Two solutions for Dirichlet double phase problems with variable exponents

Eleonora Amoroso; Gabriele Bonanno; Giuseppina D’Aguì; Patrick Winkert

doi:10.1515/ans-2023-0134

Article Open Access

Two solutions for Dirichlet double phase problems with variable exponents

Eleonora Amoroso , Gabriele Bonanno , Giuseppina D’Aguì and Patrick Winkert

Published/Copyright: May 8, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advanced Nonlinear Studies Volume 24 Issue 3

Abstract

This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.

Keywords: bounded solutions; critical point theory; double phase operator with variable exponents; multiplicity results; superlinear reaction

1991 Mathematics Subject Classification: 35A01; 35B33; 35D30; 35J62; 35J66

1 Introduction

This paper deals with the following boundary value problem with a nonlinear differential equation involving the double phase operator with variable exponents and Dirichlet boundary condition

(P _λ) − d i v | ∇ u | p ( x ) − 2 ∇ u + μ ( x ) | ∇ u | q ( x ) − 2 ∇ u = λ f ( x , u ) in Ω , u = 0 on ∂ Ω ,

where Ω ⊆ R N , N ≥ 2, is a bounded domain with Lipschitz boundary ∂Ω, p , q ∈ C ( Ω ̄ ) such that

1 < p ( x ) < N , p ( x ) < q ( x ) < p * ( x ) for all x ∈ Ω ̄ ,

with p * ( x ) = N p ( x ) N − p ( x ) for all x ∈ Ω ̄ , 0 ≤ μ(⋅) ∈ L ^∞(Ω), λ > 0 is a parameter and f : Ω × R → R is a Carathéodory function that satisfies subcritical growth and a certain behavior at ±∞, see assumptions (H_f) in Section 2.

Problems of this type are widely studied in the literature because of their application in several topics. The double phase operator, i.e.

d i v | ∇ u | p ( x ) − 2 ∇ u + μ ( x ) | ∇ u | q ( x ) − 2 ∇ u , u ∈ W 0 1 , H ( Ω ) ,

is related to the following two-phase integral functional

u → ∫ Ω | ∇ u | p ( x ) + μ ( x ) | ∇ u | q ( x ) d x , u ∈ W 0 1 , H ( Ω ) ,

where W 0 1 , H ( Ω ) denotes a Musielak–Orlicz Sobolev space that will be introduced in an appropriate way in Section 2. This functional changes ellipticity depending on the set where the weight function μ(⋅) is zero, shifting in two different phases of elliptic behavior. Zhikov [1] was the first who studied this functional for constant exponents to describe the behavior of strongly anisotropic materials; indeed, in the elasticity theory, μ(⋅) contains the information on the geometry of composites made of two different materials with power hardening exponents p(⋅) and q(⋅), see also Zikhov [2]. Furthermore, the double phase operator has several applications also in mathematical topics, as the Lavrentiev gap phenomenon and the duality theory. A first mathematical treatment of such two-phase integrals has been done by Baroni–Colombo–Mingione [3], [4], [5], Colombo–Mingione [6], [7] and Ragusa–Tachikawa [8], see also the work of De Filippis–Mingione [9] for nonautonomous integrals.

Many authors have shown existence and multiplicity results for double phase problems with constant exponents, see, for example, the papers of Biagi–Esposito–Vecchi [10], Colasuonno–Squassina [11], Gasiński–Winkert [12], Gasiński–Papageorgiou [13], Liu–Dai [14], Papageorgiou–Rădulescu–Repovš [15], Perera–Squassina [16], Zeng–Bai–Gasiński–Winkert [17] and the references therein. Whereas, in the variable exponent case there are only few results, we refer to Aberqi–Bennouna–Benslimane–Ragusa [18], Bahrouni–Rădulescu–Winkert [19], Cen–Kim–Kim–Zeng [20], Crespo–Blanco–Gasiński–Harjulehto–Winkert [21], Leonardi–Papageorgiou [22], Liu–Pucci [23], Kim–Kim–Oh–Zeng [24], Vetro–Winkert [25] and Zeng–Rădulescu–Winkert [26].

Motivated by the large interest on this differential operator in the current literature, our aim is to apply a two critical point theorem due to Bonanno–D’Aguì [27] to a more general class of problems. We observe that Theorem 2.1 of [27] gives the existence of two nontrivial critical points, one of local minimum type and the other of mountain pass type, without using standard techniques such as upper and lower methods and regularity theory. Our work extends the recent papers of Chinnì–Sciammetta–Tornatore [28] if μ ≡ 1 and of Sciammetta–Tornatore–Winkert [29] if the exponents p and q are constants. Indeed, if inf Ω ̄ μ > 0 , the double phase operator reduces to the (p(⋅), q(⋅))-Laplacian and if μ ≡ 0 to the p(⋅)-Laplacian. Moreover, instead of supposing the typical Ambrosetti–Rabinowitz condition on the function f as required in Ref. [28], we only assume that the nonlinear term on the right-hand side of (P _λ) is (q ₊ − 1)-superlinear at ±∞ (see (H_f)(iii)) and satisfies another appropriate behavior at ±∞. However, these conditions are weaker than the Ambrosetti–Rabinowitz condition. Under these assumptions, we are going to prove the existence of two weak solutions for problem (P_λ ) that have opposite energy sign and belong to L ^∞(Ω), namely they are bounded.

We also point out that the exponents p(⋅) and q(⋅) do not need to verify a condition of the type

(1.1) q ( ⋅ ) p ( ⋅ ) < 1 + 1 N

as it was needed, for example, in Kim–Kim–Oh–Zeng [24] or in Colasuonno–Squassina [11] and Liu–Dai [14] for the constant exponent case. Indeed, we only require assumption (H) (see Section 2), since Crespo–Blanco–Gasiński–Harjulehto–Winkert [21, Proposition 2.19] recently proved that our function space W 0 1 , H ( Ω ) can be equipped with the equivalent norm  ∇ ⋅ H without supposing (1.1).

The paper is organized as follows. In Section 2 we recall the main properties of the Musielak–Orlicz Sobolev spaces and the double phase operator, we present the variational framework and we formulate our assumptions. In Section 3 we present our main result (Theorem 3.2) on the existence of two nontrivial bounded weak solutions for problem (P_λ ), we consider other hypotheses in order to get nonnegative solutions (Corollary 3.3 and Theorem 3.4) and we also provide an example.

2 Variational framework and preliminaries

In this section we present the main preliminaries in order to study problem (P_λ ) and recall first the main results concerning variable exponent Lebesgue and Sobolev spaces as well as properties of Musielak–Orlicz Sobolev space and the corresponding double phase operator. These results can be mainly found in the books of Diening–Harjulehto–Hästö–R u ̊ žička [30], Harjulehto–Hästö [31] and Musielak [32], see also the papers of Colasuonno–Squassina [11], Crespo–Blanco–Gasiński–Harjulehto–Winkert [21] and Liu–Dai [14]. To this end, let Ω ⊆ R N , N ≥ 2 be a bounded domain with Lipschitz boundary ∂Ω. For 1 ≤ r ≤ ∞ we denote by L ^r(Ω) the usual Lebesgue spaces endowed with the norm ‖ ⋅ ‖_r and for 1 ≤ r < ∞, W ^1,r(Ω) and W 0 1 , r ( Ω ) indicate the Sobolev spaces endowed with the usual norms ‖ ⋅‖_1,r and ‖ ⋅ ‖_1,r,0 = ‖∇ ⋅ ‖_r, respectively. First, we introduce the Lebesgue and Sobolev spaces with variable exponents and some properties that will be useful in the sequel. For any r ∈ C ( Ω ̄ ) , we put

r + ≔ max x ∈ Ω ̄ r ( x ) and r − ≔ min x ∈ Ω ̄ r ( x ) ,

and we define

C + ( Ω ̄ ) = { r ∈ C ( Ω ̄ ) : r − > 1 } .

For any r ∈ C + ( Ω ̄ ) , we define the modular by

ρ r ( ⋅ ) ( u ) = ∫ Ω | u | r ( x ) d x ,

and the variable exponent Lebesgue space is given by

L r ( ⋅ ) ( Ω ) = { u : Ω → R measurable ∣ ρ r ( ⋅ ) ( u ) < ∞ } ,

equipped with the related Luxemburg norm

u r ( ⋅ ) = inf τ > 0 : ρ r ( ⋅ ) u τ ≤ 1 .

We can also introduce, for any r ∈ C + ( Ω ̄ ) , the corresponding Sobolev space with variable exponent, denoted by

W 1 , r ( ⋅ ) ( Ω ) = u ∈ L r ( ⋅ ) ( Ω ) : | ∇ u | ∈ L r ( ⋅ ) ( Ω ) ,

endowed with the usual norm

u 1 , r ( ⋅ ) = u r ( ⋅ ) + ∇ u r ( ⋅ ) ,

where ∇ u r ( ⋅ ) = | ∇ u | r ( ⋅ ) . Furthermore, we denote by W 0 1 , r ( ⋅ ) ( Ω ) the closure of C 0 ∞ ( Ω ) in W ^1,r(⋅)(Ω) for which we know that a Poincaré inequality holds and we can equip the space with the equivalent norm  u 1 , r ( ⋅ ) , 0 = ∇ u r ( ⋅ ) . In particular, these spaces are uniformly convex, separable and reflexive Banach spaces and we refer to Diening–Harjulehto–Hästö–R u ̊ žička [30, Theorems 3.2.7, 3.4.7, 3.4.9, 8.1.6, 8.1.13 and Corollary 3.4.5].

Next, we recall some properties about the relation between the norm and the corresponding modular that will be needed in the sequel, see Fan–Zhao [33].

Proposition 2.1.

Let r ∈ C + ( Ω ̄ ) , u ∈ L ^r(⋅)(Ω) and λ ∈ R . Then the following hold:

If u ≠ 0, then ‖ u ‖ r ( ⋅ ) = λ ⟺ ρ r ( ⋅ ) ( u λ ) = 1 ;
‖u‖_r(⋅) < 1 (resp. > 1 , = 1) ⇔ ρ _r(⋅)(u) < 1 (resp. > 1 , = 1);
If ‖ u ‖ r ( ⋅ ) < 1 ⇒ ‖ u ‖ r ( ⋅ ) r + ≤ ρ r ( ⋅ ) ( u ) ≤ ‖ u ‖ r ( ⋅ ) r − ;
If ‖ u ‖ r ( ⋅ ) > 1 ⇒ ‖ u ‖ r ( ⋅ ) r − ≤ ρ r ( ⋅ ) ( u ) ≤ ‖ u ‖ r ( ⋅ ) r + ;
‖u‖_r(⋅) → 0 ⇔ ρ _r(⋅)(u) → 0;
‖u‖_r(⋅) → +∞ ⇔ ρ _r(⋅)(u) → +∞.

Now, we introduce the Musielak–Orlicz space, the Musielak–Orlicz Sobolev space and the double phase operator, recalling some main and useful properties. We assume the following hypotheses:

p , q ∈ C ( Ω ̄ ) such that 1 < p(x) < N and p(x) < q(x) < p*(x) for all x ∈ Ω ̄ , where p * ( ⋅ ) = N p ( ⋅ ) N − p ( ⋅ ) is the critical Sobolev exponent to p(⋅), and μ ∈ L ^∞(Ω), with μ(⋅) ≥ 0.

Let H : Ω × [ 0 , ∞ ] → [ 0 , ∞ ] be the nonlinear function defined by

H ( x , t ) = t p ( x ) + μ ( x ) t q ( x ) for all ( x , t ) ∈ Ω × [ 0 , ∞ ] ,

and let ρ H ( ⋅ ) be the corresponding modular defined by

ρ H ( u ) = ∫ Ω H ( x , | u | ) d x = ∫ Ω | u | p ( x ) + μ ( x ) | u | q ( x ) d x .

Then, we denote by L H ( Ω ) the Musielak–Orlicz space, given by

L H ( Ω ) = u : Ω → R measurable ∣ ρ H ( u ) < + ∞ ,

endowed with the Luxemburg norm

‖ u ‖ H = inf τ > 0 : ρ H u τ ≤ 1 .

Similar to Proposition 2.1, we have a certain relationship between the modular ρ H ( ⋅ ) and the norm  ‖ ⋅ ‖ H , see Crespo–Blanco–Gasiński–Harjulehto–Winkert [21, Proposition 2.13] for a detailed proof.

Proposition 2.2.

Let (H) be satisfied, u ∈ L H ( Ω ) and λ ∈ R . Then the following hold:

If u ≠ 0, then ‖ u ‖ H = λ ⟺ ρ H ( u λ ) = 1 ;
‖ u ‖ H < 1 (resp. > 1 , = 1) ⟺ ρ H ( u ) < 1 (resp. > 1 , = 1);
If ‖ u ‖ H < 1 ⇒ ‖ u ‖ H q + ≤ ρ H ( u ) ≤ ‖ u ‖ H p − ;
If ‖ u ‖ H > 1 ⇒ ‖ u ‖ H p − ≤ ρ H ( u ) ≤ ‖ u ‖ H q + ;
‖ u ‖ H → 0 ⟺ ρ H ( u ) → 0 ;
‖ u ‖ H → + ∞ ⟺ ρ H ( u ) → + ∞ ;
‖ u ‖ H → 1 ⟺ ρ H ( u ) → 1 ;
If u _n → u in L H ( Ω ) , then ρ H ( u n ) → ρ H ( u ) .

We denote by W 1 , H ( Ω ) the Musielak–Orlicz Sobolev space defined by

W 1 , H ( Ω ) = u ∈ L H ( Ω ) : | ∇ u | ∈ L H ( Ω ) ,

equipped with the norm

‖ u ‖ 1 , H = ‖ ∇ u ‖ H + ‖ u ‖ H ,

with ‖ ∇ u ‖ H = ‖ | ∇ u | ‖ H and by W 0 1 , H ( Ω ) we indicate the completion of C 0 ∞ ( Ω ) in W 1 , H ( Ω ) . In Ref. [21, Proposition 2.12] the authors prove that L H ( Ω ) , W 1 , H ( Ω ) and W 0 1 , H ( Ω ) are uniformly convex, so reflexive Banach spaces. Moreover, the following embedding results can be found in Crespo–Blanco–Gasiński–Harjulehto–Winkert [21, Proposition 2.16].

Proposition 2.3.

Let (H) be satisfied. Then the following embeddings hold:

L H ( Ω ) ↪ L r ( ⋅ ) ( Ω ) , W 0 1 , H ( Ω ) ↪ W 0 1 , r ( ⋅ ) ( Ω ) are continuous for all r ∈ C ( Ω ̄ ) with 1 ≤ r(x) ≤ p(x) for all x ∈ Ω ̄ ;
W 0 1 , H ( Ω ) ↪ L r ( ⋅ ) ( Ω ) is compact for all r ∈ C ( Ω ̄ ) with 1 ≤ r(x) < p*(x) for all x ∈ Ω ̄ .

Moreover, from Proposition 2.18 in [21], we have that W 0 1 , H ( Ω ) is compactly embedded in L H ( Ω ) , so we can equip the space W 0 1 , H ( Ω ) with the equivalent norm

‖ u ‖ 1 , H , 0 = ‖ ∇ u ‖ H .

Now, for any r ∈ C ( Ω ̄ ) for which the continuous embedding W 0 1 , H ( Ω ) ↪ L r ( ⋅ ) ( Ω ) hold (see Proposition 2.3), we denote by c ̃ r the best constant for which one has

(2.1) u r ( ⋅ ) ≤ c ̃ r u 1 , H , 0 .

Finally, we introduce the assumptions on the perturbation in problem (P_λ ) and suppose the following hypotheses:

Let f : Ω × R → R and F ( x , t ) = ∫ 0 t f ( x , s ) d s be such that the following hold:
1. f is a Carathéodory function, that is, x → f(x, t) is measurable for all t ∈ R and t → f(x, t) is continuous for almost all (a.a.) x ∈ Ω;
2. there exist ℓ ∈ C + ( Ω ̄ ) with ℓ + < ( p − ) * and κ ₁ > 0 such that

| f ( x , t ) | ≤ κ 1 1 + | t | ℓ ( x ) − 1

for a.a. x ∈ Ω and for all t ∈ R ;

lim t → ± ∞ F ( x , t ) | t | q + = + ∞
uniformly for a.a. x ∈ Ω;
there exists ζ ∈ C + ( Ω ̄ ) with
ζ − ∈ ( ℓ + − p − ) N p − , ℓ +
and ζ ₀ > 0 such that
0 < ζ 0 ≤ lim inf t → ± ∞ f ( x , t ) t − q + F ( x , t ) | t | ζ ( x )
uniformly for a.a. x ∈ Ω.

Remark 2.4.

It should be noted that the condition on ζ in (H_f)(iv) is well defined since from (H_f)(ii) we have ℓ + < ( p − ) * and so it holds

( ℓ + − p − ) N p − = ℓ + N p − − ( p − ) * N − p − p − < ℓ + N p − − ℓ + N − p − p − = ℓ + .

The differential operator in (P_λ ) is the so-called double phase operator with variable exponents given by

− d i v | ∇ u | p ( x ) − 2 ∇ u + μ ( x ) | ∇ u | q ( x ) − 2 ∇ u for u ∈ W 0 1 , H ( Ω ) .

It is well known that u ∈ W 0 1 , H ( Ω ) is called a weak solution of problem (P_λ ) if

(2.2) ∫ Ω | ∇ u | p ( x ) − 2 ∇ u + μ ( x ) | ∇ u | q ( x ) − 2 ∇ u ⋅ ∇ v d x = λ ∫ Ω f ( x , u ) v d x ,

for all v ∈ W 0 1 , H ( Ω ) . In order to establish results on the existence of two nontrivial weak solution for (P_λ ), we define the functionals Φ , Ψ , I λ : W 0 1 , H ( Ω ) → R by

Φ ( u ) = ∫ Ω | ∇ u | p ( x ) p ( x ) + μ ( x ) | ∇ u | q ( x ) q ( x ) d x , Ψ ( u ) = ∫ Ω F ( x , u ( x ) ) d x ,

and

I λ ( u ) = Φ ( u ) − λ Ψ ( u ) ,

where I _λ is the so-called energy functional. We know that Φ and Ψ are Gâteaux differentiable with its derivatives

⟨ Φ ′ ( u ) , v ⟩ = ∫ Ω | ∇ u | p ( x ) − 2 ∇ u + μ ( x ) | ∇ u | q ( x ) − 2 ∇ u ⋅ ∇ v d x , ⟨ Ψ ′ ( u ) , v ⟩ = ∫ Ω f ( x , u ) v d x , ⟨ I λ ′ ( u ) , v ⟩ = ∫ Ω | ∇ u | p ( x ) − 2 ∇ u + μ ( x ) | ∇ u | q ( x ) − 2 ∇ u ⋅ ∇ v d x − λ ∫ Ω f ( x , u ) v d x ,

for all u , v ∈ W 0 1 , H ( Ω ) , where ⟨⋅, ⋅⟩ is the duality pairing between W 0 1 , H ( Ω ) and its dual space W 0 1 , H ( Ω ) * . Hence, from (2.2) it follows that u is a weak solution of (P_λ ) if and only if u is a critical point of I _λ, i.e., ⟨ I λ ′ ( u ) , v ⟩ = 0 for all v ∈ W 0 1 , H ( Ω ) . In the next proposition we summarize the properties of the operator Φ ′ : W 0 1 , H ( Ω ) → W 0 1 , H ( Ω ) * , see Ref. [21, Theorem 3.3] which is a generalization of Ref. [14, Proposition 3.1] in the variable exponent case.

Proposition 2.5.

Let hypotheses (H) be satisfied. Then, the operator Φ′ is bounded, continuous, strictly monotone and of type (S ₊), that is,

if u n ⇀ u in W 0 1 , H ( Ω ) and lim sup n → ∞ ⟨ Φ ′ ( u n ) , u n − u ⟩ ≤ 0 ,

then u _n → u in W 0 1 , H ( Ω ) .

The main tool of our investigation is a two critical point theorem established by Bonanno–D’Aguì [27, Theorem 2.1 and Remark 2.2], which is a nontrivial consequence of a local minimum theorem due to Bonanno [34, Theorem 2.3] in combination with the Ambrosetti–Rabinowitz Theorem. Here we recall the definition of the Cerami condition that will be needed. In the following, for X being a Banach space, we denote by X* its topological dual space.

Definition 2.6.

Given L ∈ C ¹(X), we say that L satisfies the Cerami-condition, (C)-condition for short, if every sequence { u n } n ∈ N ⊆ X such that

{ L ( u n ) } n ≥ 1 ⊆ R is bounded,
1 + ‖ u n ‖ X L ′ ( u n ) → 0 in X* as n → ∞,

admits a strongly convergent subsequence in X.

For the reader’s convenience, we restate Theorem 2.1 [27] taking into account Remark 2.2 [27].

Theorem 2.7.

Let X be a real Banach space and let Φ , Ψ : X → R be two continuously Gâteaux differentiable functionals such that

inf X Φ = Φ ( 0 ) = Ψ ( 0 ) = 0 .

Assume that Φ is coercive and there exist r ∈ R and u ̃ ∈ X , with 0 < Φ ( u ̃ ) < r , such that

(2.3) sup u ∈ Φ − 1 ( [ − ∞ , r ] ) Ψ ( u ) r < Ψ ( u ̃ ) Φ ( u ̃ )

and, for each λ ∈ [ Φ ( u ̃ ) Ψ ( u ̃ ) , r sup u ∈ Φ − 1 ( ] − ∞ , r ] ) Ψ ( u ) ] , the functional I _λ = Φ − λΨ satisfies the (C)-condition and it is unbounded from below. Then, for each λ ∈ [ Φ ( u ̃ ) Ψ ( u ̃ ) , r sup u ∈ Φ − 1 ( ] − ∞ , r ] ) Ψ ( u ) ] , the functional I _λ admits at least two nontrivial critical points u _λ,1, u _λ,2 such that I _λ(u _λ,1) < 0 < I _λ(u _λ,2).

3 Main result

In this section, we present our main result on the existence of two nontrivial solutions for the Dirichlet double phase problem with variational exponents given in (P_λ ). For this purpose, let

R ≔ sup x ∈ Ω dist ( x , ∂ Ω ) .

Then there exists x ₀ ∈ Ω such that the ball with center x ₀ and radius R > 0 belongs to Ω, that is,

B ( x 0 , R ) ⊆ Ω .

We indicate with ω _R the Lebesgue measure of B(x ₀, R) in R N given by

ω R ≔ | B ( x 0 , R ) | = π N 2 Γ 1 + N 2 R N ,

and we put

δ ≔ min R p − , R q + p − max 1 , μ ∞ ω R ( 2 N − 1 ) 2 q + + 1 − N .

Furthermore, for any r , η ∈ R + , we define

(3.1) α ( r ) ≔ κ 1 c ̃ 1 max q + r 1 p − , q + r 1 q + + c ̄ ℓ max q + r ℓ + p − , q + r ℓ − q + r ,

(3.2) β ( η ) ≔ δ ∫ B x 0 , R 2 F ( x , η ) d x max η p − , η q + ,

where c ̄ ℓ = max c ̃ ℓ ℓ − , c ̃ ℓ ℓ + and c ̃ 1 , c ̃ ℓ , κ ₁, ℓ are given in (2.1) and (H_f)(ii), respectively. From now on, we put

X = W 0 1 , H ( Ω ) and ⋅ X = ⋅ 1 , H , 0 = ∇ ⋅ H .

First, we present the following Lemma that we will use in the proof of the main result.

Lemma 3.1.

Let the assumptions (H) and (H_f) be satisfied. Then, the functional I _λ satisfies the (C)-condition for all λ > 0.

Proof.

Let { u n } n ∈ N ⊆ W 0 1 , H ( Ω ) be a sequence such that (C₁) and (C₂) from Definition 2.6 hold. We provide the proof in three steps.

Claim 1.

{ u n } n ∈ N is bounded in L ζ − ( Ω ) .

First, it is easy to show that using (H_f)(i), (ii) and (iv) we get that

(3.3) f ( x , t ) t − q + F ( x , t ) ≥ c 1 | t | ζ − − c 2 for a.a. x ∈ Ω and for all t ∈ R ,

with some constants c ₁, c ₂ > 0. Moreover, from (C₁) we have that there exists a constant M > 0 such that for all n ∈ N one has |I _λ(u _n)| ≤ M, so

∫ Ω | ∇ u n | p ( x ) p ( x ) + μ ( x ) | ∇ u n | q ( x ) q ( x ) d x − λ ∫ Ω F ( x , u n ) d x ≤ M ,

which, multiplying by q ₊, leads to

(3.4) ρ H ( ∇ u n ) − λ ∫ Ω q + F ( x , u n ) d x ≤ c 3 ,

for some c ₃ > 0 and for all n ∈ N . Besides, from (C₂), there exists { ε n } n ∈ N with ɛ _n → 0⁺ such that

(3.5) ⟨ I λ ′ ( u n ) , v ⟩ ≤ ε n v X 1 + u n X for all n ∈ N and for all v ∈ X .

Choosing v = u _n, one has

∫ Ω | ∇ u n | p ( x ) + μ ( x ) | ∇ u n | q ( x ) d x − λ ∫ Ω f ( x , u n ) u n d x < ε n ,

which implies

(3.6) − ρ H ( ∇ u n ) + λ ∫ Ω f ( x , u n ) u n d x < ε n

for all n ∈ N . Adding (3.4) and (3.6) we obtain

∫ Ω f ( x , u n ) u n − q + F ( x , u n ) d x < c 4

for all n ∈ N with some constant c ₄ > 0. Using this along with (3.3) we derive

∫ Ω c 1 | u n | ζ − − c 2 d x < c 4 ,

which gives

u n ζ − ζ − < c 5 for all n ∈ N

with some c ₅ > 0. Hence, { u n } n ∈ N is bounded in L ζ − ( Ω ) and so Claim 1 is proved.

Claim 2.

{ u n } n ∈ N is bounded in X.

From (H_f)(ii) and (iv), we have that

ζ − < ℓ + < ( p − ) * .

Hence, there exists s ∈ (0, 1) such that

(3.7) 1 ℓ + = s ( p − ) * + 1 − s ζ − ,

and using the interpolation inequality, see Papageorgiou–Winkert [35, Proposition 2.3.17, p. 116], one has

u n ℓ + ≤ u n ( p − ) * s u n ζ − 1 − s for all n ∈ N .

From Claim 1, it follows that

(3.8) u n ℓ + ≤ c 6 u n ( p − ) * s

for some c ₆ > 0 and for all n ∈ N . Again, from (3.5) with v = u _n, we get

(3.9) ρ H ( ∇ u n ) − λ ∫ Ω f ( x , u n ) u n d x < ε n .

We may assume u n X ≥ 1 for all n ∈ N , otherwise we are done. Then, using Proposition 2.2(iv), (3.9), (H_f)(ii) and (3.8), we derive that

u n X p − ≤ ρ H ( ∇ u n ) < ε n + λ ∫ Ω f ( x , u n ) u n d x ≤ λ κ 1 u n 1 + ρ ℓ ( ⋅ ) ( u n ) + ε n ≤ c 7 1 + c 6 ℓ + u n ( p − ) * s ℓ + + ε n

with c ₇ > 0. Hence, taking the embedding X ↪ L ( p − ) * ( Ω ) into account, we have

u n X p − ≤ c 8 1 + u n X s ℓ + + ε n ,

for all n ∈ N and for some c ₈ > 0. From (3.7) and (H_f)(iv), it follows that

s ℓ + = ( p − ) * ( ℓ + − ζ − ) ( p − ) * − ζ − = N p − ( ℓ + − ζ − ) N p − − N ζ − + ζ − p − < N p − ( ℓ + − ζ − ) N p − − N ζ − + p − ( ℓ + − p − ) N p − = p − ,

and this shows our second claim.

Claim 3.

u _n → u in X up to a subsequence.

Since { u n } n ∈ N ⊂ X is bounded (Claim 2) and X is a reflexive space, there exists a subsequence, not relabeled, that converges weakly in X and strongly in L ℓ + ( Ω ) , that is,

u n ⇀ u in X and u n → u in L ℓ + ( Ω ) .

Using this to (3.5) with v = u _n − u and passing to the limit as n → ∞, we obtain

⟨ Φ ′ ( u n ) , u n − u ⟩ → 0 as n → ∞ .

Since Φ′ satisfies the (S ₊)-property, see Proposition 2.5, the proof is complete. □

Now, we state our main result.

Theorem 3.2.

Assume that (H) and (H_f) hold. Furthermore, suppose that there exist two positive constants r, η satisfying

(3.10) max η p − , η q + < δ r ,

such that

F(x, t) ≥ 0 for a.a. x ∈ Ω and for all t ∈ [0, η];
α(r) < β(η),

as defined in (3.1) and (3.2). Then, for each λ ∈ Λ, where

Λ ≔ [ 1 β ( η ) , 1 α ( r ) ] ,

problem (P_λ ) admits at least two nontrivial bounded weak solutions u _λ,1, u λ , 2 ∈ W 0 1 , H ( Ω ) with opposite energy sign.

Proof.

Our aim is to apply Theorem 2.7. Let ( X , ⋅ X ) , Φ , Ψ be as in Section 2 and note that they already fulfill the required assumptions needed in Theorem 2.7. In particular, from Proposition 2.2(vi) and (H_f)(iii) follows that Φ is coercive and I _λ is unbounded from below, respectively.

Now, fix λ ∈ Λ, which is nonempty because of (H₂), and consider u ̃ ∈ X defined by

u ̃ ( x ) = 0 if x ∈ Ω \ B ( x 0 , R ) , 2 η R ( R − | x − x 0 | ) if x ∈ B ( x 0 , R ) \ B x 0 , R 2 , η if x ∈ B x 0 , R 2 .

Clearly, u ̃ ∈ X . We show that 0 < Φ ( u ̃ ) < r . Indeed, using (3.10), it follows

Φ ( u ̃ ) = ∫ B ( x 0 , R ) \ B x 0 , R 2 1 p ( x ) 2 η R p ( x ) + μ ( x ) q ( x ) 2 η R q ( x ) d x ≤ 2 q + p − ∫ B ( x 0 , R ) \ B x 0 , R 2 η R p ( x ) + μ ( x ) η R q ( x ) d x ≤ 2 q + p − max 1 , μ ∞ min R p − , R q + max η p − , η q + ⋅ 2 ⋅ ω R − ω R 2 = 1 δ max η p − , η q + < r .

Now, we prove (2.3). From assumption (H₁), we obtain

Ψ ( u ̃ ) = ∫ B x 0 , R 2 F ( x , η ) d x + ∫ B ( x 0 , R ) \ B x 0 , R 2 F x , 2 η R ( R − | x − x 0 | ) d x ≥ ∫ B x 0 , R 2 F ( x , η ) d x .

Hence,

(3.11) Ψ ( u ̃ ) Φ ( u ̃ ) ≥ δ ∫ B x 0 , R 2 F ( x , η ) d x max η p − , η q + .

On the other hand, fix u ∈ X such that Φ(u) ≤ r. Then, one has

q + r ≥ q + Φ ( u ) > ρ H ( ∇ u ) ≥ min u X p − , u X q + ,

which implies that

Φ − 1 ] − ∞ , r ] ⊆ u ∈ X : u X < max ( q + r ) 1 p − , ( q + r ) 1 q + .

Furthermore, we have

sup u ∈ Φ − 1 ( ] − ∞ , r ] ) Ψ ( u ) ≤ sup u ∈ Φ − 1 ( ] − ∞ , r ] ) κ 1 ∫ Ω | u | + | u | ℓ ( x ) d x = sup u ∈ Φ − 1 ( ] − ∞ , r ] ) κ 1 u 1 + ρ ℓ ( ⋅ ) ( u ) ≤ sup u ∈ Φ − 1 ( ] − ∞ , r ] ) κ 1 u 1 + max u ℓ ( ⋅ ) ℓ − , u ℓ ( ⋅ ) ℓ + ≤ sup u ∈ Φ − 1 ( ] − ∞ , r ] ) κ 1 c ̃ 1 u X + max c ̃ ℓ ℓ − , c ̃ ℓ ℓ + max u X ℓ − , u X ℓ + ≤ κ 1 c ̃ 1 max ( q + r ) 1 p − , ( q + r ) 1 q + + c ̄ ℓ max ( q + r ) ℓ + p − , ( q + r ) ℓ − q + .

Then, taking (H₂) and (3.11) into account, we get

sup u ∈ Φ − 1 ( ] − ∞ , r ] ) Ψ ( u ) r ≤ κ 1 c ̃ 1 max ( q + r ) 1 p − , ( q + r ) 1 q + + c ̄ ℓ max ( q + r ) ℓ + p − , ( q + r ) ℓ − q + r < δ ∫ B x 0 , R 2 F ( x , η ) d x max η p − , η q + ≤ Ψ ( u ̃ ) Φ ( u ̃ ) ,

namely hypothesis (2.3) is satisfied. Hence, along with Lemma 3.1, Theorem 2.7 ensures the existence of two nontrivial weak solutions u _λ,1, u λ , 2 ∈ W 0 1 , H ( Ω ) such that I _λ(u _λ,1) < 0 < I _λ(u _λ,2). Finally, from Crespo–Blanco–Winkert [36, Theorem 3.1] it follows that u _λ,1, u _λ,2 belong to L ^∞(Ω). This finishes the proof. □

Corollary 3.3.

Suppose that all assumptions of Theorem 3.2 are satisfied. Moreover, assume that f(x, 0) ≥ 0 and f(x, t) = f(x, 0) for a.a. x ∈ Ω and for all t < 0. Then, problem (P_λ ) admits at least two nontrivial and nonnegative bounded weak solutions u _λ,1, u λ , 2 ∈ W 0 1 , H ( Ω ) with opposite energy sign.

Proof.

Since all the assumptions are satisfied, we can apply Theorem 3.2. We only need to prove that the solutions u _λ,1, u _λ,2 are nonnegative. Since u _λ,1 is a weak solution of (P_λ ), from (2.2) one has ⟨ I λ ′ ( u λ , 1 ) , v ⟩ = 0 for every v ∈ X. Choosing v = − u λ , 1 − = − max { − u λ , 1 , 0 } ∈ W 0 1 , H ( Ω ) , see Ref. [21, Proposition 2.17(iii)], we have

∫ Ω | ∇ u λ , 1 | p ( x ) − 2 ∇ u λ , 1 + μ ( x ) | ∇ u λ , 1 | q ( x ) − 2 ∇ u λ , 1 ⋅ ∇ − u λ , 1 − d x = λ ∫ Ω f ( x , u λ , 1 ) − u λ , 1 − d x ,

which leads to

− ρ H ∇ u λ , 1 − ≥ 0 .

But the previous inequality implies that

min u λ , 1 − X p − , u λ , 1 − X q + ≤ ρ H ∇ u λ , 1 − ≤ 0 ,

which gives ‖ u λ , 1 − ‖ X = 0 . Then, u λ , 1 − = 0 and u _λ,1 ≥ 0. With the same argument we obtain u _λ,2 ≥ 0 and the proof is complete. □

Now we consider the special case when the nonlinear term is nonnegative.

Theorem 3.4.

Assume that (H) and (H_f) hold. Furthermore, suppose that f is nonnegative and

(H ₃) lim sup t → 0 + inf x ∈ Ω F ( x , t ) t p − = + ∞ .

Then, for each λ ∈ [0, λ*], with

λ * = sup r > 0 1 α ( r ) ,

where α(r) is given in (3.1), problem (P_λ ) admits at least two nontrivial and nonnegative bounded weak solutions u _λ,1, u λ , 2 ∈ W 0 1 , H ( Ω ) with opposite energy sign.

Proof.

We observe that (3) implies that

(3.12) lim sup η → 0 + β ( η ) = lim sup η → 0 + δ ∫ B x 0 , R 2 F ( x , η ) d x max η p − , η q + ≥ δ ω R 2 lim sup η → 0 + inf x ∈ Ω F ( x , η ) η p − = + ∞ .

Then, fixing λ ∈ [0, λ*], there exists r > 0 such that

λ < 1 α ( r ) = r κ 1 c ̃ 1 max q + r 1 p − , q + r 1 q + + c ̄ ℓ max q + r ℓ + p − , q + r ℓ − q + .

Moreover, from (3.12), there is η > 0 small enough such that

δ ω R 2 inf x ∈ Ω F ( x , η ) η p − > 1 λ ,

implying that α(r) < β(η). Applying Theorem 3.2 and arguing as in the proof of Corollary 3.3, we achieve our goal. □

Finally, we provide an example of a function that satisfies our assumptions.

Example 3.5.

Consider f : Ω × R → R defined by

f ( x , t ) = | t | α ( x ) − 2 t if | t | < 1 , | t | β ( x ) − 2 t log | t | + 1 if | t | ≥ 1 ,

where α , β ∈ C ( Ω ̄ ) such that q + < β ( x ) < ( p − ) * for all x ∈ Ω ̄ and

β + p − − β − N < 1 .

Then, f satisfies assumptions (H_f) with ζ(x) = β(x) for all x ∈ Ω ̄ and ℓ(x) = β(x) + σ for all x ∈ Ω ̄ , with σ > 0 small enough such that

ℓ + < ( p − ) * , ℓ + p − − β − N < 1 .

Moreover, we can apply Theorem 3.4 at f ₊(x, t) = |f(x, t)| for every ( x , t ) ∈ Ω × R , requiring also that α(x) < p ₋ for all x ∈ Ω ̄ .

Corresponding author: Eleonora Amoroso, Department of Engineering, University of Messina, 98166 Messina, Italy, E-mail: eleonora.amoroso@unime.it

Funding source: PRIN 2022 – Progetti di Ricerca di rilevante Interesse Nazionale

Award Identifier / Grant number: 2022ZXZTN2

Acknowledgment

The authors wish to thank the reviewers for their useful comments in order to improve the paper. The first three authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The first author thanks the University of Technology Berlin for the kind hospitality during a research stay.

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 paper is supported by PRIN 2022 –Progetti di Ricerca di rilevante Interesse Nazionale, “Nonlinear differential problems with applications to real phenomena” (2022ZXZTN2).
Data availability: Not applicable.

References

[1] V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 50, no. 4, pp. 675–710, 1986.Search in Google Scholar

[2] V. V. Zhikov, “On variational problems and nonlinear elliptic equations with nonstandard growth conditions,” J. Math. Sci., vol. 173, no. 5, pp. 463–570, 2011. https://doi.org/10.1007/s10958-011-0260-7.Search in Google Scholar

[3] P. Baroni, M. Colombo, and G. Mingione, “Harnack inequalities for double phase functionals,” Nonlinear Anal., vol. 121, pp. 206–222, 2015, https://doi.org/10.1016/j.na.2014.11.001.Search in Google Scholar

[4] P. Baroni, M. Colombo, and G. Mingione, “Nonautonomous functionals, borderline cases and related function classes,” St. Petersburg Math. J., vol. 27, no. 3, pp. 347–379, 2016. https://doi.org/10.1090/spmj/1392.Search in Google Scholar

[5] P. Baroni, M. Colombo, and G. Mingione, “Regularity for general functionals with double phase,” Calc. Var. Part. Differ. Equ., vol. 57, no. 2, p. 48, 2018. https://doi.org/10.1007/s00526-018-1332-z.Search in Google Scholar

[6] M. Colombo and G. Mingione, “Bounded minimisers of double phase variational integrals,” Arch. Ration. Mech. Anal., vol. 218, no. 1, pp. 219–273, 2015. https://doi.org/10.1007/s00205-015-0859-9.Search in Google Scholar

[7] M. Colombo and G. Mingione, “Regularity for double phase variational problems,” Arch. Ration. Mech. Anal., vol. 215, no. 2, pp. 443–496, 2015. https://doi.org/10.1007/s00205-014-0785-2.Search in Google Scholar

[8] M. A. Ragusa and A. Tachikawa, “Regularity for minimizers for functionals of double phase with variable exponents,” Adv. Nonlinear Anal., vol. 9, no. 1, pp. 710–728, 2020. https://doi.org/10.1515/anona-2020-0022.Search in Google Scholar

[9] C. De Filippis and G. Mingione, “Lipschitz bounds and nonautonomous integrals,” Arch. Ration. Mech. Anal., vol. 242, no. 2, pp. 973–1057, 2021. https://doi.org/10.1007/s00205-021-01698-5.Search in Google Scholar

[10] S. Biagi, F. Esposito, and E. Vecchi, “Symmetry and monotonicity of singular solutions of double phase problems,” J. Differ. Equ., vol. 280, pp. 435–463, 2021, https://doi.org/10.1016/j.jde.2021.01.029.Search in Google Scholar

[11] F. Colasuonno and M. Squassina, “Eigenvalues for double phase variational integrals,” Ann. Mat. Pura Appl., vol. 195, no. 6, pp. 1917–1959, 2016.10.1007/s10231-015-0542-7Search in Google Scholar

[12] L. Gasiński and P. Winkert, “Existence and uniqueness results for double phase problems with convection term,” J. Differ. Equ., vol. 268, no. 8, pp. 4183–4193, 2020. https://doi.org/10.1016/j.jde.2019.10.022.Search in Google Scholar

[13] L. Gasiński and N. S. Papageorgiou, “Constant sign and nodal solutions for superlinear double phase problems,” Adv. Calc. Var., vol. 14, no. 4, pp. 613–626, 2021. https://doi.org/10.1515/acv-2019-0040.Search in Google Scholar

[14] W. Liu and G. Dai, “Existence and multiplicity results for double phase problem,” J. Differ. Equ., vol. 265, no. 9, pp. 4311–4334, 2018. https://doi.org/10.1016/j.jde.2018.06.006.Search in Google Scholar

[15] N. S. Papageorgiou, V. D. Rădulescu, and D. D. Repovš, “Double-phase problems and a discontinuity property of the spectrum,” Proc. Am. Math. Soc., vol. 147, no. 7, pp. 2899–2910, 2019. https://doi.org/10.1090/proc/14466.Search in Google Scholar

[16] K. Perera and M. Squassina, “Existence results for double-phase problems via Morse theory,” Commun. Contemp. Math., vol. 20, no. 2, p. 14, 2018. https://doi.org/10.1142/s0219199717500237.Search in Google Scholar

[17] S. Zeng, Y. Bai, L. Gasiński, and P. Winkert, “Existence results for double phase implicit obstacle problems involving multivalued operators,” Calc. Var. Part. Differ. Equ., vol. 59, no. 5, p. 176, 2020. https://doi.org/10.1007/s00526-020-01841-2.Search in Google Scholar

[18] A. Aberqi, J. Bennouna, O. Benslimane, and M. A. Ragusa, “Existence results for double phase problem in Sobolev–Orlicz spaces with variable exponents in complete manifold,” Mediterr. J. Math., vol. 19, no. 4, p. 158, 2022. https://doi.org/10.1007/s00009-022-02097-0.Search in Google Scholar

[19] A. Bahrouni, V. D. Rădulescu, and P. Winkert, “Double phase problems with variable growth and convection for the Baouendi–Grushin operator,” Z. Angew. Math. Phys., vol. 71, no. 6, p. 14, 2020. https://doi.org/10.1007/s00033-020-01412-7.Search in Google Scholar

[20] J. Cen, S. J. Kim, Y.-H. Kim, and S. Zeng, “Multiplicity results of solutions to the double phase anisotropic variational problems involving variable exponent,” Adv. Differ. Equ., vol. 28, nos. 5–6, pp. 467–504, 2023. https://doi.org/10.57262/ade028-0506-467.Search in Google Scholar

[21] Á. Crespo-Blanco, L. Gasiński, P. Harjulehto, and P. Winkert, “A new class of double phase variable exponent problems: existence and uniqueness,” J. Differ. Equ., vol. 323, pp. 182–228, 2022, https://doi.org/10.1016/j.jde.2022.03.029.Search in Google Scholar

[22] S. Leonardi and N. S. Papageorgiou, “Anisotropic Dirichlet double phase problems with competing nonlinearities,” Rev. Mat. Complut., vol. 36, no. 2, pp. 469–490, 2023. https://doi.org/10.1007/s13163-022-00432-3.Search in Google Scholar

[23] J. Liu and P. Pucci, “Existence of solutions for a double-phase variable exponent equation without the Ambrosetti–Rabinowitz condition,” Adv. Nonlinear Anal., vol. 12, no. 1, pp. 1–18, 2023. https://doi.org/10.1515/anona-2022-0292.Search in Google Scholar

[24] I. H. Kim, Y.-H. Kim, M. W. Oh, and S. Zeng, “Existence and multiplicity of solutions to concave-convex-type double-phase problems with variable exponent,” Nonlinear Anal. R. World Appl., vol. 67, pp. 1–25, 2022. https://doi.org/10.1016/j.nonrwa.2022.103627.Search in Google Scholar

[25] F. Vetro and P. Winkert, “Constant sign solutions for double phase problems with variable exponents,” Appl. Math. Lett., vol. 135, pp. 1–7, 2023. https://doi.org/10.1016/j.aml.2022.108404.Search in Google Scholar

[26] S. Zeng, V. D. Rădulescu, and P. Winkert, “Double phase obstacle problems with variable exponent,” Adv. Differ. Equ., vol. 27, nos. 9–10, pp. 611–645, 2022. https://doi.org/10.57262/ade027-0910-611.Search in Google Scholar

[27] G. Bonanno and G. D’Aguì, “Two non-zero solutions for elliptic Dirichlet problems,” Z. Anal. Anwendungen, vol. 35, no. 4, pp. 449–464, 2016. https://doi.org/10.4171/zaa/1573.Search in Google Scholar

[28] A. Chinnì, A. Sciammetta, and E. Tornatore, “Existence of non-zero solutions for a Dirichlet problem driven by (p(x), q(x))-Laplacian,” Appl. Anal., vol. 101, no. 15, pp. 5323–5333, 2022. https://doi.org/10.1080/00036811.2021.1889523.Search in Google Scholar

[29] A. Sciammetta, E. Tornatore, and P. Winkert, “Bounded weak solutions to superlinear Dirichlet double phase problems,” Anal. Math. Phys., vol. 13, no. 2, p. 18, 2023. https://doi.org/10.1007/s13324-023-00783-0.Search in Google Scholar

[30] L. Diening, P. Harjulehto, P. Hästö, and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Heidelberg, Springer, 2011.10.1007/978-3-642-18363-8Search in Google Scholar

[31] P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Cham, Springer, 2019.10.1007/978-3-030-15100-3Search in Google Scholar

[32] J. Musielak, Orlicz Spaces and Modular Spaces, Berlin, Springer-Verlag, 1983.10.1007/BFb0072210Search in Google Scholar

[33] X. Fan and D. Zhao, “On the spaces Lp(x)(Ω) and Wm,p(x)(Ω),” J. Math. Anal. Appl., vol. 263, no. 2, pp. 424–446, 2001. https://doi.org/10.1006/jmaa.2000.7617.Search in Google Scholar

[34] G. Bonanno, “Relations between the mountain pass theorem and local minima,” Adv. Nonlinear Anal., vol. 1, no. 3, pp. 205–220, 2012. https://doi.org/10.1515/anona-2012-0003.Search in Google Scholar

[35] N. S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis, Berlin, De Gruyter, 2018.10.1515/9783110532982Search in Google Scholar

[36] Á. Crespo-Blanco and P. Winkert, “Nehari manifold approach for superlinear double phase problems with variable exponents,” Ann. Mat. Pura Appl. (4), vol. 203, no. 2, pp. 605–634, 2024.10.1007/s10231-023-01375-2Search in Google Scholar

Received: 2023-03-15

Accepted: 2024-03-24

Published Online: 2024-05-08

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

Articles in the same Issue

https://doi.org/10.1515/ans-2023-0134

Keywords for this article

bounded solutions; critical point theory; double phase operator with variable exponents; multiplicity results; superlinear reaction

Creative Commons

BY 4.0