Parametric singular double phase Dirichlet problems

Yunru Bai; Nikolaos S. Papageorgiou; Shengda Zeng

doi:10.1515/anona-2023-0122

Article Open Access

Parametric singular double phase Dirichlet problems

Yunru Bai , Nikolaos S. Papageorgiou and Shengda Zeng

Published/Copyright: December 31, 2023

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From the journal Advances in Nonlinear Analysis Volume 12 Issue 1

Abstract

We consider a parametric (with two parameters μ , λ > 0 ) Dirichlet problem driven by the double phase differential operator and a reaction which has the competing effect of a singular term and of a superlinear perturbation. We prove a bifurcation-type result in the parameter λ > 0 , when the other parameter μ > 0 is large.

Keywords: singular term; double phase operator; Orlicz-Sobolev space; positive solution; existence, and multiplicity result

MSC 2010: 35J60; 35J66; 35J75

1 Introduction

Let Ω ⊆ R N be a bounded domain with a C 2 boundary ∂ Ω . In this work, we study the following parametric (with two parameters μ , λ > 0 ) singular Dirichlet problem:

(1.1) − Δ p a u − Δ q u = μ u − η + λ f ( z , u ) in Ω , u = 0 on ∂ Ω , u > 0 in Ω ,

with μ , λ > 0 , 1 < q < p < N , and η ∈ ( 0 , 1 ) . For a ∈ L ∞ ( Ω ) \ { 0 } with a ( z ) ≥ 0 for a.a. z ∈ Ω , Δ p a denotes the weighted p -Laplace differential operator defined by

Δ p a u ≔ div ( a ( z ) ∣ D u ∣ p − 2 D u ) .

If a ≡ 1 , then we recover the standard p -Laplacian. Problem (1.1) is driven by the sum of two such operators with different exponents Δ p a u + Δ q u . So, the differential operator of problem (1.1) is not homogeneous. This operator is associated with the so-called double phase integral functional

u ↦ ∫ Ω a ( z ) ∣ D u ∣ p + ∣ D u ∣ q d z .

In the present study, we do not assume that a ( ⋅ ) is bounded away from zero (i.e., we do not require that essinf Ω a > 0 ). So, the density of the above integral functional θ ( z , t ) = a ( z ) t p + t q for all z ∈ Ω , all t ≥ 0 , exhibits unbalanced growth, namely, the following inequalities hold

t q ≤ θ ( z , t ) ≤ c 0 ( 1 + t p ) for all z ∈ Ω , all t ≥ 0 , for some c 0 > 0 .

Such functionals were first considered by Marcellini [22,23] and Zhikov [37,38], in the context of problems of calculus of variations and of nonlinear elasticity including the Lavrentiev gap phenomenon. More recently, the interest for such functionals was revived and there have been efforts to develop a regularity theory for the solutions of such problems. We mention the survey papers of Marcellini [24] and Mingione-Rǎdulescu [25] and references therein. So far, we have only local regularity results for minimizers of such functions. A global regularity theory (i.e., regularity up to the boundary) remains elusive. For balanced growth problems (e.g., driven by the ( p , q )-Laplacian), such a global theory exists and can be found in the study of Lieberman [17]. The lack of such a theory eliminates many effective tools from consideration and makes the study of double phase boundary value problems difficult. A characteristic example of such a tool is the equivalence between the local Hölder and Sobolev minimizers. This result was first observed by Brézis and Nirenberg [4] for semilinear problems driven by the Laplacian and later extended to the p -Laplacian by Garcia Azorero et al. [8] and to more general anisotropic operators by Papageorgiou et al. [27]. This result turned out to be a very fruitful tool in proving multiplicity theorems for balanced growth problems.

There have been very few works on singular double phase problems. We mention the papers of Arora et al. [1,2], Liu and Papageorgiou [20], Liu and Winkert [21], Liu et al. [19], Farkas and Winkert [7], and Papageorgiou et al. [28] for strongly singular problems. Multiplicity results are proved only in [20] and [21], for special kinds of forcing terms in which the perturbation of the singularity is of the power type. Moreover, the multiplicity results in [20] and [21] are not global with respect to the parameter λ > 0 . Here, we prove an existence and multiplicity result, which is global in λ > 0 (a bifurcation-type theorem), when the other parameter μ > 0 is large. Finally, we also mention the recent double phase works of Crespo-Blanco et al. [6] (anisotropic problems) and of Bahrouni et al. [3] (problems driven by Grushin operator).

Problem (1.1) for equations with the p -Laplacian and μ = 1 was first studied by Papageorgiou and Smyrlis [29] and by Papageorgiou and Winkert [32]. Their results were extended to ( p , q )-equations by Papageorgiou and Winkert [33] (balanced growth problems). Their methods of proof relied heavily on the existing global regularity theory. As we already mentioned earlier, this is no longer available for double phase problems. So, we have to come up with new techniques.

2 Mathematical background and hypotheses

One consequence of the unbalanced growth is that we have to abandon the convenient functional framework of standard Sobolev spaces and use generalized Orlicz-Sobolev spaces. A comprehensive presentation of the theory of these spaces can be found in the book of Harjulehto and Hästo [13].

Our hypotheses on the weight function a ( ⋅ ) and the exponents p , q , η are as follows:

H ̲ 0 : a ∈ L ∞ ( Ω ) ‒ { 0 } , a ( z ) ≥ 0 for all z ∈ Ω ¯ , 1 < q < p < N , p q < 1 + 1 N , 0 < η < q N .

Remark 2.1

The assumption on a ( ⋅ ) guarantees that the Poincaré inequality holds on the corresponding generalized Orlicz-Sobolev space (Crespo-Blanco et al. [6]). The condition p q < 1 + 1 N is common in Dirichlet double phase problems and it implies that p < q * = N q N − q , which in turn leads to compact embeddings of some relevant spaces.

Let L 0 ( Ω ) be the space of all measurable functions u : Ω → R . As usual we identify two such functions which differ only on a Lebesgue-null subset of Ω . Recall η ( z , t ) is the double phase density defined by

η ( z , t ) ≔ a ( z ) t p + t q for all z ∈ Ω , all t ≥ 0 .

Then, the generalized Orlicz-Lebesgue space L η ( Ω ) is defined by

L η ( Ω ) ≔ u ∈ L 0 ( Ω ) ∣ ρ η ( u ) ≔ ∫ Ω η ( z , u ) d z < + ∞ .

The function ρ η ( ⋅ ) is known as the modular function corresponding to η . We equip L η ( Ω ) with the so-called “Luxemburg norm” ‖ ⋅ ‖ η defined by

‖ u ‖ η ≔ inf λ : ρ η u λ ≤ 1 for all u ∈ L η ( Ω ) .

With this norm, L η ( Ω ) becomes a Banach space which is separable and reflexive (in fact uniformly convex). Using L η ( Ω ) we can define the corresponding generalized Orlicz-Sobolev space W 1 , η ( Ω ) by

W 1 , η ( Ω ) ≔ { u ∈ L η ( Ω ) ∣ ∣ D u ∣ ∈ L η ( Ω ) } ,

where D u denotes the weak gradient of u . We equip W 1 , η ( Ω ) with the norm ‖ ⋅ ‖ 1 , η defined by

‖ u ‖ 1 , η ≔ ‖ u ‖ η + ‖ D u ‖ η for all u ∈ W 1 , η ( Ω ) ,

where ‖ D u ‖ η = ‖ ∣ D u ∣ ‖ η . Also, we define

W 0 1 , η ( Ω ) ≔ C c ∞ ( Ω ) ¯ ‖ ⋅ ‖ 1 , η ,

with C c ∞ ( Ω ) ≔ { u ∈ C ∞ ( Ω ) ∣ u ( ⋅ ) has compact support in Ω } . Both spaces W 1 , η ( Ω ) and W 0 1 , η ( Ω ) are Banach spaces which are separable and reflexive (in fact uniformly convex). As we already mentioned, hypotheses H 0 imply that on W 0 1 , η ( Ω ) the Poincaré inequality holds, namely, there exists c ˆ ≔ c ˆ ( Ω ) > 0 such that

‖ u ‖ η ≤ c ˆ ‖ D u ‖ η for all u ∈ W 0 1 , η ( Ω ) .

So, on W 0 1 , η ( Ω ) , we can consider the equivalent norm ‖ ⋅ ‖ defined by

‖ u ‖ ≔ ‖ D u ‖ η for all u ∈ W 0 1 , η ( Ω ) .

There is a close relation between the modular function ρ η ( ⋅ ) and the norm ‖ ⋅ ‖ .

Proposition 2.2

The following statements hold:

‖ u ‖ = λ ⇔ ρ η D u λ = 1 .
‖ u ‖ < 1 (resp. = 1 , > 1 ) ⇔ ρ η ( D u ) < 1 (resp. = 1 , > 1 ).
‖ u ‖ < 1 ⇒ ‖ u ‖ p ≤ ρ η ( D u ) ≤ ‖ u ‖ q .
‖ u ‖ > 1 ⇒ ‖ u ‖ q ≤ ρ η ( D u ) ≤ ‖ u ‖ p .
‖ u ‖ → 0 (resp. → ∞ ) ⇔ ρ η ( D u ) → 0 (resp. → ∞ ).

The following embeddings will be helpful in our arguments.

Proposition 2.3

The following statements hold.

L η ( Ω ) ↪ L r ( Ω ) , W 0 1 , η ( Ω ) ↪ W 0 1 , r ( Ω ) continuously for all 1 ≤ r ≤ q .
W 0 1 , q ( Ω ) ↪ L r ( Ω ) continuously for all 1 ≤ r ≤ q * and compactly for all 1 ≤ r < q * .
L p ( Ω ) ↪ L η ( Ω ) continuously.

Let V : W 0 1 , η ( Ω ) → W 0 1 , η ( Ω ) * be the nonlinear operator defined by

⟨ V ( u ) , h ⟩ ≔ ∫ Ω ( a ( z ) ∣ D u ∣ p − 2 D u + ∣ D u ∣ q − 2 D u , D h ) R N d z

for all u , h ∈ W 0 1 , η ( Ω ) . This operator has the following properties (Liu and Dai [18, Proposition 3.1]).

Proposition 2.4

The operator V : W 0 1 , η ( Ω ) → W 0 1 , η ( Ω ) * is bounded (i.e., it maps bounded sets to bounded sets), continuous, strictly monotone (thus maximal monotone too), and of type ( S ) + , that is,

“if u n ⟶ w u in W 0 1 , η ( Ω ) and limsup n → ∞ ⟨ V ( u n ) , u n − u ⟩ ≤ 0 , then u n → u in W 0 1 , η ( Ω ) .”

We also introduce the following modular function:

ρ a ( D u ) ≔ ∫ Ω a ( z ) ∣ D u ∣ p d z for all u ∈ W 0 1 , η ( Ω ) .

This function is continuous, convex, hence weakly lower semicontinuous (by Mazur’s lemma).

If u ∈ L 0 ( Ω ) , then we set u ± ( z ) = max { ± u ( z ) , 0 } for all z ∈ Ω . We have u = u + − u − , ∣ u ∣ = u + + u − . Also, if u ∈ W 0 1 , η ( Ω ) , then u ± ∈ W 0 1 , η ( Ω ) . For u ∈ L 0 ( Ω ) , we write 0 ≺ u , if for all K ⊂ Ω compact, we have

0 < c K ≤ u ( z ) for a.a. z ∈ K .

Evidently, such a function satisfies 0 < u ( z ) for a.a. z ∈ Ω . If u , v ∈ L 0 ( Ω ) and u ( z ) ≤ v ( z ) for a.a. z ∈ Ω , then

[ u , v ] ≔ { h ∈ W 0 1 , η ( Ω ) ∣ u ( z ) ≤ h ( z ) ≤ v ( z ) for a.a. z ∈ Ω } .

The Banach space C 0 1 ( Ω ¯ ) is ordered with order (positive cone) C 0 1 ( Ω ¯ ) + ≔ { u ∈ C 0 1 ( Ω ¯ ) ∣ 0 ≤ u ( z ) for all z ∈ Ω ¯ } . This cone has a nonempty interior given by

int C 0 1 ( Ω ¯ ) + = u ∈ C 0 1 ( Ω ¯ ) + ∣ u ( z ) > 0 for all z ∈ Ω , ∂ u ∂ n ∂ Ω < 0 ,

with n ( ⋅ ) being the outward unit normal on ∂ Ω .

To overcome the lack of a global regularity theory, we will use critical groups (Morse theory). Let X be a Banach space and φ ∈ C 1 ( X ) . We set

K φ ≔ { u ∈ X ∣ φ ′ ( u ) = 0 } (the critical set of φ ) .

We say that φ ( ⋅ ) satisfies the “C-condition,” if the following property holds:

“every sequence { u n } ⊂ X such that { φ ( u n ) } ⊂ R is bounded, ( 1 + ‖ u n ‖ X ) φ ′ ( u n ) → 0 in X * as n → ∞ , admits a strongly convergent subsequence.”

Consider a topological pair ( Y 1 , Y 2 ) such that Y 2 ⊂ Y 1 ⊂ X . For k ∈ N 0 by H k ( Y 1 , Y 2 ) , we denote the k th-relative singular homology group with integer coefficients. Let u ∈ K φ be isolated and c = φ ( u ) . We set φ c ≔ { u ∈ X ∣ φ ( u ) ≤ c } . Then, the critical groups of φ ( ⋅ ) at u are defined by

C k ( φ , u ) ≔ H k ( φ c ∩ U , φ c ∩ U \ { u } ) for all k ∈ N 0 ,

with U being an open neighborhood of u such that K φ ∩ φ c ∩ U = { u } . The excision property of singular homology theory implies that this definition is independent of the isolating neighborhood U . Suppose that φ ∈ C 1 ( X ) satisfies the C-condition and that − ∞ < inf φ ( K φ ) . Consider c < φ ( K φ ) . The critical groups of φ ( ⋅ ) at infinity are defined by

C k ( φ , ∞ ) ≔ H k ( X , φ c ) for all k ∈ N 0 .

By the second deformation theorem (Papageorgiou et al. [26], p. 386), this definition is independent of the choice of the level c < inf φ ( K φ ) .

To exploit the properties of critical groups, we will need the following notion. Suppose X is a finite dimensional Banach space and g : X → R . We say that g ( ⋅ ) is locally Lipschitz, if for every K ⊂ X compact, g ∣ K is Lipschitz with Lipschitz constant θ K > 0 , that is,

∣ g ( u ) − g ( v ) ∣ ≤ θ K ‖ u − v ‖ X for all u , v ∈ K .

If g ˆ : Ω × X → R , then we say that g ˆ is an L ∞ -locally Lipschitz integrand, if for all x ∈ R , z ↦ g ˆ ( z , x ) is measurable and for a.a. z ∈ Ω , the function x ↦ g ˆ ( z , x ) is locally Lipschitz with Lipschitz constant θ K ( z ) such that θ K ∈ L ∞ ( Ω ) . Such a function is jointly measurable (Papageorgiou and Winkert [31], p. 106).

Our hypotheses on the perturbation f ( z , x ) are as follows:

H 1 ̲ : f : Ω × R → R is an L ∞ -locally Lipschitz integrand, f ( z , 0 ) = 0 for a.a. z ∈ Ω , and

f ( z , x ) ≤ a ( z ) ˆ ( 1 + x r − 1 ) for a.a. z ∈ Ω , all x ≥ 0 with a ˆ ∈ L ∞ ( Ω ) + , p < r < q * ;
if F ( z , x ) ≔ ∫ 0 x f ( z , s ) d s , then lim x → + ∞ F ( z , x ) x p = + ∞ uniformly for a.a. z ∈ Ω ;
there exists τ ∈ ( r − q ) N q , q * such that
0 < β ˆ 0 ≤ liminf x → + ∞ f ( z , x ) x − p F ( z , x ) x τ
uniformly for a.a. z ∈ Ω ;
for every s > 0 , there exists k s > 0 such that
k s ≤ f ( z , x ) for a.a. z ∈ Ω , all s ≤ x
and for every ρ > 0 , there exists ξ ˆ ρ > 0 such that for a.a. z ∈ Ω , the function x ↦ f ( z , x ) + ξ ˆ ρ x p − 1 is nondecreasing on [ 0 , ρ ] .

Remark 2.5

Since we are looking for positive solutions of problem (1.1) and the above hypotheses concern the positive semiaxis R + ≔ [ 0 , + ∞ ) , we may assume that f ( z , x ) = 0 for a.a. z ∈ Ω , all x ≤ 0 . Hypotheses H 1 (ii) and (iii) imply that

lim x → + ∞ f ( z , x ) x p − 1 = + ∞ uniformly for a.a. z ∈ Ω .

So, the perturbation of the singular term is ( p − 1 ) -superlinear, but we do not use the usual for superlinear problems Ambrosetti-Rabinowitz (AR) condition (Willem [35], p. 46). Instead, we use the weaker hypothesis H 1 (iii) which incorporates in our framework superlinear nonlinearities, which exhibit “slower” growth as x → + ∞ . For example, consider the function

f ( x ) ≔ ( x + ) θ − 1 if x ≤ 1 , x p − 1 ln ( x ) + x p − 1 if 1 < x , with 1 < θ .

This function satisfies hypotheses H 1 but fails to satisfy the AR condition.

3 Purely singular problem

The difficulty we face when dealing with singular problems is that on account of the singular term, the energy functional of the problem is not C 1 and so we cannot use the minimax theorems of the critical point theory. For this reason, we need to find a way to bypass the singularity and to deal with C 1 -functionals. To this end, in this section, we consider the following purely singular problem:

(3.1) − Δ p a u − Δ q u = μ u − η in Ω , u = 0 on ∂ Ω , u > 0 in Ω .

The solution of this problem will help us to fulfill the above task.

As always by a solution of (3.1), we mean a “weak solution,” that is, a function u ∈ W 0 1 , η ( Ω ) such that u − η h ∈ L 1 ( Ω ) for all h ∈ W 0 1 , η ( Ω ) and

⟨ V ( u ) , h ⟩ = μ ∫ Ω ( u − η ) h d z for all h ∈ W 0 1 , η ( Ω ) .

Proposition 3.1

If hypotheses H 0 hold and μ > 0 large, then problem (3.1) admits a unique solution

u ¯ μ ∈ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) w i t h 0 ≺ u ¯ μ .

Proof

For ε > 0 , we consider the following regularized version of (3.1):

(3.2) − Δ p a u − Δ q u = μ ( u + ε ) − η in Ω , u = 0 on ∂ Ω , u > 0 in Ω .

The idea is to solve (3.2) and then pass to the limit as ε → 0 + in order to produce a solution of (3.1).

To solve (3.2), we use a fixed point argument. So, let θ ∈ L p ( Ω ) and consider the following Dirichlet problem:

(3.3) − Δ p a u − Δ q u = μ ( ∣ θ ∣ + ε ) − η in Ω , u = 0 on ∂ Ω , u > 0 in Ω .

Equivalently, we can write (3.3) as the following operator equation:

V ( u ) = μ ( ∣ θ ∣ + ε ) − η in W 0 1 , η ( Ω ) * ,

From Proposition 2.4, we know that V : W 0 1 , η ( Ω ) → W 0 1 , η ( Ω ) * is continuous and strictly monotone (thus maximal monotone too) and

⟨ V ( u ) , u ⟩ = ρ η ( D u ) .

This means that V ( ⋅ ) is coercive (Proposition 2.2). But a maximal monotone, coercive operator is surjective ([26], p. 135). Since ( ∣ θ ∣ + ε ) − η ∈ L ∞ ( Ω ) ↪ W 0 1 , η ( Ω ) * , we can find u ε θ ∈ W 0 1 , η ( Ω ) such that

V ( u ε θ ) = μ ( ∣ θ ∣ + ε ) − η in W 0 1 , η ( Ω ) * .

The strict monotonicity of V ( ⋅ ) implies that this solution is unique. Since ( ∣ θ ∣ + ε ) − η ∈ L ∞ ( Ω ) + \ { 0 } , we have u ε θ ≥ 0 and u ε θ ≠ 0 . From Theorem 3.1 of Gasiński and Winkert [11], we have u ε θ ∈ L ∞ ( Ω ) . Finally, Proposition 2.4 of Papageorgiou et al. [30] implies that 0 ≺ u ε θ .

Consider the solution map γ ε : L p ( Ω ) → L p ( Ω ) defined by

γ ε ( θ ) = u ε θ .

We show that γ ε ( ⋅ ) is continuous. Let { θ n } n ∈ N ⊂ L p ( Ω ) be such that θ n → θ in L p ( Ω ) . We set u ε n ≔ u ε θ n for all n ∈ N . We have

(3.4) ⟨ V ( u ε n ) , h ⟩ = ∫ Ω μ h ( ∣ θ n ∣ + ε ) η d z for all h ∈ W 0 1 , η ( Ω ) and n ∈ N .

In (3.4), we choose the test function h = u ε n ∈ W 0 1 , η ( Ω ) and obtain

ρ η ( D u ε n ) ≤ ∫ Ω μ u ε n ε η d z ≤ c 1 μ ε η ‖ u ε n ‖

for some c 1 > 0 , all n ∈ N (Proposition 2.3). This means that { u ε n } ⊂ W 0 1 , η ( Ω ) is bounded (Proposition 2.2). On account of Proposition 2.3, we may assume that

(3.5) u ε n ⟶ w u ˆ ε in W 0 1 , η ( Ω ) and u ε n ⟶ w u ˆ ε in L p ( Ω ) .

In (3.4), we use the test function h = u ε n − u ˆ ε ∈ W 0 1 , η ( Ω ) , pass to the limit as n → ∞ , and use (3.5) and Proposition 2.4 to find

(3.6) lim n → ∞ ⟨ V ( u ε n ) , u ε n − u ˆ ε ⟩ = 0 ⇒ u ε n → u ˆ ε in W 0 1 , η ( Ω ) .

So, if in (3.4), we pass to the limit as n → ∞ and use (3.6) to have

⟨ V ( u ˆ ε ) , h ⟩ = ∫ Ω μ h ( ∣ θ ∣ + ε ) η d z for all h ∈ W 0 1 , η ( Ω ) ⇒ u ˆ ε = γ ε ( θ ) .

This reveals that γ ε ( ⋅ ) is continuous.

Recall that

ρ η ( u ε θ ) ≤ c 1 μ ‖ u ε θ ‖ for some c 1 independent of θ ,

so, from Proposition 2.2, we have

‖ u ε θ ‖ ≤ c 2 μ for some c 2 > 0 , all θ ∈ L p ( Ω ) .

This points out that γ ε ( L p ( Ω ) ) ⊂ W 0 1 , η ( Ω ) is bounded. Keeping in mind that W 0 1 , η ( Ω ) ↪ L p ( Ω ) compactly (by Proposition 2.3 and recall p < q * ). We infer that γ ε ( L p ( Ω ) ) ¯ ‖ ⋅ ‖ p ⊂ L p ( Ω ) is compact. From Theorem 4.3.21, p. 298, of Papageorgiou et al. [26], we know that γ ε ( ⋅ ) has a fixed point, that is, we can find u ε * ∈ W 0 1 , η ( Ω ) such that

γ ε ( u ε * ) = u ε * .

As before, we have that

u ε * ∈ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) and 0 ≺ u ε * .

Moreover, on account of the strict monotonicity of V ( ⋅ ) (or alternatively, using the fact that x ↦ ( x + ε ) η is strictly decreasing on R + ≔ [ 0 , + ∞ ) ), we see that the fixed point u ε * is unique.

Claim: 0 < ε ′ ≤ ε implies u ε * ≤ u ε ′ * .

Note that

(3.7) − Δ p a u ε ′ * − Δ q u ε ′ * = μ ( u ε ′ * + ε ′ ) − η ≥ μ ( u ε ′ * + ε ) − η in Ω .

We introduce the Carathéodory function l ε ( z , x ) defined by

(3.8) l ε ( z , x ) ≔ μ ( x + + ε ) η if x ≤ u ε ′ * ( z ) , μ ( u ε ′ * ( z ) + ε ) η if x > u ε ′ * ( z ) .

We set L ε ( z , x ) ≔ ∫ 0 x l ε ( z , s ) d s and consider the C 1 -functional β ˆ ε : W 0 1 , η ( Ω ) → R defined by

β ˆ ε ( u ) ≔ 1 p ρ a ( D u ) + 1 q ‖ D u ‖ q q − ∫ Ω L ε ( z , u ) d z

for all u ∈ W 0 1 , η ( Ω ) . From (3.8), it is clear that β ˆ ε ( ⋅ ) is coercive. Also, using Proposition 2.3, we see that β ˆ ε ( ⋅ ) is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find u ˜ ε * ∈ W 0 1 , η ( Ω ) such that

(3.9) β ˆ ε ( u ˜ ε * ) = inf { β ˆ ε ( u ) ∣ u ∈ W 0 1 , η ( Ω ) } .

Let u ∈ C 0 1 ( Ω ¯ ) ≔ { v ∈ C 1 ( Ω ¯ ) ∣ v ∣ ∂ Ω = 0 } be such that u ( z ) ≥ 0 for all z ∈ Ω ¯ and u ≠ 0 . For any t ∈ ( 0 , 1 ) , we have

β ˆ ε ( t u ) = t p p ρ a ( D u ) + t q q ‖ D u ‖ q q − μ ( 1 − η ) t ∫ { t u ≤ u ε ′ * } ( t u + ε ) 1 − η d z − μ ∫ { t u > u ε ′ * } t u − u ε ′ * ( u ε ′ * + ε ) η d z (see 3.8) ≤ t p p ρ a ( D u ) + t q q ‖ D u ‖ q q − μ ( 1 − η ) t η ∫ { t u ≤ u ε ′ * } u + ε t 1 − η d z .

If by ∣ ⋅ ∣ N , we denote the Lebesgue measure on R N , then as t → 0 + we see that ∣ { t u ≤ u ε ′ * } ∣ N → ∣ Ω ∣ N and so

1 ( 1 − η ) t η ∫ { t u ≤ u ε ′ * } u + ε t 1 − η d z → + ∞ as t → 0 + .

Therefore, for t ∈ ( 0 , 1 ) small, we have

β ˆ ε ( t u ) < 0 ⇒ β ˆ ε ( u ˜ ε * ) < 0 = β ˆ ε ( 0 ) (see (3.8)) ⇒ u ˜ ε * ≠ 0 .

From (3.9), we have

(3.10) ⟨ β ˆ ε ′ ( u ˜ ε * ) , h ⟩ = 0 ⇒ ⟨ V ( u ˜ ε * ) , h ⟩ = ∫ Ω l ε ( z , u ˜ ε * ) h d z

for all h ∈ W 0 1 , η ( Ω ) .

In (3.10), we, first, choose h = − ( u ε * ) − ∈ W 0 1 , η ( Ω ) to obtain

ρ η ( D ( u ˜ ε * ) − ) ≤ 0 ⇒ u ˜ ε * ≥ 0 and u ˜ ε * ≠ 0 (see Proposition 2.2) .

Next in (3.10), we use the test function h = ( u ˜ ε * − u ε ′ * ) + ∈ W 0 1 , η ( Ω ) to obtain

⟨ V ( u ˜ ε * ) , ( u ˜ ε * − u ε ′ * ) + ⟩ = ∫ Ω μ ( u ˜ ε * − u ε ′ * ) + ( u ε ′ * + ε ) η d z (see (3.8)) ≤ ∫ Ω μ ( u ˜ ε * − u ε ′ * ) + ( u ε ′ * + ε ′ ) η d z (see (3.7)) = ⟨ V ( u ε ′ * ) , ( u ˜ ε * − u ε ′ * ) ⟩ .

The latter combined with Proposition 2.4 implies u ˜ ε * ≤ u ε ′ * .

So, we have proved that

(3.11) u ˜ ε * ∈ [ 0 , u ε ′ * ] and u ˜ ε * ≠ 0 .

From (3.8), (3.10), and (3.11), we can see that u ˜ ε * solves (3.2). Hence,

u ˜ ε * = u ε * ≤ u ε ′ * .

This proves the claim.

Now, let ε n → 0 + and set u n * ≔ u ε n * for all n ∈ N . Then, by the Claim, we know that { u n * } ⊂ W 0 1 , η ( Ω ) is increasing and

(3.12) ⟨ V ( u n * ) , h ⟩ = ∫ Ω μ h ( u n * + ε n ) η d z for all h ∈ W 0 1 , η ( Ω ) a n d n ∈ N .

In (3.12), we use the test function h = u n * ∈ W 0 1 , η ( Ω ) to obtain

ρ η ( D u n * ) ≤ ∫ Ω μ ( u n * ) 1 − η d z ≤ c 3 ‖ u n ‖ for some c 3 > 0 and for all n ∈ N ,

where we have used Theorem 13.17, p. 196, of Hewitt and Stromberg [14] and Proposition 2.3. Then, from Proposition 2.2, we infer that

{ u n * } ⊂ W 0 1 , η ( Ω ) is bounded.

So, we may assume that

(3.13) u n * ⟶ w u ¯ μ in W 0 1 , η ( Ω ) and u n * → u ¯ μ in L p ( Ω ) .

Let d ˆ ( z ) ≔ d ( z , ∂ Ω ) for all z ∈ Ω ¯ . By Lemma 14.16, p. 355, of Gilbarg and Trudinger [12], we know that there exists ε 0 > 0 such that

d ˆ ∈ C 2 ( Ω ε 0 ) ,

where Ω ε 0 ≔ { z ∈ Ω ¯ ∣ d ˆ ( z ) < ε 0 } . For δ ∈ ( 0 , ε 0 2 ) , we define

σ ( z ) ≔ d ˆ ( z ) if d ˆ ( z ) < δ , δ + ∫ δ d ˆ ( z ) 2 δ − t δ 1 q − 1 d t if δ ≤ d ˆ ( z ) ≤ 2 δ , δ + ∫ δ 2 δ 2 δ − t δ 1 q − 1 d t if 2 δ < d ˆ ( z ) .

Note that σ ∈ int C 0 1 ( Ω ¯ ) + . Also ∣ ∇ d ˆ ∣ = 1 in Ω ε 0 , so, we have

− Δ p a σ − Δ q σ = − Δ a + 1 d ˆ on { d ˆ < δ } , − Δ p a σ − Δ q σ = − 2 δ − d ˆ 2 1 + 2 δ − d ˆ δ p − q p q Δ a + 1 d ˆ on { δ < d ˆ < 2 δ } , − Δ p a σ − Δ q σ = 0 on { d ˆ > 2 δ } .

Therefore, given ε > 0 , if μ > 0 is large, then we have

(3.14) − Δ p a σ − Δ q σ ≤ μ ( σ + ε ) η in Ω .

We introduce the Carathéodory function ζ ε ( z , x ) defined by

(3.15) ζ ε ( z , x ) ≔ μ ( σ + ε ) η if x ≤ σ ε ( z ) , μ ( x + ε ) η if σ ε ( z ) < x .

We set Z ε ( z , x ) ≔ ∫ 0 x ζ ε ( z , s ) d s and consider the C 1 -functional γ ε : W 0 1 , η ( Ω ) → R defined by

γ ε ( u ) ≔ 1 p ρ a ( D u ) + 1 q ‖ D u ‖ q q − ∫ Ω Z ε ( z , u ) d z for all u ∈ W 0 1 , η ( Ω ) .

Evidently, γ ε ( ⋅ ) is coercive and sequentially weakly lower semicontinuous. So, we can find u ˜ ε * ∈ W 0 1 , η ( Ω ) such that

γ ε ( u ˜ ε * ) = inf { γ ε ( u ) ∣ u ∈ W 0 1 , η ( Ω ) } ⇒ ⟨ γ ε ′ ( u ˜ ε * ) , h ⟩ = 0 for all h ∈ W 0 1 , η ( Ω ) .

This means that

⟨ V ( u ˜ ε * ) , h ⟩ = ∫ Ω ζ ε ( z , u ˜ ε * ) h d z for all h ∈ W 0 1 , η ( Ω ) .

Let σ ε = σ + ε and choose h = [ σ ε − u ˜ ε * ] + ∈ W 0 1 , η ( Ω ) for the equality to find

⟨ V ( u ˜ ε * ) , ( σ ε − u ˜ ε * ) + ⟩ = ∫ Ω μ ( σ + ε ) η ( σ ε − u ˜ ε * ) + d z (see (3.5)) ≥ ⟨ V ( σ ε ) , ( σ ε − u ˜ ε * ) + ⟩ (see (3.14)) .

So, we use Proposition 2.4 to obtain σ ε ≤ u ˜ ε * . Therefore, u ˜ ε * is a solution of problem (3.2), hence u ˜ ε * = u ε * . So, we have proved that

(3.16) σ ε ≤ u n * for all n ∈ N ⇒ σ ≤ u n * for all n ∈ N .

Since σ ∈ int C 0 1 ( Ω ¯ ) + , we can find c 4 > 0 such that

(3.17) d ˆ ≤ c 4 σ ⇒ d ˆ ≤ c 4 u n * for all n ∈ N (see (3.16)) .

For every h ∈ W 0 1 , q ( Ω ) , we have

∫ Ω ∣ h ∣ ( u n * ) η q d z ≤ ∫ Ω c 4 η ∣ h ∣ d ˆ η q d z (see (3.17)) = c 4 q η ∫ Ω d ˆ 1 − η ∣ h ∣ d ˆ q d z ≤ c 5 ∫ Ω ∣ h ∣ d ˆ q d z for some c 5 > 0 ≤ c 6 ‖ D h ‖ q q for some c 6 > 0 ,

where the last inequality is obtained by using Hardy’s inequality [26, p. 66]. Taking h = u n * − u ¯ μ ∈ W 0 1 , η ( Ω ) , we see that

u n * − u ¯ μ ( u n * ) η ⊂ L q ( Ω ) is bounded (see (3.13)) .

So, we can observe that u n * − u ¯ μ ( u n * ) η is uniformly integrable. Also, from (3.13), at least for a subsequence, we have

u n * − u ¯ μ ( u n * ) η → 0 for a.a. z ∈ Ω .

So, by Vitali’s theorem (Papageorgiou and Winkert [31], p. 124), we have

∫ Ω u n * − u ¯ μ ( u n * ) η d z → 0 .

From (3.12) with h = u n * − u ¯ μ ∈ W 0 1 , η ( Ω ) , we have

⟨ V ( u n * ) , u n * − u ¯ n ⟩ = ∫ Ω u n * − u ¯ μ ( u n * + ε n ) η d z ≤ ∫ Ω u n * − u ¯ μ ( u n * ) η d z .

Passing to the upper limit as n → ∞ for the inequality above, it yields

limsup n → ∞ ⟨ V ( u n * ) , u n * − u ¯ n ⟩ ≤ 0 .

The latter combined with Proposition 2.4 implies

(3.18) u n * → u ¯ μ in W 0 1 , η ( Ω ) .

So, if in (3.12), we pass to the limit as n → ∞ and use (3.18) to obtain

⟨ V ( u ¯ μ ) , h ⟩ = ∫ Ω h u ¯ μ η d z for all h ∈ W 0 1 , η ( Ω ) ,

namely, u ¯ μ is a solution of (3.1).

Recall that { u n * } is increasing. We have

u 1 * ≤ u ¯ μ ⇒ 0 ≺ u ¯ μ .

But, from (3.17), it gives

(3.19) 1 c 4 d ˆ ≤ u ¯ μ ⇒ u ¯ μ − η ≤ c 4 η d ˆ − η .

Flattening ∂ Ω and using a partition of unity as in the proof of the lemma in Lazer and Mckenna [16], we obtain that

d ˆ − η ∈ L s ( Ω ) for all s ∈ [ 1 , 1 η ) .

Hypotheses H 0 imply N q < 1 η and this from Colasuonno and Squassina [5] implies that u ¯ μ ∈ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) (see (3.19)).

Finally as before, the strict monotonicity of V ( ⋅ ) implies that u ¯ μ is unique.□

4 Positive solutions

We say that u ∈ W 0 1 , η ( Ω ) is a weak solution of (1.1), if u − η h ∈ L 1 ( Ω ) for all h ∈ W 0 1 , η ( Ω ) with u ( z ) ≥ 0 for a.a. z ∈ Ω , u ≠ 0 , and

⟨ V ( u ) , h ⟩ = ∫ Ω [ μ u − η + λ f ( z , u ) ] h d z for all h ∈ W 0 1 , η ( Ω ) .

We also introduce the following two sets:

ℒ ≔ { λ > 0 ∣ (1.1) has a positive solution } , S λ = { set of positive solutions of (1.1) corresponding to λ > 0 } .

Proposition 4.1

If hypotheses H 0 and H 1 hold, and μ > 0 is large, then ℒ ≠ ∅ and for all λ ∈ ℒ ,

∅ ≠ S λ ⊂ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) .

Proof

Consider the following auxiliary problem:

(4.1) − Δ p a u − Δ q u = μ u ¯ μ − η + 1 in Ω , u = 0 on ∂ Ω , u > 0 in Ω .

From the proof of Proposition 3.1, we know that for μ > 0 large, we have

u ¯ μ − η ∈ L s ( Ω ) for all s ∈ 1 , 1 η .

By hypothesis N q < 1 η ( H 0 ) and from Gasiński and Papageorgiou [9, p. 141], we have

( q * ) ′ = q * q * − 1 < N q < 1 η ⇒ u − η ∈ L ( q * ) ′ ( Ω ) ↪ W − 1 , q ( Ω ) ↪ W 0 1 , η ( Ω ) * .

Recall that V : W 0 1 , η ( Ω ) → W 0 1 , η ( Ω ) * is maximal monotone and coercive, hence it is surjective. So, we can find u ˜ ∈ W 0 1 , η ( Ω ) \ { 0 } such that

V ( u ˜ ) = u ¯ μ + 1 in W 0 1 , η ( Ω ) * .

Moreover, on account of the strict monotonicity of V ( ⋅ ) , this solution is unique. As before, since N q < 1 η , we have

u ˜ ∈ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω )

and

⟨ V ( u ˜ ) , ( u ¯ μ − u ˜ ) + ⟩ = ∫ Ω [ u ¯ μ − η + 1 ] ( u ¯ μ − u ˜ ) + d z ≥ ∫ Ω u ¯ μ − η ( u ¯ μ − u ˜ ) + d z = ⟨ V ( u ¯ μ ) , ( u ¯ μ − u ˜ ) + ⟩ (see Proposition 3.1) .

Hence, one has

(4.2) u ¯ μ ≤ u ˜ (see Proposition 2.4) and 1 c 4 d ˆ ≤ u ˜ (so, 0 ≺ u ˜ ) .

Hypothesis H 1 (i) implies that

f ( ⋅ , u ˜ ( ⋅ ) ) ∈ L ∞ ( Ω ) .

So, we can find λ 0 > 0 small such that

(4.3) λ f ( z , u ˜ ( z ) ) ≤ 1 for a.a. z ∈ Ω , all 0 < λ ≤ λ 0 .

Then, by (4.2) and (4.3), for 0 < λ ≤ λ 0 , we have

(4.4) − Δ p a u ˜ − Δ q u ˜ = u ¯ μ − η + 1 ≥ u ˜ − η + λ f ( z , u ˜ ) in Ω .

We introduce the Carathéodory function k λ ( z , x ) defined by

(4.5) k λ ( z , x ) ≔ μ u ¯ μ − η + λ f ( z , u ¯ μ ) if x < u ¯ μ ( z ) , μ x − η + λ f ( z , x ) if u ¯ μ ( z ) ≤ x ≤ u ˜ ( z ) , μ u ˜ − η + λ f ( z , u ˜ ) if u ˜ ( z ) < x .

We set K λ ( z , x ) ≔ ∫ 0 x k λ ( z , s ) d s and consider the functional ψ λ : W 0 1 , η ( Ω ) → R defined by

ψ λ ( u ) ≔ 1 p ρ a ( D u ) + 1 q ‖ D u ‖ q q − ∫ Ω K λ ( z , u ) d z for all u ∈ W 0 1 , η ( Ω ) .

It is clear from Papageorgiou and Smyrlis [29] that ψ λ ( ⋅ ) is C 1 and coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u λ ∈ W 0 1 , η ( Ω ) such that

ψ λ ( u λ ) = inf { ψ λ ( u ) ∣ u ∈ W 0 1 , η ( Ω ) } .

Hence,

(4.6) ⟨ ψ ′ ( u λ ) , h ⟩ = 0 ⇒ ⟨ V ( u λ ) , h ⟩ = ∫ Ω k λ ( z , u λ ) h d z

for all h ∈ W 0 1 , η ( Ω ) .

In (4.6), we, first, choose h = ( u ¯ μ − u λ ) + ∈ W 0 1 , η ( Ω ) to obtain

⟨ V ( u λ ) , ( u ¯ μ − u λ ) + ⟩ = ∫ Ω [ μ u ¯ μ − η + λ f ( z , u ¯ μ ) ] ( u ¯ μ − u λ ) + d z (see (4.5)) ≥ ∫ Ω μ u ¯ μ − η ( u ¯ μ − u λ ) + d z (since f ≥ 0 ) = ⟨ V ( u ¯ μ ) , ( u ¯ μ − u λ ) + ⟩ (see Proposition 3.1) .

This implies u ¯ μ ≤ u λ .

Next in (4.6), we use the test function h = ( u λ − u ˜ ) + ∈ W 0 1 , η ( Ω ) to obtain

⟨ V ( u λ ) , ( u λ − u ˜ ) + ⟩ = ∫ Ω [ μ u ˜ − η + λ f ( z , u ˜ ) ] ( u λ − u ˜ ) + d z (see (4.5)) ≤ ∫ Ω [ μ u ¯ μ − η + 1 ] ( u λ − u ˜ ) + d z (see (4.4)) = ⟨ V ( u ˜ ) , ( u λ − u ˜ ) + ⟩ .

This indicates that u λ ≤ u ˜ . So, we have proved that

(4.7) u λ ∈ [ u ¯ μ , u ˜ ] .

Form (4.5), (4.6), and (4.7), we see that u λ ∈ S λ and so

( 0 , λ 0 ] ⊂ ℒ ≠ ∅ .

Finally as in Lemma 3.2 of Kumar et al. [15], we have that S λ ⊂ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) .□

Next, we produce a lower bound for the elements of S λ .

Proposition 4.2

If hypotheses H 0 and H 1 hold, μ > 0 is large and λ ∈ ℒ , then u ¯ μ ≤ u for all u ∈ S λ .

Proof

Let u ∈ S λ . For every h ∈ W 0 1 , η ( Ω ) with h ≥ 0 , we have

⟨ V ( u ) − V ( u ¯ μ ) , h ⟩ − ∫ Ω μ ( u − η − u ¯ μ − η ) h d z = ∫ Ω f ( z , u ) h d z ≥ 0 .

Then, as in Lemma 2.3 of Zhang [36], we conclude that u ¯ μ ≤ u .□

The next proposition shows that ℒ of admissible parameters is connected.

Proposition 4.3

If hypotheses H 0 and H 1 hold, μ > 0 is large, λ ∈ ℒ and 0 < θ < λ , then θ ∈ ℒ .

Proof

Since λ ∈ ℒ , we can find u λ ∈ S λ ⊂ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) . From Proposition 4.2, we know that u ¯ μ ≤ u λ . So, we introduce the Carathéodory function g θ ( z , x ) defined

(4.8) g θ ( z , x ) ≔ μ u ¯ μ − η + θ f ( x , u ¯ μ ) if x < u ¯ μ ( z ) , μ x − η + θ f ( x , x ) if u ¯ μ ( z ) ≤ x ≤ u λ ( z ) , μ u λ − η + θ f ( x , u λ ) if u λ ( z ) < x .

We set G θ ( z , x ) ≔ ∫ 0 x g θ ( z , s ) d s and consider the C 1 -functional φ ˆ θ : W 0 1 , η ( Ω ) → R defined by

φ ˆ θ ( u ) ≔ 1 p ρ a ( D u ) + 1 q ‖ D u ‖ q q − ∫ Ω G θ ( z , u ) d z .

Evidently, φ ˆ θ ( ⋅ ) is coercive (see (4.8)) and sequentially weakly lower semicontinuous (by using Proposition 2.3). So, we can find u θ ∈ W 0 1 , η ( Ω ) such that

φ ˆ θ ( u θ ) ≔ inf { φ ˆ θ ( u ) ∣ u ∈ W 0 1 , η ( Ω ) } .

Then, we have

(4.9) ⟨ φ ′ ( u θ ) , h ⟩ = 0 ⇒ ⟨ V ( u θ ) , h ⟩ = ∫ Ω g θ ( z , u θ ) h d z

for all h ∈ W 0 1 , η ( Ω ) .

As in the proof of Proposition 4.1, using the test functions h = ( u ¯ μ − u θ ) + ∈ W 0 1 , η ( Ω ) and h = ( u θ − u λ ) + ∈ W 0 1 , η ( Ω ) , we show that

u θ ∈ [ u ¯ μ , u λ ] .

Then, we have u θ ∈ S θ and so θ ∈ ℒ due to (4.8) and (4.9).□

Embedded in the above proof is the following corollary.

Corollary 4.4

If hypotheses H 0 and H 1 hold, μ > 0 is large and λ ∈ ℒ , u λ ∈ S λ , and 0 < θ < λ , then θ ∈ ℒ and there exists u θ ∈ S θ such that u θ ≤ u λ .

Let λ * = sup ℒ .

Proposition 4.5

If hypotheses H 0 and H 1 hold, and μ > 0 is large, then λ * < + ∞ .

Proof

On account of hypotheses H 1 , we can find λ ˜ such that

(4.10) x p − 1 ≤ λ ˜ f ( z , x ) for a.a. z ∈ Ω , all x ≥ 0 .

Consider λ > λ ˜ and suppose that λ ∈ ℒ . Then, we can find u λ ∈ S λ ⊂ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) , 0 ≺ u λ (Proposition 4.2). Consider Ω 0 ⊂ ⊂ Ω and let m 0 ≔ ess inf Ω 0 u λ > 0 . For δ ∈ ( 0 , 1 ) small, we set m 0 δ = m 0 + δ . Let ρ ≔ ‖ u λ ‖ ∞ and ξ ˆ ρ > 0 be as postulated by hypothesis H 1 (iv). We have

(4.11) − Δ p a m 0 δ − Δ q m 0 δ + λ ξ ˆ ρ ( m 0 δ ) p − 1 − μ ( m 0 δ ) − η ≤ λ ξ ˆ ρ m 0 p − 1 + χ ( δ ) with χ ( δ ) → 0 + as δ → 0 + ≤ [ λ ξ ˆ ρ + 1 ] m 0 p − 1 + χ ( δ ) ≤ λ ξ ˆ ρ m 0 p − 1 + λ ˜ f ( z , m 0 ) + χ ( δ ) (see (4.10)) = λ [ f ( z , m 0 ) + ξ ˆ ρ m 0 p − 1 ] − ( λ − λ ˜ ) f ( z , m 0 ) + χ ( δ ) ≤ λ [ f ( z , m 0 ) + ξ ˆ ρ m 0 p − 1 ] − ( λ − λ ˜ ) k m 0 + χ ( δ ) (see hypothesis H 1 (iv)) ≤ λ [ f ( z , u λ ) + ξ ˆ u λ p − 1 ] for δ ∈ ( 0 , 1 ) small = − Δ p a u λ − Δ q u λ + λ ξ ˆ ρ u λ p − 1 − μ u λ − η in Ω 0 .

On (4.11), we act with ( m 0 δ − u λ ) + ∈ W 1 , η ( Ω 0 ) and obtain

⟨ V ( m 0 δ ) − V ( u λ ) , ( m 0 δ − u λ ) + ⟩ ≤ ∫ Ω λ ξ ˆ ρ ( u λ p − 1 − ( m 0 δ ) p − 1 ) ( m 0 δ − u λ ) + d z + ∫ Ω μ ( ( m 0 δ ) − η − u λ − η ) ( m 0 δ − u λ ) + d z ≤ 0 (since x ↦ x − η is decreasing on ( 0 , + ∞ ) ) .

Hence, we have m 0 δ ≤ u λ ( z ) for a.a. z ∈ Ω 0 with δ ∈ ( 0 , 1 ) small. This contracts the definition of m 0 > 0 . Therefore, λ ∉ ℒ and so λ * ≤ λ ˜ < ∞ .□

For λ ∈ ( 0 , λ * ) , we have multiplicity of positive solutions of (1.1) ( μ > 0 large so that ℒ ≠ ∅ ).

Proposition 4.6

If hypotheses H 0 and H 1 hold, μ > 0 is large and λ ∈ ( 0 , λ * ) , then problem (1.1) has at least two positive solutions

u 0 , u ˆ ∈ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) .

Proof

Let β ∈ ( λ , λ * ) . We know that β ∈ ℒ and we can find u β ∈ S β . According to Corollary 4.4, we can find u 0 ∈ S λ such that

u ¯ μ ≤ u 0 ≤ u β .

We introduce the following Carathéodory functions:

(4.12) γ λ ( z , x ) ≔ μ u ¯ μ − η + λ f ( z , u ¯ μ ) if x ≤ u ¯ μ ( z ) , μ x − η + λ f ( z , x ) if u ¯ μ ( z ) < x ,

and

(4.13) γ ˆ λ ( z , x ) ≔ γ λ ( z , x ) if x ≤ u β ( z ) , γ λ ( z , u β ( z ) ) if u β ( z ) < x .

We set Γ λ ( z , x ) ≔ ∫ 0 x γ λ ( z , s ) d s , Γ ˆ λ ( z , x ) ≔ ∫ 0 x γ ˆ λ ( z , s ) d s and consider the C 1 -functionals σ λ , σ ˆ λ : W 0 1 , η ( Ω ) → R defined by

σ λ ( u ) ≔ 1 p ρ a ( D u ) + 1 q ‖ D u ‖ q q − ∫ Ω Γ λ ( z , u ) d z , σ ˆ λ ( u ) = 1 p ρ a ( D u ) + 1 q ‖ D u ‖ q q − ∫ Ω Γ ˆ λ ( z , u ) d z

for all u ∈ W 0 1 , η ( Ω ) .

From (4.12) and (4.13), we see that σ ˆ λ ( ⋅ ) is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u ˜ 0 ∈ W 0 1 , η ( Ω ) such that

(4.14) σ ˆ λ ( u ˜ 0 ) = inf { σ ˆ λ ( u ) ∣ u ∈ W 0 1 , η ( Ω ) } .

Using (4.12) and (4.13), as before (see the proof of Proposition 4.1), we can check that

K σ ˆ λ ⊂ [ u ¯ , u β ] ∩ L ∞ ( Ω ) .

Therefore, we may assume that K σ ˆ λ = { u 0 } or otherwise, we already have a second bounded solution (see (4.13)) and so we are done. From (4.14), we see that

(4.15) u ˜ 0 = u 0 ⇒ C k ( σ ˆ λ , u 0 ) = δ k , 0 Z for all k ∈ N 0 .

Claim: C k ( σ ˆ λ , u 0 ) = C k ( σ λ , u 0 ) for all k ∈ N 0 .

Let M > ‖ u β ‖ ∞ and let γ λ M ( z , x ) be the M -truncation of γ λ ( z , ⋅ ) , that is, γ λ M ( z , x ) is the Carathéodory function defined by

(4.16) γ λ M ( z , x ) ≔ γ λ ( z , x ) if x ≤ M , γ λ ( z , M ) if M < x .

We set Γ λ M ( z , x ) ≔ ∫ 0 x γ λ M ( z , s ) d s and consider the C 1 -functional σ λ M : W 0 1 , η ( Ω ) → R defined by

σ λ M ( u ) ≔ 1 p ρ a ( D u ) + 1 q ‖ D u ‖ q q − ∫ Ω Γ λ M ( z , u ) d z

for all u ∈ W 0 1 , η ( Ω ) . Since Γ λ M ∣ [ u ¯ μ , u β ] = Γ ˆ λ ∣ [ u ¯ μ , u β ] and u 0 ∈ [ u ¯ μ , u β ] , we have

(4.17) ∣ σ λ M ( u ) − σ ˆ λ ( u ) ∣ ≤ ∫ Ω ∣ Γ λ M ( z , u ) − Γ ˆ λ ( z , u ) ∣ d z ≤ ∫ Ω ∣ Γ λ M ( z , u ) − Γ λ M ( z , u 0 ) ∣ d z + ∫ Ω ∣ Γ λ M ( z , u 0 ) − Γ ˆ λ ( z , u ) ∣ d z .

We estimate the two terms in the right-hand side of (4.17). Using (4.12), (4.13), and (4.16), we have

(4.18) ∫ Ω ∣ Γ λ M ( z , u ) − Γ λ M ( z , u 0 ) ∣ d z ≤ ∫ { u ¯ μ ≤ u ≤ M } μ 1 − η ( u 1 − η − u 0 1 − η ) + μ ( u − u 0 ) u ¯ μ − η d z + ∫ { u ¯ μ ≤ u ≤ M } λ ( F ( z , u ) − F ( z , u 0 ) + ( u − u 0 ) f ( z , u ¯ μ ) ) d z + ∫ { 0 ≤ u ≤ u ¯ μ } μ ( u − u ¯ μ ) ( u ¯ μ − η + f ( z , u ¯ μ ) ) + μ 1 − η ( u 0 1 − η − u ¯ μ 1 − η ) + λ ( F ( z , u 0 ) − F ( z , u ¯ μ ) ) d z + ∫ { M < u } μ 1 − η ( M 1 − η − u 0 1 − η ) + λ ( F ( z , M ) − F ( z , u 0 ) ) + ( μ M − η + λ f ( z , M ) ) ( u − M ) d z .

We make the following remarks:

the function x ↦ x 1 − η on R + is concave, thus, locally Lipschitz;
on { 0 ≤ u < u ¯ μ } , we have u 0 1 − η − u ¯ μ 1 − η ≤ u 0 1 − η − u 1 − η ;
on { M < u } , we have M 1 − η − u 0 1 − η ≤ u 1 − η − u 0 1 − η ;
on account of hypothesis H 1 (iv), the primitive F ( z , ⋅ ) is increasing on R + and so on { M < u } , we have
F ( z , M ) − F ( z , u 0 ) ≤ F ( z , u ) − F ( z , u 0 ) ;
since f ( z , x ) is an L ∞ -locally Lipschitz integrand, so, is the primitive F ( z , x ) .

Using these observations in (4.21), we obtain

∫ Ω ∣ Γ λ M ( z , u ) − Γ λ M ( z , u 0 ) ∣ d z ≤ c 5 ‖ u − u 0 ‖ for some c 5 > 0 .

Similarly, we show that

∫ Ω ∣ Γ ˆ λ ( z , u 0 ) − Γ ˆ λ ( z , u ) ∣ d z ≤ c 6 ‖ u − u 0 ‖ for some c 6 > 0 .

Returning to (4.17), we obtain

(4.19) ∣ σ λ M ( u ) − σ ˆ λ ( u ) ∣ ≤ c 7 ‖ u − u 0 ‖ for some c 7 > 0 .

Next, let h ∈ W 0 1 , η ( Ω ) . We have

(4.20) ∣ ⟨ ( σ λ M ) ′ ( u ) − σ ˆ ′ ( u ) , h ⟩ ∣ ≤ ∫ Ω ∣ γ λ M ( z , u ) − γ ˆ λ ( z , u ) ∣ ∣ h ∣ d z ≤ ∫ Ω ∣ γ λ M ( z , u ) − γ λ M ( z , u 0 ) ∣ ∣ h ∣ d z + ∫ Ω ∣ γ ˆ λ ( z , u 0 ) − γ ˆ λ ( z , u ) ∣ ∣ h ∣ d z ,

where we have used the facts M > ‖ u β ‖ ∞ , (4.12), (4.13), and (4.16). We estimate the two terms in the right-hand side of (4.20) to obtain

(4.21) ∫ Ω ∣ γ λ M ( z , u ) − γ λ M ( z , u 0 ) ∣ ∣ h ∣ d z ≤ ∫ { u ¯ μ ≤ u ≤ M } μ ∣ u − η − u 0 − η ∣ ∣ h ∣ d z + ∫ { u ¯ μ ≤ u ≤ M } λ ∣ f ( z , u ) − f ( z , u 0 ) ∣ ∣ h ∣ d z + ∫ { 0 ≤ u ≤ u ¯ μ } ( μ ∣ u ¯ μ − η − u 0 − η ∣ + λ ∣ f ( z , u ¯ μ ) − f ( z , u 0 ) ∣ ) ∣ h ∣ d z + ∫ { M < u } ( μ ∣ M − η − u 0 − η ∣ + λ ∣ f ( z , M ) − f ( z , u 0 ) ∣ ) ∣ h ∣ d z ,

where we have applied (4.12), (4.13), and (4.15). Moreover, we make the following remarks:

the function x ↦ x − η is concave on ( 0 , + ∞ ) , thus, it is locally Lipschitz
on { 0 ≤ u ≤ u ¯ μ } , we have
∣ u ¯ μ − η − u 0 − η ∣ = u ¯ μ − η − u 0 − η ≤ u − η − u 0 − η (since u ≤ u ¯ μ ≤ u 0 )
and
∣ f ( z , u ¯ μ ) − f ( z , u 0 ) ∣ ≤ ∣ f ( z , u ¯ μ ) − f ( z , u ) ∣ + ∣ f ( z , u ) − f ( z , u 0 ) ∣ ≤ c 8 ( u 0 − u ) (see hypotheses H 1 ) ,
for some c 8 > 0 ;
on { M < u } , we have
∣ M − η − u 0 − η ∣ = u 0 − η − M − η ≤ u 0 − η − u − η (recall ‖ u 0 ‖ ∞ < M < u ) ,
and
∣ f ( z , M ) − f ( z , u 0 ) ∣ ≤ ∣ f ( z , M ) − f ( z , u ) ∣ + ∣ f ( z , u ) − f ( z , u 0 ) ∣ ≤ c 9 ( u − u 0 ) (see hypotheses H 1 )
for some c 9 > 0 .

Using these remarks in (4.21), we have

∫ Ω ∣ γ λ M ( z , u ) − γ λ M ( z , u 0 ) ∣ ∣ h ∣ d z ≤ c 10 ‖ u − u 0 ‖ ‖ h ‖

for some c 10 > 0 . Similarly, we show that

∫ Ω ∣ γ ˆ λ ( z , u 0 ) − γ ˆ λ ( z , u ) ∣ ∣ h ∣ d z ≤ c 11 ‖ u − u 0 ‖ ‖ h ‖ for some c 11 > 0 .

Returning to (4.20) and using the above estimations, we obtain

∣ ⟨ ( σ λ M ) ′ ( u ) − σ ˆ ′ ( u ) , h ⟩ ∣ ≤ c 12 ‖ u − u 0 ‖ ‖ h ‖ for some c 12 > 0 .

Hence, it holds

(4.22) ‖ ( σ λ M ) ′ ( u ) − σ ˆ ′ ( u ) ‖ * ≤ c 12 ‖ u − u 0 ‖ .

Then, from (4.19) and (4.22), we see that given ε > 0 there exists δ ∈ ( 0 , 1 ) small such that

‖ ( σ λ M ) ′ ( u ) − σ ˆ ′ ( u ) ‖ C 1 ( B ¯ σ ( u 0 ) ) ≤ ε

with B ¯ δ ( u 0 ) ≔ { u ∈ W 0 1 , η ( Ω ) ∣ ‖ u − u 0 ‖ ≤ δ } . The functional σ ˆ λ ( ⋅ ) is coercive. Hence, it satisfies the C -condition (see [26], p. 369). Also, σ λ ( ⋅ ) satisfies the C -condition [27]. So, we can use the C 1 -continuity property of critical groups (Gasiński and Papageorgiou [10], Theorem 5.126, p. 836) and conclude that

C k ( σ λ M , u 0 ) = C k ( σ ˆ λ , u 0 ) for all k ∈ N 0 .

Since M > ‖ u β ‖ ∞ is arbitrary, we let M → + ∞ and obtain

C k ( σ λ , u 0 ) = C k ( σ ˆ λ , u 0 ) for all k ∈ N 0 .

This proves the Claim.

The Claim and (4.15) imply that

(4.23) C k ( σ λ , u 0 ) = δ k , 0 Z for all k ∈ N 0 .

Reasoning as in the Claim in the proof of Proposition 3.8 of Papageorgiou and Winkert [33], we show that

σ λ ( ⋅ ) satisfies the C -condition .

So, we can compute the critical groups of σ λ at infinity and from Proposition 4.8 of Papageorgiou and Winkert [34], we have

(4.24) C k ( σ λ , ∞ ) = 0 for all k ∈ N 0 .

From (4.23) and (4.24) and Morse’ relation ([26], p. 492), we infer that there exists u ˆ ∈ K σ λ , u ˆ ≠ u 0 . But from (4.12), we see that

K σ λ ⊂ { u ∈ W 0 1 , η ( Ω ) ∣ u ¯ μ ≤ u ( z ) for a.a. z ∈ Ω } ∩ L ∞ ( Ω ) .

Therefore, u ˆ is the second positive solution of problem (1.1) and 0 ≺ u ˆ .□

It remains to decide about the admissibility of the critical parameter λ * .

Proposition 4.7

If hypotheses H 0 and H 1 hold, and μ > 0 is large, then λ * ∈ ℒ .

Proof

Let { λ n } ⊂ ( 0 , λ * ) be such that λ n → ( λ * ) − . From the proof of Proposition 4.6, we know that we can find u n ∈ S λ n ⊂ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) such that

σ λ n ( u n ) ≤ σ λ n ( u ¯ μ ) for all n ∈ N .

Since f ≥ 0 , we have

σ λ n ( u ¯ μ ) ≤ 1 q ρ η ( D u ¯ μ ) − ∫ Ω μ u ¯ μ 1 − η d z ( note 1 < 1 1 − η ) = 1 q − 1 ρ η ( D u ¯ μ ) (see Proposition 3.1) < 0 .

This reveals that

(4.25) σ λ n ( u n ) < 0 for all n ∈ N .

Also, we have

(4.26) σ ′ ( u n ) = 0 in W 0 1 , η ( Ω ) * for all n ∈ N .

From (4.25), (4.26), and reasoning as in the Claim in the proof of Proposition 3.8 of Papageorgiou and Winkert [33], we infer that at least for a subsequence of { u n } , we have

(4.27) u n → u * in W 0 1 , η ( Ω ) .

We have u ¯ μ ≤ u * (Proposition 4.2). From (4.26), we have

⟨ V ( u n ) , h ⟩ = ∫ Ω [ μ u n − η + λ f ( z , u n ) ] h d z for all h ∈ W 0 1 , η ( Ω ) , all n ∈ N .

Passing to the limit as n → ∞ and using (4.27), we have

⟨ V ( u * ) , h ⟩ = ∫ Ω [ μ u * − η + λ f ( z , u * ) ] h d z for all h ∈ W 0 1 , η ( Ω ) .

This means that u * ∈ S λ * and λ * ∈ ℒ .□

Therefore, ℒ = ( 0 , λ * ] . We can state the following global in the parameter λ > 0 , existence, nonexistence, and multiplicity theorem for problem (1.1).

Theorem 4.8

If hypotheses H 0 and H 1 hold, then for all μ > 0 large, we can find λ * = λ * ( μ ) > 0 such that

for all λ ∈ ( 0 , λ * ) , problem (1.1) has at least two positive solutions
u 0 , u ˆ ∈ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) a n d 0 ≺ u 0 , u ˆ ;
for λ = λ * , problem (1.1) has at least one positive solution u * ∈ W 0 1 , η ( Ω ) ∩ L ∞ ( Ω ) and 0 ≺ u * ;
for all λ > λ * , problem (1.1) has no positive solution.

Remark 4.9

Following the analysis above, it is not difficult to prove that [ 0 , + ∞ ) ∋ μ ↦ λ * ( μ ) is increasing.

Acknowledgment

The authors wish to thank the two knowledgeable referees for their constructive comments and remarks.

Funding information: This project has received funding from the Natural Science Foundation of Guangxi Grant Nos GKAD21220144, 2021GXNSFFA196004, and GKAD23026237; NNSF of China Grant Nos 12001478, 12101143, and 12371312; the China Postdoctoral Science Foundation funded project No. 2022M721560; the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH; and the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2023ZK13. It is also supported by the project cooperation between Guangxi Normal University and Yulin Normal University.
Author contributions: The authors contributed equally to this paper.
Conflict of interest: There is no conflict of interests.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

References

[1] R. Arora, A. Fiscella, T. Mukherjee, and P. Winkert, On double phase Kirchhoff problems with singular nonlinearity, Adv. Nonlinear Anal. 12 (2023), 24pp. 10.1515/anona-2022-0312Search in Google Scholar

[2] R. Arora, A. Fiscella, T. Mukherjee, and P. Winkert, On critical double phase Kirchhoff problems with singular nonlinearity, Rend. Circolo Mate. Palermo Ser. 2 271 (2022), 1079–1106. 10.1007/s12215-022-00762-7Search in Google Scholar

[3] A. Bahrouni, V. D. Rǎdulescu, and D. D. Repovš, Nonvariational and singular double phase problems for the Baouendi-Grushin operator, J. Differential Equations 303 (2021), 645–666. 10.1016/j.jde.2021.09.033Search in Google Scholar

[4] H. Brézis and L. Nirenberg, H1 versus C1 local minimizers, CRAS Paris 317 (1993), 465–472. Search in Google Scholar

[5] F. Colasuonno and M. Squassina, Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl. 195 (2016), 1917–1959. 10.1007/s10231-015-0542-7Search in Google Scholar

[6] A. Crespo-Blanco, L. Gasiński, P. Harjulehto, and P. Winkert, A new class of double phase variable exponent problems: Existence and uniqueness, J. Differential Equations 323 (2022), 182–228. 10.1016/j.jde.2022.03.029Search in Google Scholar

[7] C. Farkas and P. Winkert, An existence result for singular Finsler double phase problems, J. Differential Equations 286 (2021), 455–473. 10.1016/j.jde.2021.03.036Search in Google Scholar

[8] J. Garcia-Azorero, I. P. Alonso, and J. J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math. 2 (2000), 385–404. 10.1142/S0219199700000190Search in Google Scholar

[9] L. Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Chapman, Hall/CRC, Boca, Raton, 2006. Search in Google Scholar

[10] L. Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis, Springer, Cham, 2016. 10.1007/978-3-319-27817-9Search in Google Scholar

[11] L. Gasiński and P. Winkert, Constant sign solutions for double phase problems with superlinear nonlinearity, Nonlinear Anal. 195 (2020), 111739. 10.1016/j.na.2019.111739Search in Google Scholar

[12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2015. Search in Google Scholar

[13] P. Harjulehto and P. Hästö, Orlicz Spaces and Generalized Orlicz Spaces, Springer, Cham, 2019. 10.1007/978-3-030-15100-3Search in Google Scholar

[14] E. Hewitt and K. Stromberg Real and Abstract Analysis, Springer, New York, 1975. Search in Google Scholar

[15] D. Kumar, V. D. Rǎdulescu, and K. Sreenadh, Singular elliptic problems with unbalanced growth and critical exponent, Nonlinearity 33 2020, 3336–3369. 10.1088/1361-6544/ab81edSearch in Google Scholar

[16] 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

[17] G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Diff. Equ. 16 (1991), 311–361. 10.1080/03605309108820761Search in Google Scholar

[18] W. Liu and G. Dai, Existence and multiplicity results for double phase problem, J. Differential Equations 265 (2018), 4311–4334. 10.1016/j.jde.2018.06.006Search in Google Scholar

[19] W. Liu, G. Dai, N. S. Papageorgiou, and P. Winkert, Existence of solutions for singular double phase problems via the Nehari manifold method, Anal. Math. Phys. 12 (2022), 25. 10.1007/s13324-022-00686-6Search in Google Scholar

[20] Z. H. Liu and N. S. Papageorgiou, Singular double phase equations, Acta Math. Sci. 43B (2023), 1–14. 10.1007/s10473-023-0304-3Search in Google Scholar

[21] W. Liu and P. Winkert, Combined effects of singular and superlinear nonlinearities in singular double phase problems in RN, J. Math. Anal. Appl. 507 (2022), 125762. 10.1016/j.jmaa.2021.125762Search in Google Scholar

[22] P. Marcellini, Regularity and existence of solutions of elliptic equations with p,q-growth conditions, J. Differential Equations 90 (1991), 1–30. 10.1016/0022-0396(91)90158-6Search in Google Scholar

[23] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions, Arch. Rational Mech. Anal. 105 (1989), 267–284. 10.1007/BF00251503Search in Google Scholar

[24] P. Marcellini, Growth conditions and regularity for weak solutions to nonlinear elliptic PDEs, J. Math. Anal. Appl. 501 (2021), 124408. 10.1016/j.jmaa.2020.124408Search in Google Scholar

[25] G. Mingione and V. D. Rǎdulescu, Recent developments in problems with nonstandard growth and nonuniform ellipticity, J. Math. Anal. Appl. 501 (2021), 125197. 10.1016/j.jmaa.2021.125197Search in Google Scholar

[26] N. S. Papageorgiou, V. D. Rǎdulescu, and D. D. Repovš, Nonlinear Analysis Theory and Methods, Springer, Switzerland, 2019. 10.1007/978-3-030-03430-6Search in Google Scholar

[27] N. S. Papageorgiou, V. D. Rǎdulescu, and Y. Zhang, Anisotropic singular double phase Dirichlet problems, Discr. Cont. Dyn. Syst.-S 12 (2021), 4465–4502. 10.3934/dcdss.2021111Search in Google Scholar

[28] N. S. Papageorgiou, V. D. Rǎdulescu, and Y. Zhang, Strongly singular double phase problems, Medit. J. Math. 19 (2022), 82. 10.1007/s00009-022-02013-6Search in Google Scholar

[29] N. S. Papageorgiou and G. Smyrlis, A bifurcation-type theorem for singular nonlinear elliptic equations, Methods Appl. Anal. 22 (2015), 147–170. 10.4310/MAA.2015.v22.n2.a2Search in Google Scholar

[30] N. S. Papageorgiou, C. Vetro, and F. Vetro, Multiple solutions for parametric double phase Dirichlet problems, Comm. Contemp. Math. 23 (2021), 2050006, 18pp. 10.1142/S0219199720500066Search in Google Scholar

[31] N. S. Papageorgiou and P. Winkert, Applied Nonlinear Functional Analysis, De Gruyter, Berlin, 2018. 10.1515/9783110532982Search in Google Scholar

[32] N. S. Papageorgiou and P. Winkert, Singular p-Laplacian equations with superlinear perturbation, J. Differential Equ. 266 (2019), 1462–1487. 10.1016/j.jde.2018.08.002Search in Google Scholar

[33] N. S. Papageorgiou and P. Winkert, Singular Dirichlet (p,q)-equations, Medit. J. Math. 18 (2021), 1–20. 10.1007/s00009-021-01780-ySearch in Google Scholar

[34] N. S. Papageorgiou and P. Winkert, Bull. Sci. Math. 141 (2017), 443–488. 10.1016/j.bulsci.2017.05.003Search in Google Scholar

[35] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. 10.1007/978-1-4612-4146-1Search in Google Scholar

[36] Q. Zhang, A strong maximum principle for differential equations with nonstandard p(x)-growth conditions, J. Math. Anal. Appl. 312 (2005), 24–32. 10.1016/j.jmaa.2005.03.013Search in Google Scholar

[37] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 29 (1987), 33–66. 10.1070/IM1987v029n01ABEH000958Search in Google Scholar

[38] V. V. Zhikov, On Lavrentievas phenomenon, Russian J. Math. Phys. 3 (1995), 249–269. Search in Google Scholar

Received: 2023-06-16

Revised: 2023-09-17

Accepted: 2023-10-26

Published Online: 2023-12-31

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

Articles in the same Issue

https://doi.org/10.1515/anona-2023-0122

Keywords for this article

singular term; double phase operator; Orlicz-Sobolev space; positive solution; existence, and multiplicity result

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