fractional gradient; positive weak solution

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