Singular quasilinear convective elliptic systems in ℝN

Umberto Guarnotta; Salvatore Angelo Marano; Abdelkrim Moussaoui

doi:10.1515/anona-2021-0208

Open Access Published by De Gruyter January 6, 2022

Singular quasilinear convective elliptic systems in ℝ^N

Umberto Guarnotta , Salvatore Angelo Marano and Abdelkrim Moussaoui

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2021-0208

Abstract

The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.

Keywords: quasilinear elliptic system; gradient dependence; singular term; entire solution; strong solution

MSC 2010: 35J47; 35J75; 35B08; 35D35

1 Introduction and main result

In this paper, we deal with the problem

(P) −Δpu=f(x,u,v,∇u,∇v)inRN,−Δqv=g(x,u,v,∇u,∇v)inRN,u,v>0inRN,

where N≥3,2−1N<p,q<N,Δrz:=div(|∇z|r−2∇z) denotes the r-Laplacian of z for 1 < r < +∞, while f,g:RN×(0,+∞)2×R2N→(0,+∞) are Carathéodory functions satisfying assumptions H1–H₃ below.

Problem (P) exhibits three interesting features:

The reaction terms f and g can be singular at zero.
f , g depend on the gradient of solutions.
Equations are set in the whole space R^N.
However, they give rise to some nontrivial difficulties, such as the loss of variational structure and the lack of compactness for Sobolev embedding. This work continues the study started in [32], whose setting was R^N and convective terms did not appear, along the very recent papers [6, 23, 24, 31], which address analogous questions, but concerning a bounded domain.

Primarily, we need an appropriate functional framework where to treat the problem, mainly because the integrability properties of solutions and their gradients may differ at infinity, as the example in [20, p. 80] shows. Accordingly, one is led to employ the so-called Beppo Levi (or homogeneous Sobolev) spaces D01,r(RN), systematically studied for the first time by Deny and Lions [15]. The monographs [20, 29, 38] provide an exhaustive introduction on the topic.

Let X:=D01,p(RN)×D01,q(RN) and let r^' denote the conjugate exponent of r > 1. A pair (u, v) ∈ X such that u, v > 0 a.e. in R^N is called:

distributional solution to (P) if for every (φ1,φ2)∈C0∞(RN)2 one has

(1.1) ∫RN|∇u|p−2∇u∇φ1dx=∫RNf(⋅,u,v,∇u,∇v)φ1dx,∫RN|∇v|q−2∇v∇φ2dx=∫RNg(⋅,u,v,∇u,∇v)φ2dx;

(weak) solution of (P) when (1.1) holds for all (φ₁, φ₂) ∈ X;
‘strong’ solution to (P) if |∇u|p−2∇u,|∇v|q−2∇v∈Wloc1,2(RN) and the differential equations are satisfied a.e. in R^N.

Obviously, both 2) and 3) force 1), whilst reverse implications turn out generally false; see also Remark 4.5. Moreover, as observed at p. 48 of [38], problems in unbounded domains may admit strong solutions that are not weak or vice-versa. So, the search for strong solutions appears of some interest in this context.

Roughly speaking, our technical approach proceeds as follows. We first solve an auxiliary problem (P^ε), ε > 0, obtained by shifting variables of reactions, which avoids singularities. To do this, nonlinear regularity theory, a priori estimates,Moser’s iteration, trapping region, and fixed point arguments are employed. Unfortunately, bounds from above alone do not allow to get a solution of (P): treating singular terms additionally requires some estimates from below. Theorem 3.1 in [14] ensures that solutions to (P^ε) turn out locally greater than a positive constant regardless of ε. Thus, under hypotheses H₁–H₃ below, we can construct a sequence {(u_ε , v_ε)} ⊆ X such that (u_ε , v_ε) solves (P^ε) for all ε > 0 and whose weak limit as ε → 0⁺ is a distributional solution to (P); cf. Lemma 4.1. Next, a localization-regularization reasoning (see Lemma 4.2) shows that

(u, v) distributional solution ⇒ (u, v) weak solution.

Through a recent differentiability result [10, Theorem 2.1] one then has

(u, v) distributional solution ⇒ (u, v) strong solution;

cf. Lemma 4.3. Further, (u,v)∈Cloc1,α(RN) once condition H₃ is slightly strengthened; see Remark 4.4.

Singular elliptic problems, either in bounded domains or in R^N, have a long history, that traces back to [12, 28] and [8, 9, 13, 25, 27] for semi-linear equations. More recent results, involving also systems, can be found in [17, 32, 33, 36] and the references therein. A very fruitful approach has been developed in [3, 4]; see also [7] and [34]. Existence, regularity, and qualitative properties of the solutions have been investigated, e.g., in [1, 2, 11, 19, 30, 40].

Henceforth, the assumptions below will be posited. If 1 < r < N then, by definition, r∗:=NrN−r.

H₁(f)There exist α1∈(−1,0],β1,δ1∈[0,q−1),γ1∈[0,p−1),m1,mˆ1>0, and a1∈Llocsp(RN), with sp>p′N,

such that

m1a1(x)s1α1s2β1≤f(x,s1,s2,t1,t2)≤mˆ1a1(x)s1α1s2β1+|t1|γ1+|t2|δ1

in R^N × (0,+∞)² × R^2N. Moreover, essBρinfa1>0 for all ρ > 0.

H¹(g)There exist β2∈(−1,0],α2,γ2∈[0,p−1),δ2∈[0,q−1),m2,mˆ2>0, and a2∈Llocsq(RN), with sq>q′N,

such that

m2a2(x)s1α2s2β2≤g(x,s1,s2,t1,t2)≤mˆ2a2(x)s1α2s2β2+|t1|γ2+|t2|δ2

in R^N × (0,+∞)² × R^2N. Moreover, ess infBρa2>0 for all ρ > 0.

H₁(a)There exist ζ₁, ζ₂ ∈ (N, +∞] such that ai∈L1(RN)∩Lζi(RN),i=1,2, where

1ζ1<1−pp∗−θ1,1ζ2<1−qq∗−θ2,

with

θ1:=maxβ1q∗,γ1p,δ1q<1−pp∗,θ2:=maxα2p∗,γ2p,δ2q<1−qq∗.

H₂ If η₁ := max{β₁, δ₁} and η₂ := max{α₂, γ₂} then

η1η2<(p−1−γ1)(q−1−δ2).

H₃ One has

1sp+maxγ1p,δ1q≤12,1sq+maxγ2p,δ2q≤12.

Example 1.1

H₁(a) is fulfilled once a1,a2∈L1(RN)∩L∞(RN) and

maxβ1q∗,γ1p,δ1q<1−pp∗,maxα2p∗,γ2p,δ2q<1−qq∗.

In fact, it suffices to choose ζ₁ := ζ₂ := +∞.

Remark 1.2

By interpolation (see, e.g., [32, Proposition 2.1]), condition H₁(a) entails ai∈Lσi,j(RN),i=1,2, where:

(i) σ1,j:=11−tj,j=1,2,3,4, with

t1=α1+1p∗+β1q∗,t2=1p∗+β1q∗,t3=1p∗+γ1p,t4=1p∗+δ1q;

(ii) σ2,j:=11−tj,j=1,2,3,4, with

t1=β2+1q∗+α2p∗,t2=1q∗+α2p∗,t3=1q∗+γ2p,t4=1q∗+δ2q.

The aim of this paper is to prove the following

Theorem 1.3

Under hypotheses H₁–H₃, problem (P) admits a weak and strong solution (u, v) ∈ X.

2 Preliminaries

Let Z be a Hausdorff topological space and let T : Z → Z be continuous. Following [22, p. 2], the operator T is called compact when T(Z)‾ turns out a compact subset of Z. If Z is a normed space, {z_n}⊆ Z, and z ∈ Z then z_n → z in Z means that the sequence {z_n} strongly converges to z,while z_n z stands for weak convergence. As usual, Z^* denotes the topological dual of Z and Z² := Z × Z.

Hereafter, N ≥ 3 is a fixed integer, B(x, ρ) indicates the open ball in R^N of radius ρ > 0 centered at x ∈ R^N and B_ρ := B(0, ρ), while |E| stands for the Lebesgue measure of E.

Let Z := Z(Ω) be a real-valued function space on a nonempty measurable set Ω ⊆ R^N. If z₁, z₂ ∈ Z and z₁(x) < z₂(x) a.e. in Ω then we simply write z₁ < z₂. The meaning of z₁ ≤ z₂, etc. is analogous. Put

Z+:={z∈Z:z>0}.

Given {z_n}⊆ Z and z ∈ Z, the symbol z_n ↑ z signifies that {z_n} is monotone increasing and z_n(x) → z(x) for almost every x ∈ Ω. Moreover,

z±:=max{±z,0},suppz:={x∈Ω:z(x)≠0}‾.

When Ω := R^N, we write z ∈ Z_loc(R^N) if for every nonempty compact subset K of R^N the restriction z_K belongs to Z(K). Similarly, a sequence {zn}⊆Zloc(RN) is called bounded in Z_loc(R^N) once the same holds for {z_nK} in Z(K), with any K as above.

Let 1 < r < N and let z : R^N → R be a measurable function. Throughout the paper, r′:=rr−1,r∗:=NrN−r,

∥z∥r:=∫RN|z(x)|rdx1/r,∥z∥∞:=esssupx∈RN|z(x)|.

We now recall the notion and some relevant properties of Beppo Levi’s spaceD01,r(RN), addressing the reader to [20, Chapter II] for a complete treatment. Put

D1,r:=z∈Lloc1(RN):|∇z|∈Lr(RN)

and denote by R the equivalence relation that identifies two elements in D^1,r whose difference is a constant. The quotient set D˙1,r, endowed with the norm

∥z∥1,r:=∫RN|∇z(x)|rdx1/r,

turns out complete. Write D01,r(RN) for the subspace of D˙1,r defined as the closure of C0∞(RN) under · _1,r, namely

D01,r(RN):=C0∞(RN)‾∥⋅∥1,r.

D01,r(RN), usually called Beppo Levi space, is reflexive and continuously embeds in Lr∗(RN), i.e.,

(2.1) D01,r(RN)↪Lr∗(RN).

Consequently, if z∈D01,r(RN) then z vanishes at infinity, meaning that the set {x ∈ R^N : |z(x)| ≥ ε} has finite measure for any ε > 0. In fact, by Chebichev’s inequality and (2.1), one has

|{x∈RN:|z(x)|≥ε}|≤ε−r∗∥z∥r∗r∗≤(cε−1∥z∥1,r)r∗<+∞,

where c > 0 is the best constant related to (2.1); see the seminal paper [39].

Hereafter, c, c_ε, and c_ε(·) will denote generic positive constants, which may change explicit value from line to line. Subscripts and/or arguments emphasize their dependence on a given variable.

To avoid cumbersome expressions, define

X:=D01,p(RN)×D01,q(RN),‖(u,v)‖:=‖u‖1,p+‖v‖1,q∀(u,v)∈X,C+1:=X+∩Cloc1(RN)2,C+1,α:=X+∩Cloc1,α(RN)2.

C+1 and C+1,α will be endowed with the topology induced by that of X.

The following a priori estimate will play a basic role in the sequel.

Lemma 2.1

Let 2−1N<r<+∞, let ζ > N, and let h∈L1(RN)∩Lζ(RN).Ifz∈D01,r(RN)∩Cloc1(RN) is a weak solution to −Δ_rz = h(x) in R^N then there exists c > 0, independent of z, such that

∥∇z∥∞r−1≤cinfR>0R1−Nζ∥h∥ζ+R−Nr′∥∇z∥rr−1.

Proof. Pick any x ∈ R^N and R > 0. Via [26, Theorem 1.1], when r ≥ 2, or [18, Theorem 1.1], if 2−1N<r<2, with Ω := B(x, R) and μ := hdx, as well as Hölder’s inequality, we easily get

|∇z(x)|r−1≤c∫0Rρ−N∫B(x,ρ)|h|dydρ+R−N∫B(x,R)|∇z|dyr−1≤c∥h∥ζ∫0Rρ−Nζdρ+R−r−1rN∥∇z∥rr−1≤cR1−Nζ∥h∥ζ+R−Nr′∥∇z∥rr−1,

where c > 0 does not depend on z, h, x, and R; see [18, Remark 1.3]. Taking the infimum in R > 0 on the right and the supremum in x ∈ R^N on the left yields the conclusion.

3 The regularized system

3.1 ‘Freezing’ the right-hand side

Fix w:=w1,w2∈C+1,ε>0 and define

fw,ε:=f⋅,w1+ε,w2,∇w,gw,ε:=g⋅,w1,w2+ε,∇w,

where ∇w := (∇w₁,∇w₂). We first focus on the auxiliary problem

(P^E_W) −Δpu=fw,ε(χ) in RN.−Δqv=gw,ε(χ) in RN,

Lemma 3.1

If H₁ holds then PWε admits a unique solution (u,v)∈C+1,α, for a suitable α ∈ (0, 1).

Proof. Hypothesis H₁ and (2.1) guarantee that (f_w_,ε , g_w_,ε) ∈ X^*. Hence, by Minty-Browder’s Theorem [5, Theorem 5.16], problem PWε possesses a unique solution (u, v) ∈ X. Thanks to H₁ again one has

fw,ε,gw,ε∈Lloc spRN×Lloc sqRN.

Thus, standard results from nonlinear regularity theory [16, p. 830] entail (u,v)∈Cloc 1,αRN2. Testing the first equation in PWε with u⁻ we next obtain

−∇u−pp=∫RNfw,εu−dx≥0,

because f is non-negative, which forces u ≥ 0. Likewise, v ≥ 0. The strong maximum principle [35, Theorem 1.1.1] finally yields (u, v) ∈ X₊.

Throughout this sub-section, (u, v) will denote the solution to PWε given by Lemma 3.1.

Lemma 3.2

Let H₁ be satisfied. Then there exists L_ε > 0 such that

∥∇u∥pp−1≤Lε1+∇w1pγ1+∇w2qη1,∥∇v∥qq−1≤Lε1+∇w1pη2+∇w2qδ2,

where η₁ := max{β₁, δ₁} and η₂ := max{α₂, γ₂}.

Proof. Test the first equation in PWε with u and exploit H₁(f), H₁(a), besides (2.1), to achieve

(3.1) ∥∇u∥pp=∫RNf⋅,w1+ε,w2,∇w1,∇w2u dx≤mˆ1∫RNa1w1+εα1w2β1+∇w1γ1+∇w2δ1u dx≤mˆ1∫RNa1max1,εα1w2β1+∇w1γ1+∇w2δ1u dx≤cE∥u∥p∗w2q⋆β1+∇w1pγ1+∇w2qδ1≤cε∥∇u∥p∇w2qβ1+∇w1pγ1+∇w2qδ1≤Lε∥∇u∥p1+∇w1pγ1+∇w2qη1,

because

(3.2) ∇w2qβ1+∇w2qδ1≤21+∇w2qη1.

This shows the first inequality. The other is verified similarly.

Lemma 3.3

Under H₁, there exists Mε:=Mε∇w1p,∇w2q>0 such that

max∥u∥∞,∥v∥∞≤Mε∇w1p,∇w2q.

Proof. The proof can be made by adapting the one of Lemma 3.3 in [32]. So, we will briefly focus the key-points only. Fix any ξ1∈1,p∗p such that

(3.3) 1ζ1<1−1ξ1−θ1,

where ζ₁ and θ₁ come from H₁(a). Set u_K := min{u, K}, K > 1, and test Pwε with φ:=uKkp+1,k≥0. Fatou’s Lemma, Hölder’s inequality joined to H₁(a), Sobolev’s embedding (2.1), and (3.2) produce

kp+1(k+1)p∥u∥(k+1)p∗(k+1)p≤kp+1(k+1)plim infK→+∞uK(k+1)p⋆(k+1)p≤cmax1,εα1∫RNa1w2β1+∇w1γ1+∇w2δ1ukp+1 dx≤cε∇w2qβ1+∇w1pγ1+∇w2qδ1∥u∥(kp+1)ξ1kp+1≤cε1+∇w1pγ1+∇w2qη1∥u∥(kp+1)ξ1kp+1;

cf. [32, pp. 1587–1588], but replacing ξ1′withξ1. Moreover,

(k+1)p⋆>(kp+1)ξ1∀k≥0

as ξ1<p⋆p. Hence, Moser’s iteration can start, and we obtain u∞ ≤ M_ε, where

Mε:=cε1+∇w1pγ1+∇w2qη1τ

for some τ > 0; details can be read in [32, pp. 1588–1590], replacing ξ1′withξ1 as above. A similar argument applies to v.

Lemma 3.4

If H₁ holds and maxwi∞,∇wi∞<+∞,i=1,2, then

∥∇u∥∞p−1≤Nε∇w1p,∇w2q,w2∞1+∇w1∞γ1+∇w2∞δ1,∥∇v∥∞q−1≤Nε∇w1p,∇w2q,w1∞1+∇w1∞γ2+∇w2∞δ2

with suitable constants Nε∇w1p,∇w2q,wi∞>0,i=1,2.

Proof. Lemma 3.1 ensures that u∈D01,pRN∩Cloc1RN, while H₁ entails fw,ε∈L1RN∩Lζ1RN. By Lemma 2.1, besides H₁ again, we thus have

∥∇u∥∞p−1≤cfw,εζ1+∥∇u∥pp−1≤cmax1,εα1a1ζ1w2∞β1+∇w1∞γ1+∇w2∞δ1+∥∇u∥pp−1≤cεw2∞β1+∇w1∞γ1+∇w2∞δ1+∥∇u∥pp−1.

Now, using Lemma 3.2 yields

∥∇u∥∞p−1≤cεw2∞β1+∇w1∞γ1+∇w2∞δ1+∇w1pγ1+∇w2qη1+1≤Nε∇w1p,∇w2q,w2∞1+∇w1∞γ1+∇w2∞δ1,

where

Nε∇w1p,∇w2q,w2∞:=cε1+w2∞β1+∇w1pγ1+∇w2qη1.

This shows the first inequality. The other is analogous.

3.2 Regularizing the right-hand side

Let H₁ be satisfied. Given ε > 0, define

Rε:=w1,w2∈C+1:∇w1p≤A1,∇w2q≤A2wi∞≤Bi,∇wi∞≤Ci,i=1,2,

with A_i , B_i , C_i > 0, i = 1, 2, such that

(3.4) A1p−1≥Lε1+A1γ1+A2η1,A2q−1≥Lε1+A1η2+A2δ2,B1,B2≥MεA1,A2,C1p−1≥NεA1,A2,B21+C1γ1+C2δ1,C2q−1≥NεA1,A2,B11+C1γ2+C2δ2,

and L_ε, Mε(⋅,⋅),Nε(⋅,⋅⋅,⋅) stemming from Lemmas 3.2–3.4. Apropos, system (3.4) admits solutions. In fact, by H₁, we can pick

(3.5) 1<σ<(p−1)(q−1)η1η2.

If A1:=K1η2andA2:=Kσq−1 then the first two inequalities of (3.4) become

Kp−1η2≥Lε1+Kγ1η2+Kση1q−1,Kσ≥Lε1+K+Kσδ2q−1,

which, due to (3.5), are true for any sufficiently large K > 0. Next, choose

B1:=B2:=MεK1η2,Kσq−1.

With A_i, B_i as above, set C1:=H1η2andC2:=Hσq−1 The last two inequalities in (3.4) rewrite as

Hp−1η2≥NεA1,A2,B21+Hγ1η2+Hσδ1q−1,Hσ≥NεA1,A2,B11+Hγ2η2+Hσδ2q−1.

Thanks to (3.5) again, they hold for every H > 0 big enough.

On the trapping region R_ε we will consider the topology induced by that of X. Let us now investigate the regularized problem

−Δpu=f(x,u+ε,v,∇u,∇v) in RN,−Δqv=g(x,u,v+ε,∇u,∇v) in RN,u,v>0 in RN,

where ε ≥ 0. Evidently, (P^ε) reduces to (P) once ε = 0.

Lemma 3.5

Under H₁, for every ε > 0 problem (P^ε) possesses a solution uε,vε∈C+1,α.

Proof. Fix ε > 0 and define, provided w ∈ R_ε,

Tε(w):=(u,v),with(u,v)being the unique solution toPwε;

cf. Lemma 3.1. From Lemmas 3.2–3.4, besides (3.4), it follows TεRε⊆Rε. Claim 1. TεRε is relatively compact in X.

To see this, pick wn⊆Rε, put

wn:=w1,n,w2,n,un,vn:=Tεwn,n∈N,

and understand any convergence up to sub-sequences. Since Tεwn⊆Rε while X is reflexive, un,vn weakly converges to a point (u, v) ∈ X. Taking any ρ > 0, if Y_ρ := L^p(B_ρ)× L^q(B_ρ), then one has

(3.6) X↪W1,pBρ×W1,qBρ↪Yρ.

Actually, the first embedding in (3.6) is continuous by (2.1) and the continuity of the restriction map L^r(R^N) → L^r(B_ρ), while the other one is compact due to Rellich-Kondrakov’s theorem [5, Theorem 9.16]. Thus, X↪Yρ compactly, which yields (u_n , v_n) → (u, v) in Y_ρ. Let us next verify that

(3.7) un(x),vn(x)→(u(x),v(x)) for almost every x∈RN.

In fact, (u_n , v_n) → (u, v) in Y₁ yields a sub-sequence un(1),vn(1)ofun,vn such that

un(1)(x),vn(1)(x)→(u(x),v(x)) for almost all x∈B1.

Since un(1),vn(1)→(u,v)inY2 we can extract a sub-sequence un(2),vn(2)fromun(1),vn(1) fulfilling

un(2)(x),vn(2)(x)→(u(x),v(x)) for almost every x∈B2. .

By induction, to each k ≥ 2 there corresponds a sub-sequence un(k),vn(k)ofun(k−1),vn(k−1) such that

un(k)(x),vn(k)(x)→(u(x),v(x)) for almost all x∈Bk.

Now, Cantor’s diagonal procedure leads to un(n),vn(n)→(u,v) a.e. in R^N, because ⋃k=1∞Bk=RN, and (3.7) follows.

Through H₁(f), besides the inclusion wn⊆Rε, we get

(3.8) ∫RN∇unp−2∇un∇un−udx=∫RNf⋅,w1,n+ε,w2,n,∇wnun−udx≤∫RNf⋅,w1,n+ε,w2,n,∇wnun−udx≤cε∫RNa1un−udx∀n∈N,

with cε:=mˆ1εα1B2β1+C1γ1+C2δ1. Using TεRε⊆Rε and (3.7) one has

a1un−u≤2B1a1∈L1RN,n∈N.

So, by (3.7)–(3.8), Lebesgue’s Theorem entails

lim supn→∞∫RN∇unp−2∇un∇un−udx≤cεlimn→∞∫RNa1un−udx=0.

Now, recall (cf., e.g., [32, Proposition 2.2]) that the operator −Δp,D01,pRN is of type (S)₊ to achieve u_n → u in D01,pRN. A similar reasoning applies to {v_n}.

Claim 2. Tε:Rε→Rε is continuous.

Let wn⊆Rε and w∈Rε satisfy w_n → w in X. Thanks to (2.1), Theorem 4.9 of [5] provides

(3.9) wn(x)→w(x) and ∇wn(x)→∇w(x) for almost every x∈RN.

Morever, if (u_n , v_n) := T_ε(w_n), n ∈ N, then there exists a point (u, v) ∈ X such that (u_n , v_n) → (u, v) in X; see the proof of Claim 1. Arguing as before, we obtain

(3.10) un(x)→u(x) and ∇un(x)→∇u(x) for almost every x∈RN.

Since ∇unp≤A1 whatever n, the sequence ∇unp−2∇un⊆Lp′RN turns out bounded. Due to (3.10) and [5, Exercise 4.16], this yields

(3.11) limn→∞∫RN∇unp−2∇un∇φdx=∫RN|∇u|p−2∇u∇φdx,φ∈D01,pRN.

On the other hand,

(3.12) limn→∞∫RNf(⋅,w1,n+ε,w2,n,∇wn)φdx=∫RNf(⋅,w1+ε,w2,∇w)φdx

by Lebesgue’s Theorem jointly with (3.9) and the inequality

f(⋅,w1,n+ε,w2,n,∇wn)|φ|≤cεa1|φ|∈L1(RN)∀n∈N,

which easily arises from H₁(f) besides the choice of R_ε. Finally,

(3.13) ∫RN|∇un|p−2∇un∇φdx=∫RNf(⋅,w1,n+ε,w2,n,∇wn)φdx,n∈N,

because (u_n , v_n) solves (Pwnε). Gathering (3.11)–(3.13) together we have

∫RN|∇u|p−2∇u∇φdx=∫RNf(⋅,w1+ε,w2,∇w)φdx∀φ∈D01,p(RN).

The same is evidently true for v. So, (u, v) turns out a solution to (Pwε). Uniqueness forces (u, v) = T_ε(w), whence T_ε(w_n) → T_ε(w).

Now, Theorem 3.2 in [22, p. 119] can be applied, and T_ε admits a fixed point (uε,vε)∈Rε. By definition of T_ε, the pair (u_ε , v_ε) solves problem (P^ε), while Lemma 3.1 gives (uε,vε)∈C+1,α.

Lemma 3.6

If H₁–H₂ hold then there exists a constant L > 0, independent of ε ≥ 0, such that (u, v)≤ L for every solution (u, v) ∈ X₊ to (P^ε).

Proof. Pick ε ≥ 0 and suppose (u, v) ∈ X₊ solves (P^ε). Via H₁ and (2.1) one has

(3.14) ∥∇u∥pp=∫RNf(⋅,u+ε,v,∇u,∇v)udx≤mˆ1∫RNa1[(u+ε)α1vβ1+|∇u|γ1+|∇v|δ1]udx≤mˆ1∫RNa1(uα1+1vβ1+|∇u|γ1u+|∇v|δ1u)dx≤c∥u∥p∗α1+1∥v∥q∗β1+∥∇u∥pγ1∥u∥p∗+∥∇v∥qδ1∥u∥p∗≤c∥∇u∥pα1+1∥∇v∥qβ1+∥∇u∥pγ1+1+∥∇v∥qδ1∥∇u∥p≤cmax{1,∥∇u∥pγ1+1}max{1,∥∇v∥qη1}.

Likewise,

(3.15) ∥∇v∥qq≤cmax{1,∥∇v∥qδ2+1}max{1,∥∇u∥pη2}.

It should be noted that the constant c does not depend on (u, v) and ε. If either ∥∇v∥q≤1 or∥∇u∥p≤1 then (3.14)–(3.15) directly lead to the conclusion, because γ₁ +1 < p and δ₂ +1 < q; see H₁. Hence, we may assume min{∥∇u∥p,∥∇v∥q}>1. Dividing (3.14)–(3.15) by ∥∇u∥pγ1+1 and∥∇v∥qδ2+1, respectively, yields

∥∇u∥pp−γ1−1≤c∥∇v∥qη1,∥∇v∥qq−δ2−1≤c∥∇u∥pη2.

This clearly entails

∥∇u∥pp−γ1−1≤c∥∇u∥pη1η2q−δ2−1,∥∇v∥qq−δ2−1≤c∥∇v∥qη1η2p−γ1−1.

The conclusion now follows from H₂.

Lemma 3.7

Let H₁–H₂ be satisfied. Then there exists M > 0, independent of ε ≥ 0, such that

max∥u∥∞,∥v∥∞≤M

for every solution (u,v)∈X+to Pε.

Proof. The proof is similar to that one of Lemma 3.3. With the same notation, fix ε ≥ 0, suppose (u, v) ∈ X₊ solves (P^ε), and define Ω₁ := {x ∈ R^N : u(x) ≥ 1}. Moreover, given z ∈ L^r(R^N), r > 1, write z_r in place of ∥z∥LrΩ1 when no confusion can arise. Exploiting H₁(f) one has

∫Ω1|∇u|p−2∇u∇φdx≤mˆ1∫Ω1a1vβ1+|∇u|γ1+|∇v|δ1φdx

for all φ∈D01,pRN+; cf. [32, Lemma 3.2]. If φ:=uKkp+1,k≥0, then Fatou’s Lemma, Hölder’s inequality combined with H₁(a), Sobolev’s embedding (2.1), and Lemma 3.6 produce

kp+1(k+1)p∥u∥(k+1)p∗(k+1)p≤kp+1(k+1)plim infK→+∞uK(k+1)p∗(k+1)p≤c∫Ω1a1vβ1+|∇u|γ1+|∇v|δ1ukp+1 dx≤c∥∇v∥qβ1+∥∇u∥pγ1+∥∇v∥qδ1∥u∥(kp+1)ξ1kp+1≤c∥u∥(kp+1)ξ1kp+1,

where ξ1∈1,p⋆p fulfills (3.3) while c does not depend on (u, v) and ε. We now proceed exactly as in the proof of Lemma 3.3, getting ∥u∥∞≤M. The other inequality is analogous.

Lemma 3.8

Assume H₁–H₂. Then to every ρ > 0 there corresponds σ_ρ > 0 such that

(3.16) minessinfBρu,essinfBρv≥σρ

for all (u, v) ∈ X₊ distributional solution of (P^ε), with 0 ≤ ε ≤ 1.

Proof. Fix ρ > 0. Conditions H₁(f)–H₁(g), besides Lemma 3.7, entail

f(⋅,u+ε,v,∇u,∇v)≥m1essinfBρa1(M+1)α1vβ1,g(⋅,u,v+ε,∇u,∇v)≥m2essinfBρa2(M+1)β2uα2

a.e. in B_ρ. From [14, Theorem 3.1] it thus follows

essinfBρup−1≥cρBρ∫Bρvβ1 dx≥cρ(essinfv)β1,essinfBρvq−1≥cρBρ∫Bρuα2 dx≥cρessinfBρuα2,

which easily give

essinfBρu≤CρessinfBρu(p−1)(q−1)α2β1,assinfBρv≤CρessinfBρv(p−1)(q−1)α2β1.

Now, (3.16) is a simple consequence of H₁, because α₂β₁ < (p − 1)(q − 1).

4 Proof of the main result

Lemma 4.1

Under H₁–H₃, problem (P) possesses a distributional solution (u, v) ∈ X₊.

Proof. Let εn:=1n,n∈N. Lemma 3.5 furnishes a sequence un,vn⊆C+1 such that (u_n , v_n) solves Pεn for all n ∈ N. Since X is reflexive, by Lemma 3.6 one has (u_n , v_n) (u, v) in X, where a sub-sequence is considered when necessary. As before (cf. the proof of Lemma 3.5), this forces (3.7). Moreover, (u, v) ∈ X₊ because, thanks to Lemma 3.8, to each ρ > 0 there corresponds σ_ρ > 0 satisfying

(4.1) mininfBρun,infBρvn≥σρ∀n∈N.

Claim. For every ρ > 0, and along a sub-sequence if necessary, one has

(4.2) un,vn→(u,v) in W1,pBρ×W1,qBρ.

Likewise the proof of (3.9), this will force

(4.3) ∇un,∇vn→(∇u,∇v) a.e. in RN.

Let ρ > 0. Hypothesis H₁, (4.1), Lemma 3.7, and H₃ yield

(4.4) f⋅,un+1/n,vn,∇un,∇vn≤mˆ1a1un+1/nα1vnβ1+∇unγ1+∇vnδ1 in B2ρ≤mˆ1σ2ρα1Mβ1+∇unγ1+∇vnδ1a1∈L2B2ρ

whatever n. So, [10, Theorem 2.1] combined with Lemma 3.6 ensures that ∇unp−2∇un turns out bounded in W^1,2(B_ρ). Since p>2−1N, by Rellich-Kondrakov’s theorem [5, Theorem 9.16], the embedding W1,2Bρ↪ Lp′Bρ is compact. Thus, up to sub-sequences,

(4.5) ∇unp−2∇un→U in Lp′Bρ.

Next, observe that the linear operator

z∈D01,pRN↦∇z Bρ∈LpBρ

turns out well-defined and continuous in the strong topologies. Therefore,

(4.6) ∇un→∇u in LpBρ;

cf. [5, Theorem 3.10]. Gathering [5, Proposition 3.5] and (4.5)–(4.6) together gives

limn→∞∫Bρ∇unp−2∇un∇un−udx=0.

Since −Δp,W1,pBρ enjoys the (S)₊-property, we easily achieve un→u in W1,pBρ. A similar conclusion holds for {v_n}, which shows (4.2).

Now, to verify that (u, v) is a distributional solution of (P), pick any φ1,φ2∈C0∞RN2 and choose ρ > 0 fulfilling

suppφ1∪suppφ2⊆Bρ.

By (4.2), [5, Theorem 4.9] furnishes (h,k)∈LpBρ×LqBρ such that

∇un≤h,∇vn≤ka.e. inBρand for alln∈N,

whence

f⋅,un+1/n,vn,∇un,∇vnφ1≤cρ1+hγ1+kδ1a1φ1∈L1RN,n∈N,

through (4.4). So, thanks to (3.7) and (4.3), Lebesgue’s Theorem entails

limn→∞∫RNf⋅,un+1/n,vn,∇un,∇vnφ1 dx=∫RNf(⋅,u,v,∇u,∇v)φ1 dx.

On account of (4.5) and (4.3), we then get

limn→∞∫RN∇unp−2∇un∇φ1 dx=∫RN|∇u|p−2∇u∇φ1 dx.

Recalling that each (u_n , v_n) weakly solves Pεn produces

∫RN|∇u|p−2∇u∇φ1 dx=∫RNf(⋅,u,v,∇u,∇v)φ1 dx.

Likewise,

∫RN|∇v|p−2∇v∇φ2 dx=∫RNg(⋅,u,v,∇u,∇v)φ2 dx,

and the assertion follows.

Lemma 4.2

Let H₁–H₂ be satisfied and let (u, v) ∈ X₊ be a distributional solution to problem (P). Then (u, v) weakly solves (P).

Proof. We evidently have, for any φ∈D01,pRN,

(4.7) φ=φ+−φ−.

Due to the nature of φ⁺, a localization-regularization procedure will be necessary. With this aim, fix θ ∈ C∞([0,+∞)) such that

(4.8) θ(t)=1 if 0≤t≤1,0 when t≥2,θ is decreasing in (1,2)

and a sequence ρk⊆C0∞RN of standard mollifiers [5, p. 108]. Define, for every n, k ∈ N,

θn(⋅):=θ(|⋅|/n)∈C0∞RN,φn:=θnφ+∈D01,pRN,ψk⋅n:=ρk∗φn∈C0∞RN.

Using (4.8) we easily get φn↑φ+. Moreover, limk→∞ψk,n=φninD01,pRN which entails

(4.9) limk→∞∫RN|∇u|p−2∇u∇ψk,n dx=∫RN|∇u|p−2∇u∇φn dx,n∈N.

If, to shorten notation,fˆ:=f(⋅,u,v,∇u,∇v) then the linear functional

ψ∈D01,pRN↦∫B2n+2fˆψdx

turns out continuous. In fact, Lemmas 3.7 – 3.8, Hölder’s inequality combined with H₁(a), and (2.1) produce

∫B2n+2a1uα1vβ1|ψ|dx≤σ2n+2α1Mβ1a1p∗′∥ψ∥p∗≤cn∥∇ψ∥p.

Now, the assertion follows from H₁(f), because convection terms can be estimated as already made in (3.14). Observe next that

suppψk,n⊆suppρk+suppφn‾⊆B1+B2n‾⊆B2n+2∀n,k∈N;

see [5, Proposition 4.18]. Hence,

(4.10) suppψk,n⊆suppρk+suppφn‾⊆B1+B2n‾⊆B2n+2∀n,k∈N.

On the other hand, the hypothesis (u, v) ∈ X₊ distributional solution to (P) evidently forces

∫RN|∇u|p−2∇u∇ψk,n dx=∫RNfˆψk,n dx,k,n∈N.

Letting k → +∞ and exploiting (4.9)–(4.10) we thus achieve

(4.11) ∫RN|∇u|p−2∇u∇φn dx=∫RNfˆφn dx∀n∈N.

Claim. φn→φ+in D01,pRN.

In fact, for every n ∈ N one has

(4.12) ∫RN∇φn−∇φ+p dx=∫RNφ+∇θn+θn∇φ+−∇φ+p dx≤C∫RN1−θnp∇φ+p dx+∫B2n∖Bn∇θnpφ+p dx≤C∫RN1−θnp∇φ+p dx+C∫B2n∖Bn∇θnpp⋆p⋆−p dx1−pp⋆∫B2n∖Bnφ+p⋆dxpp⋆=C∫RN1−θnp∇φ+p dx+c∇θnNp∫B2n∖Bnφ+p⋆dxpp⋆.

Recall that φ+∈D01,pRN. By (4.8), Lebesgue’s Theorem yields

(4.13) limn→∞∫RN1−θnp∇φ+p dx=0

while, on account of (2.1),

(4.14) limn→∞∫B2n∖Bnφ+p∗ dx=0.

Since, due to (4.8) again,

∫RN∇θnN dx=1nN∫RNθ′|x|nN dx=∫RNθ′(|x|)N dx<+∞∀n∈N,

gathering (4.12)–(4.14) together shows the claim.

Consequently,

(4.15) limn→∞∫RN|∇u|p−2∇u∇φn dx=∫RN|∇u|p−2∇u∇φ+dx.

From φn↑φ+and fˆ≥0 it then follows

(4.16) limn→∞∫RNfˆφn dx=∫RNfˆφ+dx

by Beppo Levi’s Theorem. Through (4.11), (4.15)–(4.16) we thus arrive at

∫RN|∇u|p−2∇u∇φ+dx=∫RNfˆφ+dx.

Likewise, one has

∫RN|∇u|p−2∇u∇φ−dx=∫RNfˆφ−dx,

whence (cf. (4.7))

∫RN|∇u|p−2∇u∇φdx=∫RNf(⋅,u,v,∇u,∇v)φdx∀φ∈D01,pRN.

An analogous argument applies to the second equation in (P).

Lemma 4.3

Let H₁–H₃ be satisfied and let (u, v) ∈ X₊ be a distributional solution of (P). Then (u, v) strongly solves (P).

Proof. Reasoning as before (see (4.4)) provides f(⋅,u,v,∇u,∇v)∈Lloc2RN. Thanks to [10, Theorem 2.1], this implies |∇u|p−2∇u∈Wloc 1,2RN. Moreover,

−Δpu(x)=f(x,u(x),v(x),∇u(x),∇v(x)) a.e. in RN

because of [5, Corollary 4.24]. Similarly about v and the other equation.

Proof of Theorem 1.3.

Lemmas 4.1–4.3 directly give the conclusion.

Remark 4.4

If H₃ is replaced by the stronger condition

H3′_ One has

1sp+maxγ1p,δ1q<1p′N,1sq+maxγ2p,δ2q<1q′N

then any distributional solution (u, v) ∈ X₊ to (P) actually lies in C+1,α, with α ∈ (0, 1). To show this, pick Sˆp,Sˆq>0 such that

1sp+maxγ1p,δ1q≤1sˆp<1p′N,1sq+maxγ2p,δ2q≤1sˆq<1q′N.

As in the proof of (4.4), for every ρ > 0 one has

f(⋅,u,v,∇u,∇v)≤cρa11+|∇u|γ1+|∇v|δ1∈LsˆpBρ,g(⋅,u,v,∇u,∇v)≤cρa21+|∇u|γ2+|∇v|δ2∈LsˆqBρ.

Hence, known nonlinear regularity results [16, p. 830] entail (u,v)∈C+1,α.

Remark 4.5

Unfortunately, we were not able to find in the literature a definition of strong solution for elliptic equations driven by non-linear operators in divergence form. The one adopted here represents a quite natural extension of the semi-linear case p = 2, where it is asked that the solution u∈Wloc2,2(RN) and satisfies the differential equation a.e. in R^N; cf. [21, p. 219] and [38, pp. 7–8]. We cannot expect u∈Wloc2,q(RN) for some q > 1, as the example of [10, Remark 2.7] shows. Nevertheless, if (u,v)∈C+1 is a distributional solution to (P) then u,v∈Wloc2,2(RN) once 1 < p, q < 3; see [37, p. 2]. On the other hand, each strong solution turns out distributional. So, our notion of strong solution should be read as a distributional solution with an extra differentiability property on the fields |∇u|p−2∇u and |∇v|q−2∇v.

Acknowledgments

The authors thank S.J.N. Mosconi for helpful and stimulating discussions.

U.Guarnotta and S.A. Marano were supported by the following research projects: 1) PRIN 2017 ‘Nonlinear Differential Problems via Variational, Topological and Set-valued Methods’ (Grant No. 2017AYM8XW) of MIUR; 2) PRA 2020–2022 Linea 2 ‘MO.S.A.I.C.’ of the University of Catania.

A. Moussaoui was supported by the Directorate-General of Scientific Research and Technological Development (DGRSDT).

Conflict of interest

Conflict of interest statement: Authors state no conflict of interest.

References

[1] C. Azizieh, P. Clément, and E. Mitidieri, Existence and a priori estimates for positive solutions of p-Laplace systems, J. Differential Equations 184 (2002), 422–442.10.1006/jdeq.2001.4149Search in Google Scholar

[2] S. Biagi, F. Esposito, and E. Vecchi, Symmetry and monotonicity of singular solutions of double phase problems, J. Differential Equations 280 (2021), 435–463.10.1016/j.jde.2021.01.029Search in Google Scholar

[3] L. Boccardo, A Dirichlet problem with singular and supercritical nonlinearities, Nonlinear Anal. 75 (2012), 4436–4440.10.1016/j.na.2011.09.026Search in Google Scholar

[4] L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial Differential Equations 37 (2010), 363–380.10.1007/s00526-009-0266-xSearch in Google Scholar

[5] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.10.1007/978-0-387-70914-7Search in Google Scholar

[6] P. Candito, R. Livrea, and A. Moussaoui, Singular quasilinear elliptic systems involving gradient terms, Nonlinear Anal. Real World Appl. 55 (2020), 103142.10.1016/j.nonrwa.2020.103142Search in Google Scholar

[7] A. Canino, B. Sciunzi, and A. Trombetta, Existence and uniqueness for p-Laplace equations involving singular nonlinearities, NoDEA Nonlinear Differential Equations Appl. 23 (2016), Paper No. 8, 18 pp.10.1007/s00030-016-0361-6Search in Google Scholar

[8] J. Chabrowski, Existence results for singular elliptic equations, Hokkaido Math. J. 20 (1991), 465–475.10.14492/hokmj/1381413980Search in Google Scholar

[9] J. Chabrowski and M. König, On entire solutions of elliptic equations with a singular nonlinearity, Comment. Math. Univ. Carolin. 31 (1990), 643–654.Search in Google Scholar

[10] A. Cianchi and V.G. Maz’ya, Second-order two-sided estimates in nonlinear elliptic problems, Arch. Ration. Mech. Anal. 229 (2018), 569–599.10.1007/s00205-018-1223-7Search in Google Scholar

[11] P. Clément, J. Fleckinger, E. Mitidieri, and F. De Thélin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations 166 (2000), 455–477.10.1006/jdeq.2000.3805Search in Google Scholar

[12] M.G. Crandall, P.H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193–222.10.1080/03605307708820029Search in Google Scholar

[13] R. Dalmasso, Solutions d’équations elliptiques semi-linéaires singulières, Ann. Mat. Pura Appl. 153 (1988), 191–201.10.1007/BF01762392Search in Google Scholar

[14] L. D’Ambrosio and E. Mitidieri, Entire solutions of quasilinear elliptic systems on Carnot groups, Reprint of Tr. Mat. Inst. Steklova 283 (2013), 9–24, Proc. Steklov Inst. Math. 283 (2013), 3–19.10.1134/S0081543813080026Search in Google Scholar

[15] J. Deny and J.L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier (Grenoble) 5 (1955), 305–370.10.5802/aif.55Search in Google Scholar

[16] E. DiBenedetto, C^1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827–850.10.1016/0362-546X(83)90061-5Search in Google Scholar

[17] P. Drabek and L. Sankar, Singular quasilinear elliptic problems on unbounded domains, Nonlinear Anal. 109 (2014), 148–155.10.1016/j.na.2014.07.002Search in Google Scholar

[18] F. Duzaar and G. Mingione, Gradient estimates via linear and nonlinear potentials, J. Funct. Anal. 259 (2010), 2961–2998.10.1016/j.jfa.2010.08.006Search in Google Scholar

[19] F. Esposito and B. Sciunzi, On the Höpf boundary lemma for quasilinear problems involving singular nonlinearities and applications, J. Funct. Anal. 278 (2020), Paper No. 108346, 25 pp.10.1016/j.jfa.2019.108346Search in Google Scholar

[20] G.P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems, 2nd ed., Springer Monographs in Mathematics, Springer, New York, 2011.10.1007/978-0-387-09620-9Search in Google Scholar

[21] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001.10.1007/978-3-642-61798-0Search in Google Scholar

[22] A. Granas and J. Dugundji, Fixed point theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.10.1007/978-0-387-21593-8Search in Google Scholar

[23] U. Guarnotta and S.A. Marano, Infinitely many solutions to singular convective Neumann systems with arbitrarily growing reactions, J. Differential Equations 271 (2021), 849–863.10.1016/j.jde.2020.09.024Search in Google Scholar

[24] U. Guarnotta, S.A. Marano, and D. Motreanu, On a singular Robin problem with convection terms, Adv. Nonlinear Stud. 20 (2020), 895–909.10.1515/ans-2020-2093Search in Google Scholar

[25] T. Kusano and C.A. Swanson, Entire positive solutions of singular semilinear elliptic equations, Japan. J. Math. (N.S.) 11 (1985), 145–155.10.4099/math1924.11.145Search in Google Scholar

[26] T. Kuusi and G. Mingione, Linear potentials in nonlinear potential theory, Arch. Ration. Mech. Anal. 207 (2013), 215–246.10.1007/s00205-012-0562-zSearch in Google Scholar

[27] A.V. Lair and A.W. Shaker, Entire solution of a singular semilinear elliptic problem, J. Math. Anal. Appl. 200 (1996), 498–505.10.1006/jmaa.1996.0218Search in Google Scholar

[28] A.C. Lazer and P.J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc. 111 (1991), 721–730.10.1090/S0002-9939-1991-1037213-9Search in Google Scholar

[29] E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, 2nd ed., American Mathematical Society, Providence, 2001.10.1090/gsm/014Search in Google Scholar

[30] G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.10.1016/0362-546X(88)90053-3Search in Google Scholar

[31] Z. Liu, D. Motreanu, and S. Zeng, Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient, Calc. Var. Partial Differential Equations 58 (2019), Paper No. 28, 22 pp.10.1007/s00526-018-1472-1Search in Google Scholar

[32] S.A. Marano, G. Marino, and A. Moussaoui, Singular quasilinear elliptic systems in R^N Ann. Mat. Pura Appl. 198 (2019), 1581–1594.10.1007/s10231-019-00832-1Search in Google Scholar

[33] A. Moussaoui, B. Khodja, and S. Tas, A singular Gierer-Meinhardt system of elliptic equations in R^N Nonlinear Anal. 71 (2009), 708–716.10.1016/j.na.2008.10.103Search in Google Scholar

[34] F. Oliva and F. Petitta, On singular elliptic equations with measure sources, ESAIM Control Optim. Calc. Var. 22 (2016), 289–308.10.1051/cocv/2015004Search in Google Scholar

[35] P. Pucci and J. Serrin, The maximum principle, Prog. Nonlinear Differential Equations Appl. 73, Birkhäuser Verlag, Basel, 2007.10.1007/978-3-7643-8145-5Search in Google Scholar

[36] C.A. Santos, R. Lima Alves, M. Reis, and J. Zhou, Maximal domains of the λ, μ-parameters to existence of entire positive solutions for singular quasilinear elliptic systems, J. Fixed Point Theory Appl. 22 (2020), Paper No. 54, 30 pp.10.1007/s11784-020-00783-8Search in Google Scholar

[37] B. Sciunzi, Regularity and comparison principles for p-Laplace equations with vanishing source term, Commun. Con-temp. Math. 16 (2014), 1450013, 20 pp.10.1142/S0219199714500138Search in Google Scholar

[38] C.G. Simader and H. Sohr, The Dirichlet problem for the Laplacian in bounded and unbounded domains. A new approach to weak, strong and (2 + k-solutions in Sobolev-type spaces, Pitman Research Notes in Mathematics Series 360, Long-man, Harlow, 1996.Search in Google Scholar

[39] G. Talenti, Best constants in Sobolev inequalities, Ann. Mat. Pura Appl. 110 (1976), 353–372.10.1007/BF02418013Search in Google Scholar

[40] E. Teixeira, Regularity for quasilinear equations on degenerate singular sets, Math. Ann. 358 (2014), 241–256.10.1007/s00208-013-0959-5Search in Google Scholar

Received: 2021-03-10

Accepted: 2021-09-17

Published Online: 2022-01-06

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