ASYMPTOTIC ANALYSIS OF A NONLINEAR PARTIAL DIFFERENTIAL EQUATION IN A SEMICYLINDER. U + q(u)u = H in C + (1.1) = 0 on Ω (0, ). (1.

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1 Proceedings of Equadiff-11 25, pp ISBN ASYMPTOTIC ANALYSIS OF A NONLINEAR PARTIAL DIFFERENTIAL EQUATION IN A SEMICYLINDER PETER RAND Absrac. Small soluions of a nonlinear parial differenial equaion in a semi-infinie cylinder will be sudied. We consider he asympoic behaviour of hese soluions a infiniy under Neumann boundary condiion as well as Dirichle boundary condiion. In he Neumann case i can be shown ha any soluion small enough eiher vanishes a infiniy or ends o a nonzero periodic soluion of a nonlinear ordinary differenial equaion. In he Dirichle case every soluion small enough vanishes. 1. Inroducion. Le Ω be a bounded domain in R n 1 wih C 2 -boundary. We define he semi-infinie cylinder C + = {x = (x, x n : x Ω, x n > }. In Secion 2 we consider bounded soluions of he equaion under he boundary condiion U + q(uu = H in C + (1.1 U ν = on Ω (,. (1.2 The subjec is o describe he asympoic behaviour as x n of soluions U of problem (1.1, (1.2 subjec o U(x Λ for x C +, (1.3 where Λ is a posiive consan. We assume ha q(u > if u. Moreover, q is coninuous s, Λ q(ss q( C Λ s (1.4 wih C Λ < λ 1. Here, λ 1 is he firs posiive eigenvalue of he Neumann problem for he operaor n 1 = 2 x 2 k=1 k in Ω. We se C = Ω (, + 1 define L r loc (C +, 1 r, as he space of funcions which belong o L r (C for every. We also suppose H L p loc (C + where (1 + s H L p (C s ds <, (1.5 { p > n/2 if n 4 p = 2 if n = 2, 3. (1.6 Deparmen of Mahemaics, Linköping Universiy, SE Linköping, Sweden, (peran@mai.liu.se 377

2 378 P. R The main resul of Secion 2 is Theorem 2.1, which saes ha one of wo alernaives is valid: 1. U admis he asympoic represenaion U(x = u h (x n + w(x as x n +, where u h is a nonzero periodic soluion of u h + q(u h u h = w as x n. An esimae for he remainder erm w is given. 2. U as x n. An esimae for U is given in he heorem. In Secion 3 we consider soluions U of (1.1 subjec o (1.3 under he Dirichle boundary condiion Now we suppose ha q is coninuous ha U = on Ω (,. (1.7 q(v C Λ if v Λ, where C Λ < λ D. Here, λ D is he firs eigenvalue of he Dirichle problem for in Ω. We assume also ha H Lp (C, wih p as in (1.6, is a bounded funcion of for. We ge Theorem 3.1 which gives an explici bound for U L (C in erms of he funcion H L p (C. This implies in paricular ha U L (C as if he same is valid for H L p (C. If H = q( =, hen he esimae from Theorem 3.1 implies ha U(x, x n C ε e λ D ε x n, (1.8 where ε > is arbirary. In Secion 4 we sudy all bounded soluions of (1.1, (1.7. Here we have some addiional assumpions on Ω q, among oher hings ha Ω is sar-shaped wih respec o he origin. We presen a heorem which saes ha, under hese assumpions, every bounded soluion of (1.1, (1.7 wih H = saisfies (1.8. Some examples of funcions q saisfying he addiional assumpions are given. The work considered in his paper is a brief review of he firs par of he auhor s docoral hesis [12]. Complee proofs of he menioned heorems can be found here. The problem (1.1 under he boundary condiions (1.2 or (1.7 wih q(u = U p 1, p > 1, has been sudied in Kozlov [11]. There i is shown ha he resricion (1.3 is essenial for Theorems One of he goals of his work is o exend some resuls from [11] o he equaion (1.1. The equaion u a u q 1 u = in C +, (1.9 where q > 1, a > wih he boundary condiion (1.2 is considered in Kondraiev [8]. Furhermore, he problem Lu = in C + u ν + a u q 1 u = on Ω (,, where L is an ellipic parial differenial operaor, a > q > 1 are consans, is sudied in Kondraiev [9]. In boh hese cases i is proved ha he soluions of hese problems

3 Equadiff-11. Asympoic analysis of a nonlinear parial differenial equaion in a semicylinder 379 have asympoics of he form u(x, x n = Cx σ n wih σ >. This shows ha he minus sign in (1.9 essenially changes he asympoic behaviour of soluions a infiniy. There is a lo of research on posiive soluions of nonlinear problems in an infinie cylinder oher unbounded domains. We direc he reader o Ble Essén [2], Beresycki [3], Beresycki, Caffarelli Nirenberg [4], Beresycki, Larrouurou Roquejoffre [5], Beresycki Nirenberg [6] Kondraiev [1] where also furher references can be found. Small global soluions of he equaion u + λu + f(u, u x, u y = in a wo-dimensional srip wih homogeneous Dirichle boundary condiions are sudied in Amick, Tol [1] Kirchgässner, Scheurle [7]. 2. The Neumann problem. Assume ha p is subjec o (1.6. We sudy he asympoic behaviour as x n of soluions U W 2,p loc (C + of he problem U + q(uu = H in C + U ν = on Ω (, (2.1 saisfying (1.3. Here denoes he Laplace operaor in R n ν denoes he ouward uni normal o he curved par of C +. We assume ha q is coninuous posiive for u saisfies (1.4. We suppose furher ha H L p loc (C + is subjec o (1.5. We have he following heorem concerning he asympoic behaviour of soluions of (2.1 subjec o (1.3: Theorem 2.1. Suppose ha U W 2,p loc (C +, where p saisfies (1.6, is a soluion of (2.1 subjec o (1.3. Suppose also ha q is coninuous, q(u > if u ha he Lipschiz condiion (1.4 is fulfilled. Finally, assume ha H L p loc (C + saisfies (1.5. Then one of he following alernaives is valid: 1. U(x = u h (x n + w(x, where u h is a nonzero periodic soluion of w L (C C + u h + q(u h u h = s H L p (C s ds e λ 1 C Λ( s H Lp (C s ds + e λ 1 C Λ for 1. The righ-h side ends o as. 2. U(, x n L (Ω as x n. Furhermore, U(x = u (x n + w(x, where u( (u ( 2 + q(vv dv 2 C H Lp (C s ds + for 1. e λ 1 C Λ( s H Lp (C s ds + e λ 1 C Λ w L (C C e λ 1 C Λ s H L p (C s ds + e λ 1 C Λ

4 38 P. R Since he proof of his heorem is long, i is jus oulined here. We begin by working in he cylinder C = {(x, x n : x Ω x n R} use hen a smooh cu-off funcion in order o rever o he original problem in C +. So le us sudy a soluion u W 2,p loc (C of he problem u + q(uu = h in C u ν = on C (2.2 saisfying sup u(x Λ, (2.3 x C where Λ is he same consan as in (1.3 p is subjec o (1.6. h L p loc (C ha We assume ha (1 + s h Lp (C s ds <. We see immediaely ha is an eigenvalue of he Neumann problem for in Ω wih he consan φ = Ω 1/2 as normalized eigenfuncion in L 2 (Ω. We se u o he orhogonal projecion of u ono he subspace of L 2 (Ω spanned by φ, ha is define v(x by he equaliy u(x n = 1 Ω Ω u(x, x n dx, u(x = u(x n + v(x. Afer a few calculaions we obain he equaion where u (x n + f(u(x n = h(x n + f(u(x n f(u(x n, (2.4 f(u = q(uu, h(x n = 1 h(x, x n dx Ω Ω f(u(x n = 1 f(u(x, x n dx Ω Ω ogeher wih he problem v = f(u f(u + h h in C v ν = on C. (2.5

5 Equadiff-11. Asympoic analysis of a nonlinear parial differenial equaion in a semicylinder 381 We have hus splied he problem (2.2 ino he equaion (2.4 he problem (2.5. I is possible o prove he following wo nonrivial lemmas. Lemma 2.2. Le ξ be a bounded soluion of he equaion where g is subjec o ξ ( + q(ξ(ξ( = g(, 1 sg(s ds <. (2.6 Then one of he wo following alernaives occurs: 1. ξ( = ξ h ( + w(, where ξ h ( is a nonzero periodic soluion of he equaion ξ ( + q(ξ(ξ( = w( + w ( = O sg(s ds as. 2. Boh ξ( ξ ( end o as (ξ ( ξ( q(vv dv = O g(s ds. Lemma 2.3. The funcion v in (2.5 saisfies he esimae where C depends on p, n, Ω, Λ C Λ. v L (C C e λ 1 C Λ s h Lp (C s ds, Afer checking ha he righ-h side of (2.4 saisfies condiion (2.6, we obain he following resul from Lemma 2.2 Lemma 2.3. Lemma 2.4. Le u W 2,p loc (C be a soluion of (2.2 subjec o (2.3. Then eiher 1. u(x = u h (x n + w(x, where u h is a nonzero periodic soluion of u h + q(u h u h = w L (C C s h L p (C s ds + e λ 1 C Λ( s h Lp (C s ds (2.7 or for 1

6 382 P. R 2. u(, x n L (Ω as x n. If u = u + v as before, hen u( (u ( 2 + q(vv dv 2 C h Lp (C s ds + e λ 1 C Λ( s h Lp (C s ds (2.8 v L (C C e λ 1 C Λ s h Lp (C s ds. (2.9 The righ-h sides of (2.7, (2.8 (2.9 end o as. Finally, by seing u(x = η(x n U(x, where η is a smooh cu-off funcion equal o for x n 1 1 for x n 2, we ge Theorem The Dirichle problem. Here we sudy bounded soluions of he Dirichle problem { U + q(uu = H in C+ U = on Ω (,. (3.1 We assume ha sup U(x Λ (3.2 x C + for some posiive consan Λ ha q is coninuous. Le λ D be he firs eigenvalue of he Dirichle problem for in Ω. We suppose ha here exiss a consan C Λ < λ D such ha v Λ q(v C Λ. (3.3 Finally, we assume ha H L p loc (C +, where p saisfies (1.6, ha H Lp (C is a bounded funcion of for. We ge he following heorem concerning he asympoic behaviour of soluions U of (3.1. Theorem 3.1. Assume ha U W 2,p loc (C +, where p saisfies (1.6, is a soluion of (3.1 subjec o (3.2. Assume furher ha q is coninuous ha (3.3 is saisfied. Also, assume ha H L p loc (C + ha H L p (C is a bounded funcion of for. Then U L (C C e λ D C Λ s H Lp (C s ds + e λ D C Λ, (3.4 where C is independen of. In paricular, if H = q( =, hen ( U L (C = O e λ D ε, (3.5 where ε > is arbirary. Remark. If H L p (C as, hen i follows ha he righ-h side of (3.4 ends o as.

7 Equadiff-11. Asympoic analysis of a nonlinear parial differenial equaion in a semicylinder 383 As for he Neumann problem in Secion 2, we sudy he problem corresponding o (3.1 in he infinie cylinder C when proving Theorem 3.1. However, a crucial difference is ha is no an eigenvalue of he Dirichle problem for in Ω. This means ha we do no need o make he specral spliing u = u + v as in Secion 2 bu can immediaely apply a resul similar o Lemma 2.3 in order o obain Theorem 3.1. The equaliy (3.5 is merely a consequence of (3.4. Namely, for T large enough, we consider (3.1 in Ω (T, insead of C The case of a sar-shaped cross-secion. We presen a heorem which saes ha under some special assumpions on Ω q, every bounded soluion of (3.1 wih H = will saisfy (3.5. This is a generalizaion of [11, Theorem 2(iii] where he case q(u = U p 1 is sudied. Theorem 4.1. Suppose ha n 4 Ω is sar-shaped wih respec o he origin has C 2 -boundary. Also assume ha q is coninuous wih q( =, q(u > oherwise, ha n 3 q(uu 2 (n 1 2 u q(vv dv ε q(uu 2 (4.1 for some ε >. Then every bounded soluion of (3.1 wih H = is subjec o (3.5. We do no prove he heorem here bu check insead ha all funcions of he form q(u = f( u u a+δ (4.2 wih a = 4/(n 3, δ > f being a nondecreasing funcion saisfy (4.1. Obviously, he funcion q in (4.2 is even. Therefore also boh sides of he inequaliy (4.1 are even we can assume ha u. We have u by using his inequaliy we obain where n 3 q(uu 2 (n 1 2 q(vv dv f(uua+2+δ a δ ε = u q(vv dv εf(uu a+2+δ, δ(n 3 2(a δ. Hence (4.1 is fulfilled. Here are some examples of funcions saisfying (4.2: (i q(u = u p, p > 4 n 3. (ii q(u = u p e u, p > 4 n 3. (iii q(u = u p (e u 1, p > 7 n n 3. (iv Linear combinaions wih posiive coefficiens of funcions from (i (iii. The imporan poin of Theorem 4.1 is ha he soluion is jus supposed o be bounded, wihou any dependence of λ D.

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