On a Singular Robin Problem with Convection Terms

Umberto Guarnotta; Salvatore A. Marano; Dumitru Motreanu

doi:10.1515/ans-2020-2093

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On a Singular Robin Problem with Convection Terms

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Published/Copyright: May 29, 2020

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From the journal Advanced Nonlinear Studies Volume 20 Issue 4

Abstract

In this paper, the existence of smooth positive solutions to a Robin boundary-value problem with non-homogeneous differential operator and reaction given by a nonlinear convection term plus a singular one is established. Proofs chiefly exploit sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity theory, as well as fixed point arguments. A uniqueness result is also presented.

Keywords: Robin Problem; Quasilinear Elliptic Equation; Gradient Dependence; Singular Term

MSC 2010: 35J60; 35J62; 35J92

1 Introduction

Let Ω⊆ℝN (N≥3) be a bounded domain with a C2-boundary ∂⁡Ω and let f:Ω×ℝ×ℝN→[0,+∞), g:Ω×(0,+∞)→[0,+∞) be two Carathéodory functions. In this paper, we study existence and uniqueness of solutions to the following Robin problem:

(1.1) { - div ⁡ a ⁢ ( ∇ ⁡ u ) = f ⁢ ( x , u , ∇ ⁡ u ) + g ⁢ ( x , u ) in ⁢ Ω , u > 0 in ⁢ Ω , ∂ ⁡ u ∂ ⁡ ν a + β ⁢ u p - 1 = 0 on ⁢ ∂ ⁡ Ω ,

where a:ℝN→ℝN denotes a continuous strictly monotone map having suitable properties, which basically stem from Liebermann’s nonlinear regularity theory [12] and Pucci–Serrin’s maximum principle [19]; see Section 2 for details. Moreover, β>0, 1<p<+∞, while ∂∂⁡νa denotes the co-normal derivative associated with a.

This problem gathers together several hopefully interesting technical features, namely:

The involved differential operator appears in a general form that includes non-homogeneous cases.
f depends on the solution and its gradient. So, the reaction exhibits nonlinear convection terms.
g can be singular at zero, i.e., lims→0+⁡g⁢(x,s)=+∞.
Robin boundary conditions are imposed instead of (much more frequent) Dirichlet ones.

All these things have been extensively investigated, although separately. For instance, both differential operator and Robin conditions already appear in [8], where, however, the problem has a fully variational structure, whilst [15] falls inside non-variational settings. The paper [4] addresses the presence of convection terms; see also [14, 15, 20], which exhibit more general contexts. Last but not least, singular problems were considered especially after the seminal works of Crandall, Rabinowitz and Tartar [2], and Lazer and McKenna [10]. Among recent contributions on this subject, we mention [7, 16]. Finally, [13] treats a p-Laplacian Dirichlet problem whose right-hand side has the same form as that in (1.1). It represented the starting point of our research.

Several issues arise when passing from Dirichlet to Robin boundary conditions. Accordingly, here, we try to develop some useful tools in this direction, including the location of solutions to an auxiliary variational problem inside an opportune sublevel of its energy functional, constructed for preserving some compactness and semicontinuity properties (cf. Section 3).

Our main result, Theorem 3.19, establishes the existence of a regular solution to (1.1) chiefly via sub-super-solution and truncation techniques, set-valued analysis, recursive methods, nonlinear regularity theory, as well as Schaefer’s fixed point theorem. Uniqueness is also addressed, but only when p=2 (vide Section 4).

Usually, linear problems possess only one solution, whereas multiplicity is encountered in nonlinear phenomena. Hence, it might be of interest to seek hypotheses on f and g that yield uniqueness even if p≠2. As far as we know, this is still an open problem.

Let us finally note that replacing the constant β with a nontrivial non-negative function β∈L∞⁢(Ω) does not invalidate our results.

2 Preliminaries

Let X be a set and let C⊆X. We denote by χC the characteristic function of C. If C≠∅ and Γ:C→C, then

Fix ⁡ ( Γ ) := { x ∈ C : x = Γ ⁢ ( x ) }

is the fixed point set of Γ. The following result, usually called Schaefer’s theorem [6, p. 827] or Leray–Schauder’s alternative principle, will play a basic role in the sequel.

Theorem 2.1.

Let X be a Banach space, let C⊆X be nonempty convex, and let Γ:C→C be continuous. Suppose Γ maps bounded sets into relatively compact sets. Then either {x∈C:x=t⁢Γ⁢(x)⁢ for some ⁢t∈(0,1)} turns out unbounded or Fix⁡(Γ)≠∅.

Given a partially ordered set (X,≤), we say that X is downward directed when, for every x1,x2∈X, there exists x∈X such that x≤xi, i=1,2. The notion of upward directed set is analogous.

If Y is a real function space on a set Ω⊆ℝN and u,v∈Y, then u≤v means u⁢(x)≤v⁢(x) for almost every x∈Ω. Moreover, Y+:={u∈Y:u≥0}, Ω(u≤v):={x∈Ω:u(x)≤v(x)}, etc.

Let X,Y be two metric spaces and let 𝒮:X→2Y. The multifunction 𝒮 is called lower semicontinuous when for every xn→x in X, y∈𝒮⁢(x), there exists a sequence {yn}⊆Y having the following properties:

y n → y in ⁢ Y , y n ∈ 𝒮 ⁢ ( x n ) for all ⁢ n ∈ ℕ .

Finally, if X is a Banach space and J∈C1⁢(X), then

Crit ⁡ ( J ) := { x ∈ X : J ′ ⁢ ( x ) = 0 }

is the critical set of J.

The monograph [1] represents a general reference on these topics.

Given any s>1, the symbol s′ will indicate the conjugate exponent of s, namely, s′:=ss-1.

Henceforth, for 1<p<+∞, β>0, Ω as in the Introduction, and u:Ω¯→ℝ appropriate, the notation below will be adopted:

∥ u ∥ ∞ := ess ⁢ sup x ∈ Ω | u ( x ) | , ∥ u ∥ C 1 ⁢ ( Ω ¯ ) := ∥ u ∥ ∞ + ∥ ∇ ⁡ u ∥ ∞ ,

∥ u ∥ p := ( ∫ Ω | u | p d x ) 1 p , ∥ u ∥ p , ∂ ⁡ Ω := ( ∫ ∂ ⁡ Ω | u | p d σ ) 1 p ,

∥ u ∥ 1 , p := ( ∥ u ∥ p p + ∥ ∇ ⁡ u ∥ p p ) 1 p , ∥ u ∥ β , 1 , p := ( β ⁢ ∥ u ∥ p , ∂ ⁡ Ω p + ∥ ∇ ⁡ u ∥ p p ) 1 p .

Here, σ denotes the (N-1)-dimensional Hausdorff measure on ∂⁡Ω. If ν⁢(x) is the outward unit normal vector to ∂⁡Ω at its point x, then ∂∂⁡νa stands for the co-normal derivative associated with a, defined by extending the map u↦〈a⁢(∇⁡u),ν〉 from C1⁢(Ω¯) to W1,p⁢(Ω).

Remark 2.2.

The trace inequality ensures that ∥u∥p,∂⁡Ω makes sense whenever u∈W1,p⁢(Ω); see, for instance, [3] or [9].

Remark 2.3.

It is known that (see [5])

int ⁡ ( C 1 ⁢ ( Ω ¯ ) + ) = { u ∈ C 1 ⁢ ( Ω ¯ ) : u ⁢ ( x ) > 0 ⁢ for all ⁢ x ∈ Ω ¯ } .

Remark 2.4.

∥ ⋅ ∥ β , 1 , p is a norm on W1,p⁢(Ω) equivalent to ∥⋅∥1,p. In particular, there exists c1=c1⁢(p,β,Ω)∈(0,1) such that

(2.1) c 1 ⁢ ∥ u ∥ 1 , p ≤ ∥ u ∥ β , 1 , p ≤ 1 c 1 ⁢ ∥ u ∥ 1 , p for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) .

For the proof we refer to [17].

Let ω∈C1⁢(0,+∞) satisfy

C 1 ≤ t ⁢ ω ′ ⁢ ( t ) ω ⁢ ( t ) ≤ C 2 , C 3 ⁢ t p - 1 ≤ ω ⁢ ( t ) ≤ C 4 ⁢ ( 1 + t p - 1 )

in (0,+∞), with Ci suitable positive constants.

Assumption 2.5.

The operator a:ℝN→ℝN has the following properties:

a ⁢ ( ξ ) = a 0 ⁢ ( | ξ | ) ⁢ ξ for all ξ∈ℝN, where a0:(0,+∞)→(0,+∞) is C1, t↦t⁢a0⁢(t) turns out strictly increasing, and

lim t → 0 + ⁡ t ⁢ a 0 ⁢ ( t ) = 0 , lim t → 0 + ⁡ t ⁢ a 0 ′ ⁢ ( t ) a 0 ⁢ ( t ) > - 1 .
| D ⁢ a ⁢ ( ξ ) | ≤ C 5 ⁢ ω ⁢ ( | ξ | ) | ξ | in ℝN∖{0}.
〈 D ⁢ a ⁢ ( ξ ) ⁢ y , y 〉 ≥ ω ⁢ ( | ξ | ) | ξ | ⁢ | y | 2 for every y,ξ∈ℝN, ξ≠0.

Example 2.6.

Various differential operators comply with Assumption 2.5. Three classical examples are listed below.

The so-called p-Laplacian: Δp⁢u:=div⁡(|∇⁡u|p-2⁢∇⁡u), which stems from a0⁢(t):=tp-2.
The (p,q)-Laplacian: Δp⁢u+Δq⁢u, where 1<q<p<+∞. In this case, a0⁢(t):=tp-2+tq-2.
The generalized p-mean curvature operator:

u ↦ div ⁡ [ ( 1 + | ∇ ⁡ u | 2 ) p - 2 2 ⁢ ∇ ⁡ u ] ,

corresponding to a0⁢(t):=(1+t2)p-22.

Finally, define

G 0 ⁢ ( t ) := ∫ 0 t s ⁢ a 0 ⁢ ( s ) ⁢ d s for all ⁢ t ∈ ℝ , G ⁢ ( ξ ) := G 0 ⁢ ( | ξ | ) for all ⁢ ξ ∈ ℝ N .

Proposition 2.7.

Under Assumption 2.5, there exists c2∈(0,1) such that

| a ⁢ ( ξ ) | ≤ 1 c 2 ⁢ ( 1 + | ξ | p - 1 ) 𝑎𝑛𝑑 c 2 ⁢ | ξ | p ≤ 〈 a ⁢ ( ξ ) , ξ 〉 ≤ 1 c 2 ⁢ ( 1 + | ξ | p )

for all ξ∈RN. In particular,

c 2 ⁢ | ξ | p ≤ G ⁢ ( ξ ) ≤ 1 c 2 ⁢ ( 1 + | ξ | p ) , ξ ∈ ℝ N .

Proof.

See [8, Lemmas 2.1–2.2] or [17, Lemma 2.2 and Corollary 2.3]. ∎

3 Existence

Throughout this section, the convection term f and the singularity g will fulfill the assumptions below, where, to avoid unnecessary technicalities, ‘for all x’ takes the place of ‘for almost all x’.

Assumption 3.1.

f : Ω × ℝ × ℝ N → [ 0 , + ∞ ) is a Carathéodory function. Moreover, to every M>0, there correspond cM,dM>0 such that

f ⁢ ( x , s , ξ ) ≤ c M + d M ⁢ | s | p - 1 for all ⁢ ( x , s , ξ ) ∈ Ω × ℝ × ℝ N , with ⁢ | ξ | ≤ M .

Assumption 3.2.

g : Ω × ( 0 , + ∞ ) → [ 0 , + ∞ ) is a Carathéodory function having the following properties:

g ⁢ ( x , ⋅ ) turns out nonincreasing on (0,1] whenever x∈Ω, and g⁢(⋅,1)≢0.
There exist c,d>0 such that

g ⁢ ( x , s ) ≤ c + d ⁢ s p - 1 for all ⁢ ( x , s ) ∈ Ω × ( 1 , + ∞ ) .
With appropriate θ∈int⁢(C1⁢(Ω¯)+) and ε0>0, the map x↦g⁢(x,ε⁢θ⁢(x)) belongs to Lp′⁢(Ω) for any ε∈(0,ε0).

The paper [13] contains meaningful examples of functions g that satisfy Assumption 3.2. A very simple case is g⁢(x,s):=s-γ for all (x,s)∈Ω×(0,+∞), where γ>0 and θ⁢(⋅)≡1.

Fix w∈C1⁢(Ω¯). We first focus on the singular problem (without convection terms)

(3.1) { - div ⁡ a ⁢ ( ∇ ⁡ u ) = f ⁢ ( x , u , ∇ ⁡ w ) + g ⁢ ( x , u ) in ⁢ Ω , u > 0 in ⁢ Ω , ∂ ⁡ u ∂ ⁡ ν a + β ⁢ u p - 1 = 0 on ⁢ ∂ ⁡ Ω .

Definition 3.3.

u ∈ W 1 , p ⁢ ( Ω ) is called a subsolution to (3.1) when

∫ Ω 〈 a ( ∇ u ) , ∇ v 〉 d x + β ∫ ∂ ⁡ Ω | u | p - 2 u v d σ ≤ ∫ Ω [ f ( ⋅ , u , ∇ w ) + g ( ⋅ , u ) ] v d x

for all v∈W1,p⁢(Ω)+. The set of subsolutions will be denoted by U¯w.

We say that u∈W1,p⁢(Ω) is a supersolution to (3.1) if

(3.2) ∫ Ω 〈 a ( ∇ u ) , ∇ v 〉 d x + β ∫ ∂ ⁡ Ω | u | p - 2 u v d σ ≥ ∫ Ω [ f ( ⋅ , u , ∇ w ) + g ( ⋅ , u ) ] v d x

for every v∈W1,p⁢(Ω)+, and indicate with U¯w the supersolution set.

Finally, u∈W1,p⁢(Ω) is called a solution of (3.1), provided

∫ Ω 〈 a ( ∇ u ) , ∇ v 〉 d x + β ∫ ∂ ⁡ Ω | u | p - 2 u v d σ = ∫ Ω [ f ( ⋅ , u , ∇ w ) + g ( ⋅ , u ) ] v d x

for all v∈W1,p⁢(Ω)+. The corresponding solution set will be denoted by Uw. Obviously, Uw=U¯w∩U¯w.

Lemma 3.4.

If u1,u2∈U¯w(resp. u1,u2∈U¯w), then min⁡{u1,u2}∈U¯w(resp. max⁡{u1,u2}∈U¯w). In particular, the set U¯w(resp. U¯w) is downward (resp. upward) directed.

Proof.

This proof is patterned after that of [13, Lemma 10] (see also [1]). Thus, we only sketch it. Pick u1,u2∈U¯w, set u:=min⁡{u1,u2}, and define, for every t∈ℝ,

η ε ⁢ ( t ) := { 0 if ⁢ t < 0 , t ε if ⁢ 0 ≤ t ≤ ε , 1 if ⁢ t > ε ,

where ε>0. Further, to shorten the notation, write η¯ε⁢(x):=ηε⁢(u2⁢(x)-u1⁢(x)). Evidently, we have both η¯ε∈W1,p⁢(Ω)+ and

∇ ⁡ η ¯ ε = η ε ′ ⁢ ( u 2 - u 1 ) ⁢ ∇ ⁡ ( u 2 - u 1 ) .

Let v^∈C1⁢(Ω¯)+. Since ui fulfills (3.2), one has

∫ Ω 〈 a ( ∇ u i ) , ∇ v 〉 d x + β ∫ ∂ ⁡ Ω | u i | p - 2 u i v d σ ≥ ∫ Ω [ f ( ⋅ , u i , ∇ w ) + g ( ⋅ , u i ) ] v d x

whenever v∈W1,p⁢(Ω)+. Choosing v:=η¯ε⁢v^ if i=1, v:=(1-η¯ε)⁢v^ if i=2, and adding them term by term produces

∫ Ω 〈 a ⁢ ( ∇ ⁡ u 1 ) - a ⁢ ( ∇ ⁡ u 2 ) , ∇ ⁡ ( u 2 - u 1 ) 〉 ⁢ η ε ′ ⁢ ( u 2 - u 1 ) ⁢ v ^ ⁢ d x + ∫ Ω 〈 a ⁢ ( ∇ ⁡ u 1 ) , ∇ ⁡ v ^ 〉 ⁢ η ¯ ε ⁢ d x + ∫ Ω 〈 a ⁢ ( ∇ ⁡ u 2 ) , ∇ ⁡ v ^ 〉 ⁢ ( 1 - η ¯ ε ) ⁢ d x

+ β ( ∫ ∂ ⁡ Ω | u 1 | p - 2 u 1 η ¯ ε v ^ d σ + ∫ ∂ ⁡ Ω | u 2 | p - 2 u 2 ( 1 - η ¯ ε ) v ^ d σ )

(3.3) ≥ ∫ Ω [ f ⁢ ( ⋅ , u 1 , ∇ ⁡ w ) + g ⁢ ( ⋅ , u 1 ) ] ⁢ η ¯ ε ⁢ v ^ ⁢ d x + ∫ Ω [ f ⁢ ( ⋅ , u 2 , ∇ ⁡ w ) + g ⁢ ( ⋅ , u 2 ) ] ⁢ ( 1 - η ¯ ε ) ⁢ v ^ ⁢ d x .

The strict monotonicity of a combined with ηε′⁢(u2-u1)⁢v^≥0 lead to

∫ Ω 〈 a ⁢ ( ∇ ⁡ u 1 ) - a ⁢ ( ∇ ⁡ u 2 ) , ∇ ⁡ ( u 2 - u 1 ) 〉 ⁢ η ε ′ ⁢ ( u 2 - u 1 ) ⁢ v ^ ⁢ d x ≤ 0 .

For almost every x∈Ω, we have

∇ ⁡ u ⁢ ( x ) = { ∇ ⁡ u 1 ⁢ ( x ) if ⁢ u 1 ⁢ ( x ) < u 2 ⁢ ( x ) , ∇ ⁡ u 2 ⁢ ( x ) otherwise,

as well as

lim ε → 0 + ⁡ η ¯ ε ⁢ ( x ) = χ Ω ( u 1 < u 2 ) ⁢ ( x ) .

Hence, letting ε→0+ and using the dominated convergence theorem, inequality (3.3) becomes

∫ Ω 〈 a ( ∇ u ) , ∇ v ^ 〉 d x + β ∫ ∂ ⁡ Ω | u | p - 2 u v ^ d σ ≥ ∫ Ω [ f ( ⋅ , u , ∇ w ) + g ( ⋅ , u ) ] v ^ d x ,

see [13, Lemma 10] for more details. Since v^∈C1⁢(Ω¯)+ was arbitrary, by density, one arrives at u∈U¯w. ∎

Lemma 3.5.

Let Assumptions 3.1–3.2 be satisfied. Then there exists a subsolution u¯∈int⁢(C1⁢(Ω¯)+) to (3.1) independent of w and such that ∥u¯∥∞≤1.

Proof.

Given any δ>0, consider the problem

(3.4) { - div ⁡ a ⁢ ( ∇ ⁡ u ) = g ~ ⁢ ( x , u ) in ⁢ Ω , ∂ ⁡ u ∂ ⁡ ν a + β ⁢ | u | p - 2 ⁢ u = 0 on ⁢ ∂ ⁡ Ω ,

where

(3.5) g ~ ⁢ ( x , s ) := min ⁡ { g ⁢ ( x , s ) , δ } , ( x , s ) ∈ Ω × ( 0 , + ∞ ) .

Standard arguments yield a nontrivial solution u¯∈W1,p⁢(Ω) to (3.4), because g~ is bounded. Testing with -u¯-, we get

- ∫ Ω 〈 a ( ∇ u ¯ ) , ∇ u ¯ - 〉 d x - β ∫ Ω | u ¯ | p - 2 u ¯ u ¯ - d σ = - ∫ Ω g ~ ( x , u ¯ ) u ¯ - d x ≤ 0 ,

whence, by Proposition 2.7,

c 2 ⁢ ∥ u ¯ - ∥ β , 1 , p p ≤ ∫ Ω 〈 a ⁢ ( ∇ ⁡ u ¯ - ) , ∇ ⁡ u ¯ - 〉 ⁢ d x + β ⁢ ∫ Ω ( u ¯ - ) p ⁢ d σ ≤ 0 .

Therefore, u¯≥0. Regularity up to the boundary [12] and the strong maximum principle [19] then force u¯∈int⁢(C1⁢(Ω¯)+). Now, if uδ∈C1,α⁢(Ω¯)+ satisfies

(3.6) { - div ⁡ a ⁢ ( ∇ ⁡ u ) = δ in ⁢ Ω , ∂ ⁡ u ∂ ⁡ ν a + β ⁢ | u | p - 2 ⁢ u = 0 on ⁢ ∂ ⁡ Ω ,

then, by compactness of the embedding C1,α(Ω¯))↪C1(Ω¯), we can find u∈C1⁢(Ω¯) such that limδ→0+⁡uδ=u in C1⁢(Ω¯) up to subsequences. One evidently has u≡0, because uδ solves (3.6). Thus, 0≤uδ≤1 once δ is small enough. Using (3.5), the comparison principle finally entails

(3.7) ∥ u ¯ ∥ ∞ ≤ ∥ u δ ∥ ∞ ≤ 1 .

Let θ and ε0 be as in Assumption 3.2 (iii). Since u¯,θ∈int⁢(C1⁢(Ω¯)+), there exists ε∈(0,ε0) such that u¯-ε⁢θ∈int⁢(C1⁢(Ω¯)+). From Assumption 3.2 (i), (iii), and (3.7), we thus infer

(3.8) 0 ≤ g ⁢ ( ⋅ , u ¯ ) ≤ g ⁢ ( ⋅ , ε ⁢ θ ) ∈ L p ′ ⁢ ( Ω ) .

The conclusion is achieved by verifying that u¯∈U¯w for any w∈C1⁢(Ω¯). Pick such a w, test (3.4) with v∈W1,p⁢(Ω)+, and recall (3.5), to arrive at

∫ Ω 〈 a ⁢ ( ∇ ⁡ u ¯ ) , ∇ ⁡ v 〉 ⁢ d x + β ⁢ ∫ ∂ ⁡ Ω u ¯ p - 1 ⁢ v ⁢ d σ = ∫ Ω g ~ ⁢ ( ⋅ , u ¯ ) ⁢ v ⁢ d x ≤ ∫ Ω g ⁢ ( ⋅ , u ¯ ) ⁢ v ⁢ d x ≤ ∫ Ω [ f ⁢ ( ⋅ , u , ∇ ⁡ w ) + g ⁢ ( ⋅ , u ¯ ) ] ⁢ v ⁢ d x ,

as desired. ∎

Remark 3.6.

This proof shows that the subsolution u¯ constructed in Lemma 3.5 enjoys further the property:

(3.9) ∫ Ω 〈 a ( ∇ u ¯ ) , ∇ v 〉 d x + β ∫ ∂ ⁡ Ω | u ¯ | p - 2 u ¯ v d σ ≤ ∫ Ω g ( ⋅ , u ¯ ) v d x for all v ∈ W 1 , p ( Ω ) + .

Given w∈C1⁢(Ω¯), consider the truncated problem

(3.10) { - div ⁡ a ⁢ ( ∇ ⁡ u ) = f ^ ⁢ ( x , u ) + g ^ ⁢ ( x , u ) in ⁢ Ω , u > 0 in ⁢ Ω , ∂ ⁡ u ∂ ⁡ ν a + β ⁢ u p - 1 = 0 on ⁢ ∂ ⁡ Ω ,

where

(3.11) f ^ ⁢ ( x , s ) := { f ⁢ ( x , u ¯ ⁢ ( x ) , ∇ ⁡ w ⁢ ( x ) ) if ⁢ s ≤ u ¯ ⁢ ( x ) , f ⁢ ( x , s , ∇ ⁡ w ⁢ ( x ) ) otherwise,

(3.12) g ^ ⁢ ( x , s ) := { g ⁢ ( x , u ¯ ⁢ ( x ) ) if ⁢ s ≤ u ¯ ⁢ ( x ) , g ⁢ ( x , s ) otherwise.

The energy functional corresponding to (3.10) reads

ℰ w ( u ) := 1 p ∫ Ω G ( ∇ u ) d x + β p ∫ ∂ ⁡ Ω | u | p d σ - ∫ Ω F ^ ( ⋅ , u ) d x - ∫ Ω G ^ ( ⋅ , u ) d x

for all u∈W1,p⁢(Ω), with

F ^ ⁢ ( x , s ) := ∫ 0 s f ^ ⁢ ( x , t ) ⁢ d t , G ^ ⁢ ( x , s ) := ∫ 0 s g ^ ⁢ ( x , t ) ⁢ d t .

Assumptions 3.1–3.2 ensure that ℰw is of class C1 and weakly sequentially lower semicontinuous; see, e.g., [8, Lemma 3.1]. Under the additional condition

(3.13) d M + d < c 1 p ⁢ c 2 for all ⁢ M > 0 ,

it turns out also coercive, as the next lemma shows.

Lemma 3.7.

Let B be a nonempty bounded set in C1⁢(Ω¯). If Assumptions 3.1–3.2 and condition (3.13) hold true, then there exist α1∈(0,1), α2>0 such that

ℰ w ⁢ ( u ) ≥ α 1 p ⁢ ∥ u ∥ 1 , p p - α 2 ⁢ ( 1 + ∥ u ∥ 1 , p ) for all ⁢ ( u , w ) ∈ W 1 , p ⁢ ( Ω ) × ℬ .

Proof.

Put M^:=supw∈ℬ∥w∥C1⁢(Ω¯). By (3.11)–(3.12), Proposition 2.7 entails

ℰ w ⁢ ( u ) ≥ c 2 p ⁢ ∥ ∇ ⁡ u ∥ p p + β p ⁢ ∥ u ∥ p , ∂ ⁡ Ω p - ∫ Ω [ f ⁢ ( ⋅ , u ¯ , ∇ ⁡ w ) + g ⁢ ( ⋅ , u ¯ ) ] ⁢ u ¯ ⁢ d x

- ∫ Ω ( u > u ¯ ) ( ∫ u ¯ u f ( ⋅ , t , ∇ w ) d t ) d x - ∫ Ω ( u > u ¯ ) ( ∫ u ¯ u g ( ⋅ , t ) d t ) d x .

Assumption 3.1 along with Hölder’s inequality imply

∫ Ω ( u > u ¯ ) ( ∫ u ¯ u f ( ⋅ , t , ∇ w ) d t ) d x ≤ ∫ Ω ( u > u ¯ ) ( ∫ 0 u f ( ⋅ , t , ∇ w ) d t ) d x

≤ c M ^ ⁢ | Ω | 1 p ′ ⁢ ∥ u ∥ p + d M ^ p ⁢ ∥ u ∥ p p ≤ c M ^ ⁢ | Ω | 1 p ′ ⁢ ∥ u ∥ 1 , p + d M ^ p ⁢ ∥ u ∥ 1 , p p .

Exploiting (3.7), Assumption 3.2 (i), (ii), and Hölder’s inequality again, we have

∫ Ω ( u > u ¯ ) ( ∫ u ¯ u g ( ⋅ , t ) d t ) d x ≤ ∫ Ω ( u > u ¯ ) ( ∫ u ¯ 1 g ( ⋅ , t ) d t ) d x + ∫ Ω ( u > 1 ) ( ∫ 1 u g ( ⋅ , t ) d t ) d x

≤ ∫ Ω ( u > u ¯ ) g ( ⋅ , u ¯ ) d x + ∫ Ω ( u > 1 ) ( ∫ 1 u ( c + d t p - 1 ) d t ) d x

≤ ∫ Ω g ⁢ ( ⋅ , u ¯ ) ⁢ d x + c ⁢ | Ω | 1 p ′ ⁢ ∥ u ∥ p + d p ⁢ ∥ u ∥ p p

≤ ∫ Ω g ⁢ ( ⋅ , u ¯ ) ⁢ d x + c ⁢ | Ω | 1 p ′ ⁢ ∥ u ∥ 1 , p + d p ⁢ ∥ u ∥ 1 , p p .

Hence, through (2.1) we easily arrive at

ℰ w ⁢ ( u ) ≥ c 2 p ⁢ ∥ u ∥ β , 1 , p p - d M ^ + d p ⁢ ∥ u ∥ 1 , p p - ( c M ^ + c ) ⁢ | Ω | 1 p ′ ⁢ ∥ u ∥ p - K

≥ c 1 p ⁢ c 2 - d M ^ - d p ⁢ ∥ u ∥ 1 , p p - ( c M ^ + c ) ⁢ | Ω | 1 p ′ ⁢ ∥ u ∥ 1 , p - K

≥ c 1 p ⁢ c 2 - d M ^ - d p ⁢ ∥ u ∥ 1 , p p - max ⁡ { ( c M ^ + c ) ⁢ | Ω | 1 p ′ , K } ⁢ ( 1 + ∥ u ∥ 1 , p ) ,

where

K := ∫ Ω [ f ⁢ ( ⋅ , u ¯ , ∇ ⁡ w ) + g ⁢ ( ⋅ , u ¯ ) ] ⁢ u ¯ ⁢ d x + ∫ Ω g ⁢ ( ⋅ , u ¯ ) ⁢ d x

≤ ∫ Ω ( c M ^ + d M ^ ) ⁢ d x + 2 ⁢ ∫ Ω g ⁢ ( ⋅ , ε ⁢ θ ) ⁢ d x

≤ ( c M ^ + d M ^ ) ⁢ | Ω | + 2 ⁢ ∥ g ⁢ ( ⋅ , ε ⁢ θ ) ∥ p ′ ⁢ | Ω | 1 p ,

due to Assumption 3.1 and (3.7)–(3.8). Now, the conclusion follows from (3.13). ∎

Remark 3.8.

A standard application of Moser’s iteration technique [11] shows that any solution to (3.10) lies in L∞⁢(Ω). By Liebermann’s regularity theory [12], it actually is Hölder continuous up to the boundary.

Lemma 3.9.

Let Assumptions 3.1–3.2 and condition (3.13) be satisfied. Then

∅ ≠ Crit ⁢ ( ℰ w ) ⊆ U w ∩ { u ∈ C 1 ⁢ ( Ω ¯ ) : u ≥ u ¯ } .

Proof.

Since ℰw is coercive (cf. Lemma 3.7), the Weierstrass–Tonelli theorem produces Crit⁢(ℰw)≠∅. Pick any u∈Crit⁢(ℰw), test (3.10) with (u¯-u)+, and exploit (3.11)–(3.12), besides (3.9), to achieve

∫ Ω 〈 a ( ∇ u ) , ∇ ( u ¯ - u ) + 〉 d x + β ∫ ∂ ⁡ Ω | u | p - 2 u ( u ¯ - u ) + d σ = ∫ Ω [ f ^ ( ⋅ , u ) + g ^ ( ⋅ , u ) ] ( u ¯ - u ) + d x

≥ ∫ Ω g ^ ⁢ ( ⋅ , u ) ⁢ ( u ¯ - u ) + ⁢ d x = ∫ Ω g ⁢ ( ⋅ , u ¯ ) ⁢ ( u ¯ - u ) + ⁢ d x

≥ ∫ Ω 〈 a ( ∇ u ¯ ) , ∇ ( u ¯ - u ) + 〉 d x + β ∫ ∂ ⁡ Ω | u ¯ | p - 2 u ¯ ( u ¯ - u ) + d σ .

Rearranging terms, we get

∫ Ω 〈 a ( ∇ u ¯ ) - a ( ∇ u ) , ∇ ( u ¯ - u ) + 〉 d x + β ∫ ∂ ⁡ Ω ( | u ¯ | p - 2 u ¯ - | u | p - 2 u ) ( u ¯ - u ) + d σ ≤ 0 .

The strict monotonicity of a, combined with [18, Lemma A.0.5], entail

∇ ( u ¯ - u ) + = 0 in Ω , ( u ¯ - u ) + = 0 on ∂ Ω .

So, ∥(u¯-u)+∥β,1,p=0, which means u≥u¯. Finally, by (3.11)–(3.12), one has u∈Uw, while u∈C1⁢(Ω¯) according to Remark 3.8. ∎

For every w∈C1⁢(Ω¯), we define

𝒮 ⁢ ( w ) := { u ∈ C 1 ⁢ ( Ω ¯ ) : u ∈ U w , u ≥ u ¯ , ℰ w ⁢ ( u ) < 1 } .

Lemma 3.10.

Under Assumptions 3.1–3.2 and condition (3.13), the multifunction S:C1⁢(Ω¯)→2C1⁢(Ω¯) takes nonempty values, and maps bounded sets into relatively compact sets.

Proof.

If w∈C1⁢(Ω¯), then there exists u^w∈Crit⁢(ℰw) such that

u ^ w ∈ C 1 ⁢ ( Ω ¯ ) , u ^ w ≥ u ¯ , ℰ w ⁢ ( u ^ w ) = inf W 1 , p ⁢ ( Ω ) ⁡ ℰ w ≤ ℰ w ⁢ ( 0 ) = 0 < 1 ,

cf. the proof of Lemma 3.9. Hence, 𝒮⁢(w)≠∅, because u^w∈𝒮⁢(w). Let ℬ⊆C1⁢(Ω¯) be nonempty and bounded. From Lemma 3.7, it follows that

α 1 p ⁢ ∥ u ∥ 1 , p p - α 2 ⁢ ( 1 + ∥ u ∥ 1 , p ) ≤ ℰ w ⁢ ( u ) < 1 for all ⁢ u ∈ 𝒮 ⁢ ( w ) , w ∈ ℬ ,

whence 𝒮⁢(ℬ) turns out bounded in W1,p⁢(Ω). By nonlinear regularity theory [12], the same holds when C1,α⁢(Ω¯), with suitable α∈(0,1), replaces W1,p⁢(Ω). Recalling that C1,α⁢(Ω¯)↪C1⁢(Ω¯) compactly yields the conclusion. ∎

To see that 𝒮 is lower semicontinuous, we shall employ the next technical lemma.

Lemma 3.11.

Let α,β,γ>0, let 1<p<+∞, and let {ak}⊆[0,+∞) satisfy the recursive relation

(3.14) α ⁢ a k p ≤ β ⁢ a k + γ ⁢ a k - 1 p for all ⁢ k ∈ ℕ .

If γ<α, then the sequence {ak} is bounded.

Proof.

Using the obvious inequality

a k ≤ T + T 1 - p ⁢ a k p , T > 0 ,

(3.14) becomes

( α - β ⁢ T 1 - p ) ⁢ a k p ≤ β ⁢ T + γ ⁢ a k - 1 p for all ⁢ k ∈ ℕ .

Since σ:=1/p<1, this entails

( α - β ⁢ T 1 - p ) σ ⁢ a k ≤ ( β ⁢ T + γ ⁢ a k - 1 p ) σ ≤ ( β ⁢ T ) σ + γ σ ⁢ a k - 1

or, equivalently,

(3.15) a k ≤ ( β ⁢ T α - β ⁢ T 1 - p ) σ + ( γ α - β ⁢ T 1 - p ) σ ⁢ a k - 1 , k ∈ ℕ ,

provided T>0 is large enough. Choosing T>(βα-γ)1p-1, the coefficient of ak-1 turns out strictly less than 1. A standard computation based on (3.15) completes the proof. ∎

Lemma 3.12.

Suppose Assumptions 3.1–3.2 hold and, moreover,

(3.16) d M + d < c 1 p ⁢ c 2 p for all ⁢ M > 0 .

Then the multifunction S:C1⁢(Ω¯)→2C1⁢(Ω¯) is lower semicontinuous.

Proof.

The proof is patterned after that of [13, Lemma 20]. So, some details will be omitted. Let

(3.17) w n → w in ⁢ C 1 ⁢ ( Ω ¯ ) .

We claim that to each u~∈𝒮⁢(w), there corresponds a sequence {un}⊆C1⁢(Ω¯) enjoying the following properties:

u n ∈ 𝒮 ⁢ ( w n ) , n ∈ ℕ , u n → u ~ in ⁢ C 1 ⁢ ( Ω ¯ ) .

Fix u~∈𝒮⁢(w). For every n∈ℕ, consider the auxiliary problem

(3.18) P ( u ~ , w n ) = { - div ⁡ a ⁢ ( ∇ ⁡ u ) = f ⁢ ( x , u ~ , ∇ ⁡ w n ) + g ^ ⁢ ( x , u ~ ) in ⁢ Ω , u > 0 in ⁢ Ω , ∂ ⁡ u ∂ ⁡ ν a + β ⁢ u p - 1 = 0 on ⁢ ∂ ⁡ Ω ,

with g^⁢(x,s) given by (3.12). One has g^⁢(x,u~)=g⁢(x,u~), because u~∈𝒮⁢(w), while the associated energy functional reads

ℰ u ~ , w n ( u ) := 1 p ∫ Ω G ( ∇ u ) d x + β ∫ ∂ ⁡ Ω | u | p d σ - ∫ Ω f ( x , u ~ , ∇ w n ) u d x - ∫ Ω g ^ ( x , u ~ ) u d x , u ∈ W 1 , p ( Ω ) .

Since ℰu~,wn turns out strictly convex, the same argument exploited to show Lemma 3.9 yields here a unique solution un0∈int⁢(C1⁢(Ω¯)+) of (3.18) such that

(3.19) ℰ u ~ , w n ⁢ ( u n 0 ) ≤ 0 .

Via (3.17)–(3.19), reasoning as in Lemmas 3.7 and 3.10 (but for ℰu~,w instead of ℰw and ℬ:={wn:n∈ℕ}), we deduce that {un0}⊆C1⁢(Ω¯) is relatively compact. Consequently, un0→u0 in C1⁢(Ω¯), where a subsequence is considered when necessary. By (3.17) again and Lebesgue’s dominated convergence theorem, u0 solves problem P⁢(u~,w). Thus, a fortiori, u0=u~, because P⁢(u~,w) possesses one solution at most. An induction procedure provides now a sequence {unk} such that unk solves problem P⁢(unk-1,wn), the inequality ℰunk-1,wn⁢(unk)≤0 holds, and

(3.20) lim n → + ∞ ⁡ u n k = u ~ in ⁢ C 1 ⁢ ( Ω ¯ ) ⁢ for all ⁢ k ∈ ℕ .

Claim: {unk}k∈N⊆C1⁢(Ω¯) is relatively compact. In fact, recalling (3.17), pick M=supn∈ℕ∥wn∥C1⁢(Ω¯). Through Hölder’s and Young’s inequalities, besides (3.8), we obtain

(3.21) 1 p ∫ Ω G ( ∇ u n k ) d x + β p ∫ ∂ ⁡ Ω | u n k | p d σ ≥ c 1 p ⁢ c 2 p ∥ u n k ∥ 1 , p p ,

∫ Ω f ( ⋅ , u n k - 1 , ∇ w n ) u n k d x ≤ c M | Ω | 1 p ′ ∥ u n k ∥ p + d M ∫ Ω | u n k - 1 | p - 1 | u n k | d x

(3.22) ≤ c M ⁢ | Ω | 1 p ′ ⁢ ∥ u n k ∥ p + d M ⁢ ( 1 p ′ ⁢ ∥ u n k - 1 ∥ p p + 1 p ⁢ ∥ u n k ∥ p p ) ,

as well as

∫ Ω g ^ ⁢ ( ⋅ , u n k - 1 ) ⁢ u n k ⁢ d x = ∫ Ω ( u n k - 1 ≤ 1 ) g ^ ⁢ ( ⋅ , u n k - 1 ) ⁢ u n k ⁢ d x + ∫ Ω ( u n k - 1 > 1 ) g ^ ⁢ ( ⋅ , u n k - 1 ) ⁢ u n k ⁢ d x

≤ ∫ Ω ( u n k - 1 ≤ 1 ) g ⁢ ( ⋅ , u ¯ ) ⁢ u n k ⁢ d x + ∫ Ω ( u n k - 1 > 1 ) g ⁢ ( ⋅ , u n k - 1 ) ⁢ u n k ⁢ d x

≤ ( ∥ g ( ⋅ , u ¯ ) ∥ p ′ + c | Ω | 1 p ′ ) ∥ u n k ∥ p + d ∫ Ω | u n k - 1 | p - 1 | u n k | d x

(3.23) ≤ ( ∥ g ⁢ ( ⋅ , u ¯ ) ∥ p ′ + c ⁢ | Ω | 1 p ′ ) ⁢ ∥ u n k ∥ p + d ⁢ ( 1 p ′ ⁢ ∥ u n k - 1 ∥ p p + 1 p ⁢ ∥ u n k ∥ p p ) .

Since ℰunk-1,wn⁢(unk)≤0, estimates (3.21)–(3.23) entail

c 1 p ⁢ c 2 - d M - d p ⁢ ∥ u n k ∥ 1 , p p ≤ ( ∥ g ⁢ ( ⋅ , u ¯ ) ∥ p ′ + ( c M + c ) ⁢ | Ω | 1 p ′ ) ⁢ ∥ u n k ∥ 1 , p + d M + d p ′ ⁢ ∥ u n k - 1 ∥ 1 , p p

for all k∈ℕ. Thanks to (3.16), Lemma 3.11 applies, and the sequence {unk}k∈ℕ turns out bounded in W1,p⁢(Ω). Standard arguments involving regularity up to the boundary (cf. the proof of Lemma 3.10) yield the claim.

We may thus assume there exists {un}⊆C1⁢(Ω¯) fulfilling

(3.24) lim k → ∞ ⁡ u n k = u n in ⁢ C 1 ⁢ ( Ω ¯ )

whenever n∈ℕ. By (3.24) and Lebesgue’s dominated convergence theorem, one has un∈Uwn. Moreover, as in the proof of Lemma 3.9, un≥u¯. Due to (3.20) and (3.24), the double limit lemma [6, Proposition A.2.35] gives

(3.25) u n → u ~ in ⁢ C 1 ⁢ ( Ω ¯ ) .

Thus, it remains to show that ℰwn⁢(un)<1. From (3.17), we easily infer ℰwn⁢(u~)→ℰw⁢(u~). Since ℰwn is of class C1, via (3.17) and (3.25), one arrives at

lim n → + ∞ ⁡ ( ℰ w n ⁢ ( u n ) - ℰ w ⁢ ( u ~ ) ) = 0 ,

namely ℰwn⁢(un)→ℰw⁢(u~). This completes the proof, because u~∈𝒮⁢(w), whence ℰw⁢(u~)<1. ∎

Lemma 3.13.

Under Assumptions 3.1–3.2 and condition (3.13), the set S⁢(w), w∈C1⁢(Ω¯), is downward directed.

Proof.

Let u1,u2∈𝒮⁢(w) and let u^:=min⁡{u1,u2}. By Lemma 3.4, we have u^∈U¯w. Consider the problem

(3.26) { - div ⁡ a ⁢ ( ∇ ⁡ u ) = h ⁢ ( x , u ) in ⁢ Ω , u > 0 in ⁢ Ω , ∂ ⁡ u ∂ ⁡ ν a + β ⁢ u p - 1 = 0 on ⁢ ∂ ⁡ Ω ,

where

h ⁢ ( x , s ) = { f ⁢ ( x , u ¯ ⁢ ( x ) , ∇ ⁡ w ⁢ ( x ) ) + g ⁢ ( x , u ¯ ⁢ ( x ) ) if ⁢ s ≤ u ¯ ⁢ ( x ) , f ⁢ ( x , s , ∇ ⁡ w ⁢ ( x ) ) + g ⁢ ( x , s ) if ⁢ u ¯ ⁢ ( x ) < s < u ^ ⁢ ( x ) , f ⁢ ( x , u ^ ⁢ ( x ) , ∇ ⁡ w ⁢ ( x ) ) + g ⁢ ( x , u ^ ⁢ ( x ) ) if ⁢ s ≥ u ^ ⁢ ( x ) .

The associated energy functional reads

ℰ ~ w ( u ) := 1 p ∫ Ω G ( ∇ u ) d x + β ∫ ∂ ⁡ Ω | u | p d x - ∫ Ω d x ∫ 0 u h ( ⋅ , t ) d t , u ∈ W 1 , p ( Ω ) .

Arguing as in Lemma 3.10 produces a solution u~∈C1⁢(Ω¯) to (3.26) such that ℰ~w⁢(u~)≤0. Next, adapt the proof of Lemma 3.9 and exploit the fact that u^ is a supersolution of (3.26) to achieve u¯≤u~≤u^. Consequently, u~∈Uw and

ℰ w ⁢ ( u ~ ) = ℰ ~ w ⁢ ( u ~ ) ≤ 0 < 1 .

This forces u~∈𝒮⁢(w), besides u~≤min⁡{u1,u2}. ∎

Lemma 3.14.

If Assumptions 3.1–3.2 and condition (3.13) hold true, then for every w∈C1⁢(Ω¯), the set S⁢(w) possesses absolute minimum.

Proof.

Fix w∈C1⁢(Ω¯). We already know (see Lemma 3.13) that 𝒮⁢(w) turns out downward directed. If 𝒞⊆𝒮⁢(w) is a chain in 𝒮⁢(w), then there exists a sequence {un}⊆𝒮⁢(w) satisfying

lim n → ∞ ⁡ u n = inf ⁡ 𝒞 .

On account of Lemma 3.10 and up to subsequences, one has un→u^ in C1⁢(Ω¯). Thus, u^=inf⁡𝒞. By Zorn’s lemma, 𝒮⁢(w) admits a minimal element uw. It remains to show that uw=min⁡𝒮⁢(w). Pick any u∈𝒮⁢(w). Through Lemma 3.13 we get u~∈𝒮⁢(w) such that u~≤min⁡{uw,u}. The minimality of uw entails uw=u~. Therefore, uw≤u, as desired. ∎

Remark 3.15.

This proof is patterned after the one in [13, Theorem 23].

Lemma 3.14 allows to consider the function Γ:C1⁢(Ω¯)→C1⁢(Ω¯) given by

Γ ⁢ ( w ) := min ⁡ 𝒮 ⁢ ( w ) for all ⁢ w ∈ C 1 ⁢ ( Ω ¯ ) .

Lemma 3.16.

Under Assumptions 3.1–3.2 and condition (3.16), Γ is continuous and maps bounded sets into relatively compact sets.

Proof.

The proof is analogous to that of [13, Lemma 24]. So, we will omit details. Let ℬ⊆C1⁢(Ω¯) be bounded. Since Γ⁢(ℬ)⊆𝒮⁢(ℬ) and 𝒮⁢(ℬ) turns out relatively compact (cf. Lemma 3.10), Γ⁢(ℬ) enjoys the same property. Next, suppose wn→w in C1⁢(Ω¯). Setting un:=Γ⁢(wn), one evidently has un→u in C1⁢(Ω¯), where a subsequence is considered when necessary. The function u complies with u≥u¯ and ℰw⁢(u)<1 (see the proof of Lemma 3.12). Via the Lebesgue dominated convergence theorem, from un∈Uwn, it follows u∈Uw. Plugging them all together, we get u∈𝒮⁢(w). It remains to verify that u=Γ⁢(w). Lemma 3.12 provides a sequence {vn}⊆C1⁢(Ω¯) fulfilling both vn∈𝒮⁢(wn) for all n∈ℕ and vn→Γ⁢(w) in C1⁢(Ω¯). The choice of Γ entails un=Γ⁢(wn)≤vn, besides Γ⁢(w)≤u. Letting n→+∞, we thus arrive at

Γ ⁢ ( w ) ≤ u = lim n → + ∞ ⁡ u n ≤ lim n → + ∞ ⁡ v n = Γ ⁢ ( w ) ,

i.e., u=Γ⁢(w), which completes the proof. ∎

To establish our main result, the stronger version below of Assumption 3.1 will be employed.

Assumption 3.17.

f : Ω × ℝ × ℝ N → [ 0 , + ∞ ) is a Carathéodory function such that

f ⁢ ( x , s , ξ ) ≤ c 3 + c 4 ⁢ | s | p - 1 + c 5 ⁢ | ξ | p - 1 for all ⁢ ( x , s , ξ ) ∈ Ω × ℝ × ℝ N ,

with appropriate c3,c4,c5>0.

Condition (3.13) is substituted by

(3.27) c 4 + ( 2 ⁢ p - 1 ) ⁢ c 5 + d < c 1 p ⁢ c 2 .

Remark 3.18.

Assumption 3.17 clearly implies Assumption 3.1, with cM:=c3+c5⁢Mp-1 and dM:=c4. Likewise, (3.27) forces (3.13) while (3.16) reads as

(3.28) c 4 + d < c 1 p ⁢ c 2 p .

Theorem 3.19.

Let Assumptions 3.17, 3.2 and conditions (3.27)–(3.28) be satisfied. Then problem (1.1) possesses a solution u∈int⁢(C1⁢(Ω¯)+). The set of solutions to (1.1) is compact in C1⁢(Ω¯).

Proof.

Define

Λ ⁢ ( Γ ) := { u ∈ C 1 ⁢ ( Ω ¯ ) : u = τ ⁢ Γ ⁢ ( u ) ⁢ for some ⁢ τ ∈ ( 0 , 1 ) } .

Claim : Λ⁢(Γ) is bounded in W1,p⁢(Ω). To see this, pick any u∈Λ⁢(Γ). Since uτ=Γ⁢(u)∈𝒮⁢(u), one has ℰu⁢(uτ)<1. Assumption 3.17, combined with Young’s and Hölder’s inequalities, produces

∫ Ω ( u τ > u ¯ ) ( ∫ u ¯ u τ f ( ⋅ , t , ∇ u ) d t ) d x ≤ ∫ Ω ( ∫ 0 u τ ( c 3 + c 4 t p - 1 + c 5 | ∇ u | p - 1 ) d t ) d x

≤ c 3 ∥ u τ ∥ 1 + c 4 p ∥ u τ ∥ p p + c 5 ∫ Ω | ∇ u | p - 1 | u τ | d x

≤ c 3 ⁢ | Ω | 1 p ′ ⁢ ∥ u τ ∥ p + c 4 p ⁢ ∥ u τ ∥ p p + c 5 ⁢ ( ∥ u τ ∥ p p p + ∥ ∇ ⁡ u ∥ p p p ′ )

≤ c 3 ⁢ | Ω | 1 p ′ ⁢ ∥ u τ ∥ 1 , p + c 4 + c 5 p ⁢ ∥ u τ ∥ 1 , p p + c 5 p ′ ⁢ ∥ u ∥ 1 , p p .

Analogously, on account of (3.7),

∫ Ω f ⁢ ( ⋅ , u ¯ , ∇ ⁡ u ) ⁢ u ¯ ⁢ d x ≤ ∫ Ω ( c 3 + c 4 ⁢ u ¯ p - 1 + c 5 ⁢ | ∇ ⁡ u | p - 1 ) ⁢ u ¯ ⁢ d x

≤ ( c 3 + c 4 + c 5 p ) ⁢ | Ω | + c 5 p ′ ⁢ ∥ ∇ ⁡ u ∥ p p

≤ ( c 3 + c 4 + c 5 p ) ⁢ | Ω | + c 5 p ′ ⁢ ∥ u ∥ 1 , p p .

Reasoning as in Lemma 3.7 and recalling that τ∈(0,1), we thus achieve

1 > ℰ u ⁢ ( u τ ) ≥ c 1 p ⁢ c 2 - c 4 - ( 2 ⁢ p - 1 ) ⁢ c 5 - d p ⁢ ∥ u τ ∥ 1 , p p - ( c 3 + c ) ⁢ | Ω | 1 p ′ ⁢ ∥ u τ ∥ 1 , p - K ′ ,

where

K ′ := ( c 3 + c 4 + c 5 p ) ⁢ | Ω | + 2 ⁢ ∥ g ⁢ ( ⋅ , ε ⁢ θ ) ∥ p ′ ⁢ | Ω | 1 p .

Thanks to (3.27), the above inequalities force

∥ u ∥ 1 , p ≤ ∥ u τ ∥ 1 , p ≤ K * ,

with K*>0 independent of u and τ. Thus, the claim is proved.

By regularity [12], the set Λ⁢(Γ) turns out bounded in C1⁢(Ω¯). Hence, due to Lemma 3.16, Theorem 2.1 applies, which entails Fix⁡(Γ)≠∅. Let u∈Fix⁡(Γ). From u=Γ⁢(u)∈𝒮⁢(u), we deduce both u≥u¯ and u∈Uu. Accordingly,

f ^ ⁢ ( ⋅ , u ) = f ⁢ ( ⋅ , u , ∇ ⁡ u ) , g ^ ⁢ ( ⋅ , u ) = g ⁢ ( ⋅ , u ) ,

namely, the function u solves problem (1.1). Further, u∈int⁢(C1⁢(Ω¯)+) because of the strong maximum principle.

Finally, arguing as in Lemma 3.8 ensures that each solution to (1.1) lies in C1,α⁢(Ω¯). Since C1,α⁢(Ω¯)↪C1⁢(Ω¯) compactly and the solution set of (1.1) is closed in C1⁢(Ω¯), the conclusion follows. ∎

Remark 3.20.

The same techniques can be applied for finding solutions to the Neumann problem

{ - div ⁡ a ⁢ ( ∇ ⁡ u ) + | u | p - 2 ⁢ u = f ⁢ ( x , u , ∇ ⁡ u ) + g ⁢ ( x , u ) in ⁢ Ω , u > 0 in ⁢ Ω , ∂ ⁡ u ∂ ⁡ ν a = 0 on ⁢ ∂ ⁡ Ω .

In fact, it is enough to replace the norm ∥⋅∥β,1,p with the standard one ∥⋅∥1,p.

4 Uniqueness (for p=2)

Throughout this section, p=2, the operator a fulfills Assumption 2.5, while the nonlinearities f and g comply with Assumptions 3.1 and 3.2, respectively. The following further conditions will be posited:

There exists c6∈(0,1] such that

〈 a ⁢ ( ξ ) - a ⁢ ( η ) , ξ - η 〉 ≥ c 6 ⁢ | ξ - η | 2 for all ⁢ ξ , η ∈ ℝ N .
With appropriate c7,c8>0, one has

(4.1) [ f ⁢ ( x , s , ξ ) - f ⁢ ( x , t , ξ ) ] ⁢ ( s - t ) ≤ c 7 ⁢ | s - t | 2 ,

(4.2) | f ⁢ ( x , t , ξ ) - f ⁢ ( x , t , η ) | ≤ c 8 ⁢ | ξ - η |

in Ω×ℝ×ℝN.
There exists c9>0 such that

(4.3) [ g ⁢ ( x , s ) - g ⁢ ( x , t ) ] ⁢ ( s - t ) ≤ c 9 ⁢ | s - t | 2 for all ⁢ x ∈ Ω , s , t ∈ [ 1 , + ∞ ) .

Moreover,

(4.4) g ⁢ ( x , s ) ≤ g ⁢ ( x , 1 ) in ⁢ Ω × ( 1 , + ∞ ) .

Example 4.1.

The parametric (2,q)-Laplacian Δ+μ⁢Δq, where 1<q<2, μ≥0, satisfies Assumption 2.5 and (C1), cf. [18, Lemma A.0.5].

Theorem 4.2.

Under the above assumptions, problem (1.1) admits a unique solution, provided

(4.5) c 7 + c 1 ⁢ c 8 + c 9 < c 1 2 ⁢ c 6 .

Proof.

Suppose u,v solve (1.1). Test with u-v and subtract to arrive at

∫ Ω 〈 a ( ∇ u ) - a ( ∇ v ) , ∇ ( u - v ) 〉 d x + β ∫ ∂ ⁡ Ω | u - v | 2 d σ

(4.6) = ∫ Ω [ f ⁢ ( ⋅ , u , ∇ ⁡ u ) - f ⁢ ( ⋅ , v , ∇ ⁡ v ) ] ⁢ ( u - v ) ⁢ d x + ∫ Ω [ g ⁢ ( ⋅ , u ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x .

The left-hand side of (4.6) can easily be estimated from below via (C1) as follows:

(4.7) ∫ Ω 〈 a ( ∇ u ) - a ( ∇ v ) , ∇ ( u - v ) 〉 d x + β ∫ ∂ ⁡ Ω | u - v | 2 d σ ≥ c 6 ∥ u - v ∥ β , 1 , 2 2 .

Using (4.1)–(4.2) and Hölder’s inequality, we get

∫ Ω [ f ⁢ ( ⋅ , u , ∇ ⁡ u ) - f ⁢ ( ⋅ , v , ∇ ⁡ v ) ] ⁢ ( u - v ) ⁢ d x

= ∫ Ω [ f ⁢ ( ⋅ , u , ∇ ⁡ u ) - f ⁢ ( ⋅ , v , ∇ ⁡ u ) ] ⁢ ( u - v ) ⁢ d x + ∫ Ω [ f ⁢ ( ⋅ , v , ∇ ⁡ u ) - f ⁢ ( ⋅ , v , ∇ ⁡ v ) ] ⁢ ( u - v ) ⁢ d x

≤ c 7 ∫ Ω | u - v | 2 d x + c 8 ∫ Ω | ∇ u - ∇ v | | u - v | d x

≤ c 7 ⁢ ∥ u - v ∥ 2 2 + c 8 ⁢ ∥ ∇ ⁡ ( u - v ) ∥ 2 ⁢ ∥ u - v ∥ 2

(4.8) ≤ c 7 c 1 2 ⁢ ∥ u - v ∥ β , 1 , 2 2 + c 8 c 1 ⁢ ∥ u - v ∥ β , 1 , 2 2 .

Observe now that

∫ Ω [ g ⁢ ( ⋅ , u ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x = ∫ Ω ( max { u , v } ≤ 1 ) [ g ⁢ ( ⋅ , u ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x

+ ∫ Ω ( min { u , v } > 1 ) [ g ⁢ ( ⋅ , u ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x

+ ∫ Ω ( u ≤ 1 < v ) [ g ⁢ ( ⋅ , u ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x

(4.9) + ∫ Ω ( v ≤ 1 < u ) [ g ⁢ ( ⋅ , u ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x .

By Assumption 3.2 (i), one has

(4.10) ∫ Ω ( max { u , v } ≤ 1 ) [ g ⁢ ( ⋅ , u ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x ≤ 0 .

Inequality (4.3) entails

(4.11) ∫ Ω ( min { u , v } > 1 ) [ g ⁢ ( ⋅ , u ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x ≤ c 9 ⁢ ∥ u - v ∥ 2 2 ≤ c 9 c 1 2 ⁢ ∥ u - v ∥ β , 1 , 2 2 .

Thanks to Assumption 3.2 (i) again and (4.4), we obtain

(4.12) ∫ Ω ( u ≤ 1 < v ) [ g ⁢ ( ⋅ , u ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x ≤ ∫ Ω ( u ≤ 1 < v ) [ g ⁢ ( ⋅ , 1 ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x ≤ 0 .

Likewise,

(4.13) ∫ Ω ( v ≤ 1 < u ) [ g ⁢ ( ⋅ , u ) - g ⁢ ( ⋅ , v ) ] ⁢ ( u - v ) ⁢ d x ≤ 0 .

Plugging (4.10)–(4.13) into (4.9) and (4.7)–(4.9) into (4.6) yields

c 6 ⁢ ∥ u - v ∥ β , 1 , 2 2 ≤ ( c 7 c 1 2 + c 8 c 1 + c 9 c 1 2 ) ⁢ ∥ u - v ∥ β , 1 , 2 2 .

On account of (4.5), this directly leads to u=v, as desired. ∎

Remark 4.3.

The conditions that guarantee existence or uniqueness, namely, (3.27), (3.28) and (4.5), represent a balance between data (growth or variation of reaction terms) and structure (driving operator and domain) of the problem .

Remark 4.4.

The choice p=2 directly stems from the technical approach adopted in proving Theorem 4.2. To treat the general case, a natural attempt is to replace both |ξ-η|2 and |s-t|2 by |ξ-η|p and |s-t|p, respectively, in conditions (C1)–(C3). However, if p>2, then (4.1)–(4.3) imply that f⁢(x,⋅,⋅) as well as g⁢(x,⋅) are constants, whereas even the p-Laplacian would not meet (C1) for 1<p<2.

Communicated by Patrizia Pucci

Funding statement: This work is performed within PTR 2018–2020 - Linea di intervento 2: “Metodi Variazionali ed Equazioni Differenziali” of the University of Catania and partly funded by Research project of MIUR (Italian Ministry of Education, University and Research) Prin 2017 “Nonlinear Differential Problems via Variational, Topological and Set-valued Methods” (Grant Number 2017AYM8XW).

Acknowledgements

The authors wish to thank the referee for insightful remarks that helped to improve the paper.

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Received: 2020-04-22

Revised: 2020-05-09

Accepted: 2020-05-10

Published Online: 2020-05-29

Published in Print: 2020-11-01

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

Articles in the same Issue

https://doi.org/10.1515/ans-2020-2093

Keywords for this article

Robin Problem; Quasilinear Elliptic Equation; Gradient Dependence; Singular Term

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