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 −푀( ∫

Ω

∣∇푢∣2)Δ푢 = 푓(푥, 푢, ∇푢) with zero Dirichlet boundary conditions on a bounded and smooth domain of ℝ푛,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 −푀( ∫

Ω

∣∇푢∣2)Δ푢 = 푓(푥, 푢, ∇푢) in Ω, 푢 = 0

• n ∂Ω,

(1.1) where Ω ⊂ ℝ푁,푁 ≥ 3 is a bounded smooth domain, 푓 : Ω푥ℝ푥ℝ푁 → 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 휌푢푡푡 − [ 푃0 ℎ + 퐸 2퐿 ∫ 퐿 푢2

푥푑푥

] 푢푥푥 = 0 in (0, 퐿) × (0, ∞), 푢(0, 푡) = 0 = 푢(퐿, 푡)

• n (0, 푇),

푢(푥, 0) = 푢0(푥), 푢푡(푥, 0) = 푢1(푥) in (0, 퐿). (1.2) 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) (nonlocal because of the presence of the term 푀 ( ∫

Ω ∣∇푢∣2푑푥

) ) 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 −푀( ∫

Ω

∣∇푤∣2)Δ푢 = 푓(푥, 푢, ∇푤) in Ω, 푢 = 0

• n ∂Ω,

(1.3) for each 푤 ∈ 퐻1

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 (H.1) (i)A typical assumption for 푀 ∈ 퐶1(0, +∞) is that there exists 푚0 > 0 such that 푀(푡) ≥ 푚0 for all 푡 ∈ [0, +∞[ (ii) There exists 푚1 > 푚0 such that 푀(푡) = 푚1 ∀푡 ≥ 푡0 for some 푡0 > 0 (H.2) We suppose that 푓 : Ω푥ℝ푥ℝ푁 → ℝ is a locally Lipchitz continuous (i)lim푡→0

푓(푥,푡,휉) 푡

= 0 uniformly for all 푥 ∈ Ω, 휉 ∈ ℝ푁 (ii)There exist constants 푎1 > 0 and 푝 ∈ ( 1, 푁+2

푁−2

) such that ∣푓(푥, 푡, 휉)∣ ≤ 푎1(1 + ∣푡∣푝) ∀휉 ∈ ℝ푁, 푡 ∈ ℝ, 휉 ∈ ℝ푁 (iii)There exists constant 휃 > max { 2, 2 푚0 } and 푇 > 0 such that 0 < 휃퐹(푥, 푡, 휉) ≤ 푡푓(푥, 푡, 휉) ∀휉 ∈ ℝ푁, ∣푡∣ ≥ 푇 ∈ ℝ, 휉 ∈ ℝ푁 where 퐹(푥, 푡, 휉) = ∫ 푡 푓(푥, 푠, 휉)푑푠 (iv)There exist constant 푎2, 푎3 > 0 such that 퐹(푥, 푡, 휉) ≥ 푎2∣푡∣휃 − 푎3 for all 푥 ∈ Ω, 푡 ∈ ℝ, 휉 ∈ ℝ푁 Remark 2.1. From (푖) and (푖푖) it follows that 휃 ≤ 푝 + 1 (H.3) The function f satisfies (푖) ∣푓(푥, 푡′, 휉) − 푓(푥, 푡′′, 휉)∣ ≤ 퐿1∣푡′ − 푡′′∣ ∀푥 ∈ Ω, 푡′, 푡′′ ∈ [0, 휌1[, ∣휉∣ ≤ 휌2 (푖푖) ∣푓(푥, 푡, 휉′) − 푓(푥, 푡, 휉′′)∣ ≤ 퐿2∣휉′ − 휉′′∣ ∀푥 ∈ Ω, 푡 ∈ [0, 휌1[, ∣휉′∣, ∣휉′′∣ ≤ 휌2 where 휌1 and 휌2 depend explicitly on 푝, 푁, 휃, 푎1, 푎2, 푎3 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 ∈ 퐻1

0(Ω) such that

∫

Ω

∇푢 ⋅ ∇휑푑푥 = ∫

Ω

푓(푥, 푢, ∇푢) 푀( ∫

Ω ∣∇푢∣2)휑푑푥,

for all 휑 ∈ 퐻1

0(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 푐1, 푐2 such that for each 푤 ∈ 퐻1

0(Ω) then problem (1.3) has one solution 푢푤

such that 푐1 ≤ ∥푢푤∥ ≤ 푐2 (2.1) where ∥푢∥ = (∫

Ω ∣∇푢∣2)1/2.Moreover ,under the above hypotheses,problem (1.3)

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 4푀2푐2

2 + 퐿2휆−1/2 1

푚0 − 퐿1휆−1

1

< 1 where 휆1 is the first eigenvalue of −Δ and 푀2 = max{∣푀 ′(푟)∣; 0 ≤ 푟 ≤ 푐2

2} .More-

• ver the solutions obtained are of the class 퐶2.
• 3. Proof of theorem 1.

The weak solutions of (1.3) are precisely the critical points of the functional 퐼푤(푢) = 1 2∥푢∥2 − ∫

Ω

퐻(푥, 푢, ∇푤)푑푥 (3.1) where 퐻(푥, 푢, ∇푤) =

퐹 (푥,푢,∇푤) 푀( ∫

Ω ∣∇푤∣2)

We will prove ,by steps,that 퐼푤 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 ∈ 퐻1

0(Ω) .Then there exists positive numbers 휌, 훼 > 0 which are

independent of 푤 such that 퐼푤(푢) ≥ 훼 ∀푢 ∈ 퐻1

0(Ω) : ∥푢∥ = 휌

(3.2)

• Proof. By (H.2)(i),given any 휖 > 0there exists 훿 > 0 such that

∣퐻(푥, 푡, 휁)∣ < 휖푡2 2푚0 ∀ ∣푡∣ ≤ 훿 and,by (H.2)(ii) ,there exists 퐾 = 퐾훿 > 0 such that ∣퐻(푥, 푡, 휁)∣ < 퐾∣푡∣푝+1 ∀ ∣푡∣ ≥ 훿 So ,using Sobolev Embedding Theorem ,we get 퐼푤(푢) ≥ (1 2 − 휖 푚0휆1 ) ∥푢∥2 − 푘휖∥푢∥푝+1 with 푘휖 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 ∈ 퐻1

0(Ω).Fix 푣0 ∈ 퐻1 0(Ω) with ∥푣0∥ = 1 .Then there is a 푇 >

0,independent of w,such that 퐼푤(푡푣0) ≤ 0 for all 푡 ≥ 푇 (3.3)

• Proof. First ,we observe that ,from (H.1)(ii) and (H.2)(iii)

∣퐻(푥, 푡, 휁)∣ ≥ 퐶1∣푡∣휃 − 퐶2, for all 푡 > 푡0 (3.4) So,it follows from (3.4) that 퐼푤(푡푣0) =1 2푡2∥푣0∥2) − ∫

Ω

퐻(푥, 푡푣0, ∇푤)푑푥 ≤ 1 2푡2 − 푐∣푡∣휃 + 퐶 → −∞ as 푡 → +∞ due to 휃 > 2 .So,we obtain independent of 푣0 and also 푤 that (3.3) holds. □

• Step3. Let{푢푛} be a Palais-Smale sequence in 퐻1

0(Ω) that is 퐼푤 (푢푛) → 푐 and

퐼′

푤 (푢푛) → 0.Then

푐 + ∥푢푛∥ ≥퐼푤 (푢푛) − 1 휃⟨퐼′

푤 (푢푛) 푢푛⟩

= (1 2 − 1 휃 ) ∥푢푛∥2 − ∫

Ω

( 퐻(푥, 푢푛, ∇푤) − 1 휃 푓(푥, 푢푛, ∇푤) 푀( ∫

Ω ∣∇푤∣2)푢푛

) 푑푥

• 퐿

(3.5) Here, we claim that L is bounded.Indeed, we consider Ω푛 = {푥 ∈ Ω : ∥푢푛(푥)∥ > 푇} with 푇 given in (H.2)(iii), then 퐿 = ∫

Ω푛

( 퐻(푥, 푢푛, ∇푤) − 1 휃 푓(푥, 푢푛, ∇푤) 푀( ∫

Ω ∣∇푤∣2)푢푛

) 푑푥

• 퐿1

+ ∫

Ω∖Ω푛

( 퐻(푥, 푢푛, ∇푤) − 1 휃 푓(푥, 푢푛, ∇푤) 푀( ∫

Ω ∣∇푤∣2)푢푛

) 푑푥

• 퐿2

(3.6) But 퐿1 ≤ 0 and ∣퐿2∣ ≤ 푎1∣Ω∣ 푚0 [ (1 + ∣푇∣푝) + 1 휃(∣푇∣ + ∣푇∣푝+1 푝 + 1 ) ] = 퐾 Hence 퐿 ≤ 퐾.So {푢푛} is bounded in 퐻1

0(Ω),and it admits a weakly convergence sub-

sequence.From the Rellich-Kondrachov Theorem and results of weak convergence ,standard argument shows that {푢푛} admits a strongly convergence subsequence.

• Step4. From Steps 1-3 and 퐼푤(0) = 0 ,I satisfies the conditions of the mountain

EXISTENCE OF SOLUTIONS TO NONLOCAL PROBLEM 5

pass theorem.So 퐼푤 admits at least one nontrivial critical point 푢푤, at an inf max level,which is a weak solution of (1.3),that is 푎)퐼′

푤(푢푤) = 0

푏)퐼푤(푢푤) = inf

훾∈Γ max 푡∈[0,1] 퐼푤(훾(푡)) ≥ 훼

(3.7) where Γ = {훾 ∈ 퐶0 ( [0, 1], 퐻1

0(Ω)

) : 훾(0) = 0, 훾(1) = 푇푣0} ,for some 푣0 and 푇 as in Step 2. From now on we fix such a 푣0 and such a 푇.

• Step5. Let 푤 ∈ 퐻1

0(Ω).There exists a positive constant 푐1,independent of 푤 such

that ∥푢푤∥ ≥ 푐1 (3.8) for all solution푢푤 obtained in Step 4.

• Proof. From the equation (1.3) one gets

∫

Ω

∣∇푢푤∣2푑푥 = ∫

Ω

푓(푥, 푢푤, ∇푤) 푀( ∫

Ω ∣∇푤∣2)푢푤푑푥

(3.9) By (H.2)(i)-(ii), given 휖 > 0 ,there exists 푐휖 > 0 independent of 푤 ,such that ∣푓(푥, 푡, ∇푤)∣ ≤ 휖∣푡∣ + 푐휖∣푡∣푝 So,we get ∫

Ω

∣∇푢푤∣2푑푥 ≤ 휖 푚0 ∫

Ω

∣푢푤∣2푑푥 + 푐′

휖

∫

Ω

∣푢푤∣푝+1푑푥 Hence we have ( 1 − 휖 휆1푚0 ) ∥푢푤∥ ≤ ˜ 푐휖∥푢푤∥푝+1 which implies (3.8) choosing 휖 < 휆1푚0, since 푝 + 1 > 2 □ Step6.There exists a positive constant 푐2 independent of 푤 such that ∥푢푤∥ ≤ 푐2

• Proof. From the infmax characterization of 푢푤 in Step4 ,choosing the path in Γ as

the segment line joining 0 and 푣0,we obtain 퐼푤(푢푤) ≤ max

푡≥0 퐼푤(푡푣0)

and from (H.3)(iv) we have max

푡≥0 퐼푤(푡푣0) ≤ max 푡≥0

{푡2 2 ∥푣0∥2 − 푎2∣푡∣휃 ∫

Ω

∣푣0∣휃 + 푎3∣Ω∣ } = 푐2 (3.10) Therefore we have obtained that 퐼푤(푢푤) ≤ 푐2 Here ,using the criticality of 푢푤 for 퐼푤,(3.10),(H.2)(iii),one has 1 2∥푢푤∥2 ≤ ˆ 푐2 + 1 휃 ∫

Ω

푓(푥, 푢푤, ∇푤)푢푤 푀( ∫

Ω ∣∇푤∣2)

Therefore (1 2 − 1 푚0휃 ) ∥푢푤∥2 ≤ 푐표푛푠푡.

6

• V. E.CARRERA B., J. BERNUI B.,B. GODOY T.

The positivity of 푢푤 it derives from standard arguments.That is one replaces 푓 by ˆ 푓 defined as ˆ 푓(푥, 푡, 휉) = { 푓(푥, 푡, 휉) si 푡 ≥ 0 si 푡 < 0 Here ,we observe that ˆ 푓 still verifies (H.2)(iii)-(iv)( also we take 푣0 > 0 in step2).So we find a critical point of mountain-pass type for the corresponding functional ˆ 퐼푤 that is solution of the problem −푀( ∫

Ω

∣∇푤∣2)Δ푢푤 = ˆ 푓(푥, 푢푤, ∇푤) in Ω, 푢푤 = 0

• n ∂Ω,

Multiplying the equation by 푢−

푤 and integrating by parts ,we conclude that 푢− 푤 = 0.

Hence 푢−

푤 is positive.

□ Remark 3.1. In Step 4 we have obtained a weak solution 푢푤 of (1.3) for each given 푤 ∈ 퐻1

0(Ω).Since 푝 < 푁 + 2

푁 − 2 a standard bootstrap argument ,using regularity the-

• ry,shows that 푢푤 ∈ 퐶0,훼 for some 훼 ∈ (0, 1).Now,if 푤 have the additional regularity

푤 ∈ 퐶1(¯ Ω),using the the Schauder regularity theory,we show that 푢푤 ∈ 퐶2,훼(Ω).As a consequence of the Sobolev embedding theorems and Step6 we conclude that,there exist positive constants 휌1, 휌2 such that the solution 푢푤satisfies ∥푢푤∥퐶0 ≤ 휌1, ∥∇푢푤∥퐶0 ≤ 휌2

• 4. Proof of theorem 2.

By applying in an iterative way Theorem 1,we construct a sequence {푢푛} ⊂ 퐻1

0(Ω) where푢푛 is a solution of the problem

−푀( ∫

Ω

∣∇푢푛−1∣2)Δ푢푛 = 푓(푥, 푢푛, ∇푢푛−1) in Ω, 푢푛 = 0

• n ∂Ω,

(4.1)

• btained by the mountain pass theorem in theorem 1.We start from as arbitrary

푢0 ∈ 퐻1

0(Ω) ∩ 퐶1(Ω).

By Remark 3.1,we see that ∥푢푛∥퐶0 ≤ 휌1, ∥∇푢푛∥퐶0 ≤ 휌2 Now using (4.1) for 푢푛 we get 푀(∥푢푛∥2)∥푢푛+1 − 푢푛∥2 = [ 푀(∥푢푛−1∥2) − 푀(∥푢푛∥2) ] ∫

Ω

∇푢푛.(∇푢푛+1 − ∇푢푛) + ∫

Ω

[푓(푥, 푢푛+1, ∇푢푛) − 푓(푥, 푢푛, ∇푢푛)](푢푛+1 − 푢푛) + ∫

Ω

[푓(푥, 푢푛, ∇푢푛) − 푓(푥, 푢푛, ∇푢푛−1)](푢푛+1 − 푢푛) hence,from (H.1),(H.3),Cauchy-Schwarz and Poincar´ e inequalities we have 푚0∥푢푛+1 − 푢푛∥2 ≤4푀2푐2

2∥푢푛 − 푢푛−1∥∥푢푛+1 − 푢푛∥

+ 퐿1휆−1

1 ∥푢푛+1 − 푢푛∥2 + 퐿2휆−1/2 1

∥푢푛+1 − 푢푛∥∥푢푛 − 푢푛−1∥

EXISTENCE OF SOLUTIONS TO NONLOCAL PROBLEM 7

Therefore ,we conclude that ∥푢푛+1 − 푢푛∥ ≤4푀2푐2

2 + 퐿2휆−1/2 1

푚0 − 퐿1휆−1

1

∥푢푛 − 푢푛−1∥ =: 푘∥푢푛 − 푢푛−1∥ Since the coefficient 푘 < 1, we have that {푢푛} is a Cauchy sequence in 퐻1

0,and

so,{푢푛} strongly converges in 퐻1

0 to some function 푢 ∈ 퐻1 0.

Since ∥푢푛∥ ≥ 푐1,it follows that 푢 ∕= 0 .Hence we find that 푢 is a nontrivial solution

• f (1.1).By the same argument as in Step6 we have that 푢 > 0 in Ω. □

References

[1] C. O. Alves & F. J. S. A. Corrˆ ea, On existence of solutions for a class of problem involving a nonlinear operator, Communications on applied nonlinear analysis, 8(2001), N. 2, 43-56. [2] C. O. Alves & D. G. de Figueiredo, Nonvariational elliptic systems via Galerkin methods, Function Spaces, Differential Operators and Nonlinear Analysis - The Hans Triebel Anniver- sary Volume, Ed. Birkhauser, Switzerland, 47-57, 2003. [3] A. Ambrosetti, H. Brezis & G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122(1994), 519-543. [4] D.G. de Figueiredo, M. Girardi, M. Matzeu, Semilinear elliptic equations with dependence

• n the gradient via Mountain Pass techniques, Differ. Integ. Eq.,17 (2004) 119-126.

[5] G.M. Figueiredo, Quasilinear equations with dependence on the gradient via Mountain Pass techniques in 푅푁, Applied Mathematics and Computation 203 (2008) 14-18. [6] M. Chipot & J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, Mathemat- ical Modelling and Numerical Analysis, Vol. 26, No. 3, 1992, 447-468. [7] J. L. Lions, Quelques M´ ethodes de r´ esolution des probl´ emes aux limites non lin´ eaires, Dunod, Gauthier-Vill [8] C. O. Alves & F. J. S. A. Corrˆ ea, On existence of solutions for a class of problem involving a nonlinear operator, Communications on applied nonlinear analysis, 8(2001), N. 2, 43-56. [9] C. O. Alves & D. G. de Figueiredo, Nonvariational elliptic systems via Galerkin methods, Function Spaces, Differential Operators and Nonlinear Analysis - The Hans Triebel Anniver- sary Volume, Ed. Birkhauser, Switzerland, 47-57, 2003. [10] A. Ambrosetti, H. Brezis & G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122(1994), 519-543. [11] M. Chipot & B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Analysis, T.M.A., Vol. 30, No. 7,(1997), 4619-4627. [12] M. Chipot & J. F. Rodrigues, On a class of nonlocal nonlinear elliptic problems, Mathemat- ical Modelling and Numerical Analysis, Vol. 26, No. 3, 1992, 447-468. [13] J. G. Eisley, Nonlinear vibrations of beams and rectangular plates, Z. Anger. Math. Phys. 15,(1964)167-175. [14] J. Limaco & L. A. Medeiros, Kirchhoff-Carrier elastic strings in noncylindrical domains, Portugaliae Mathematica, Vol. 14. N. 04,(1999)464-500. [15] J. L. Lions, Quelques M´ ethodes de r´ esolution des probl´ emes aux limites non lin´ eaires, Dunod, Gauthier-Villars, Paris, 1969. Victor E.Carrera Barrantes Instituto de Investigaci´

• n de Matem´

atica, Facultad de Ciencias Matem´ aticas, Univer- sidad Nacional Mayor de San Marcos, Lima, Per´ u E-mail address: vacrrerab@yahoo.com Juan B.Bernui B. Instituto de Investigaci´

• n de Matem´

atica, Facultad de Ciencias Matem´ aticas, Univer- sidad Nacional del Callao, Lima, Per´ u E-mail address: jbernuib@yahoo.com

8

• V. E.CARRERA B., J. BERNUI B.,B. GODOY T.

Benigno Godoy Torres. Instituto de Investigaci´

• n de Matem´

atica, Facultad de Ciencias Matem´ aticas, Univer- sidad Nacional de Mayor de San Marcos , Lima, Per´ u E-mail address: bgodoy@unmsm.edu.pe