A generalized p-Laplacian problem with parameters

Jiabin Zuo; Shapour Heidarkhani; Shahin Moradi; José Vanterler da Costa Sousa

doi:10.1515/dema-2025-0135

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A generalized p-Laplacian problem with parameters

Jiabin Zuo , Shapour Heidarkhani , Shahin Moradi and José Vanterler da Costa Sousa

Published/Copyright: May 22, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In recent years, research into the multiplicity of solutions to the p -Laplace operator problem has attracted attention, and several important results have been investigated and others still remain open. Problems involving a critical point are indeed interesting and relevant, especially challenging. Motivated by such questions, in this article, we are interested, through a critical point theorem, to investigate the existence of at least three distinct weak solutions for a generalized p -Laplacian problem with parameters under appropriate hypotheses, applicable in physics, for instance, in fluid mechanics, and in Newtonian fluids. In this sense, as a direct consequence of the main result, we finish the work with two other results of weak solutions.

Keywords: three solutions; p-Laplacian type; critical point; variational methods

MSC 2010: 35J25; 35J62

1 Introduction

We are going to study the p -Laplacian problem with parameters given by

− div A ( t , ∇ u ( t ) ) = λ h ( t , u ( t ) ) + μ e ( t , u ( t ) ) , in Ω , u = 0 , on ∂ Ω , ( P λ , μ h , e )

where the bounded domain Ω ⊂ R N ( N ≥ 2 ) has a smooth boundary ∂ Ω , A : Ω × R N → R N is a function admitting a potential and satisfying some natural conditions such that the differential operator div A ( t , ∇ u ) including the usual p -Laplacian ( p > 1 ) , the parameters λ > 0 and μ ≥ 0 , while h , e : Ω × R → R are suitable Carathéodory functions.

In the last decades, this type of equation ( P λ , μ h , e ) has been deliberated due to its wide applications in physics, for instance, in fluid mechanics, and in Newtonian fluids [1].

In recent years, the research on the multiplicity of solutions to the p -Laplace operator problem has captivated the attention of many scholars (we refer the reader to [2–14] and references therein). For example, Bonanno et al. [4], applying variational methods, obtained some existence results for the problem ( P λ , μ h , e ). Kristály et al. [13] obtained the existence of three weak solutions for a general elliptic problem of divergence form (in particular, a p -Laplace operator) with the help of an abstract critical point theorem of Bonanno [3]. Colasuonno et al. [7] using two recent theorems proposed by Arcoya and Carmona [2] to study the homogeneous single parameter problem and the non-homogeneous two-parameter problem under different boundary conditions. Nápoli and Mariani [14], employing mountain pass lemma, secured the existence of at least one solution and proved the existence of infinitely many solutions of a parameterized version of the problem under further assumptions. When μ = 0 , Heidarkhani et al. [12] obtained the existence of at least three weak solutions for the problem ( P λ , μ h , e ), in the case μ = 0 , by using the critical point theory. Furthermore, in [10], the existence of at least one weak solution and infinitely many weak solutions for the problem ( P λ , μ h , e ) was discussed via variational methods.

Here, influenced by [7] and [13], we study the problem ( P λ , μ h , e ) when A admits a potential A : Ω ¯ × R N → R , such that

A = A ( t , ε ) is continuous on Ω ¯ × R N , with a continuous derivative with respect to ε and A = ∂ ε A . Moreover,
A ( t , 0 ) = 0 and A ( t , ε ) = A ( t , − ε ) for each t ∈ Ω and ε ∈ R N .
A ( t , ⋅ ) is strictly convex in R N for each t ∈ Ω .
There exist two constants a 1 and a 2 , with 0 < a 1 ≤ a 2 such that
(1.1) A ( t , ε ) . ε ≥ a 1 ∣ ε ∣ p and ∣ A ( t , ε ) ∣ ≤ a 2 ∣ ε ∣ p − 1 ,
for each ( t , ε ) ∈ Ω × R N .

Employing the three critical points theorem [15, Theorem 3.1], we guarantee the existence of at least three weak solutions for the problem ( P λ , μ h , e ) for appropriate values of the parameters λ and μ belonging to real intervals (Theorem 3.1). In fact, in Theorem 3.1, we establish the existence of two intervals of positive real parameters λ and an interval of positive real parameter μ for which the problem ( P λ , μ h , e ) possesses three weak solutions, whose norms are uniformly bounded with respect to λ belonging to one of the two intervals. In Theorem 3.1, in addition, we need an asymptotic condition on the function e . In Remark 3.2, we point out a simple consequence of Theorem 3.1. In Corollary 3.5, we study the autonomous version of Theorem 3.1. Theorem 3.6 is a special case of Theorem 3.1. Our article supplies new contributions and accomplishes some previous results in the literature for the problem ( P λ , μ h , e ). For other interesting works on variational methods and critical point theory, see [16,17].

2 Preliminaries

Our essential tool to study the problem ( P λ , μ h , e ) is [15, Theorem 3.1]. Some applications for this theorem can be found in [18,19].

Let Ω be a bounded domain of R N , 1 < p < N , W 0 1 , p ( Ω ) is the Sobolev space equipped with the norm

‖ v ‖ = ‖ ∇ v ‖ p ,

and W − 1 , p ′ ( Ω ) is its dual space, where 1 p + 1 p ′ = 1 . If 1 < p < N and p * = N p N − p , there is a constant σ = σ ( N , p ) such that

(2.1) ‖ v ‖ p * ≤ σ ‖ v ‖ ,

for each v ∈ W 0 1 , p ( Ω ) with

σ = π − 1 2 N − 1 p p − 1 N − p 1 − 1 p Γ ( 1 + N 2 ) Γ ( N ) Γ N p Γ 1 + N − N p 1 N ,

where Γ ( ⋅ ) is the gamma function [20]. Obviously, (2.1), together with the Hölder’s inequality, infers that for each l ∈ [ 1 , p * ] ,

‖ v ‖ l ≤ σ ( meas ( Ω ) ) ( p * − l ) ( p * l ) ‖ v ‖ ,

∀ v ∈ W 0 1 , p ( Ω ) , where meas ( Ω ) is the Lebesgue measure of Ω and the embedding W 0 1 , p ( Ω ) ↪ L l ( Ω ) is compact, provided l ∈ [ 1 , p * [ .

Suppose that A : Ω × R N → R N has a smooth potential A : Ω ¯ × R N → R . Using the conditions ( P ) ( i ) and ( P ) ( i i i ) yields that

(2.2) a 1 ∣ ε ∣ p ≤ p A ( t , ε ) ≤ a 2 ∣ ε ∣ p ,

for each ( t , ε ) ∈ Ω × R N [7].

Let h : Ω × R → R be a Carathéodory function, and let a be a positive function in L β ( Ω ) with β > N ⁄ p . Given that 1 < q ≤ p , we say that h is of type ( G h ) if it satisfies the following growth condition [4]:

( G h ) there exist K 1 > 0 and K 2 > 0 (constants) such that
∣ h ( t , x ) ∣ ≤ a ( t ) ∣ ( K 1 + K 2 ∣ x ∣ q − 1 ) ,
for a.e. ( t , x ) ∈ Ω × R .

Definition 2.1

We mean by a (weak) solution of the BVP ( P λ , μ h , e ), any function ζ ∈ W 0 1 , p such that

∫ Ω A ( t , ∇ ζ ( t ) ) ∇ n ( t ) d t − λ ∫ Ω h ( t , ζ ( t ) ) n ( t ) d t − μ ∫ Ω e ( t , ζ ( t ) ) n ( t ) d t = 0 ,

for each n ∈ W 0 1 , p ( Ω ) .

3 Main results

In this section, we state our main results for the problem ( P λ , μ h , e ). Let

P ( t , x ) = ∫ 0 x h ( t , ε ) d ε and ℰ ( t , x ) = ∫ 0 x e ( t , ε ) d ε ,

for each ( t , x ) ∈ Ω × R . In what follows, β ′ is the conjugate of β , i.e., 1 β + 1 β ′ = 1 .

Let R : Ω → [ 0 , ∞ [ be the function defined by R ( t ) = d ( t , ∂ Ω ) for every t ∈ Ω . Thus, for each t 0 ∈ Ω , one has B ( t 0 , R ( t 0 ) ) = { t ∈ Ω : ∣ t − t 0 ∣ < R ( t 0 ) } ⊆ Ω . For our convenience, set

(3.1) ℰ θ = ∫ Ω sup ∣ ε ∣ ≤ θ ℰ ( t , ε ) d t , for each θ > 0 ,

and

ℰ η = inf Ω × R ℰ ( t , η ) for each η > 0 .

If e is sign-changing, then clearly, ℰ θ ≥ 0 and ℰ η ≤ 0 . Let us use the following notation B = B (0, 1). Fixing t 0 ∈ Ω , θ > 0 and η > 0 (constants), put

D θ = K 1 ‖ a ‖ β σ ( meas ( Ω ) ) ( p * − β ′ ) ( p * β ′ ) ( meas ( Ω ) ) p p * σ p θ p 1 p + K 2 q ‖ a ‖ β σ q ( meas ( Ω ) ) ( p * − β ′ q ) ( p * β ′ ) ( meas ( Ω ) ) p p * σ p θ p q p , δ ̲ λ , e = min a 1 ( meas ( Ω ) ) p p * θ p − λ p σ p D θ p σ p ℰ θ , a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p − λ R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ ℰ η − ℰ θ ,

where a ∈ L β ( Ω ) (with a > N ⁄ p ), and

(3.2) δ ¯ λ , e = min δ ̲ λ , e , 1 max 0 , p σ p a 1 ( meas ( Ω ) ) p p * limsup ∣ x ∣ → + ∞ sup t ∈ Ω ℰ ( t , x ) x p .

Theorem 3.1

If ( P ) ( i ) – ( P ) ( i i i ) hold, and h is of type ( G h ) , then under the existence of t 0 ∈ Ω , θ > 0 and η > 0 satisfying

θ < σ p ( meas ( Ω ) ) p p * R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p p ,

along with the following conditions:

h ( t , x ) ≥ 0 for each ( t , x ) ∈ Ω × R ,
D θ satisfies
D θ < a 1 ( meas ( Ω ) ) p p * θ p 2 σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) ,
the asymptotic condition limsup ∣ x ∣ → ∞ P ( t , x ) x p < Θ 1 holds uniformly for t ∈ Ω , where Θ 1 is given by
Θ 1 = max D θ a 1 ( meas ( Ω ) ) p p * p σ p θ p , a 1 ( meas ( Ω ) ) p p * θ p σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ K a 1 ( meas ( Ω ) ) p p * p σ p θ p ,
with K > 1 .

Then, for each

λ ∈ Λ 1 = a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ , a 1 ( meas ( Ω ) ) p p * p σ p θ p D θ ,

and for each continuous function e : Ω × R → R fulfilling the condition

(3.3) limsup ∣ x ∣ → + ∞ sup t ∈ Ω ℰ ( t , x ) x p < + ∞ ,

∃ δ ¯ λ , e > 0 given by (3.2) such that, for each μ ∈ [ 0 , δ ¯ λ , e ) , the problem ( P λ , μ h , e ) possesses at least three weak solutions in W 0 1 , p and, moreover, for each K > 1 , there exist an open interval

Λ 2 ⊆ 0 , K a 1 ( meas ( Ω ) ) p p * p σ p θ p a 1 ( meas ( Ω ) ) p p * θ p σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ

and a positive real number σ , for each λ ∈ Λ 2 , and for each continuous function e : Ω × R → R fulfilling the condition (3.3), for each

μ ∈ 0 , 1 max 0 , p σ p a 1 ( meas ( Ω ) ) p p * limsup ∣ x ∣ → + ∞ sup t ∈ Ω ℰ ( t , x ) x p ,

the problem ( P λ , μ h , e ) possesses at least three weak solutions in W 0 1 , p bounded by σ .

Proof

Here, we will use [15, Theorem 3.1]. Let the functional I : W 0 1 , p ( Ω ) → R be defined by

(3.4) I ( ζ ) = ∫ Ω A ( t , ∇ ζ ( t ) ) d t ,

which is convex, weakly lower semicontinuous, and of class C 1 in W 0 1 , p ( Ω ) , being

I ′ ( ζ ) ( s ) = ∫ Ω A ( t , ∇ ζ ( t ) ) ∇ s ( t ) d t ,

for each ζ , s ∈ W 0 1 , p ( Ω ) . Moreover, I ′ : W 0 1 , p ( Ω ) → W − 1 , p ′ ( Ω ) satisfies the ( S + ) condition, i.e., for each sequence { ζ n } in W 0 1 , p ( Ω ) such that ζ n ⇀ ζ weakly in W 0 1 , p ( Ω ) and

limsup n → ∞ ∫ Ω A ( t , ∇ ζ n ) . ( ∇ ζ n − ∇ ζ ) d t ≤ 0 ,

then ζ n ⇀ ζ strongly in W 0 1 , p ( Ω ) (see [7, Lemma 2.5]). The condition (2.2) assures the following:

(3.5) a 1 p ‖ ζ ‖ p ≤ I ( ζ ) ≤ a 2 p ‖ ζ ‖ p ,

for each ζ ∈ W 0 1 , p . The first inequality in (3.5) gives

lim ‖ ζ ‖ → + ∞ I ( ζ ) = ∞ ,

namely, I is coercive. Now, we claim that I ′ admits a continuous inverse on ( W 0 1 , p ) * . It is obvious that

I ′ ( ζ ) ( ζ ) = ∫ Ω A ( t , ∇ ζ ( t ) ) ∇ ζ ( t ) d t ≥ a 1 ‖ ζ ‖ p ,

namely, I ′ is coercive. On account of our assumptions on the data, one has

⟨ I ′ ( ζ ) − I ′ ( s ) , m − s ⟩ = ∫ Ω A ( t , ( ∇ ζ ( t ) − ∇ s ( t ) ) ) ( ∇ ζ ( t ) − ∇ s ( t ) ) d t ≥ a 1 ‖ m − s ‖ p > 0 ,

for each ζ , s ∈ W 0 1 , p , which denotes I ′ is strictly monotone. Additionally, since W 0 1 , p is reflexive, for ζ n ⟶ ζ strongly in W 0 1 , p as n → + ∞ , one has I ′ ( ζ n ) → I ′ ( ζ ) weakly in ( W 0 1 , p ) * as n → ∞ . Hence, I ′ is demicontinuous, so by [21, Theorem 26.A(d)], the inverse operator I ′ − 1 of I ′ exists, and it is continuous. Because, letting z n be a sequence of ( W 0 1 , p ) * such that z n → z strongly in ( W 0 1 , p ) * as n → ∞ , and letting ζ n and ζ in W 0 1 , p such that I ′ − 1 ( z n ) = ζ n and I ′ − 1 ( z ) = ζ , giving consideration to that I ′ is coercive, we can say the sequence ζ n is bounded, so for a suitable subsequence, ζ n → ζ ˆ weakly in W 0 1 , p as n → ∞ , which concludes

⟨ I ′ ( ζ n ) − I ′ ( ζ ) , ζ n − ζ ˆ ⟩ = ⟨ z n − z , ζ n − ζ ˆ ⟩ = 0 .

Observing ζ n → ζ ˆ weakly in W 0 1 , p as n → + ∞ and I ′ ( ζ n ) → I ′ ( ζ ˆ ) strongly in ( W 0 1 , p ) * as n → + ∞ , one has ζ n → ζ ˆ strongly in W 0 1 , p as n → + ∞ , and since I ′ is continuous, we have ζ n → ζ ˆ weakly in W 0 1 , p as n → + ∞ and I ′ ( ζ n ) → I ′ ( ζ ˆ ) = I ′ ( ζ ) strongly in ( W 0 1 , p ) * as n → + ∞ . Hence, since I ′ is an injection, we have ζ = ζ ˆ , and it is continuous. Now, let the functional J : W 0 1 , p ( Ω ) → R be defined by

(3.6) J ( ζ ) = ∫ Ω P ( t , ζ ( t ) ) d t + μ λ ∫ Ω ℰ ( t , ζ ( t ) ) d t ,

which is of class C 1 being

J ′ ( ζ ) ( s ) = ∫ Ω h ( t , ζ ( t ) ) s ( t ) d t + μ λ ∫ Ω e ( t , ζ ( t ) ) s ( t ) d t .

Moreover, the operator J ′ : W 0 1 , p ( Ω ) → W − 1 , p ′ ( Ω ) is compact and sequentially weakly continuous in W 0 1 , p ( Ω ) (see [7, Lemma 3.2]). In Lemma 3.2 of [7], the compactness of J ′ is proved when 1 < q < p , but the arguing in the same way guarantees that the same property holds in the case q = p occurs. Thus, the regularity assumptions on I and J , as requested in Theorem [15, Theorem 3.1], are verified. Note that the operator Q λ = I − λ J is a C 1 ( W 0 1 , p , R ) functional and the critical points of Q λ are weak solutions of the problem ( P λ , μ h , e ). We now search for the existence of critical points of the functional Q λ in W 0 1 , p . Then, our aim is to justify ( a 1 ) , ( a 2 ) , and ( a 3 ) . To this end, since

μ < 1 max 0 , p σ p a 1 ( meas ( Ω ) ) p p * limsup ∣ x ∣ → + ∞ sup t ∈ Ω ℰ ( t , x ) x p ,

fix l > 0 such that

limsup ∣ x ∣ → + ∞ sup t ∈ Ω ℰ ( t , x ) x p < l and μ l < p σ p a 1 ( meas ( Ω ) ) p p * .

Consequently, ∃ ϱ ∈ R function so that

(3.7) ℰ ( t , x ) ≤ l x p + ϱ

for each ( t , x ) ∈ Ω × R . Additionally, fixing λ > 0 , from ( A 3 ) , there exist two constants γ , τ ∈ R with γ < 1 λ Θ 1 a 1 p − μ l such that

1 Θ 1 P ( t , x ) < γ x p + τ , for all ( t , x ) ∈ Ω × R .

Fix ζ ∈ W 0 1 , p . Then,

(3.8) P ( t , ζ ( t ) ) < Θ 1 ( γ ∣ ζ ( t ) ∣ p + τ ) , for all t ∈ Ω .

Now, to prove the coercivity of the functional Q λ , first, we assume that γ > 0 . Operating (3.7) and (3.8) yields that

Q λ ( ζ ) ≥ a 1 p ‖ ζ ‖ p − λ ∫ Ω P ( t , ζ ( t ) ) + μ λ ℰ ( t , ζ ( t ) ) d t ≥ a 1 p ‖ ζ ‖ p − λ Θ 1 γ ∫ Ω ∣ ζ ( t ) ∣ p d t + τ − μ l ∫ Ω ∣ ζ ( t ) ∣ p d t + ϱ ≥ a 1 p − λ Θ 1 γ − μ l ‖ ζ ‖ p − λ Θ 1 τ − μ ϱ

and so

lim ‖ ζ ‖ → + ∞ Q λ ( ζ ) = + ∞ .

On the other hand, if γ ≤ 0 , clearly, we obtain lim ‖ ζ ‖ → + ∞ Q λ ( ζ ) = + ∞ . Both cases generate the coercivity of functional Q λ . Now, we define

(3.9) w ( t ) = 0 , t ∈ Ω \ B ¯ ( t 0 , R ( t 0 ) ) , 2 η R ( t 0 ) ( R ( t 0 ) − ∣ t − t 0 ∣ ) , t ∈ B ( t 0 , R ( t 0 ) ) \ B ¯ ( t 0 , R ( t 0 ) ⁄ 2 ) , η , t ∈ B ( t 0 , R ( t 0 ) ⁄ 2 ) .

Evidently, w ∈ W 0 1 , p . Then, we have Φ ( 0 ) = Ψ ( 0 ) = 0 . A direct computation using (3.5) shows that

I ( w ) ≥ a 1 p 2 p [ R ( t 0 ) ] p ∣ B ( t 0 , R ( t 0 ) ) \ B ¯ ( t 0 , R ( t 0 ) ⁄ 2 ) ∣ η p = a 1 p 2 p [ R ( t 0 ) ] p ∣ B ∣ ( [ R ( t 0 ) ] N − [ R ( t 0 ) ⁄ 2 ] N ) η p = a 1 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p .

Moreover,

Φ ( w ) ≤ a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p .

In addition, because of the assumption ( A 1 ) , one has

J ( w ) = ∫ Ω P ( t , w ( t ) ) d t + μ λ ∫ Ω ℰ ( t , w ( t ) ) d t ≥ ∫ B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) d t + μ λ ∫ Ω ℰ ( t , w ( t ) ) d t ≥ ∣ B ( t 0 , R ( t 0 ) ⁄ 2 ) ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) + μ λ ∫ Ω ℰ ( t , w ( t ) ) d t ≥ R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) + μ λ ℰ η .

Put

r = a 1 ( meas ( Ω ) ) p p * p σ p θ p .

Thanks to θ < σ p ( meas ( Ω ) ) p p * R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p p , we have I ( w ) > r . From the definition of I and in view of (3.5) for each r > 0 , we have

I − 1 ( − ∞ , r ) = { ζ ∈ W 0 1 , p , Φ ( ζ ) < r } ⊆ ζ ∈ W 0 1 , p , ‖ ζ ‖ ≤ p r a 1 1 p ⊆ ζ ∈ W 0 1 , p , ‖ ζ ‖ ≤ ( meas ( Ω ) ) p p * σ p θ p 1 p .

Hence, it yields

sup I ( ζ ) ≤ r J ( ζ ) ≤ K 1 ‖ a ‖ β σ ( meas ( Ω ) ) ( p * − β ′ ) ( p * β ′ ) ( meas ( Ω ) ) p p * σ p θ p 1 p + K 2 q ‖ a ‖ β σ q ( meas ( Ω ) ) ( p * − β ′ q ) ( p * β ′ ) ( meas ( Ω ) ) p p * σ p θ p q p + μ λ ℰ θ .

Owing to our assumptions, we have

sup ζ ∈ I − 1 ( ] − ∞ , r [ ) ¯ w J ( ζ ) ≤ D θ + μ λ ℰ θ

and

r r + I ( w ) J ( w ) = r r + I ( w ) ∫ Ω P ( t , w ( t ) ) + μ λ ℰ ( t , w ( t ) ) d t ≥ a 1 ( meas ( Ω ) ) p p * p σ p θ p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) + μ λ ℰ η a 1 ( meas ( Ω ) ) p p * p σ p θ p + a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p ≥ 1 2 a 1 ( meas ( Ω ) ) p p * θ p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) + μ λ ℰ η σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p .

Now, we can employ [15, Theorem 3.1]. Bear in mind that

I ( w ) J ( w ) − sup u ∈ I − 1 ( ] − ∞ , r [ ) ¯ w J ( u ) ≤ a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ + μ λ ( ℰ η − ℰ θ )

and

r sup u ∈ I − 1 ( ] − ∞ , r [ ) ¯ w J ( u ) ≥ a 1 ( meas ( Ω ) ) p p * p σ p θ p D θ + μ λ ℰ θ .

Since μ < δ ̲ λ , e , one has

μ < a 1 ( meas ( Ω ) ) p p * θ p − λ p σ p D θ p σ p ℰ θ ,

which conveys

p σ p a 1 ( meas ( Ω ) ) p p * D θ + μ λ ℰ θ θ 2 < 1 λ .

Furthermore,

μ < a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p − λ R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ ℰ η − ℰ θ ,

which means

(3.10) R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ + μ λ ( ℰ η − ℰ θ ) a 1 ( meas ( Ω ) ) p p * p σ p θ p > 1 λ .

Then,

p σ p a 1 ( meas ( Ω ) ) p p * D θ + μ λ ℰ θ θ 2 < R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ + μ λ ( ℰ η − ℰ θ ) a 1 ( meas ( Ω ) ) p p * p σ p θ p .

Additionally, one has

K r r J ( w ) I ( w ) − sup u ∈ I − 1 ( ] − ∞ , r [ ) ¯ w J ( u ) ≤ K a 1 ( meas ( Ω ) ) p p * p σ p θ p ϱ = ϑ ,

where

ϱ = a 1 ( meas ( Ω ) ) p p * σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ + μ λ ( ℰ η − ℰ θ ) .

Hence, choosing u 0 = 0 and u 1 = w , [15, Theorem 3.1] follows that, for each λ ∈ Λ 1 , and there exist an open interval Λ 2 ⊆ [ 0 , ϑ ] and a real positive number σ such that, for each λ ∈ Λ 2 , the problem ( P λ , μ h , e ) possesses at least three weak solutions whose norms in W 0 1 , p are less than σ .□

Remark 3.2

In Theorem 3.1, we note that

a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ < a 1 ( meas ( Ω ) ) p p * p σ p θ p D θ .

Because, from ( A 2 ) , one has

2 σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p D θ < a 1 ( meas ( Ω ) ) p p * θ p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) ,

and since

θ < p σ p a 1 ( meas ( Ω ) ) p p * a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p ,

we obtain

a 1 ( meas ( Ω ) ) p p * θ p + σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p D θ < a 1 ( meas ( Ω ) ) p p * θ p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η )

and so

σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p D θ < a 1 ( meas ( Ω ) ) p p * θ p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ .

Hence, multiplying by 1 p σ p , it follows

a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p D θ < a 1 ( meas ( Ω ) ) p p * p σ p θ p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) D θ ,

which arrives at

a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( t , η ) − D θ < a 1 ( meas ( Ω ) ) p p * p σ p θ p D θ .

Therefore, Λ 1 ≠ ∅ .

Now, let us provide some remarks on our findings.

Remark 3.3

In the case h and e are non-negative, the weak solutions secured in Theorem 3.1 are non-negative. Indeed, let ζ 0 be a non-trivial weak solution of the problem ( P λ , μ h , e ). Proof by a contradiction, suppose L = { t ∈ Ω : ζ 0 ( t ) < 0 } is non-empty and of positive measure. Set n ¯ ( t ) = min { 0 , ζ 0 ( t ) } for all t ∈ Ω . Undoubtedly, n ¯ ∈ W 0 1 , p and one has

∫ Ω A ( t , ∇ ζ 0 ( t ) ) ∇ n ¯ ( t ) d t − λ ∫ Ω h ( t , ζ 0 ( t ) ) n ¯ ( t ) d t − μ ∫ Ω e ( t , ζ 0 ( t ) ) n ¯ ( t ) d t = 0 .

Thus, from the sign assumptions on the data, we have

0 ≤ a 1 ‖ ζ 0 ‖ L p ≤ ∫ L A ( t , ∇ ζ 0 ( t ) ) ∇ ζ 0 ( t ) d t = λ ∫ L h ( t , ζ 0 ( t ) ) ζ 0 ( t ) d t + μ ∫ L e ( t , ζ 0 ( t ) ) ζ 0 ( t ) d t ≤ 0 .

Hence, ζ 0 = 0 in L , and this is obstructive.

Remark 3.4

In the case, in Theorem 3.1, either h ( t , 0 ) ≠ 0 or e ( t , 0 ) ≠ 0 for all t ∈ Ω , or both remain true, then the secured weak solutions in the aforementioned results are undoubtedly non-trivial. On the other hand, the non-triviality of the weak solutions can be reached also in the case h ( t , 0 ) = 0 and e ( t , 0 ) = 0 for all t ∈ Ω demanding the extra condition at zero on h , i.e., there are a non-empty open set Q ⊆ Ω and a set O ⊂ Q of positive Lebesgue measure such that

limsup x → 0 + ess inf t ∈ O P ( t , x ) ∣ x ∣ p = + ∞

and

liminf x → 0 + ess inf t ∈ Q P ( t , x ) ∣ x ∣ p > − ∞

(see [10, Theorem 3.1]).

We have the following results as a direct consequence of Theorem 3.1.

Corollary 3.5

Let h : R → R be a continuous function and there exist K 1 , K 2 > 0 and q ∈ ( 1 , p ) such that

h ( x ) ≤ K 1 + K 2 ∣ x ∣ q − 1 .

Assume that there exist two positive constants θ and η with

θ < σ p ( meas ( Ω ) ) p p * R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p p ,

where t 0 ∈ Ω , such that

h ( x ) ≥ 0 for each x ∈ R ,
D ˜ θ < a 1 ( meas ( Ω ) ) p p * θ p 2 σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( η ) ,
where
D ˜ θ = K 1 σ ( meas ( Ω ) ) ( p * − β ′ ) ( p * β ′ ) ( meas ( Ω ) ) p p * σ p θ p 1 p + K 2 q σ q ( meas ( Ω ) ) ( p * − β ′ q ) ( p * β ′ ) ( meas ( Ω ) ) p p * σ p θ p q p ,
limsup ∣ x ∣ → ∞ P x x p < Θ 1 ′ where
Θ 1 ′ = max D ˜ θ a 1 ( meas ( Ω ) ) p p * p σ p θ p , a 1 ( meas ( Ω ) ) p p * θ p σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( η ) − D ˜ θ K a 1 ( meas ( Ω ) ) p p * p σ p θ p
with K > 1 .

Then, for each

λ ∈ Λ 1 ′ = a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( η ) − D ˜ θ , a 1 ( meas ( Ω ) ) p p * p σ p θ p D ˜ θ

and for each continuous function e : Ω × R → R fulfilling the condition

(3.11) limsup ∣ x ∣ → + ∞ sup t ∈ Ω E ( t , x ) x p < + ∞ ,

there exists δ ¯ λ , e > 0 such that, for every μ ∈ [ 0 , δ ¯ λ , e ) , the problem

(3.12) − div A ( t , ∇ u ( t ) ) = λ h ( u ( t ) ) + μ e ( t , u ( t ) ) , i n Ω , u = 0 , o n ∂ Ω ,

possesses at least three weak solutions in W 0 1 , p and, moreover, for each K > 1 , there exist an open interval

Λ 2 ′ ⊆ 0 , K a 1 ( meas ( Ω ) ) p p * p σ p θ p a 1 ( meas ( Ω ) ) p p * θ p σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( η ) − D ˜ θ

and a positive real number σ , for every λ ∈ Λ 2 ′ , and for each continuous function e : Ω × R → R satisfying the condition (3.11), for each

μ ∈ 0 , 1 max 0 , p σ p a 1 ( meas ( Ω ) ) p p * limsup ∣ x ∣ → + ∞ sup t ∈ Ω ℰ ( t , x ) x p ,

problem (3.12) possesses at least three weak solutions in W 0 1 , p bounded by σ .

Proof

Set

h + ( t , x ) = h ( x ) , if ( t , x ) ∈ Ω × [ 0 , ∞ ) , 0 , if ( t , x ) ∈ Ω × ( − ∞ , 0 ) ,

and P + ( t , x ) = ∫ 0 x h ( t , s ) d s for each ( t , x ) ∈ Ω × R . The functions h + and P + fit all the conditions of Theorem 3.1 with a ≡ 1 . Then, Theorem 3.1 yields the conclusion.□

We end this article by displaying the following version of Theorem 3.1.

Theorem 3.6

Assume that P ( η ) > 0 for some η > 0 and

liminf x → 0 P ( x ) x p = limsup x → ∞ P ( x ) x p = 0 .

Then, there is λ * > 0 such that for every λ > λ * and for each continuous function e : Ω × R → R satisfying the asymptotically condition

limsup ∣ x ∣ → ∞ sup t ∈ Ω ∫ 0 x e ( t , s ) d s x p < ∞ ,

there exists δ λ , e * > 0 such that, for every μ ∈ [ 0 , δ λ , e * ) , problem (3.12) possesses at least three weak solutions.

Proof

Fix

λ > λ * = a 2 p R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( η ) − D ˜ θ

for some η > 0 and t 0 ∈ Ω . From the condition

liminf x → 0 P ( x ) x p = 0 ,

there is a sequence { θ n } ⊂ ] 0 , + ∞ [ such that lim n → ∞ θ n = 0 and

lim n → ∞ sup ∣ x ∣ ≤ θ n P ( x ) θ n p = 0 .

Indeed, one has

lim n → ∞ sup ∣ x ∣ ≤ θ n P ( x ) θ n p = lim n → ∞ P ( x θ n ) x θ n p x θ n p θ n p = 0 ,

where P ( x θ n ) = sup ∣ x ∣ ≤ θ n P ( x ) . Hence, there exists θ ¯ > 0 such that

D ˜ θ ¯ < min a 1 ( meas ( Ω ) ) p p * θ ¯ p 2 σ p a 2 R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p R ( t 0 ) 2 N ∣ B ∣ ess inf B ( t 0 , R ( t 0 ) ⁄ 2 ) P ( η ) , p σ p a 1 ( meas ( Ω ) ) p p * λ θ ¯ p

and

θ ¯ < σ p ( meas ( Ω ) ) p p * R ( t 0 ) 2 N − p ( 2 N − 1 ) ∣ B ∣ η p p .

By applying Theorem 3.1, we have the result.□

Funding information: Jiabin Zuo was supported by the Guangdong Basic and Applied Basic Research Foundation (2024A1515012389).
Author contributions: All authors of this manuscript contributed equally to this work.
Conflict of interest: Dr. Jiabin Zuo is a member of the Editorial Board of the journal Demonstratio Mathematica, but was not involved in the review process of this article.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] C. Florica, D. Motreanu, and V. Rădulescu, Weak solutions of quasilinear problems with nonlinear boundary condition, Nonlinear Anal. 43 (2001), no. 5, 623–636, DOI: https://doi.org/10.1016/S0362-546X(99)00224-2. 10.1016/S0362-546X(99)00224-2Search in Google Scholar

[2] D. Arcoya and J. Carmona, A nondifferentiable extension of a theorem of Pucci and Serrin and applications, J. Differential Equations 235 (2007), no. 2, 683–700, DOI: https://doi.org/10.1016/j.jde.2006.11.022. 10.1016/j.jde.2006.11.022Search in Google Scholar

[3] G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal. 54 (2003), no. 4, 651–665, DOI: https://doi.org/10.1016/S0362-546X(03)00092-0. 10.1016/S0362-546X(03)00092-0Search in Google Scholar

[4] G. Bonanno, G. D’Aguì, and R. Livrea, Triple solutions for nonlinear elliptic problems driven by a non-homogeneous operator, Nonlinear Anal. 197 (2020), 111862, DOI: https://doi.org/10.1016/j.na.2020.111862. 10.1016/j.na.2020.111862Search in Google Scholar

[5] C. Chen, and H. Wang, Ground state solutions for singular p-Laplacian equation inRN, J. Math. Anal. Appl. 351 (2009), no. 2, 773–780, DOI: https://doi.org/10.1016/j.jmaa.2008.11.010. 10.1016/j.jmaa.2008.11.010Search in Google Scholar

[6] F. Colasuonno, and P. Pucci, Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011), no. 17, 5962–5974, DOI: https://doi.org/10.1016/j.na.2011.05.073. 10.1016/j.na.2011.05.073Search in Google Scholar

[7] F. Colasuonno, P. Pucci, and C. Varga. Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Anal. 75 (2012), no. 12, 4496–4512, DOI: https://doi.org/10.1016/j.na.2011.09.048. 10.1016/j.na.2011.09.048Search in Google Scholar

[8] F. Demengel, and E. Hebey, On some nonlinear equations involving the p-Laplacian with critical Sobolev growth, Adv. Differential Equations 3 (1998), no. 4, 533–574, DOI: https://doi.org/10.57262/ade/1366292563. 10.57262/ade/1366292563Search in Google Scholar

[9] Y. Deng, and H. Pi, Multiple solutions for p-harmonic type equations, Nonlinear Anal. 71 (2009), no. 10, 4952–4959, DOI: https://doi.org/10.1016/j.na.2009.03.067. 10.1016/j.na.2009.03.067Search in Google Scholar

[10] J. R. Graef, S. Heidarkhani, L. Kong, and S. Moradi, Existence results for a nonlinear elliptic problem driven by a non-homogeneous operator, J. Appl. Anal. Comput. 15 (2025), no. 1, 547–563.Search in Google Scholar

[11] S. Heidarkhani, Multiple solutions for a quasilinear second order differential equation depending on a parameter, Acta Math. Appl. Sin. Engl. Ser. 32 (2016), no. 1, 199–208, DOI: https://doi.org/10.1007/s10255-016-0548-y. 10.1007/s10255-016-0548-ySearch in Google Scholar

[12] S. Heidarkhani, S. Moradi, G. Caristi, and M. Ferrara, A variational approach for parametric nonlinear Dirichlet problem involving a nonhomogeneous differential operator of p-Laplacian type, preprint. Search in Google Scholar

[13] A. Kristály, H. Lisei, and C. Varga, Multiple solutions for p-Laplacian type equations, Nonlinear Anal. 68 (2008), no. 5, 1375–1381, DOI: https://doi.org/10.1016/j.na.2006.12.031. 10.1016/j.na.2006.12.031Search in Google Scholar

[14] P. Nápoli, and M. C. Mariani, Mountain pass solutions to equations of p-Laplacian type, Nonlinear Anal. 54, (2003), no. 7, 1205–1219, DOI: https://doi.org/10.1016/S0362-546X(03)00105-6. 10.1016/S0362-546X(03)00105-6Search in Google Scholar

[15] G. Bonanno, A critical points theorem and nonlinear differential problems, J. Global Optim. 28 (2004), 249–258, DOI: https://doi.org/10.1023/B:JOGO.0000026447.51988.f6. 10.1023/B:JOGO.0000026447.51988.f6Search in Google Scholar

[16] S. Heidarkhani, M. Ferrara, G. Caristi, and A. Salari, Multiplicity results for Kirchhoff-type three-point boundary value problems, Acta Appl. Math. 156 (2018), 133–157, DOI: https://doi.org/10.1007/s10440-018-0157-2. 10.1007/s10440-018-0157-2Search in Google Scholar

[17] S. Heidarkhani, M. Ferrara, and A. Salari, Infinitely many periodic solutions for a class of perturbed second-order differential equations with impulses, Acta Appl. Math. 139 (2015), no. 1, 81–94, DOI: https://doi.org/10.1007/s10440-014-9970-4. 10.1007/s10440-014-9970-4Search in Google Scholar

[18] S. Heidarkhani and G. A. Afrouzi, Some multiplicity results to the existence of three solutions for a Dirichlet boundary value problem involving the p-Laplacian, Math. Model. Anal. 16 (2011), no. 3, 390–400, DOI: https://doi.org/10.3846/13926292.2011.601770. 10.3846/13926292.2011.601770Search in Google Scholar

[19] L. Jiang and Z. Zhou, Three solutions to Dirichlet boundary value problems for-Laplacian difference equations, Adv. Differential Equations 2008 (2007), 1–10, DOI: https://doi.org/10.1155/2008/345916. 10.1155/2008/345916Search in Google Scholar

[20] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353–372, DOI: https://doi.org/10.1007/BF02418013. 10.1007/BF02418013Search in Google Scholar

[21] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II/B, Springer, Berlin, Heidelberg, New York, 1985. 10.1007/978-1-4612-5020-3Search in Google Scholar

Received: 2024-11-19

Revised: 2025-03-17

Accepted: 2025-04-29

Published Online: 2025-05-22

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

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https://doi.org/10.1515/dema-2025-0135

Keywords for this article

three solutions; p-Laplacian type; critical point; variational methods

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