Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system

Francisco Ortegón Gallego; Mohamed Rhoudaf; Hajar Talbi

doi:10.1515/ans-2023-0133

Article Open Access

Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system

Francisco Ortegón Gallego , Mohamed Rhoudaf and Hajar Talbi

Published/Copyright: April 22, 2024

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From the journal Advanced Nonlinear Studies Volume 24 Issue 3

Abstract

In this paper, we analyze the following nonlinear elliptic problem A ( u ) = ρ ( u ) | ∇ φ | 2 in Ω , div ( ρ ( u ) ∇ φ ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω . where A(u) = −div a(x, u, ∇u) is a Leray-Lions operator of order p. The second member of the first equation is only in L ¹(Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique.

Keywords: coupled system; bilateral solution; nonlinear elliptic equation; thermistor problem; penalization techniques

1 Introduction

A thermistor is a specialized resistor. Its performance is based on changing the electrical resistance of the device according to temperature changes. If u and φ denote, respectively, the steady-state temperature and electric potential inside a semiconductor device Ω ⊂ R N , then they satisfy the so-called thermistor problem, namely,

(1) − div a ( x , u , ∇ u ) = ρ ( u ) | ∇ φ | 2 in Ω , div ( ρ ( u ) ∇ φ ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω .

where ρ(u) is the temperature dependent electric conductivity. The first equation expresses the diffusion of heat due to the Joule effect, whereas the second equation describes the conservation of electric charges. In the classical thermistor problem, the field a(x, u, ∇u) is of the form a(x, u, ∇u) = κ(u)∇u where κ(u) is the temperature dependent semiconductor thermal diffusion (see Refs. [1, 2] for the steady-state thermistor problem, and Ref. [3] for the evolution case). Here, we are considering a more general setting by taking a = a(x, u, ∇u), which includes non-homogeneous and anisotropic materials.

In most practical situations, one has ρ ∈ C ( R ) and is such that ρ(s) > 0 with ρ(s) → 0 as s → ∞. Thus, the equation for φ is nonuniformly elliptic. In order to deal with this difficulty, Xu [4] introduced the concept of a capacity solution to the thermistor problem (evolution case) in the framework of Sobolev spaces with p = 2 where he proved the existence of a capacity solution. Afterwards, other existence results have been obtained by many authors, in particular those of González Montesinos and Ortegón Gallego [5, 6] where the authors have shown the existence of a capacity solution for the evolution thermistor problem in Sobolev spaces for any p ≥ 2. Also, Moussa et al. [7, 8] have studied the existence of a solution for both the steady-state and the evolution thermistor problem in the context of Orlicz-Sobolev spaces.

Recently, Ortegón Gallego, Rhoudaf and Talbi analyzed in Ref. [9] the existence of a capacity solution to a coupled nonlinear parabolic-elliptic system in the context of anisotropic Sobolev spaces. This result was generalized to the anisotropic Orlicz-Sobolev space by Ortegón Gallego, Ouyahya and Rhoudaf [10]. In all the aforementioned references, the authors have just considered the case of Sobolev space with p ≥ 2 or the Orlicz-Sobolev space with an equivalent condition, that is, the N-function M admits the representation M ( s ) = ∫ 0 | s | m ( t ) d t with m(t) ≥ t for all t ≥ 0. Due to this assumption, we may deduce that the right hand side of the first equation of (1) lies in the ‘good’ dual space and one has ρ(u)|∇φ|² = div (ρ(u)φ∇φ). This last equality is the key that leads to the introduction of the notion of a capacity solution and to solve the problem (1) in that sense. The case 1 < p < 2 has never been considered up to date.

The analysis developed in this work does not exclude the case 1 < p < 2 and we establish an existence result of a bilateral solution to the system (1), where the operator a : Ω × R × R N ↦ R N is of Leray-Lions type (for instance, the p-Laplacian operator) and the function ρ is continuous and such that ρ(s) > 0, for all s ∈ R . In order to prove an existence result of a solution for a certain approximate problem, we will make use of the so-called penalization technique. We recall that this technique was firstly introduced by Boccardo and Murat [11]. These authors have approximated the following variational inequality formulation

(2) To find u ∈ K such that 〈 A ( u ) , v − u 〉 ≥ 〈 f , v − u 〉 , for all v ∈ K ,

where the convex set K is defined by

(3) K = v ∈ W 0 1 , p ( Ω ) / | v ( x ) | ≤ 1 a.e. in Ω ,

A(u) = −Δ_p u = −div (|∇u|^p−2∇u) and f ∈ W ^−1,p′(Ω), by the sequence of problems

(4) A ( u n ) + | u n | n − 2 u n = f in D ′ ( Ω ) , u n = 0 on ∂ Ω .

Inspired by this approach, we look for certain bilateral solutions of system (1) at height M in the sense of Definition 1 given below. Notice that this work remains valid for 1 < p < ∞.

In order to study the problem (1) under the assumptions given below, we will adopt the following plan. In Section 2 we enumerate the assumptions on the data and we introduce the notion of bilateral solution adapted to our context. In Section 3 we present our main results and in Section 4 we develop their respective proofs.

2 Assumptions and definitions

We consider the following nonlinear elliptic coupled system

(5) − div a ( x , u , ∇ u ) = ρ ( u ) | ∇ φ | 2 in Ω , div ( ρ ( u ) ∇ φ ) = 0 in Ω , u = 0 , on ∂ Ω , φ = φ 0 on ∂ Ω ,

where φ ₀ is a given function, Ω is an open, bounded, connected and smooth enough set of R N , N ≥ 2 is an integer. Let p ∈ (1, ∞) and p ′ = p p − 1 .

The assumptions on the data are the following.

a : Ω × R × R N → R N is a Carathéodory function such that for all s ∈ R , ξ , ξ ′ ∈ R N and for a.e. x ∈ Ω it holds
(6) a ( x , s , ξ ) − a ( x , s , ξ ′ ) ( ξ − ξ ′ ) ≥ α | ξ − ξ ′ | 2 ( | ξ | + | ξ ′ | ) p − 2 , if 1 < p < 2 , α | ξ − ξ ′ | p , if p ≥ 2 .

(7) | a ( x , s , ξ ) | ≤ β [ b ( x ) + | s | p − 1 + | ξ | p − 1 ] ,

(8) a ( x , s , 0 ) = 0 ,
where α and β are positive constants and b is a nonnegative function in L ^p′(Ω).
ρ ∈ C ( R ) and
(9) 0 < ρ ( s ) for all s ∈ R .
φ ₀ ∈ H ¹(Ω) and is not a constant function on ∂Ω.

Remark 1.

In the case 1 < p < 2 the right hand side of the inequality (6) is understood to be equal to zero whenever (ξ, ξ′) = (0, 0). Notice that (6) together with (8) yield, in particular, the ellipticity condition, namely, for all s ∈ R , ξ ∈ R N and for a.e. x ∈ Ω, it holds

(10) a ( x , s , ξ ) ξ ≥ α | ξ | p , if p ∈ ( 1 , + ∞ ) .

For any M > 0, we denote by K M the closed and convex set in W 0 1 , p ( Ω ) given as

K M ≔ v ∈ W 0 1 , p ( Ω ) / | v ( x ) | ≤ M a.e. in Ω .

Now we introduce the definition of a bilateral solution to problem (5).

Definition 1.

Let M be a positive real number. A pair (u, φ) is called a bilateral solution of problem (5) at height M if the following conditions are fulfilled.

u ∈ K M and φ − φ 0 ∈ H 0 1 ( Ω ) .
For all v ∈ K M one has
(11) ∫ Ω a ( x , u , ∇ u ) ∇ ( u − v ) ≤ ∫ Ω ρ ( u ) | ∇ φ | 2 ( u − v ) .
For all ψ ∈ H 0 1 ( Ω ) it holds
∫ Ω ρ ( u ) ∇ φ ∇ ψ = 0 .

Remark 2.

Assume that (u, φ) is a bilateral solution at a certain height M > 0 such that, for some constant γ > 0, |u(x)| ≤ γ < M for almost everywhere x ∈ Ω. Then, (u, φ) is a weak solution to the thermistor problem (5), that is,

(12) u ∈ W 0 1 , p ( Ω ) , φ − φ 0 ∈ H 0 1 ( Ω ) , ∫ Ω a ( x , u , ∇ u ) ∇ v = ∫ Ω ρ ( u ) | ∇ φ | 2 v , for all v ∈ W 0 1 , p ( Ω ) ∩ L ∞ ( Ω ) , ∫ Ω ρ ( u ) ∇ φ ∇ ψ = 0 , for all ψ ∈ H 0 1 ( Ω ) .

Indeed, if w ∈ W 0 1 , p ( Ω ) ∩ L ∞ ( Ω ) then, for δ = (M − γ)/(1 + ‖w‖_∞), the two functions v _± = u ± δw belong to K M . Plugging v _± in (11) we readily obtain (12).

Remark 3.

In general, we cannot assure that if (u, φ) is a weak solution then u is bounded. Thus, we may interpret a bilateral solution at a given height M as the solution of the projection problem on the convex set K M given by the conditions (C ₁)–(C ₃).

3 Main results

The nature of a bilateral solution (u, φ) is by approximation. This means that (u, φ) will be obtained as the limit of the solutions of certain approximate problems. To do so, for any two integers n ≥ 1 and m ≥ 1, we first consider the following problem.

(13) − div a x , u m n , ∇ u m n + u m n M n − 2 u m n M = T m ( ρ m u m n | ∇ φ m n | 2 ) in Ω , div ( ρ m u m n ∇ φ m n ) = 0 in Ω , u m n = 0 on ∂ Ω , φ m n = φ 0 on ∂ Ω .

where the regularized function, ρ _m(s), is given by

(14) ρ m ( s ) = ρ ( T m ( s ) ) ,

and T _m(s) is the truncation function at height m: T _m(s) = min(m, max(s, − m)), for all s ∈ R .

The existence of a solution u m n , φ m n to this approximate problem is guaranteed by the following result, which is proved in the next section.

Lemma 1.

There exists u m n , φ m n ∈ W 0 1 , p ( Ω ) × H 1 ( Ω ) such that φ m n − φ 0 ∈ H 0 1 ( Ω ) ,

(15) ∫ Ω a x , u m n , ∇ u m n ∇ v + ∫ Ω u m n M n − 2 u m n M v = ∫ Ω T m ( ρ m u m n | ∇ φ m n | 2 ) v , for all v ∈ W 0 1 , p ( Ω ) ∩ L ∞ ( Ω ) and for v = u m n ,

and

(16) ∫ Ω ρ m u m n ∇ φ m n ∇ ψ = 0 for all ψ ∈ H 0 1 ( Ω ) .

Now, we fix m ≥ 1 and consider the sequences u m n n ≥ 1 ⊂ W 0 1 , p ( Ω ) and φ m n n ≥ 1 ⊂ H 1 ( Ω ) . We will also show in the next section the following result.

Lemma 2.

Let u m n , φ m n ∈ W 0 1 , p ( Ω ) × H 1 ( Ω ) be a solution to (15) and (16). Then, there exist u m ∈ K M and φ _m ∈ H ¹(Ω), with φ m − φ 0 ∈ H 0 1 ( Ω ) , and subsequences, still denoted in the same way, such that

lim n → ∞ u m n = u m in W 0 1 , p ( Ω ) , lim n → ∞ φ m n = φ m in H 1 ( Ω ) ,

and (u _m, φ _m) satisfies the approximate bilateral problem

(17) ∫ Ω a ( x , u m , ∇ u m ) ∇ ( u m − v ) ≤ ∫ Ω T m ( ρ m ( u m ) | ∇ φ m | 2 ) ( u m − v ) , for all v ∈ K M ,

(18) ∫ Ω ρ m ( u m ) ∇ φ m ∇ ψ = 0 , for all ψ ∈ H 0 1 ( Ω ) .

The main result of this work now follows.

Theorem 1.

Assume (A ₁), (A ₂) and (A ₃) and let (u _m, φ _m) be a solution to (17) and (18). Then, there exists a subsequence that converges to a bilateral solution (u, φ) to the problem (5).

Remark 4.

From this point on, we will denote by C (respectively, C _m) any positive constant, which may depend on the data of our problem and not on n or m (respectively, on n), and whose value may differ from one occurrence to another.

4 Proof of the main results

This section is devoted to develop the proof of Lemmas 1 and 2 and Theorem 1.

Proof of Lemma 1.

The proof is based on Schauder’s fixed point theorem. Let ω m n ∈ L p ( Ω ) , and consider the following elliptic problem

(19) div ( ρ m ω m n ∇ φ m n ) = 0 in Ω , φ m n = φ 0 on ∂ Ω .

Since

0 < min − m ≤ s ≤ m ρ ( s ) ≤ ρ m ω m n ≤ max − m ≤ s ≤ m ρ ( s ) ,

it follows, thanks to Lax-Milgram’s theorem, that (19) has a unique solution φ m n ∈ H 1 ( Ω ) . Moreover, by using φ m n − φ 0 ∈ H 0 1 ( Ω ) as a test function in (19), we get

∫ Ω ρ m ω m n ∇ φ m n ∇ φ m n − φ 0 = 0 .

Hence, from (9) and (14),

min − m ≤ s ≤ m ρ ( s ) ∫ Ω | ∇ φ m n | 2 ≤ ∫ Ω ρ m ω m n ∇ φ m n ∇ φ 0 ≤ max − m ≤ s ≤ m ρ ( s ) ∫ Ω | ∇ φ m n | ∇ φ 0 .

By the Cauchy-Schwarz inequality, we obtain

(20) ∫ Ω | ∇ φ m n | 2 ≤ C m , φ 0 = C m .

Now we consider the following monotone elliptic problem.

(21) − div a x , ω m n , ∇ u m n + u m n M n − 2 u m n M = T m ( ρ m ω m n | ∇ φ m n | 2 ) in Ω , u m n = 0 on ∂ Ω .

By the definition of the truncation function, T _m, it is clear that the right hand side of (21) belongs to L ^∞(Ω) ⊂ W ^−1,p′(Ω), then it is well known (see, for instance, [12]) that (21) has at least one weak solution u m n ∈ W 0 1 , p ( Ω ) , that is

(22) ∫ Ω a x , ω m n , ∇ u m n ∇ v + ∫ Ω u m n M n − 2 u m n M v = ∫ Ω T m ( ρ m ω m n | ∇ φ m n | 2 ) v , for all v ∈ W 0 1 , p ( Ω ) ∩ L ∞ ( Ω ) and v = u m n .

Moreover, since t ↦ |t|ⁿ⁻² t is non-decreasing and using the condition (6), it follows that the solution to (21) is unique. Thus, the mapping G:

ω m n ∈ L p ( Ω ) ⟼ G ω m n = u m n ∈ W 0 1 , p ( Ω ) ,

u m n being the unique solution to (22), defines an operator from L ^p(Ω) into itself. We have the following lemma which will be shown in the Appendix below.

Lemma 3.

The operator G satisfies the hypotheses of Schauder’s fixed point theorem.

By Lemma 3 and according to the Schauder’s fixed point theorem, we conclude that the operator G has at least one fixed point u m n = G u m n , which means that

(23) − div a x , u m n , ∇ u m n + u m n M n − 2 u m n M = T m ( ρ m u m n | ∇ φ m n | 2 ) .

Therefore, we have proved the existence of a solution u m n , φ m n to the approximate problem (13), where u m n belongs to the Sobolev space W 0 1 , p ( Ω ) , φ m n − φ 0 belongs to H 0 1 ( Ω ) and

div ( ρ m u m n ∇ φ m n ) = 0 .

□

Proof of Lemma 2.

We first fix m ≥ 1 and deduce some estimates on the sequences u m n n ≥ 1 and φ m n n ≥ 1 .

By taking φ m n − φ 0 ∈ H 0 1 ( Ω ) as a test function in the second equation of (13), we get

∫ Ω ρ m u m n ∇ φ m n ∇ φ m n − φ 0 = 0 .

Since ρ is continuous and by the definition of ρ _m, we get

min − m ≤ s ≤ m ρ ( s ) ∫ Ω | ∇ φ m n | 2 ≤ ∫ Ω ρ m u m n ∇ φ m n ∇ φ 0 ≤ max − m ≤ s ≤ m ρ ( s ) ∫ Ω | ∇ φ m n | ∇ φ 0 .

By the Cauchy-Schwarz inequality, we obtain

(24) ∫ Ω | ∇ φ m n | 2 ≤ C m , φ 0 = C m .

here C _m does not depends on n. Hence, there exists a function φ _m ∈ H ¹(Ω) and subsequence, still denoted in the same way, such that

(25) φ m n → φ m weakly in H 1 ( Ω ) , strongly in L 2 ( Ω ) and a.e. in Ω

As for the sequence u m n n ≥ 1 , taking v = u m n as a test function in the first equation of (13), we get

(26) ∫ Ω a x , u m n , ∇ u m n ∇ u m n + ∫ Ω M u m n M n = ∫ Ω T m ( ρ m u m n | ∇ φ m n | 2 ) u m n .

Using (10), we get

α ∫ Ω | ∇ u m n | p + ∫ Ω M u m n M n ≤ ∫ Ω T m ( ρ m u m n | ∇ φ m n | 2 ) u m n .

The positivity of the second term in the left hand side of this last equation yields

α ∫ Ω | ∇ u m n | p ≤ ∫ Ω T m ( ρ m u m n | ∇ φ m n | 2 ) u m n .

Since T _m(s) ≤ m, then

α ∫ Ω | ∇ u m n | p ≤ m ∫ Ω | u m n | .

Using Hölder’s inequality, we get for some constant C _m > 0

∫ Ω | ∇ u m n | p ≤ C m ∫ Ω | u m n | p 1 / p .

By the Poincaré inequality, we obtain for another constant C _m > 0

∫ Ω | ∇ u m n | p ≤ C m ∫ Ω | ∇ u m n | p 1 / p .

Thus

∫ Ω | ∇ u m n | p 1 / p ′ ≤ C m .

and then

(27) ∫ Ω | ∇ u m n | p ≤ C m p ′ ,

By (27), we may extract a subsequence, still denoted in the same way, such that

(28) u m n → u m weakly in W 0 1 , p ( Ω ) , strongly in L p ( Ω ) and a.e. in Ω .

From (16), (25) and the almost everywhere convergence of u m n n ≥ 1 to u _m we readily obtain the equation for φ _m, namely,

(29) ∫ Ω ρ m ( u m ) ∇ φ m ∇ ψ = 0 , for all ψ ∈ H 0 1 ( Ω ) .

Now, it is easy to deduce that the convergence of φ m n n ≥ 1 to φ _m in H ¹(Ω) is, in fact, strongly. Indeed, from (16) and (18) we have

(30) ∫ Ω ρ m u m n ∇ φ m n ∇ ψ = ∫ Ω ρ m ( u m ) ∇ φ m ∇ ψ = 0 , for all ψ ∈ H 0 1 ( Ω ) .

Taking ψ = φ m n − φ m in the first equality of (30), we get

∫ Ω ρ m u m n ∇ φ m n ∇ φ m n − φ m = ∫ Ω ρ m ( u m ) ∇ φ m ∇ φ m n − φ m .

Inserting − ρ m u m n ∇ φ m ∇ φ m n − φ m in both integrals above, we get

∫ Ω ρ m u m n | ∇ φ m n − φ m | 2 = ∫ Ω ( ρ m ( u m ) − ρ m u m n ) ∇ φ m ∇ φ m n − φ m .

Using Hölder’s inequality and (24) we deduce

(31) min − m ≤ s ≤ m ρ ( s ) ∫ Ω | ∇ φ m n − φ m | 2 ≤ C m ∫ Ω | ρ m ( u m ) − ρ m u m n | 2 | ∇ φ m | 2 .

By the continuity of ρ, we have

ρ m u m n − ρ m ( u m ) → 0 a.e. in Ω ,

and also

| ρ m ( u m ) − ρ m u m n | 2 | ∇ φ m | 2 ≤ C m | ∇ φ m | 2 .

Since ∇φ _m ∈ L ²(Ω), then by the Lebesgue convergence theorem, we have

(32) ∫ Ω | ρ m ( u m ) − ρ m u m n | 2 | ∇ φ m | 2 → 0 as n → ∞ .

Thus,

(33) ∫ Ω | ∇ φ m n − φ m | 2 → 0 as n → ∞ ,

that is, ∇ φ m n → ∇ φ m strongly in L 2 ( Ω ) .

From (26) and since a x , u m n , ∇ u m n ∇ u m n ≥ 0 , we get

M ∫ Ω u m n M n ≤ ∫ Ω T m ( ρ m u m n | ∇ φ m n | 2 ) u m n .

Thus, by the definition of T _m and using both the Hölder and the Poincaré inequalities, we get, for some different constants C _m > 0,

∫ Ω u m n M n ≤ C m ∫ Ω | u m n | p 1 / p ≤ C m ∫ Ω | ∇ u m n | p 1 / p .

Hence, using (27), we deduce

(34) 0 ≤ ∫ Ω u m n M n ≤ C m .

Lemma 4.

The weak limit u _m appearing in (28) verifies |u _m| ≤ M almost everywhere in Ω. Moreover, the convergence of u m n n to u _m is strong in W 0 1 , p ( Ω ) .

Proof

Let δ be a real number such that δ > 1. Then, by (34), we obtain

0 ≤ 1 δ ∫ u m n M ≥ δ u m n M = δ n − 1 δ n ∫ u m n M ≥ δ u m n M ≤ 1 δ n ∫ u m n M ≥ δ u m n M n ≤ C m δ n

Therefore,

(35) 0 ≤ lim n → ∞ 1 δ ∫ u m n M ≥ δ u m n M ≤ lim n → ∞ C m δ n = 0

Hence,

0 = 1 δ lim n → ∞ ∫ u m n M ≥ δ u m n M = lim n → ∞ 1 δ ∫ Ω u m n M χ u m n M ≥ δ χ u m M ≤ δ + 1 δ ∫ Ω u m n M χ u m n M ≥ δ χ u m M > δ ≥ lim n → ∞ 1 δ ∫ Ω u m n M χ u m n M ≥ δ χ u m M > δ = 1 δ ∫ Ω u m M χ u m M > δ ≥ ∫ Ω χ u m M > δ ≥ 0 ,

which implies that meas ( { u m M > δ } ) = 0 for every δ > 1. Thus, u m ≤ M almost everywhere in Ω.

It remains to prove the strong convergence of u m n to u _m in W 0 1 , p ( Ω ) . Indeed, for any h > 0, taking T h u m n − u m as a test function in the first equation of (13), we get

(36) ∫ Ω a x , u m n , ∇ u m n ∇ T h u m n − u m + ∫ Ω u m n M n − 2 u m n M T h u m n − u m = ∫ Ω T m ( ρ m u m n | ∇ φ m n | 2 ) T h u m n − u m .

We distinguish the cases p ≥ 2 and 1 < p < 2.

p ≥ 2 As long as the first term of the left hand side of (36) is concerned, using (6), we have

(37) ∫ Ω a x , u m n , ∇ u m n ∇ T h u m n − u m = ∫ Ω ( a x , u m n , ∇ u m n − a x , u m n , ∇ u m ) ∇ T h u m n − u m + ∫ Ω a x , u m n , ∇ u m ∇ T h u m n − u m ≥ α ∫ Ω | ∇ T h u m n − u m | p + ∫ Ω a x , u m n , ∇ u m ∇ T h u m n − u m

For the second term of (36), observe that u m n u m n − u m has the same sign as u m n T h u m n − u m and that u m n u m n − u m ≥ 0 on the set, P, defined by

(38) P = u m n ≥ 0 and u m n ≥ u m ∪ u m n ≤ 0 and u m n ≤ u m .

The complimentary set of P, P ̄ , is

P ̄ = u m n < 0 and u m n > u m ∪ u m n > 0 and u m n < u m = 0 < u m n < u m ∪ u m < u m n < 0 .

Thus,

(39) ∫ Ω u m n M n − 2 u m n M T h u m n − u m ≥ ∫ 0 < u m n < u m u m n M n − 2 u m n M T h u m n − u m + ∫ u m < u m n < 0 u m n M n − 2 u m n M T h u m n − u m = I h , n m , 1 + I h , n m , 2 .

Since T h u m n − u m converges to u m n − u m strongly in L ¹(Ω) as h goes to ∞, then

(40) lim h → ∞ I h , n m , 1 = ∫ 0 < u m n < u m u m n M n − 2 u m n M u m n − u m

(41) lim h → ∞ I h , n m , 2 = ∫ u m < u m n < 0 u m n M n − 2 u m n M u m n − u m

Moreover, since | u m n | < M on 0 < u m n < u m and on u m < u m n < 0 , then

(42) lim n → ∞ ∫ 0 < u m n < u m u m n M n − 2 u m n M u m n − u m = 0

(43) lim n → ∞ ∫ u m < u m n < 0 u m n M n − 2 u m n M u m n − u m = 0

Gathering (37), (39), (42) and (43) then, since T h u m n − u m converges to u m n − u m strongly in W 0 1 , p ( Ω ) as h → ∞, we get

lim sup n → ∞ α ∫ Ω | ∇ u m n − u m | p + lim sup n → ∞ ∫ Ω a x , u m n , ∇ u m ∇ u m n − u m ≤ lim sup n → ∞ ∫ Ω T m ( ρ m u m n | ∇ φ m n | 2 ) u m n − u m

Owing to (28), we get

(44) lim n → ∞ ∫ Ω a x , u m n , ∇ u m ∇ u m n − u m = 0 .

Using (28) again, since T m ( ρ m u m n | ∇ φ m n | 2 ) is bounded in L ^∞(Ω) and | ∇ φ m n | 2 → | ∇ φ m | 2 strongly in L ¹(Ω) and a.e. in Ω,

(45) T m ( ρ m u m n | ∇ φ m n | 2 ) → T m ( ρ m ( u m ) | ∇ φ m | 2 ) in L q ( Ω ) for all 1 ≤ q < ∞ .

Thus,

(46) lim n → ∞ ∫ Ω T m ( ρ m u m n | ∇ φ m n | 2 ) u m n − u m = 0 .

Gathering all these properties, we obtain

lim n → ∞ ∫ Ω | ∇ u m n − u m | p = 0 ,

that is, u m n → u m strongly in W 0 1 , p ( Ω ) .

1 < p < 2 In this case, the estimate for the first term of the left hand side of (36) becomes

(47) ∫ Ω a x , u m n , ∇ u m n ∇ T h u m n − u m = ∫ Ω ( a x , u m n , ∇ u m n − a x , u m n , ∇ u m ) ∇ T h u m n − u m + ∫ Ω a x , u m n , ∇ u m ∇ T h u m n − u m ≥ α ∫ Ω | ∇ T h u m n − u m | 2 | ∇ u m n | + | ∇ u m | p − 2 + ∫ Ω a x , u m n , ∇ u m ∇ T h u m n − u m .

On the other hand, for γ = p(2 − p)/2 and owing to Hölder’s inequality, we have

∫ Ω | ∇ T h u m n − u m | p = ∫ Ω | ∇ T h u m n − u m | p | ∇ u m n | + | ∇ u m | − γ | ∇ u m n | + | ∇ u m | γ ≤ ∫ Ω | ∇ T h u m n − u m | 2 | ∇ u m n | + | ∇ u m | p − 2 p / 2 ∫ Ω | ∇ u m n | + | ∇ u m | p ( 2 − p ) / 2 ,

and using (27) we readily obtain that, for some constant C _m > 0,

C m ∫ Ω | ∇ T h u m n − u m | p 2 / p ≤ ∫ Ω | ∇ T h u m n − u m | 2 | ∇ u m n | + | ∇ u m | p − 2 ,

and combining this last expression with (47) it yields

∫ Ω a x , u m n , ∇ u m n ∇ T h u m n − u m ≥ α C m ∫ Ω | ∇ T h u m n − u m | p 2 / p + ∫ Ω a x , u m n , ∇ u m ∇ T h u m n − u m .

From this point on, we may repeat all the steps as in the previous case p ≥ 2 and deduce again the strong convergence u m n → u m in W 0 1 , p ( Ω ) . □

It remains to show that u _m satisfies the variational inequatity (17). To do so, we follow the same arguments of Ref. [11]. We take T k u m n − θ v as a test function in (23), with v ∈ K M and where θ is a real number such that 0 < θ < 1. This yields

(48) ∫ Ω a x , u m n , ∇ u m n ∇ T k u m n − θ v + ∫ Ω u m n M n − 2 u m n M T k u m n − θ v = ∫ Ω T m ( ρ m u m n | ∇ φ m n | 2 ) T k u m n − θ v .

By Lemma 4 and (7), we have

(49) a x , u m n , ∇ u m n → a ( x , u m , ∇ u m ) strongly in L p ′ ( Ω ) N .

Thus,

(50) ∫ Ω a x , u m n , ∇ u m n ∇ T k u m n − θ v → ∫ Ω a ( x , u m , ∇ u m ) ∇ T k ( u m − θ v )

Also, using (45), then

(51) ∫ Ω T m ( ρ m u m n | ∇ φ m n | 2 ) T k u m n − θ v → ∫ Ω T m ( ρ m ( u m ) | ∇ φ m | 2 ) T k ( u m − θ v )

For the second term of (48), by the same arguments as (39), we get

∫ Ω u m n M n − 2 u m n M T k u m n − θ v ≥ ∫ 0 < u m n < θ v u m n M n − 2 u m n M T k u m n − θ v + ∫ θ v < u m n < 0 u m n M n − 2 u m n M T k u m n − θ v = I m , n k , 1 + I m , n k , 2 .

Notice that u m n M ≤ θ in both sets 0 < u m n < θ v and θ v < u m n < 0 . Hence,

0 ≥ lim n → ∞ I m , n k , 1 = lim n → ∞ ∫ 0 < u m n < θ v u m n M n − 1 T k u m n − θ v ≥ lim n → ∞ ∫ 0 < u m n < θ v | θ | n − 1 T k u m n − θ v

whenever these limits exist. On the other hand,

0 ≥ lim n → ∞ I m , n k , 2 = − lim n → ∞ ∫ θ v < u m n < 0 u m n M n − 1 T k u m n − θ v ≥ − lim n → ∞ ∫ θ v < u m n < 0 | θ | n − 1 T k u m n − θ v

if these limit exist. Since 0 < θ < 1 and T k u m n − θ v is bounded, then

lim n → ∞ ∫ θ v < u m n < 0 | θ | n − 1 T k u m n − θ v = 0 , lim n → ∞ ∫ 0 < u m n < θ v | θ | n − 1 T k u m n − θ v = 0 .

Thus, the above limits exist and

lim n → ∞ I m , n k , 1 = 0 and lim n → ∞ I m , n k , 2 = 0 .

Consequently,

(52) lim inf n → ∞ ∫ Ω u m n M n − 2 u m n M T k u m n − θ v ≥ 0

Gathering (50)–(52) we obtain:

For all k > 0 and all v ∈ K M , ∫ Ω a ( x , u m , ∇ u m ) ∇ T k ( u m − θ v ) ≤ ∫ Ω T m ( ρ m ( u m ) | ∇ φ m | 2 ) T k ( u m − θ v ) .

For k large enough, we get

(53) ∫ Ω a ( x , u m , ∇ u m ) ∇ ( u m − θ v ) ≤ ∫ Ω T m ( ρ m ( u m ) | ∇ φ m | 2 ) ( u m − θ v ) .

Finally, letting θ tend to 1, we obtain that for all v ∈ K M

(54) ∫ Ω a ( x , u m , ∇ u m ) ∇ ( u m − v ) ≤ ∫ Ω T m ( ρ m ( u m ) | ∇ φ m | 2 ) ( u m − v ) .

This ends the proof of the Lemma 2. □

Proof of the Theorem 1.

Again, we begin with the derivation of some a priori estimates for (u _m) and (φ _m), and then we show that (u _m, φ _m) converges, up to a subsequence, to a bilateral solution to the problem (5). Notice that ρ _m(u _m) = ρ(u _m) for m ≥ M. From now on, we will assume that m ≥ M.

By taking ψ = φ _m − φ ₀ as a test function in (18), we get

∫ Ω ρ ( u m ) | ∇ φ m | 2 = ∫ Ω ρ ( u m ) ∇ φ m ∇ φ 0

Moreover, owing to Lemma 4 and since ρ is continuous, then

min − M ≤ s ≤ M ρ ( s ) ∫ Ω | ∇ φ m | 2 ≤ ∫ Ω ρ ( u m ) ∇ φ m ∇ φ 0 ≤ max − M ≤ s ≤ M ρ ( s ) ∫ Ω | ∇ φ m | ∇ φ 0 .

Hence, using Hölder’s inequality, we get

(55) ∫ Ω | ∇ φ m | 2 ≤ C ,

here C does not depends on m. Hence, there exists a function φ ∈ H ¹(Ω) and a subsequence, still denoted in the same way, such that

(56) φ m → φ weakly in H 1 ( Ω ) , strongly in L 2 ( Ω ) and a.e. in Ω .

In fact, once we have established the almost everywhere convergence, for some suitable subsequence, of (u _m), we readily obtain

(57) φ m → φ strongly in H 1 ( Ω ) .

Indeed, taking v = 0 in (54), we get

∫ Ω a ( x , u m , ∇ u m ) ∇ u m ≤ ∫ Ω T m ( ρ ( u m ) | ∇ φ m | 2 ) u m .

Thus, using Lemma 4,

∫ Ω a ( x , u m , ∇ u m ) ∇ u m ≤ M ∫ Ω T m ( ρ ( u m ) | ∇ φ m | 2 ) .

From (8) and the definition of the truncation function, we obtain

α ∫ Ω | ∇ u m | p ≤ M ∫ Ω ρ ( u m ) | ∇ φ m | 2 .

Thus, owing to (55),

(58) ‖ u m ‖ W 1 , p ( Ω ) ≤ C .

Hence, there exist u ∈ W 0 1 , p ( Ω ) and a subsequence, still denoted in the same way, such that,

(59) u m → u weakly in W 1 , p ( Ω ) , strongly in L p ( Ω ) and a.e. in Ω .

Furthermore, for 0 < θ < 1, we have

(60) ∫ Ω { [ a ( x , u m , ∇ u m ) − a ( x , u m , ∇ u ) ] ∇ ( u m − u ) } θ → 0 ,

as m tends to infinity. For the proof of (60) one may repeat the same arguments as in Proposition 1 below. Thus, using (6), up to a subsequence, still denoted in the same way,

(61) ∇ u m → ∇ u a.e. in Ω .

Now, we are going to pass to the limit in (54). Owing to (58), (59) and (61), and in view of (7), we get

(62) a ( x , u m , ∇ u m ) → a ( x , u , ∇ u ) weakly in L p ′ ( Ω ) .

Combining (59) and (62) and using Fatou’s lemma, it yields

(63) ∫ Ω a ( x , u , ∇ u ) ∇ ( u − v ) ≤ lim inf m → ∞ ∫ Ω a ( x , u m , ∇ u m ) ∇ ( u m − v ) .

In view of (59) and using the fact that ρ is continuous, we get

ρ ( u m ) → ρ ( u ) a.e. in Ω .

Using (9) and the Lebesgue theorem, we get

ρ ( u m ) → ρ ( u ) in L q ( Ω ) for all q < ∞ .

Hence, using (57),

(64) ρ ( u m ) ∇ φ m → ρ ( u ) ∇ φ strongly in L 2 ( Ω ) ,

and also,

(65) T m ( ρ ( u m ) | ∇ φ m | 2 ) → ρ ( u ) | ∇ φ | 2 strongly in L 1 ( Ω ) .

Gathering (59) and (65), we can pass to the limit in the right hand side of (54) to get

(66) ∫ Ω T m ( ρ ( u m ) | ∇ φ m | 2 ) ( u m − v ) → ∫ Ω ρ ( u ) | ∇ φ | 2 ( u − v )

The estimate (63), together with (66), allows us to pass to the limit in (54). This yields

(67) ∫ Ω a ( x , u , ∇ u ) ( u − v ) ≤ ∫ Ω ρ ( u ) | ∇ φ | 2 ( u − v ) for all v ∈ K M .

Notice that (64) implies

(68) ∫ Ω ρ ( u ) ∇ φ ∇ ψ = 0 , for all ψ ∈ H 0 1 ( Ω ) .

This completes the proof of Theorem 1. □

5 Conclusions

In this work, we have extended the results obtained by González Montesinos and Ortegón Gallego [5] for the case p ≥ 2 to the case 1 < p < 2. In this setting, the search for weak solutions or even for capacity solutions are not well suited and then we need to develop another approach; this has been done by using the concept of bilateral solution at a given height M > 0. The main result of this work establishes the existence of a bilateral solution to a strongly nonlinear elliptic coupled system. The system may be regarded as a generalization of the well-known thermistor problem. The proof of this result is based on a penalization technique combined with a fixed point argument. Indeed, this kind of solution (u, φ) is obtained as the limit of solution of certain approximate problems.

Corresponding author: Francisco Ortegón Gallego, Departamento de Matemáticas, Universidad de Cádiz, Puerto Real, Spain, E-mail: francisco.ortegon@uca.es

Funding source: Ministerio de Ciencia e Innovación of the Spanish Government with the participation of the European Regional Development Fund (ERDF/FEDER).

Award Identifier / Grant number: PID2020-117201RB-C21

Acknowledgment

The authors wish to thank the anonymous reviewers for their fruitful comments and suggestions which have help us to improve this presentation.

Research ethics: Not applicable.
Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Competing interests: The authors state no conflict of interest.
Research funding: This research was partially supported by Ministerio de Ciencia e Innovación of the Spanish Government [grant PID2020-117201RB-C21] with the participation of the European Regional Development Fund (ERDF/FEDER).
Data availability: Not applicable.

Appendix

Proof of Lemma 3

First of all, we will show that the operator G has an invariant convex, close and bounded set. Indeed, let as consider v = u m n as a test function in (22), we get

∫ Ω a x , ω m n , ∇ u m n ∇ u m n + ∫ Ω u m n M n − 2 u m n M u m n = ∫ Ω T m ( ρ m ω m n | ∇ φ m n | 2 ) u m n .

Thus, by the definition of T _m, it yields

∫ Ω a x , ω m n , ∇ u m n ∇ u m n + ∫ Ω M u m n M n ≤ m ∫ Ω | u m n | .

Using (6) and (8), Hölder’s and Poincaré’s inequality, we get

α ∫ Ω | ∇ u m n | p ≤ ∫ Ω a x , ω m n , ∇ u m n ∇ u m n ≤ m meas ( Ω ) 1 / p ′ ∫ Ω | u m n | p 1 / p ≤ C m ∫ Ω | ∇ u m n | p 1 / p .

Consequently,

‖ u m n ‖ W 1 , p ( Ω ) ≤ C m ,

and, in particular

(69) ‖ u m n ‖ L p ( Ω ) ≤ C m .

Consider the set B m = v ∈ L p ( Ω ) / ‖ v ‖ L p ( Ω ) ≤ C m . From (69), for C _m > 0 large enough, the operator G transforms the set B _m into itself. It is clear that the closed set B _m is bounded and convex. On the other hand, due to the boundedness of u m n n ≥ 1 in W 0 1 , p ( Ω ) and since the embedding W 0 1 , p ( Ω ) ↪ L p ( Ω ) is compact, the operator G is compact. It remains to show that G is continuous. To this end, let ω m n , j j ⊂ B m such that

(70) ω m n , j → ω m n strongly in L p ( Ω )

and consider the corresponding functions to ω m n , j , that is, u m n , j = G ω m n , j and φ m n , j , i.e. the pair u m n , j , φ m n , j verifies the following system

(71) − div a x , ω m n , j , ∇ u m n , j + u m n , j M n − 2 u m n , j M = T m ( ρ m ω m n , j | ∇ φ m n , j | 2 ) in Ω div ( ρ m ω m n , j ∇ φ m n , j ) = 0 in Ω u m n , j = 0 , on ∂ Ω φ m n , j = φ 0 on ∂ Ω

We have to show that

u m n , j → G ω m n strongly in L p ( Ω ) .

The variational formulation of the first equation of (71) is as follows.

(72) ∫ Ω a x , ω m n , j , ∇ u m n , j ∇ v + ∫ Ω u m n , j M n − 2 u m n , j M v = ∫ Ω T m ( ρ m ω m n , j | φ m n , j | 2 ) v , for all v ∈ W 1 , p ( Ω ) ∩ L ∞ ( Ω ) and v = u m n , j .

As above, it is easy to show that

(73) ‖ u m n , j ‖ W 1 , p ( Ω ) ≤ C m ,

and that

(74) ∫ Ω u m n , j M n ≤ C m ,

where C _m does not depend on j. Hence, for a suitable subsequence, there exists v m n ∈ W 1 , p ( Ω ) such that

(75) u m n , j → v m n weakly in W 1 , p ( Ω )

(76) u m n , j → v m n strongly in L p ( Ω ) and a.e. in Ω .

Thus,

u m n , j M n − 2 u m n , j M → v m n M n − 2 v m n M a.e. in Ω .

We want to show the strong convergence of u m n , j M n − 2 u m n , j M in L ¹(Ω). To this end, we will make use of Vitali’s theorem, thus, we will prove that u m n , j M n − 2 u m n , j M is equi-integrable. Let E be a measurable set of Ω and γ be a positive real number. From (74), we get

∫ E u m n , j M n − 1 = ∫ E ∩ u m n , j M ≤ γ u m n , j M n − 1 + ∫ E ∩ u m n , j M > γ u m n , j M n − 1 ≤ γ n − 1 meas ( E ) + 1 γ ∫ u m n , j M > γ u m n , j M n ≤ γ n − 1 meas ( E ) + C γ .

Since this last quantity can be chosen arbitrarily small for γ sufficiently large and any measurable set E ⊂ Ω such that meas (E) small enough, we get

(77) u m n , j M n − 2 u m n , j M → v m n M n − 2 v m n M in L 1 ( Ω ) .

Also, as in the proof of Theorem 1, we can show that

(78) ρ m ω m n , j | ∇ φ m n , j | 2 → ρ m ω m n | ∇ φ m n | 2 strongly in L 1 ( Ω ) .

We will make use of the following proposition.

Proposition 1 .

[13]Let 1/p < θ < 1, then

∫ Ω a x , ω m n , j , ∇ u m n , j − a x , ω m n , j , ∇ v m n ∇ u m n , j − v m n θ → 0 ,

as j → ∞.

Proof.

We fix m and n, then for j ≥ 1, we put

S j = a x , ω m n , j , ∇ u m n , j − a x , ω m n , j , ∇ v m n ∇ u m n , j − v m n .

We have

∫ Ω S j θ = ∫ | v m n | ≤ k S j θ + ∫ | v m n | > k S j θ = I k , j 1 + I k , j 2 .

According to the a priori estimates proved in the previous step, (S _j) is bounded in L ¹(Ω), then using Hölder’s inequality, we get

I k , j 2 ≤ ∫ | v m n | > k S j θ meas | v m n | > k 1 − θ ,

hence

I k , j 2 ≤ C meas | v m n | > k 1 − θ = ω 1 ( k ) .

On the other hand, we can write I k , n 1 as:

I k , j 1 = ∫ | v m n | ≤ k a x , ω m n , j , ∇ u m n , j − a ( x , ω m n , j , ∇ T k v m n ) ∇ ( u m n , j − T k v m n ) θ ,

which is smaller than

∫ Ω a x , ω m n , j , ∇ u m n , j − a ( x , ω m n , j , ∇ T k v m n ) ∇ ( u m n , j − T k v m n ) θ .

Then, for any ɛ > 0, we have

I k , j 1 ≤ ∫ { | u m n , j − T k v m n | ≤ ε } a x , ω m n , j , ∇ u m n , j − a ( x , ω m n , j , ∇ T k v m n ) ∇ ( u m n , j − T k v m n ) θ + ∫ { | u m n , j − T k v m n | > ε } a x , ω m n , j , ∇ u m n , j − a ( x , ω m n , j , ∇ T k v m n ) ∇ ( u m n , j − T k v m n ) θ = J 1 + J 2 .

Notice that J ₁ can be written as

J 1 = ∫ Ω a x , ω m n , j , ∇ u m n , j − a ( x , ω m n , j , ∇ T k v m n ) ∇ T ε ( u m n , j − T k v m n ) θ .

Using Hölder’s inequality, and putting |Ω| = meas (Ω), we obtain

J 1 ≤ | Ω | 1 − θ ∫ Ω a x , ω m n , j , ∇ u m n , j − a ( x , ω m n , j , ∇ T k v m n ) ∇ T ε ( u m n , j − T k v m n ) θ .

Using T ε ( u m n , j − T k v m n ) in (72) as a test function it yields

J 1 ≤ C ∫ Ω T m ( ρ m ω m n , j | ∇ φ m n , j | 2 ) T ε ( u m n , j − T k v m n ) − ∫ Ω a ( x , ω m n , j , ∇ T k v m n ) ∇ T ε ( u m n , j − T k v m n ) − ∫ Ω u m n , j M n − 2 u m n , j M T ε ( u m n , j − T k v m n ) θ ≤ C J 1 1 − J 1 2 − J 1 3 θ .

From (78) we obtain

(79) J 1 1 ≤ ∫ Ω ρ m ω m n , j | ∇ φ m n , j | 2 ε ≤ C ε .

We have that

(80) lim j → ∞ J 1 2 = ∫ Ω a ( x , ω m n , ∇ T k v m n ) ∇ T ε ( v m n − T k v m n ) = ω 2 ( k ) ,

and also, using (77), that

(81) lim j → ∞ J 1 3 = ∫ Ω v m n M n − 2 v m n M T ε ( v m n − T k v m n ) = ω 3 ( k ) ,

Gathering (79)–(81), we obtain

(82) lim j → ∞ J 1 ≤ C ( C ε − ω 2 ( k ) − ω 3 ( k ) ) θ .

On the other hand, using Hölder’s inequality again in the integral term J ₂, we obtain

J 2 ≤ ∫ { | u m n , j − T k v m n | > ε } a x , ω m n , j , ∇ u m n , j − a ( x , ω m n , j , ∇ T k v m n ) ∇ ( u m n , j − T k v m n ) θ × meas | u m n , j − T k v m n | > ε 1 − θ .

Thus,

J 2 ≤ C meas | u m n , j − T k v m n | > ε 1 − θ .

We notice that

(83) lim j → ∞ meas | u m n , j − T k v m n | > ε = meas | v m n − T k v m n | ≥ ε = ω 4 ( k ) .

Thus

lim j → ∞ J 2 ≤ ω 4 ( k ) .

Hence we have shown that

lim j → ∞ I k , j 1 + I k , j 2 ≤ ω 1 ( k ) + C ( C ε − ω 2 ( k ) − ω 3 ( k ) ) θ + ω 4 ( k )

where ω _i(k), i = 1, …, 4, converges to zero as k tends to infinity. Letting k → ∞ then ɛ → 0, we get the desired result. □

Using Proposition 1 and (6) we get, for a certain constant C = C(m, θ) > 0,

C ∫ Ω | ∇ u m n , j − v m n | p θ ≤ ∫ Ω S j θ → 0 as j → ∞ .

Thus, for a suitable subsequence still denoted in the same way, we have

∇ u m n , j → ∇ v m n a.e. in Ω ,

which implies that

a x , ω m n , j , ∇ u m n , j → a x , ω m n , ∇ v m n a.e. in Ω .

Furthermore, in view of (7), we get

(84) a x , ω m n , j , ∇ u m n , j → a x , ω m n , ∇ v m n weakly in L p ′ ( Ω ) .

Gathering (77), (78) and (84), we can pass to the limit in (72) to get

∫ Ω a x , ω m n , ∇ v m n ∇ v + ∫ Ω v m n M n − 2 v m n M v = ∫ Ω T m ( ρ m ω m n | ∇ φ m n | 2 ) v , for all v ∈ W 0 1 , p ( Ω ) ∩ L ∞ ( Ω ) .

For v = T k v m n , as k goes to infinity, we have

∫ Ω a x , ω m n , ∇ v m n ∇ T k v m n − ∫ Ω T m ( ρ m ω m n | ∇ φ m n | 2 ) T k v m n → ∫ Ω a x , ω m n , ∇ v m n ∇ v m n − ∫ Ω T m ( ρ m ω m n | ∇ φ m n | 2 ) v m n

On the other hand, from (74), (76) and using Fatou’s lemma, we get v m n M n ∈ L 1 ( Ω ) . Since

v m n M n − 2 v m n M T k v m n → M v m n M n a.e. in Ω ,

and

v m n M n − 1 | T k v m n | ≤ M v m n M n ,

Then, by Lebesgue’s dominated convergence theorem, we obtain

v m n M n − 2 v m n M T k v m n → M v m n M n in L 1 ( Ω ) .

Thus,

∫ Ω a x , ω m n , ∇ v m n ∇ v + ∫ Ω v m n M n − 2 v m n M v = ∫ Ω T m ( ρ m ω m n | ∇ φ m n | 2 ) v , for all v ∈ W 0 1 , p ( Ω ) ∩ L ∞ ( Ω ) and v = v m n .

which means that v m n = G ω m n = u m n . □

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Received: 2023-05-11

Accepted: 2024-03-24

Published Online: 2024-04-22

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Articles in the same Issue

https://doi.org/10.1515/ans-2023-0133

Keywords for this article

coupled system; bilateral solution; nonlinear elliptic equation; thermistor problem; penalization techniques

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