ON THE EXISTENCE OF POSITIVE SOLUTION FOR AN ELLIPTIC EQUATION OF KIRCHHOFF TYPE VIA MOSER ITERATION METHOD FRANCISCO J ´ ULIO S. A. CORR ˆ EA AND GIOVANY M. FIGUEIREDO Received 18 November 2005; Revised 11 April 2006; Accepted 18 April 2006 Dedicated to our dear friend and collaborator Professor Claudianor O. Alves We investigate the questions of existence of positive solution for the nonlocal problem −M(u 2 )Δu = f (λ,u)inΩ and u = 0on∂Ω,whereΩ is a bounded smooth domain of R N ,andM and f are continuous functions. Copyright © 2006 F. J. S. A. Corr ˆ ea and G. M. Figueiredo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited. 1. Introduction In this paper, we study some questions related to the existence of positive solution for the nonlocal elliptic problem − M u 2 Δu = f (λ, u)inΩ, u = 0on∂Ω, (P) λ where Ω is a bounded smooth domain, M : R + → R is a function w hose behavior will be stated later, f : R + × R → R is a given nonlinear function, and ·is the usual norm in H 1 0 (Ω)givenby u 2 = |∇ u| 2 (1.1) and finally, through this work, u denotes the integral Ω u(x)dx. The main goal of this paper is to establish conditions on M and f under which prob- lem (P) λ possesses a positive solution. Problem (P) λ is called nonlocal because of the presence of the term M(u 2 ) which implies that the equation i n (P) λ is no longer a pointwise identity. This provokes some mathematical difficulties which make the study of such a problem particulary interesting. Hindawi Publishing Corporation Boundary Value Problems Volume 2006, Article ID 79679, Pages 1–10 DOI 10.1155/BVP/2006/79679 2 A Kirchhoff-type equation Besides, these kinds of problems have motivations in physics. Indeed, the operator M( u 2 )Δu appears in the Kirchhoff equation, by virtue of this ( P) λ ,iscalledofthe Kirchhoff type, which arises in nonlinear vibrations, namely, u tt − M u 2 Δu = f (x,u)inΩ × (0,T), u = 0on∂Ω × (0,T), u(x,0) = u 0 (x), u t (x,0) = u 1 (x) . (1.2) Hence, problem (P) λ is the stationary counterpart of the above evolution equation. Such a hyperbolic equation is a general version of the Kirchhoff equation ρ ∂ 2 u ∂t 2 − P 0 h + E 2L L 0 ∂u ∂x 2 dx ∂ 2 u ∂x 2 = 0 (1.3) presented by Kirchhoff [14]. This equation extends the classical d’Alembert’s wave equa- tion by considering the effects of the changes in the length of the strings during the vibra- tions. The parameters in (1.3) have the following meanings: L is the length of the string, h is the area of cross-section, E is the Young modulus of the material, ρ is the mass density and P 0 is the initial tension. Problem ( 1.2) began to call the attention of several researchers mainly after the work of Lions [15], where a functional analysis approach was proposed to attack it. The reader may consult [1, 2, 8, 16, 18] and the references therein, for more informa- tion on (P) λ . Actually, problem (P) λ is a particular example of a wide class of the so-called nonlocal equations whose study has deserved the attention of many researchers, mainly in recent years. Let us cite some nonlocal problems in order to emphasize the importance of their studies. First, we consider the problem −a | u| q dx Δu = H(x) f (u)inΩ, u = 0on∂Ω, (1.4) where a : R + → R + is a given function, which does not have variational structure. Such a problem appears in some physical situations related, for example, with biology in which u sometimes describes the population of bacteria, in case q = 1. In case q = 2, we get the well-known Carrier equation which is an appropriate model to study some ques- tions related to nonlinear deflections of beams. See [4–7, 10] and the references therein, for more details related to problem (1.4). Another relevant nonlocal problem is −Δu = a(x,u)u q p in Ω, u = 0on∂Ω, (1.5) F. J. S. A. Corr ˆ ea and G. M. Figueiredo 3 where a : ¯ Ω × R → R + is a known function and · q is the usual L q -norm, and its related system −Δu m =v α p in Ω, −Δv n =u β q in Ω, u = v = 0on∂Ω (1.6) comes from a parabolic phenomenon. Such problems arise in the study of the flow of a fluid through a homogeneous isotropic rigid porous medium or in studies of popula- tion dynamics. It has been suggested that nonlocal growth terms present a more realistic model of population. See [9, 11, 12, 20] and references therein. To close this series of examples, we cite the problem Δu = f (u) α f (u) β in Ω, u = 0on∂Ω, (1.7) which arises in numerous physical models such as: systems of particles in thermodynam- ical equilibrium via gravitational (Coulomb) potential, 2-D fully turbulent behavior of real flow, thermal runaway in ohmic heating , shear bounds in metal deformed under high strain rates, among others. References to these applications may be found in [21]. After these motivations, let us go back to our original problem (P) λ .Weimposethe following conditions on M and f : M is a continuous function and satisfies M(t) ≥ m 0 > 0 ∀t ≥ 0, (M 1 ) M(k) < μm 0 2 for some 2 <μ<p,foranyk>0, (M 2 ) max M(k) (2−p+q)/(p−2) ,M(k) 2/p−2 ≤ k θ (M 3 ) for any k>0, for some q ≤ p,2<p<2 ∗ ,andθ>0, where 2 ∗ = 2N/(N − 2) if N ≥ 3and 2 ∗ =∞if N = 2. We also suppose that f is a continuous function and satisfies f (λ,t) −|t| p−2 t λ : = g(t)withg(t) ≥ 0. ( f 1 ) Note that by ( f 1 ), f (λ,t) ≥ 0, for all λ>0 and assume that for all t ≥ 0, lim t→0 + g(t) t = 0. (g 1 ) Moreover, we require that there exists 2 <μ<psuch that 0 <μG(t) = t 0 g(s)ds ≤ g(t)t ∀t>0. (g 2 ) Our main result is as follows. 4 A Kirchhoff-type equation Theorem 1.1. Let us suppose that the function M satisfies (M 1 ), (M 2 ), and (M 3 ), f satisfies ( f 1 ), and g satisfies (g 1 )and(g 2 ). Then there exists λ 0 > 0 such that problem (P) λ possesses a positive solution for each λ ∈ [0,λ 0 ]. We point out that the function g(t) =|t| s−2 t with s ≥ 2 ∗ satisfies assumptions (g 1 )and (g 2 ). In the present paper, we continue the study from [2], because we consider supercriti- cal nonlinearities. In [2], the authors only consider nonlinearities with subcritical growth and so they are able to use a combination of the mountain pass theorem and an appro- priate truncation of the function M to attack problem (P) λ . In order to solve problem (P) λ , we first consider a truncated problem which involves only a subcritical Sobolev exponent. We show that positive solution of truncated problem is a positive solution of (P) λ . In Sections 2 and 3, we study the truncated problem and in Section 4,weprovean existence result for problem (P) λ . 2. The truncated problem Firstofall,wehavetonotethatbecause f has a supercritical growth, we cannot use directly the variational techniques, due to the lack of compactness of the Sobolev immer- sions. So we construct a suitable truncation of f in order to use variational methods or, more precisely, the mountain pass theorem. This truncation was used in the paper [19] (see [3, 13]). Let K>0 be a real number, whose precise value will be fixed later, and consider the function g K : R → R given by g K (t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0ift<0, g(t)if0 ≤ t ≤ K, g(K) K p−1 t p−1 if t ≥ K. (2.1) We also study the associated truncated problem −M u 2 Δu = f K (u)inΩ, u = 0on∂Ω, (T) λ where f K (t) = (t + ) p−1 + λg K (t). Such a function enjoys the following conditions: f K (t) = o(t)(ast −→ 0), ( f K,1 ) 0 <μ F K (u) ≤ f K (u)u ∀u ∈ H 1 0 (Ω), u>0, ( f K,2 ) where μ>2andF K (t) = t 0 f K (s)ds; lim t→∞ f K (t) t p−1 = 1+λ g(K) K p−1 . ( f K,3 ) F. J. S. A. Corr ˆ ea and G. M. Figueiredo 5 3. Existence of solution for the truncated problem First, we note that f K (t) ≤ C 1 |t| q−1 + C 2 |t| p−1 ,(f K,4 ) where C 1 ≥ 0, C 2 > 0, and for all q ≥ 1. This is an immediate consequence of the definition of f K . Hence, by using ( f K,3 ), ( f K,4 ), and (M 1 ), we conclude from [2, Lemma 2] that there exists θ>0suchthat u λ 2 ≤ max M u λ (2−p+q)/(p−2) ,M u λ 2 2/p−2 θ (3.1) for all classical solutions u λ of (T) λ . We now use ( f K,1 ), ( f K,2 ), ( f K,3 ), (M 1 ), (M 2 )(withμ>2 obtained from condition ( f K,2 )) and (M 3 )(withθ>0obtainedin(3.1)) to obtain, thanks to [2,Theorem5],a positive solution u λ of T 0 such that I λ (u λ ) = c λ ,wherec λ is the mountain pass level asso- ciated to the functional I λ u λ = 1 2 M u λ 2 − 1 p F K u λ (3.2) which is related to the problem T 0 ,where M(t) = t 0 M(s)ds. Furthermore, I λ u λ − 1 μ I λ u λ u λ ≥ m 0 2 − M u λ 2 μ u λ 2 + 1 μ f K u λ u λ − F K u λ ≥ m 0 2 u λ 2 + 1 μ f K u λ u λ − F K u λ . (3.3) 4. Proof of Theorem 1.1 In the proof of Theorem 1.1, we need the following estimate. Lemma 4.1. If u λ is a solution (positive) of problem T 0 , then u λ ≤C for all λ ≥ 0,where C>0 is a constant that does not depend on λ. Proof. Since F k (t) ≥ t p + /p,onehasc λ ≤ c 0 ,wherec 0 is the mountain pass level related to the functional I 0 (u) = 1 2 M u 2 − 1 p | u| p (4.1) which is associated to the problem −M u 2 Δu =|u| p−2 u in Ω, u = 0on∂Ω. (T 0 ) 6 A Kirchhoff-type equation Furthermore, c 0 ≥ c λ = I λ u λ = I λ u λ − 1 μ I λ u λ u λ (4.2) and from (3.3), c 0 ≥ m 0 2 u λ 2 + 1 μ f K u λ u λ − F K u λ . (4.3) From ( f K,2 ), we get u λ ≤ 2c 0 m 0 := C (4.4) for all λ ≥ 0. Next, we are going to use the Moser iteration method [17](see [3, 13]). Proof of Theorem 1.1. Let u λ be a solution of problem T 0 . We will show that there is K 0 such that for all K>K 0 , there exists a corresponding λ 0 for which u λ L ∞ (Ω) ≤ K ∀λ ∈ 0,λ 0 . (4.5) If this is the case, one has f K (u λ ) = u p−1 λ + λg(u λ )andsou λ is a solution of problem (P) λ for all λ ∈ [0, λ 0 ]. For the sake of simplicity, we will use the following notation: u λ := u. (4.6) For L>0, let us define the following functions: u L = ⎧ ⎨ ⎩ u if u ≤ L, L if u>L, z L = u 2(β−1) L u, w L = uu β−1 L , (4.7) where β>1 will be fixed later. Let us use z L as a test function, that is, M u 2 ∇ u∇z L = f K (u)z L (4.8) which implies M u 2 u 2(β−1) L |∇u| 2 =−2(β − 1) u 2β−3 L u∇u∇u L + f K (u)uu 2(β−1) L . (4.9) Because of the definition of u L ,wehave 2(β − 1) u 2β−3 L u∇u∇u L = 2(β − 1) {u≤L} u 2(β−1) |∇u| 2 ≥ 0 (4.10) F. J. S. A. Corr ˆ ea and G. M. Figueiredo 7 and using the fact f K (u) ≤ 1+λ g(u) K p−1 | u| p−1 (4.11) together with (M 1 ) u 2(β−1) L |∇u| 2 ≤ 1+λ g(K) K p−1 1 m 0 u p u 2(β−1) L , (4.12) we obtain u 2(β−1) L |∇u| 2 ≤ C λ,K u p u 2(β−1) L , (4.13) where C λ,K = (1+λ(g(u)/K p−1 ))(1/m 0 ). On the other hand, from the continuous Sobolev immersion, one gets w L 2 2 ∗ ≤ C 1 ∇ w L 2 = C 1 ∇ uu β−1 L 2 . (4.14) Consequently, w L 2 2 ∗ ≤ C 1 u 2(β−1) L |∇u| 2 + C 1 (β − 1) 2 u 2(β−2) L u 2 ∇ u L 2 (4.15) which gives w L 2 2 ∗ ≤ C 2 β 2 u 2(β−1) L |∇u| 2 . (4.16) From (4.13)and(4.16), we get w L 2 2 ∗ ≤ C 2 β 2 C λ,K u p u 2(β−1) L (4.17) and hence, w L 2 2 ∗ ≤ C 2 β 2 C λ,K u p−2 uu β−1 L 2 = C 2 β 2 C λ,K u p−2 w 2 L . (4.18) We now use H ¨ older i nequality, with exponents 2 ∗ /[p − 2] and 2 ∗ /[2 ∗ − (p − 2)], to ob- tain w L 2 2 ∗ ≤ C 2 β 2 C λ,K u 2 ∗ (p−2)/2 ∗ w 2.2 ∗ /[2 ∗ −(p−2)] L [2 ∗ −(p−2)]/2 ∗ , (4.19) where 2 < 2.2 ∗ /(2 ∗ − (p − 2)) < 2 ∗ . Considering the continuous Sobole v immersion H 1 0 (Ω) L q (Ω), 1 ≤ q ≤ 2 ∗ ,weobtain w L 2 2 ∗ ≤ C 2 β 2 C λ,K u p−2 w L 2 α ∗ , (4.20) 8 A Kirchhoff-type equation where α ∗ = 2.2 ∗ /(2 ∗ − (p − 2)). Using Lemma 4.1,weget w L 2 2 ∗ ≤ C 3 β 2 C λ,K C p−2 w L 2 α ∗ . (4.21) Since w L = uu β−1 L ≤ u β and supposing that u β ∈ L α ∗ (Ω), we have from (4.21)that uu β−1 L 2 ∗ 2/2 ∗ ≤ C 4 β 2 C λ,K u βα ∗ 2/α ∗ < +∞. (4.22) We now apply Fatou’s lemma with respect to the variable L to obtain |u| 2β β ·2 ∗ ≤ C 4 C λ,K β 2 |u| 2β βα ∗ (4.23) so |u| β.2 ∗ ≤ C 4 C λ,K 1/β2 β 1/β |u| βα ∗ . (4.24) Furthermore, by considering χ = 2 ∗ /α ∗ ,wehave2 ∗ = χα ∗ and βχα ∗ = 2 ∗ · β for all β>1 verifying u β ∈ L α ∗ (Ω). Let us consider two cases. Case 1. First, we consider β = 2 ∗ /α ∗ and note that u β ∈ L α ∗ (Ω). (4.25) Hence, from the Sobolev immersions, Lemma 4.1, and inequality (4.24), we get |u| (2 ∗ ) 2 /α ∗ ≤ C 4 C λ,K 1/2β β 1/β CC 5 , (4.26) so |u| χ 2 α ∗ ≤ C 6 C λ,K 1/χ2 χ 1/χ . (4.27) Case 2. We now consider β = (2 ∗ /α ∗ ) 2 and note again that u β ∈ L α ∗ (Ω). (4.28) From inequality (4.24), we obtain |u| (2 ∗ ) 3 /(α ∗ ) 2 ≤ C 6 C λ,K 1/β2 β 1/β |u| (2 ∗ ) 2/α ∗ , (4.29) which implies |u| χ 3 α ∗ ≤ C 6 C λ,K 1/χ 2 χ 2 1/χ 2 |u| χ 2 α ∗ (4.30) or |u| χ 3 α ∗ ≤ C 7 C λ,K 1/χ 2 +1/χ 2 χ 2 2/χ 2 +1/χ . (4.31) F. J. S. A. Corr ˆ ea and G. M. Figueiredo 9 An iterative process leads to |u| χ (m+1) α ∗ ≤ C 8 C λ,K m i =1 χ 2(−i) χ 2 m i =1 iχ −i . (4.32) Taking limit as m →∞,weobtain |u| L ∞ (Ω) ≤ C 8 C λ,K σ 1 χ σ 2 , (4.33) where σ 1 = ∞ i=1 χ 2(−i) and σ 2 = 2 ∞ i=1 iχ −i . In order to choose λ 0 , we consider the inequality C 8 C σ 1 λ,K χ σ 2 = C 8 1+λ g(K) K p−1 1 m 0 σ 1 χ σ 2 ≤ K, (4.34) from which 1+ λg(K) K p−1 σ 1 ≤ Km σ 1 0 χ σ 2 C 8 . (4.35) Choosing λ 0 , verifying the inequality λ 0 ≤ K 1/σ 1 m 0 C 9 − 1 K p−1 g(K) , (4.36) and fixing K such that K 1/σ 1 m 0 C 9 − 1 > 0, (4.37) we obtain u λ L ∞ (Ω) ≤ K ∀λ ∈ 0,λ 0 , (4.38) which concludes the proof. Acknowledgments We would like to thank the two anonymous referees whose suggestions improved this work. The first author was partially supported by Instituto do Mil ˆ enio-AGIMB, Brazil. References [1] C. O. Alves and F. J. S. A. Corr ˆ ea, On existence of solutions for a class of problem involving a nonlinear operator, Communications on Applied Nonlinear Analysis 8 (2001), no. 2, 43–56. [2] C.O.Alves,F.J.S.A.Corr ˆ ea, and T. F. 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Rabinowitz, Variat ional methods for nonlinear elliptic eigenvalue problems, Indiana Univer- sity Mathematics Journal 23 (1974), 729–754. [20] P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source,JournalofDifferential Equations 153 (1999), no. 2, 374–406. [21] R. Sta ´ nczy, Nonlocal elliptic equations, Nonlinear Analysis. Theory, Methods & Applications 47 (2001), no. 5, 3579–3584. Francisco J ´ ulio S. A. Corr ˆ ea: Departamento de Matem ´ atica-CCEN, Universidade Federal do Par ´ a, 66.075-110 Bel ´ em-Par ´ a, Brazil E-mail address: fjulio@ufpa.br Giovany M. Figueiredo: Departamento de Matem ´ atica-CCEN, Universidade Federal do Par ´ a, 66.075-110 Bel ´ em-Par ´ a, Brazil E-mail address: giovany@ufpa.br . ON THE EXISTENCE OF POSITIVE SOLUTION FOR AN ELLIPTIC EQUATION OF KIRCHHOFF TYPE VIA MOSER ITERATION METHOD FRANCISCO J ´ ULIO S. A. CORR ˆ EA AND GIOVANY M. FIGUEIREDO Received. Proof of Theorem 1.1 In the proof of Theorem 1.1, we need the following estimate. Lemma 4.1. If u λ is a solution (positive) of problem T 0 , then u λ ≤C for all λ ≥ 0,where C>0 is a constant. following meanings: L is the length of the string, h is the area of cross-section, E is the Young modulus of the material, ρ is the mass density and P 0 is the initial tension. Problem ( 1.2) began to