EXISTENCE OF SOLUTIONS TO NONLOCAL ELLIPTIC PROBLEM WITH DEPENDENCE - PDF document

EXISTENCE OF SOLUTIONS TO NONLOCAL ELLIPTIC PROBLEM WITH DEPENDENCE ON THE GRADIENT VIA MOUNTAIN-PASS TECHNIQUES VICTOR E.CARRERA BARRANTES, JUAN B. BERNUI BARROS. & BENIGNO GODOY TORRES. Abstract. The main goal of this work is to study the solvability of the non- local elliptic problem ∫ ∣∇ 푢 ∣ 2 )Δ 푢 = 푓 ( 푥, 푢, ∇ 푢 ) − 푀 ( Ω with zero Dirichlet boundary conditions on a bounded and smooth domain of ℝ 푛 ,with 푓 : Ω → ℝ and 푀 : ℝ → ℝ are given functions. 1. Introduction The purpose of this article is to investigate the existence of solutions for the nonlocal elliptic problem ∫ ∣∇ 푢 ∣ 2 )Δ 푢 = 푓 ( 푥, 푢, ∇ 푢 ) − 푀 ( in Ω , (1.1) Ω 푢 = 0 on ∂ Ω , where Ω ⊂ ℝ 푁 , 푁 ≥ 3 is a bounded smooth domain, 푓 : Ω 푥 ℝ 푥 ℝ 푁 → ℝ and 푀 : ℝ → ℝ are given functions. The equation ((1.1))is not variational and when 푀 ( 푡 ) = 1 was studied by several authors(See : )using topological degree,methods of sub and supersolutions,etc. So the well developed critical point theory is of no avail for a direct attack to problem (1.1).In the present work we adapt the technique explored by De Figueiredo et al. [5]:we associate with the problem(1.1)a family of semilinear elliptic problems with no dependence on the gradient of the solution;this new problems are variational and we can apply the mountain-pass techniques,then we use an iterative scheme. As it s well known,problem (1.1) is the stationary counterpart of the hyperbolic Kirchhoff equation [ ] ∫ 퐿 푃 0 ℎ + 퐸 푢 2 휌푢 푡푡 − 푥 푑푥 푢 푥푥 = 0 in (0 , 퐿 ) × (0 , ∞ ) , 2 퐿 0 (1.2) 푢 (0 , 푡 ) = 0 = 푢 ( 퐿, 푡 ) on (0 , 푇 ) , 푢 ( 푥, 0) = 푢 0 ( 푥 ) , 푢 푡 ( 푥, 0) = 푢 1 ( 푥 ) in (0 , 퐿 ) . that appeared at the first time in the work ok Kirchhoff [ ? ],in 1883.The equation in (1.2) is called Kirchhoff equation and it extends the classical D’alembert wave 2000 Mathematics Subject Classification. 35B40, 35L70, 45K05. Key words and phrases. Nonlocal elliptic problems;mountain pass,iteration methods. 1

2 V. E.CARRERA B., J. BERNUI B.,B. GODOY T. equation,by considering the effects of the changes in the length of the strings during the vibrations. The interest of the mathematicians on the so-called nonlocal problems like (1.1),(1.2) ( ∫ ) Ω ∣∇ 푢 ∣ 2 푑푥 (nonlocal because of the presence of the term 푀 ) has increased be- cause they represent a variety of relevant physical situations and requires a non- trivial apparatus to solve them. The paper is organized as follows: In section 2,we will give the existence of solutions for the system ∫ ∣∇ 푤 ∣ 2 )Δ 푢 = 푓 ( 푥, 푢, ∇ 푤 ) − 푀 ( in Ω , (1.3) Ω 푢 = 0 on ∂ Ω , for each 푤 ∈ 퐻 1 0 (Ω) .In section 3 we will study the solution for (1.1) using a iterative scheme and results of section 2. 2. Notations and Preliminaries We will denote by C the general positive constant (the exact value may change from line to line).For convenience,we give the following hypotheses (i)A typical assumption for 푀 ∈ 퐶 1 (0 , + ∞ ) is that there exists 푚 0 > 0 (H.1) such that 푀 ( 푡 ) ≥ 푚 0 for all 푡 ∈ [0 , + ∞ [ (ii) There exists 푚 1 > 푚 0 such that 푀 ( 푡 ) = 푚 1 ∀ 푡 ≥ 푡 0 for some 푡 0 > 0 (H.2) We suppose that 푓 : Ω 푥 ℝ 푥 ℝ 푁 → ℝ is a locally Lipchitz continuous 푓 ( 푥,푡,휉 ) = 0 uniformly for all 푥 ∈ Ω , 휉 ∈ ℝ 푁 (i)lim 푡 → 0 푡 ( ) 1 , 푁 +2 (ii)There exist constants 푎 1 > 0 and 푝 ∈ such that 푁 − 2 ∣ 푓 ( 푥, 푡, 휉 ) ∣ ≤ 푎 1 (1 + ∣ 푡 ∣ 푝 ) ∀ 휉 ∈ ℝ 푁 , 휉 ∈ ℝ 푁 푡 ∈ ℝ , { } 2 , 2 (iii)There exists constant 휃 > max and 푇 > 0 such that 푚 0 ∀ 휉 ∈ ℝ 푁 , 휉 ∈ ℝ 푁 0 < 휃퐹 ( 푥, 푡, 휉 ) ≤ 푡푓 ( 푥, 푡, 휉 ) ∣ 푡 ∣ ≥ 푇 ∈ ℝ , where ∫ 푡 퐹 ( 푥, 푡, 휉 ) = 푓 ( 푥, 푠, 휉 ) 푑푠 0 (iv)There exist constant 푎 2 , 푎 3 > 0 such that 퐹 ( 푥, 푡, 휉 ) ≥ 푎 2 ∣ 푡 ∣ 휃 − 푎 3 휉 ∈ ℝ 푁 for all 푥 ∈ Ω , 푡 ∈ ℝ , Remark 2.1. From ( 푖 ) and ( 푖푖 ) it follows that 휃 ≤ 푝 + 1 (H.3) The function f satisfies ∣ 푓 ( 푥, 푡 ′ , 휉 ) − 푓 ( 푥, 푡 ′′ , 휉 ) ∣ ≤ 퐿 1 ∣ 푡 ′ − 푡 ′′ ∣ ∀ 푥 ∈ Ω , 푡 ′ , 푡 ′′ ∈ [0 , 휌 1 [ , ∣ 휉 ∣ ≤ 휌 2 ( 푖 ) ∣ 푓 ( 푥, 푡, 휉 ′ ) − 푓 ( 푥, 푡, 휉 ′′ ) ∣ ≤ 퐿 2 ∣ 휉 ′ − 휉 ′′ ∣ ∀ 푥 ∈ Ω , 푡 ∈ [0 , 휌 1 [ , ∣ 휉 ′ ∣ , ∣ 휉 ′′ ∣ ≤ 휌 2 ( 푖푖 ) where 휌 1 and 휌 2 depend explicitly on 푝, 푁, 휃, 푎 1 , 푎 2 , 푎 3 given in the previous hypotheses.

EXISTENCE OF SOLUTIONS TO NONLOCAL PROBLEM 3 We recall that by a solution of (1.1) we mean a weak solution, that is, a function 푢 ∈ 퐻 1 0 (Ω) such that ∫ ∫ 푓 ( 푥, 푢, ∇ 푢 ) for all 휑 ∈ 퐻 1 ∇ 푢 ⋅ ∇ 휑푑푥 = Ω ∣∇ 푢 ∣ 2 ) 휑푑푥, 0 (Ω) . ∫ 푀 ( Ω Ω Now, we are in position to establish our main result. Theorem 1. Assume hypotheses (H.1)-(H.2) hold.Then ,there exists positive con- stants 푐 1 , 푐 2 such that for each 푤 ∈ 퐻 1 0 (Ω) then problem (1.3) has one solution 푢 푤 such that 푐 1 ≤ ∥ 푢 푤 ∥ ≤ 푐 2 (2.1) (∫ Ω ∣∇ 푢 ∣ 2 ) 1 / 2 .Moreover ,under the above hypotheses,problem (1.3) where ∥ 푢 ∥ = has a positive and negative solution. Remark 2.2. It is well known ,that if we are looking only positive solutions,we need assumptions (H.2) (iii)-(iv) only for positive t. Theorem 2. Assume (H.1)-(H.3).Then problem (1.1) has a positive and negative solution provided 2 + 퐿 2 휆 − 1 / 2 4 푀 2 푐 2 1 < 1 푚 0 − 퐿 1 휆 − 1 1 where 휆 1 is the first eigenvalue of − Δ and 푀 2 = max {∣ 푀 ′ ( 푟 ) ∣ ; 0 ≤ 푟 ≤ 푐 2 2 } .More- over the solutions obtained are of the class 퐶 2 . 3. Proof of theorem 1. The weak solutions of (1.3) are precisely the critical points of the functional ∫ 퐼 푤 ( 푢 ) = 1 2 ∥ 푢 ∥ 2 − 퐻 ( 푥, 푢, ∇ 푤 ) 푑푥 (3.1) Ω 퐹 ( 푥,푢, ∇ 푤 ) where 퐻 ( 푥, 푢, ∇ 푤 ) = 푀 ( ∫ Ω ∣∇ 푤 ∣ 2 ) We will prove ,by steps,that 퐼 푤 has the geometry of the mountain pass theo- rem,and finally that the obtained solutions have the uniform bounds stated in the theorem. Step1. Let 푤 ∈ 퐻 1 0 (Ω) .Then there exists positive numbers 휌, 훼 > 0 which are independent of 푤 such that ∀ 푢 ∈ 퐻 1 퐼 푤 ( 푢 ) ≥ 훼 0 (Ω) : ∥ 푢 ∥ = 휌 (3.2) Proof. By (H.2)(i),given any 휖 > 0there exists 훿 > 0 such that ∣ 퐻 ( 푥, 푡, 휁 ) ∣ < 휖푡 2 ∀ ∣ 푡 ∣ ≤ 훿 2 푚 0 and,by (H.2)(ii) ,there exists 퐾 = 퐾 훿 > 0 such that ∣ 퐻 ( 푥, 푡, 휁 ) ∣ < 퐾 ∣ 푡 ∣ 푝 +1 ∀ ∣ 푡 ∣ ≥ 훿 So ,using Sobolev Embedding Theorem ,we get ( 1 ) 휖 ∥ 푢 ∥ 2 − 푘 휖 ∥ 푢 ∥ 푝 +1 퐼 푤 ( 푢 ) ≥ 2 − 푚 0 휆 1 with 푘 휖 a constant independent of 푤 .Since 푝 > 1 ,the thesis easily follows. □

4 V. E.CARRERA B., J. BERNUI B.,B. GODOY T. Step2. Let 푤 ∈ 퐻 1 0 (Ω).Fix 푣 0 ∈ 퐻 1 0 (Ω) with ∥ 푣 0 ∥ = 1 .Then there is a 푇 > 0,independent of w,such that 퐼 푤 ( 푡푣 0 ) ≤ 0 for all 푡 ≥ 푇 (3.3) Proof. First ,we observe that ,from (H.1)(ii) and (H.2)(iii) ∣ 퐻 ( 푥, 푡, 휁 ) ∣ ≥ 퐶 1 ∣ 푡 ∣ 휃 − 퐶 2 , for all 푡 > 푡 0 (3.4) So,it follows from (3.4) that ∫ 퐼 푤 ( 푡푣 0 ) =1 2 푡 2 ∥ 푣 0 ∥ 2 ) − 퐻 ( 푥, 푡푣 0 , ∇ 푤 ) 푑푥 Ω ≤ 1 2 푡 2 − 푐 ∣ 푡 ∣ 휃 + 퐶 → −∞ as 푡 → + ∞ due to 휃 > 2 .So,we obtain independent of 푣 0 and also 푤 that (3.3) holds. □ Step3. Let { 푢 푛 } be a Palais-Smale sequence in 퐻 1 0 (Ω) that is 퐼 푤 ( 푢 푛 ) → 푐 and 퐼 ′ 푤 ( 푢 푛 ) → 0.Then 푐 + ∥ 푢 푛 ∥ ≥ 퐼 푤 ( 푢 푛 ) − 1 휃 ⟨ 퐼 ′ 푤 ( 푢 푛 ) 푢 푛 ⟩ ( 1 ) ∫ ( ) 2 − 1 퐻 ( 푥, 푢 푛 , ∇ 푤 ) − 1 푓 ( 푥, 푢 푛 , ∇ 푤 ) ∥ 푢 푛 ∥ 2 − = Ω ∣∇ 푤 ∣ 2 ) 푢 푛 푑푥 ∫ 푀 ( 휃 휃 Ω � �� � 퐿 (3.5) Here, we claim that L is bounded.Indeed, we consider Ω 푛 = { 푥 ∈ Ω : ∥ 푢 푛 ( 푥 ) ∥ > 푇 } with 푇 given in (H.2)(iii), then ∫ ( ) 퐻 ( 푥, 푢 푛 , ∇ 푤 ) − 1 푓 ( 푥, 푢 푛 , ∇ 푤 ) 퐿 = Ω ∣∇ 푤 ∣ 2 ) 푢 푛 푑푥 + ∫ 휃 푀 ( Ω 푛 � �� � 퐿 1 (3.6) ( ) ∫ 퐻 ( 푥, 푢 푛 , ∇ 푤 ) − 1 푓 ( 푥, 푢 푛 , ∇ 푤 ) Ω ∣∇ 푤 ∣ 2 ) 푢 푛 푑푥 ∫ 휃 푀 ( Ω ∖ Ω 푛 � �� � 퐿 2 But 퐿 1 ≤ 0 and [ ] 휃 ( ∣ 푇 ∣ + ∣ 푇 ∣ 푝 +1 ∣ 퐿 2 ∣ ≤ 푎 1 ∣ Ω ∣ (1 + ∣ 푇 ∣ 푝 ) + 1 푝 + 1 ) = 퐾 푚 0 Hence 퐿 ≤ 퐾 .So { 푢 푛 } is bounded in 퐻 1 0 (Ω),and it admits a weakly convergence sub- sequence.From the Rellich-Kondrachov Theorem and results of weak convergence ,standard argument shows that { 푢 푛 } admits a strongly convergence subsequence. Step4. From Steps 1-3 and 퐼 푤 (0) = 0 ,I satisfies the conditions of the mountain