On the First Eigensurface for the Third Order Spectrum of p-biharmonic Operator with Weight

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1 Applied Mathematical Sciences, Vol. 8, 2014, no. 89, HIKARI Ltd, On the First Eigensurface for the Third Order Spectrum of p-biharmonic Operator with Weight K. Ben Haddouch, N. Tsouli, El Miloud Hssini University Mohamed I Faculty of Sciences Department of Mathematics computer, Oujda, Morocco Z. El Allali University Mohamed I Faculty Multidisciplinary of Nador Department of Mathematics computer, Morocco Copyright c 2014 K. Ben Haddouch, N. Tsouli, El Miloud Hssini Z. El Allali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited. Abstract In this work, we will study the simplicity the isolation of the first eigensurface for the spectrum of the operator Δ 2 pu+2β. ( Δu p 2 Δu)+ β 2 Δu p 2 Δu, where β IR N under Navier boundary conditions. Mathematics Subject Classification: 35A15, 35J40 Keywords: p-biharmonic operator, Nonlinear spectral theory 1 Introduction We consider the following eigenvalue problem F ind (β,γ,u) IR N IR + X \{0} such that Δ 2 pu +2β. ( Δu p 2 Δu)+ β 2 Δu p 2 Δu =Γm u p 2 u in, u =Δu =0 on, (1)

2 4414 K. Ben Haddouch, N. Tsouli, El Miloud Hssini Z. El Allali where is a bounded smooth domain in IR N (N 1), β IR N,Δ 2 p denotes the p-biharmonic operator defined by Δ 2 p u =Δ( Δu p 2 Δu), X = W 2,p () W 1,p 0 (), m M = {m L ()/meas{x /m(x) > 0} 0}. In the case where β = 0, we obtain the eigenvalue problem F ind (Γ,u) IR + X \{0} such that Δ 2 p u =Γm u p 2 u in, u =Δu =0 on. This problem was considered by P. Drábek M. Ôtani [9] for m = 1, the authors showed that the problem (2) has a principal positive eigenvalue which is simple isolated. In [11], A. El Khalil, S. Kellati A. Touzani, have studied the spectrum of the p-biharmonic operator with weight with Dirichlet boundary conditions, they showed that this spectrum contains at least one non-decreasing sequence of positive eigenvalues. In [18] M. Talbi N. Tsouli considered the spectrum of the weighted p-biharmonic operator with weight showed that the following eigenvalue problem { Δ(ρ Δu p 2 Δu) =λm u p 2 u in, (3) u =Δu =0 on, where ρ C() ρ > 0, contains at least one non-decreasing sequence of eigenvalues studied the one dimensional case. The authors, in the same reference gave the first eigenvalue λ 1 showed that if m 0 a.e, λ 1 is simple associated with positive eigenfunction. Also they showed that if m C(), λ 1 is isolated every positive or negative eigenfunction is associated with λ 1. In [4], the authors have studied the spectrum of the problem (1), in the linear case (p = 2) recently the authors in [5], have showed that the spectrum of problem (1) in the nonlinear case, contains at least one sequence of positive eigensurfaces (Γ p n(., m)) n defined by ( β IR N ) Γ p n(β,m) = inf sup e β.x Δu p dx, K B n u K Γ p n (β,m) + as n +, where B n = {K N β : K is compact, symmetric γ(k) n} N β = {u X; me β.x u p dx =1}. The main goal of this work is to show that Γ p 1(., m) is the first eigensurface if m 0 a.e, Γ p 1(., m) is simple associated with positive eigenfunction. Furthermore if m C(), Γ p 1(., m) is isolated every positive eigenfunction is associated with Γ p 1(., m). (2)

3 On the first eigensurface for the third order spectrum Preliminaries In our further considerations we will use the stard spaces X = W 2,p () W 1,p 0 (), L p () L (), with corresponding norms u 2,p = ( Δu p p + u p p ) 1 p, u p =( u p dx) 1 p u respectively. Recall that for all f L p (), the Poisson equation associated with the Dirichlet problem { Δu = f in, (4) u =0 on, is uniquely solvable in X (cf. [12]). We denote by Λ the inverse operator of Δ :X L p (). In the following lemma we give some properties of the operator Λ (cf. [2],). Lemma 2.1 (i) (Continuity): There exists a constant C p > 0 such that: Λf 2,p C p f p holds for all p ]1, + [ f L p (). (ii) (Continuity) Given k IN, there exists a constant C p,k > 0 such that Λf W k+2,p C p,k f W k,p. (iii) (Symmetry) The following identity: Λu.vdx = u.λvdx holds for all u L p () v L p () with p ]1, + [. (iv) (Regularity) Given f L (), we have Λf C 1,α () for all α ]0, 1[; moreover, there exists C α > 0 such that Λf C 1,α C α f. (v) (Regularity Hopf-type maximum principale) Let f C() f 0 then w =Λf C 1,α (), for all α ]0, 1[ w satisfies: w>0in, w < 0 on. n (vi) (Order preserving property) Given f,g L p (), if f g in then Λf <Λg in. Let N p be the Nemytskii operator defined by: { Np (v)(x) = v(x) p 2 v(x) if v(x) 0, N p (v)(x) =0 if v(x) =0. (5) We have ( v L p ()) ( w L p ()) N p (v) =w v = N p (w). The third order eigenvalue problem of p-biharmonic operator is defined by F ind (β,γ,u) IR N IR + X \{0} such that Δ 2 pu +2β. ( Δu p 2 Δu)+ β 2 Δu p 2 Δu =Γm u p 2 u in, u =Δu =0 on,

4 4416 K. Ben Haddouch, N. Tsouli, El Miloud Hssini Z. El Allali Lemma 2.2 ([5]) The problem (1) is equivalent to the problem { F ind (v, Γ) L p () \{0} IR + such that e β.x N p (v) = ΓΛ(e β.x mn p (Λv)) in L p (). (6) Definition 2.3 We say that (u, Γ) X \{0} IR + is a solution of problem (1) if (v, Γ) where v = Δu is a solution of the problem (6) We consider the functionals F β,g β : L p () IR such that F β (v) = 1 e β.x v p dx G β (v) = 1 p p e β.x m Λv p dx. F β G β are of class C 1 on L p () for all v L p () one has F β (v) =eβ.x N p (v) G β (v) =Λ(meβ.x N p (Λv)) in L p (). Set M β = {v L p (); pg β (v) =1} Γ n = {K M β : K is symmetric, compact γ(k) n}, where γ(k) indicates the genus of K. Lemma 2.4 ([5]) (i) For all β IR N, F β satisfies condition (S +), i.e. v n vin L p () imply v n v strongly in L p (). lim sup F β (v n)(v n v)dx 0, (7) n + (ii) For all β IR N, G β is completely continuous in L p (). Theorem 2.5 ([5]) The problem (1) has at least one sequence of positive eigensurfaces (Γ p n(., m)) n 1 defined by ( β IR N ) Γ p n (β,m) = inf sup e β.x Δu p dx, K B n u K Γ p n(β,m) + as n +. Remark 2.6 The first eigensurface is Γ p 1(., m) which defined by ( β IR N ) Γ p 1(β,m) = inf pf β (v) = inf e β.x Δu p dx. v M β u N β

5 On the first eigensurface for the third order spectrum 4417 Proposition 2.7 The spectrum of problem (1) is closed. Proof 2.8 Let (Γ p n (., m)) n 1 be a sequence of eigensurfaces of problem (1) such that, for all β IR N Γ p n(β,m) Γ as n +, associated with a sequence of eigenfunctions (v n ) of L p () such that v n p =1, for a subsequence still denoted by v n, We have v n vin L p (). The fact that G β is completely continuous, imply that e β.x N p (v n )=Γ p n (β,m)λ(meβ.x N p (Λv n )) ΓΛ(me β.x N p (Λv)) in L p (). (8) Since we get By (8) (9) we deduce that v n v in L p () e β.x N p (v n ) ΓΛ(me β.x N p (Λv)) in L p () e β.x N p (S+), v n v in L p () e β.x N p (v n ) e β.x N p (v) in L p (). (9) e β.x N p (v) = ΓΛ(me β.x N p (Λv)), i.e. Γ is an eigensurface of problem (1). Remark 2.9 Analogously, we can prove that the set of eigenfunctions associated with the same eigensurface of problem (1) is compact. The main results of this work are stated in the following section. 3 On the first eigensurface In this section we will study the first eigensurface Γ p 1(., m). Using the results of P. Dràbek, M. Ôtani (cf. [9]) the abstract result of T. Idogawa, M. Ôtani (cf. [13]) we will prove the following theorem. Theorem 3.1 The eigensurface Γ p 1(., m) of problem (1) verifies the following results: (i) Γ p 1(., m) is the first positive eigensurface. (ii) If m 0 a.e, then

6 4418 K. Ben Haddouch, N. Tsouli, El Miloud Hssini Z. El Allali (a) Every eigenfunction associated with Γ p 1(., m) is positive or negative. (b) Γ p 1(., m) is simple. (iii) If m C() m 0, then Γ p 1(., m) is isolated every positive eigenfunction is associated with Γ p 1(., m). To prove the theorem (3.1) we need the following lemmas. Lemma 3.2 v 1 is an eigenfunction associated with Γ p 1(., m) if only if for any β IR N : F β (v 1 ) Γ p 1(β,m)G β (v 1 ) = 0 = min F β(v) Γ p 1(β,m)G β (v). v L p ()\{0} Proof 3.3 Assume that v 1 is an eigenfunction associated with Γ p 1(., m), without loss of generality, we can assume that v 1 M β, then the infimum in remark 2.6 is attained at v 1 i.e. Hence, for all β IR N, we obtain Γ p 1(., m) = inf v M β pf β (v) =pf β (v 1 ). F β (v 1 ) Γ p 1(β,m)G β (v 1 ) = 0 = min F β(v) Γ p 1(β,m)G β (v) v L p ()\{0} Lemma 3.4 Suppose that m 0 a.e. If v w are positive solutions of problem (6) associated with Γ p 1(., m), then the functions max(v,w) min(v,w) are also solutions of (6) associated with Γ p 1(., m). Proof 3.5 P. Drábek M. Ôtani (cf. [9]) showed that Λ max(v,w)(x) p dx + Λ min(v,w)(x) p dx Λv p dx + Λw(x) p dx. If m 0, using the similar method, we get for any β IR N m(x)e β.x Λ max(v,w)(x) p dx + m(x)e β.x Λ min(v,w)(x) p dx (10) m(x)e β.x Λv p dx + m(x)e β.x Λw(x) p dx. Then if we put, for any β IR N Φ β (v) = e β.x v p dx Γ p 1(β,m) me β.x Λv p dx, from (10) lemma 3.2 we have 0 Φ β (max(v,w)) + Φ β (min(v,w)) Φ(v)+Φ(w) =0. It follows that Φ β (max(v,w)) = Φ β (min(v,w)) = 0, hence max(v,w) min(v,w) are solutions of problem (6) associated with Γ p 1(., m)

7 On the first eigensurface for the third order spectrum 4419 Lemma 3.6 Every eigenfunction of problem (6) is in C(). Proof 3.7 If v is an eigenfunction of problem (6) associated with Γ, then v =Γ 1/(p 1) e ( 1/(p 1))β.x N p (Λ(me β.x N p (Λv))). (11) The assertion (vi) of lemma 2.1 enables us to notice that Λv Λ v v L s () s (1, + ). Then v Γ 1/(p 1) e ( 1/(p 1))β.x me β.x 1 p 1 N p (ΛN p ( Λv )). (12) In [9], the authors showed that N p (ΛN p (Λv)) C() can be done in the same way that N p (ΛN p ( Λv )) C(). Hence from (12) we deduce that v L () it follows from (11) the assertion (vi) of lemma 2.1 that v C(). Lemma 3.8 Let Γ IR + (Γp n (., m)) be a sequence of eigensurfaces of problem (6). If Γ p n (β,m) Γ for every β IR N, then there is v C()\{0} a subsequence of eigenfunctions associated with a subsequence of (Γ p n(., m)) which converges to v in C(). Moreover v is an eigenfunction associated with Γ. Proof 3.9 The fact that Γ p n (β,m) Γ, proposition 2.7 implies that Γ is an eigensurface of problem (6). Let (v n ) be a subsequence of eigenfunctions associated with (Γ p n(., m)). From lemma 3.6 we deduce that (v n ) C(). It follows that (v n ) L q () for every q (1, + ), in particular (v n ) n 1 L q () for q> N. We choose v 2 n such that v n q =1. Hence for a subsequence still denoted by v n there is v L q () such that v n vin L q () Λv n Λv in W 2,q () W 1,q 0 (). Since W 2,q () W 1,q 0 () C() with compact embedding, we have Λv n Λv in C() N p (Λv n ) N p (Λv) in C(). Therefore for any β IR N one has me β.x N p (Λv n ) me β.x N p (Λv) in L (). The assertion (v) in lemma (2.1) implies that Λ(me β.x N p (Λv n )) Λ(me β.x N p (Λv)) in C 1,α (). Since v n =(Γ p n (β,m))1/(p 1) (e β.x ) 1/(p 1) N p (Λ(me β.x N p (Λv n ))) (Γ p n(β,m)) 1/(p 1) (e β.x ) 1/(p 1) N p (Λ(me β.x N p (Λv n )))

8 4420 K. Ben Haddouch, N. Tsouli, El Miloud Hssini Z. El Allali converges to in C(), then Γ 1/(p 1) (e β.x ) 1/(p 1) N p (Λ(me β.x N p (Λv))) v n Γ 1/(p 1) (e β.x ) 1/(p 1) N p (Λ(me β.x N p (Λv))) in C(). Finally as C() L q () v n vin L q () we conclude that v n v in C() v =Γ 1/(p 1) (e β.x ) 1/(p 1) N p (Λ(me β.x N p (Λv))). Proof of theorem 3.1. (i) Let v 1 an eigenfunction associated with Γ p 1(., m) Suppose that there exists Γ (0, Γ p 1(., m)) such that Γ is an eigensurface of problem (6) with v be an eigenfunction associated with Γ. Then by lemma 3.2, for all β IR N, we have F β (v 1 ) Γ p 1(β,m)G β (v 1 )=0 F β (v) Γ p 1(β,m)G β (v) <F β (v) ΓG β (v) =0 which is impossible. Thus Γ p 1(., m) is the first eigensurface associated to problem (6). (ii) (a) Let v be an eigenfunction associated with Γ p 1(., m). We have v L p () Λv Λ v. From lemma 3.2 the fact that m 0 we have 0 e β.x v p dx Γ p 1(β,m) me β.x Λ v p dx e β.x v p dx Γ p 1(β,m) me β.x Λv p dx =0. Then e β.x v p dx Γ p 1(β,m) me β.x Λ v p dx =0 v is an eigenfonction associated with Γ p 1(., m). Therefore v =(e β.x ) 1 p 1 (Γ p 1(β,m)) 1 p 1 Np (Λ(me β.x N p (Λ v )). The assertion (vi) in lemma 2.1 implies that v > 0. Consequently, every nontrivial solution of problem (6) associated with Γ p 1(., m) is positive or negative. (b) Let v w be two positive eigenfunctions associated to Γ p 1(., m). For x 0 set k = v(x 0 )/w(x 0 ) max k (x) = max(v(x),kw(x)). Lemma 3.4 enables us to claim that max k is a solution of problem (6) associated to Γ p 1(., m). Since e β.x N p (v) =Γ p 1(β,m)Λ(me β.x N p (Λv)), e β.x N p (w) =Γ p 1(β,m)Λ(me β.x N p (Λw)) e β.x N p (max k )=Γ p 1(β,m)Λ(me β.x N p (Λmax k )),

9 On the first eigensurface for the third order spectrum 4421 lemma 2.1 lemma 3.6 imply that for any β IR N e β.x N p (v), e β.x N p (w), e β.x N p (max k ) C 1,α () e β.x N p (v), e β.x N p (w) are positive in. Then For any unit vector e, we have e β.x N p (v)/e β.x N p (w) =N p (v)/n p (w) C 1 (). N p (v)(x 0 + te) N p (v)(x 0 ) N p (max k )(x 0 + te) N p (max k )(x 0 ) N p (kw)(x 0 + te) N p (kw)(x 0 ) N p (max k )(x 0 + te) N p (max k )(x 0 ). Dividing these inequalities by t>0 t<0 letting t tend to 0 ±, we get Thus N p (v)(x 0 )= N p (max k )(x 0 )=k p 1 N p (w)(x 0 ). ( eβ.x N p (v) e β.x N p (w) )(x 0) = ( N p(v) N p (w) )(x 0) Hence = ( (N p(v))(x 0 )N p (w)(x 0 ) N p (v)(x 0 ) (N p (w))(x 0 )) (N p (w)(x 0 )) 2 = 0. N p ( v w )= N p(v) N p (w) = const = kp 1 in, v = k in. w This implies that v = kw, as wanted to prove. (iii) Now assume that m 0 m C(). First we prove that every positive eigenfunction is associated with Γ p 1(., m). Let Γ > Γ p 1(β,m), For any β IR N suppose that the problem (6) has a positives eigenfunctions w v associated with Γ Γ p 1(., m) respectively. One has e β.x N p (v) =Γ p 1(β,m)Λ(me β.x N p (Λv)) e β.x N p (w) = ΓΛ(me β.x N p (Λw)). Then from the assertion (v) in lemma 2.1, we have e β.x N p (v), e β.x N p (w) C 1,α ()

10 4422 K. Ben Haddouch, N. Tsouli, El Miloud Hssini Z. El Allali (e β.x N p (v))/ n < 0, (e β.x N p (w))/ n < 0 on. It follows that e β.x N p (v)/e β.x N p (w) =N p (v)/n p (w) C(). Set a = max N p (v)(x)/n p (w)(x). x We deduce that N p (v) an p (w). The monotonicity of N p implies that v a 1 p 1 w. Since a 1 p 1 w is also a solution of problem(6), we may assume without loss of generality that v w. Then, from the assertion (vi) in lemma 2.1 by the monotonicity of N p, we obtain for all β IR N e β.x N p (v) = Γ p 1(β,m)Λ(me β.x N p (Λv)) Γ p 1(β,m)Λ(me β.x N p (Λw)) = ΓΛ(me β.x N p (Λcw)) = e β.x N p (cw), with c =(Γ p 1(β,m)/Γ) 1/(p 1) < 1. Hence it follows by the monotonicity of N p that v < cw. Repeating this argument n times, we obtain 0 v c n w. Therefore by letting n tend to infinity, we deduce that v =0, this is a contradiction. Hence Γ = Γ p 1(., m) w is associated with Γ p 1(., m). Now we show that Γ p 1(., m) is isolated. Assume by contradiction that there is an infinite sequence of eigensurfaces (Γ p n (., m)) n 1 associated to a sequence (v n ) n 1 of eigenfunctions of problem (6) such that for any β IR N : Γ p n (β,m) Γp 1(β,m) asn +. On after foregoing, the sequences (v n ) n 1 (Λv n ) n 1 must change sign in, by lemma 3.8 there exists v 1 C() \{0} a subsequence also denoted by (v n ) n 1 such that v n v 1 in C(). Moreover v 1 is an eigenfunction associated to Γ p 1(., m) which we can suppose nonnegative. The assertion (v) in lemma 2.1 imply that Λv n Λv 1 in C 1,α () for some α ]0, 1[, Λvn Λv 1 C(). Therefore Λvn Λv 1 1inC(). Consequently, Λv n > 0 for n large enough, which is absurd. References [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, (1975).

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