Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions

Biagio Ricceri

doi:10.1515/anona-2023-0104

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Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions

Biagio Ricceri

Published/Copyright: February 6, 2024

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From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

Let Ω ⊂ R n be a smooth bounded domain. In this article, we prove a result of which the following is a by-product: Let q ∈ ] 0 , 1 [ , α ∈ L ∞ ( Ω ) , with α > 0 , and k ∈ N . Then, the problem

− tan ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x Δ u = α ( x ) u q in Ω u > 0 in Ω u = 0 on ∂ Ω ( k − 1 ) π < ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x < ( k − 1 ) π + π 2

has a unique weak solution u ˜ , which is the unique global minimum in H 0 1 ( Ω ) of the functional

u → 1 2 tan ∫ Ω ∣ ∇ u ˜ ( x ) ∣ 2 d x ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x − 1 q + 1 ∫ Ω α ( x ) ∣ u + ( x ) ∣ q + 1 d x ,

where u + = max { 0 , u } .

Keywords: discontinuous Kirchhoff function; existence; uniqueness; localization; minimization property

MSC 2010: 35J15; 35J25; 35J61; 49J35

1 Introduction

First, we stress that we have chosen the above long title just to summarize the main features and novelties of our results.

Throughout the sequel, Ω ⊂ R n is a smooth bounded domain and K is a real-valued function defined in [ 0 , + ∞ [ .

Given a function φ : Ω × R → R , we are interested in the problem

− K ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x Δ u = φ ( x , u ) in Ω u > 0 in Ω u = 0 on ∂ Ω .

We define the weak solution to this problem as any u ∈ H 0 1 ( Ω ) , with u > 0 in Ω , such that, for every v ∈ H 0 1 ( Ω ) , the function φ ( ⋅ , u ( ⋅ ) ) v ( ⋅ ) lies in L 1 ( Ω ) , and one has

K ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x ∫ Ω ∇ u ( x ) ∇ v ( x ) d x − ∫ Ω φ ( x , u ( x ) ) v ( x ) d x = 0 .

This is a Kirchhoff-type problem, with K being the Kirchhoff function. Unquestionably, it is among the most studied nonlinear problems of the last two decades. For a lucid introduction to the subject jointly with the relevant bibliography, we refer to the recent article by Pucci and Rădulescu [5].

Here is a most remarkable corollary of our main result.

Theorem 1.1

Assume that there exists an open interval I ⊆ ] 0 , + ∞ [ such that the restriction of K to I is increasing and K ( I ) = ] 0 , + ∞ [ .

Then, for each q ∈ ] 0 , 1 [ and for each α ∈ L ∞ ( Ω ) , with α > 0 , the problem

− K ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x Δ u = α ( x ) u q in Ω u > 0 in Ω u = 0 on ∂ Ω ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x ∈ I

has a unique weak solution u ˜ , which is the unique global minimum in H 0 1 ( Ω ) of the functional

(1.1) u → 1 2 K ∫ Ω ∣ ∇ u ˜ ( x ) ∣ 2 d x ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x − 1 q + 1 ∫ Ω α ( x ) ∣ u + ( x ) ∣ q + 1 d x .

Moreover, u ˜ satisfies the inequality

K ∫ Ω ∣ ∇ u ˜ ( x ) ∣ 2 d x 2 1 − q ∫ Ω ∣ ∇ u ˜ ( x ) ∣ 2 d x ≤ 2 q + 1 2 ess sup Ω α λ 1 q + 1 1 1 − q ∫ Ω α ( x ) d x ,

where

λ 1 = inf u ∈ H 0 1 ( Ω ) ⧹ { 0 } ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x ∫ Ω ∣ u ( x ) ∣ 2 d x .

The main novelties of Theorem 1.1 are the lack of continuity of K in [ 0 , + ∞ [ , the localization of the solution given by ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x ∈ I and the property that u ˜ minimizes the functional (1.1) (which depends on u ˜ itself).

Actually, as far as we know, the continuity of K in [ 0 , + ∞ [ is an assumption present in each article devoted to this subject, except the one by Ricceri [9]. More precisely, in the article by Ricceri [9], we assumed that, for some r > 0 , K is continuous and increasing in [ 0 , r [ , with lim t → r − ∫ 0 t K ( s ) d s = + ∞ .

So, for instance, given q ∈ ] 0 , 1 [ , α ∈ L ∞ ( Ω ) , with α > 0 , k ∈ N , c > 0 and s ∈ ] 0 , 1 [ , the problems

− log ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x Δ u = α ( x ) u q in Ω u > 0 in Ω u = 0 on ∂ Ω ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x > 1

− tan ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x Δ u = α ( x ) u q in Ω u > 0 in Ω u = 0 on ∂ Ω ( k − 1 ) π < ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x < ( k − 1 ) π + π 2

and

− c − ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x − s Δ u = α ( x ) u q in Ω u > 0 in Ω u = 0 on ∂ Ω ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x < c

cannot be covered by any of the results known up to now. Thanks to Theorem 1.1, each of these problems has a unique weak solution u ˜ , which minimizes the functional (1.1).

But even when I = ] 0 , + ∞ [ , Theorem 1.1 turns out to be new. Actually, the known results for the problem

− K ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x Δ u = u q in Ω u > 0 in Ω u = 0 on ∂ Ω ,

where q ∈ ] 0 , 1 [ , require that K , besides being continuous in [ 0 , + ∞ [ , is non-increasing ([1], Theorem 4; [3], Example 1) or K ( t ) = a t + b , with a , b > 0 ([10], Theorem 1.2; [11], Corollary 1.1) or with a > 0 and b ≥ 0 ([4], Theorem 5.4).

2 Results

If X is a topological space, a function f : X → R is said to be inf-compact (resp. sup-compact) provided that, for each r ∈ R , the set f − 1 ( ] − ∞ , r ] ) (resp. f − 1 ( [ r , + ∞ [ ) ) is compact.

We will obtain our main result via the following abstract theorem.

Theorem 2.1

Let X be a topological space, and let Φ : X → R , with Φ − 1 ( 0 ) ≠ ∅ , and J : X → R be two functions such that, for each λ > 0 , the function λ Φ − J is lower semicontinuous, inf-compact and admits a unique global minimum. Moreover, assume that J has no global maxima in X. Furthermore, let I ⊆ ] 0 , + ∞ [ be an open interval and Ψ : I → R be an increasing function such that Ψ ( I ) = ] 0 , + ∞ [ .

Then, there exists a unique x ˜ ∈ X such that Φ ( x ˜ ) ∈ I and

Ψ ( Φ ( x ˜ ) ) Φ ( x ˜ ) − J ( x ˜ ) = inf x ∈ X ( Ψ ( Φ ( x ˜ ) ) Φ ( x ) − J ( x ) ) .

Proof

Clearly, the function Ψ − 1 is increasing and continuous in ] 0 , + ∞ [ , with lim λ → 0 + Ψ − 1 ( λ ) = inf I . So, setting Ψ − 1 ( 0 ) = inf I , we think of Ψ − 1 as an increasing and continuous function in [ 0 , + ∞ [ . Now, consider the function φ : X × [ 0 , + ∞ [ → R defined by

φ ( x , λ ) = λ Φ ( x ) − J ( x ) − ∫ 0 λ Ψ − 1 ( t ) d t

for all ( x , λ ) ∈ X × [ 0 , + ∞ [ . Of course, for each x ∈ X , the function φ ( x , ⋅ ) is concave, while, for each λ > 0 , the function φ ( ⋅ , λ ) is lower semicontinuous, inf-compact and admits a unique global minimum. Consequently, in view of Theorem 1.1 of [8], we have

sup λ > 0 inf x ∈ X φ ( x , λ ) = inf x ∈ X sup λ > 0 φ ( x , λ ) .

But, by continuity, we have

sup λ ≥ 0 φ ( x , λ ) = sup λ > 0 φ ( x , λ ) ,

and so

inf x ∈ X sup λ ≥ 0 φ ( x , λ ) = inf x ∈ X sup λ > 0 φ ( x , λ ) ≤ sup λ ≥ 0 inf x ∈ X φ ( x , λ ) ,

from which it follows that

(2.1) sup λ ≥ 0 inf x ∈ X φ ( x , λ ) = inf x ∈ X sup λ ≥ 0 φ ( x , λ ) .

Of course, lim λ → + ∞ ∫ 0 λ Ψ − 1 ( t ) d t = + ∞ . By assumption, there is some x 0 ∈ X such that Φ ( x 0 ) = 0 , and so we have lim λ → + ∞ φ ( x 0 , λ ) = − ∞ . Hence, φ ( x 0 , ⋅ ) is sup-compact. Then, from equality (2.1), it follows that there exists ( x ˜ , λ ˜ ) ∈ X × [ 0 , + ∞ [ such that

sup λ ≥ 0 φ ( x ˜ , λ ) = φ ( x ˜ , λ ˜ ) = inf x ∈ X φ ( x , λ ˜ ) .

Consequently,

(2.2) sup λ ≥ 0 λ Φ ( x ˜ ) − ∫ 0 λ Ψ − 1 ( t ) d t = λ ˜ Φ ( x ˜ ) − ∫ 0 λ ˜ Ψ − 1 ( t ) d t

and

(2.3) inf x ∈ X ( λ ˜ Φ ( x ) − J ( x ) ) = λ ˜ Φ ( x ˜ ) − J ( x ˜ ) .

Note that λ ˜ > 0 . Indeed, otherwise, by equality (2.3), x ˜ would be a global maximum of J , against an assumption. On the other hand, by equality (2.2), it follows that Ψ − 1 ( λ ˜ ) = Φ ( x ˜ ) . Hence, Φ ( x ˜ ) ∈ I and λ ˜ = Ψ ( Φ ( x ˜ ) ) , and equality (2.3) provides the conclusion for the existence part. Now, let us prove the uniqueness of x ˜ . So, let y ˜ ∈ X be such that Φ ( y ˜ ) ∈ I and

Ψ ( Φ ( y ˜ ) ) Φ ( y ˜ ) − J ( y ˜ ) = inf x ∈ X ( Ψ ( Φ ( y ˜ ) ) Φ ( x ) − J ( x ) ) .

Arguing by contradiction, assume that y ˜ ≠ x ˜ . We claim that Φ ( y ˜ ) = Φ ( x ˜ ) . Indeed, if, for instance, Φ ( y ˜ ) < Φ ( x ˜ ) , then Ψ ( Φ ( y ˜ ) ) < Ψ ( Φ ( x ˜ ) ) , and so, by Proposition 3.1 of [8], we would have Φ ( y ˜ ) > Φ ( x ˜ ) . So, if we set λ ≔ Ψ ( Φ ( x ˜ ) ) , the points x ˜ and y ˜ would be two distinct global minima of the function λ Φ − J against an assumption. The proof is complete.□

Here is our main result.

Theorem 2.2

Assume that there exists an open interval I ⊆ ] 0 , + ∞ [ such that the restriction of K to I is increasing and K ( I ) = ] 0 , + ∞ [ . Let f : [ 0 , + ∞ [ → [ 0 , + ∞ [ be a continuous function, with f ( 0 ) = 0 , such that the function ξ → f ( ξ ) ξ is decreasing in ] 0 , + ∞ [ and lim ξ → + ∞ f ( ξ ) ξ = 0 , and lim ξ → 0 + f ( ξ ) ξ = + ∞ .

Then, for each α ∈ L ∞ ( Ω ) , with α > 0 , the problem

− K ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x Δ u = α ( x ) f ( u ) in Ω u > 0 in Ω u = 0 on ∂ Ω ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x ∈ I

has a unique weak solution u ˜ , which is the unique global minimum in H 0 1 ( Ω ) of the functional

u → 1 2 K ∫ Ω ∣ ∇ u ˜ ( x ) ∣ 2 d x ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x − ∫ Ω α ( x ) ∫ 0 u + ( x ) f ( t ) d t d x .

Proof

First of all, extend f to R putting f ( ξ ) = 0 for all ξ < 0 . We are going to apply Theorem 2.1, taking X = H 0 1 ( Ω ) endowed with the weak topology, Ψ = K , and defining Φ and J by

Φ ( u ) = ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x ,

J ( u ) = 2 ∫ Ω α ( x ) F ( u + ( x ) ) d x

for all u ∈ H 0 1 ( Ω ) , where F ( ξ ) = ∫ 0 ξ f ( t ) d t . The functionals Φ and J are C 1 with derivatives given by

Φ ′ ( u ) ( v ) = 2 ∫ Ω ∇ u ( x ) ∇ v ( x ) d x

J ′ ( u ) ( v ) = 2 ∫ Ω α ( x ) f ( u ( x ) ) v ( x ) d x

for all u , v ∈ H 0 1 ( Ω ) . Moreover, due to the sub-critical growth of f , J is sequentially weakly continuous. Fix λ > 0 and choose ε ∈ 0 , λ λ 1 2 , where λ 1 is the constant defined in Theorem 1.1. Since lim ξ → + ∞ F ( ξ ) ξ 2 = 0 , there is c ε > 0 such that

F ( ξ ) ≤ ε M ∣ ξ ∣ 2 + c ε

for all ξ ∈ R , where M = ess sup Ω α . Consequently, we have

J ( u ) ≤ 2 ε ∫ Ω ∣ u ( x ) ∣ 2 d x + 2 c ε ∫ Ω α ( x ) d x ,

and so

λ Φ ( u ) − J ( u ) ≥ λ ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x − 2 ε ∫ Ω ∣ u ( x ) ∣ 2 d x − 2 c ε ∫ Ω α ( x ) d x ≥ λ − 2 ε λ 1 ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x − 2 c ε ∫ Ω α ( x ) d x

for all u ∈ H 0 1 ( Ω ) . Hence, due to the choice of ε , we have

lim ‖ u ‖ → + ∞ ( λ Φ ( u ) − J ( u ) ) = + ∞ .

This fact, jointly with the reflexivity of H 0 1 ( Ω ) and the Eberlein-Smulyan theorem, implies that the sequentially weakly lower semicontinuous functional λ Φ − J is weakly inf-compact. We now show that it has a unique global minimum in H 0 1 ( Ω ) . Indeed, its critical points are exactly the weak solutions of the problem

− Δ u = 1 λ α ( x ) f ( u ) in Ω u = 0 on ∂ Ω .

In turn, since the right-hand side of the equation is non-negative, the non-zero weak solutions of the problem are positive in Ω . Moreover, since, for each x ∈ Ω , the function α ( x ) f ( ξ ) λ ξ is decreasing in ] 0 , + ∞ [ , Theorem 1 of [2] ensures that the problem has at most one positive weak solution, and so it has at most one non-zero weak solution. As a consequence, we infer that the functional λ Φ − J has a unique global minimum in H 0 1 ( Ω ) , since otherwise, in view of Corollary 1 of [6], it would have at least three critical points. Now, fix any positive function u ∈ C 2 ( Ω ) ∩ C 0 ( Ω ¯ ) , with u = 0 on ∂ Ω , such that

λ 1 ∫ Ω ∣ u ( x ) ∣ 2 d x = ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x .

Also, fix γ ∈ ] 0 , esssup Ω α [ . Of course, the set

Ω γ ≔ { x ∈ Ω : α ( x ) ≥ γ }

has a positive measure. Furthermore, fix M > 0 so that

M > λ λ 1 ∫ Ω ∣ u ( x ) ∣ 2 d x 2 γ ∫ Ω γ ∣ u ( x ) ∣ 2 d x .

Since lim ξ → 0 + F ( ξ ) ξ 2 = + ∞ , there is δ > 0 such that

F ( ξ ) ≥ M ξ 2

for all ξ ∈ [ 0 , δ ] . Now, set v = μ u , where μ = δ sup Ω ¯ u . Then, we have

J ( v ) ≥ 2 M ∫ Ω α ( x ) ∣ v ( x ) ∣ 2 d x ≥ 2 M γ ∫ Ω γ ∣ v ( x ) ∣ 2 > λ λ 1 ∫ Ω ∣ v ( x ) ∣ 2 d x = λ ∫ Ω ∣ ∇ v ( x ) ∣ 2 d x = λ Φ ( v ) .

This shows that 0 is not a global minimum for the functional λ Φ − J . Consequently, the global minimum of this functional agrees with its only non-zero critical point. Finally, let us show that J has no global maxima. Arguing by contradiction, suppose that u ˆ ∈ H 0 1 ( Ω ) is a global maximum of J . Clearly, J ( u ˆ ) > 0 . Consequently, the set

A ≔ { x ∈ Ω : α ( x ) f ( u ˆ ( x ) ) > 0 }

has a positive measure. Fix a closed set C ⊂ A of positive measure. Let v ∈ H 0 1 ( Ω ) be such that v ≥ 0 and v ( x ) = 1 for all x ∈ C . Then, we have

∫ Ω α ( x ) f ( u ˆ ( x ) ) v ( x ) d x ≥ ∫ C α ( x ) f ( u ˆ ( x ) ) d x > 0 ,

and so J ′ ( u ˆ ) ≠ 0 , which is absurd. Therefore, each assumption of Theorem 2.1 is satisfied. As a consequence, there exists a unique u ˜ ∈ H 0 1 ( Ω ) , with ∫ Ω ∣ ∇ u ˜ ( x ) ∣ 2 d x ∈ I , such that

K ∫ Ω ∣ ∇ u ˜ ( x ) ∣ 2 d x ∫ Ω ∣ ∇ u ˜ ( x ) ∣ 2 d x − 2 ∫ Ω α ( x ) F ( u ˜ ( x ) ) d x = inf u ∈ H 0 1 ( Ω ) K ∫ Ω ∣ ∇ u ˜ ( x ) ∣ 2 d x ∫ Ω ∣ ∇ u ( x ) ∣ 2 d x − 2 ∫ Ω α ( x ) F ( u + ( x ) ) d x .

Clearly, from what seen above, the function u ˜ satisfies the conclusion.□

Remark 2.1

In Theorem 1.1, the inequality satisfied by the solution follows directly from Theorem 1.2 of [7].

Funding information: The author has been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by the Università degli Studi di Catania, PIACERI 2020-2022, Linea di intervento 2, Progetto “MAFANE.”
Conflict of interest: The author declares that they have no conflict of interest.

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Received: 2023-03-20

Accepted: 2023-12-06

Published Online: 2024-02-06

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Keywords for this article

discontinuous Kirchhoff function; existence; uniqueness; localization; minimization property

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