NOTE ON THE NODAL LINE OF THE P-LAPLACIAN. 1. Introduction In this paper we consider the nonlinear elliptic boundary-value problem

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1 2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp ISSN: URL: or ftp ejde.math.txstate.edu (login: ftp) NOTE ON THE NODAL LINE OF THE P-LAPLACIAN ABDEL R. EL AMROUSS, ZAKARIA EL ALLALI, NAJIB TSOULI Abstract. In this paper, we prove that the length of the nodal line of the eigenfunctions associated to the second eigenvalue of the problem pu = λρ(x) u p 2 u in with the Dirichlet conditions is not bounded uniformly with respect to the weight. 1. Introduction In this paper we consider the nonlinear elliptic boundary-value problem p u = λρ(x) u p 2 u u = 0 on, in (1.1) where p u = div( u p 2 u) is the p-laplacian operator, is a bounded and smooth domain in R N (1 < p < + ) and ρ L () is an indefinite weight such that meas( + ρ ) 0 with + ρ = {x ρ(x) > 0}. Several authors have been studied the spectrum σ( p ) of p-laplacian, precisely around of the first and the second eigenvalue. In particular Anane [1] proved that the spectrum σ( p ) contains a positive non-decreasing sequence of eigenvalues (λ n ) n N such that λ n + by using the Ljusternik-Schnirelmann, where and λ 1 n = λ n (, ρ) 1 = sup inf K A n v K ρ(x) v p dx A n = {K W 1,p 0 () : K is symmetrical compact and γ(k) n}. Moreover, he showed that the first eigenvalue is simple and isolated, and that the first eigenfunction corresponding to λ 1 does not change the sign in. In [2] they have showed that the second eigenvalue of the spectrum σ( p ) is exactly λ 2. The complete determination of this spectrum remains unanswered question. It is useful to announce that in the linear case (p = 2), the spectrum is perfectly given [4, 6] Mathematics Subject Classification. 58E05, 35J65, 56J20. Key words and phrases. Nonlinear eigenvalue problem; p-laplacian; nodal line. c 2006 Texas State University - San Marcos. Published September 20,

2 156 A. R. EL AMROUSS, Z. EL ALLALI, N. TSOULI EJDE/CONF/14 Let us consider a solution (u, λ) of problem P (, ρ). We denote by Z(u) = {x : u(x) = 0} the nodal set of u, N (u) is the number of connected components of \Z(u), C(u) is the set of connected components of \Z(u), N (λ) = max{n (u) (u, λ) solution of P (, ρ)} and N (λ n ) = N (n). Recently Cuesta, de Figueiredo and Gossez proved that N (2) = 2 [3]. The main result in this paper is the generalization of the work of Kappeler and Ruf [5], in which they affirmed that the length of the nodal lines is not bounded uniformly with respect to the weights in dimension N = 2 and p = 2. In this work, we locate in the case p > N and we prove that for all real number L > 0, there exists a weights ρ L () and an eigenfunction associated to the second eigenvalue of (1.1) such that the length of Z(u) is largest than L. The proof is relatively simpler than that given by Kappeler and Ruf; in which they use the uniform convergence of the gradient. 2. On the measure of nodal sets In this section, we will extend the result of Kappeler and Ruf [5] in the case p Main result. We consider the case p > N. Let Γ be a surface of class C 1 which subdivides in two nodal components 1 and 2 such that µ 1 ν 1 (2.1) where µ 1 (respectively ν 1 ) is the first eigenvalue of P ( 1, 1) (respectively P ( 2, 1). Let v 1 (respectively w 1 ) the associated eigenfunction. For n N, let n = { x 1 : dist(x, Γ) < 1 }, n + 1 n = 1 \ n where Γ n of class C 1. Then, we denote by v1 n the eigenfunction associated to µ n 1 the first eigenvalue of (1.1) with ρ = 1. Let (a n ) n N be a sequence of decreasing positive real numbers such that a n = µn 1 ν 1 (2.2) which tends to the limit a R + (0 < a = µ1 ν 1 1). Let (ρ n ) n N be a sequence of weight functions defined by for all x, where r n > 0 such that ρ n (x) = r n 1 n (x) + a n 1 n (x) (x), (2.3) lim n + c n d n r n (p 1)/p2 = 0 with c n and d n are strictly positive constants of immersion and interpolation. Let us denote u n 2 the eigenfunction associated to the second eigenvalue λ n 2 of (1.1). Theorem 2.1. There exists a subsequence of (u n 2 ) n N such that still denoted by (u n 2 ) n N

3 EJDE/CONF/14 NODAL LINE OF THE P-LAPLACIAN 157 (i) The sequence (u n 2 ) n N converges weakly to (αv 1 + βw 1 ) in W 1,p 0 (), for some scalars α, β not all null, where v 1 (respectively w 1 ) is the extension of v 1 (respectively w 1 ) by zero in. (ii) If U and V are two opens of R N such that U 1 and V 2. Then for n enough large, we have and u n 2 change the sign on U V U Z(u n 2 ) = V Z(u n 2 ) = To prove this result, we need the following preliminary lemmas. Lemma 2.2. The following inequalities are true independently of (r n ) n N : { (i) 0 < b 1 λ 2 (, 1) λ n 1 if µ 1 < ν 1 2 ν 1, where b = a 1 if µ 1 = ν 1, (ii) p u n 2 p L p ( 2) Mνp 1, (iii) p u n 2 p L p ( n) M(a 1ν 1 ) p, where M is the Sobolev-Poincaré constant. Proof. (i) Let F 2 be the vector subspace of W 1,p 0 () spanned by {v1 n, w 1}, where v1 n (resp. w 1) is the extension by zero of v1 n (resp. w 1 ) in \ n (resp. \ 1 ). Let S 2 denote the unit sphere of F 2. For all v S 2 such that v = αv1 n + βw 1, we have α p + β p = 1 Using (2.3), we get and ρ n (x) v(x) p dx = α p a n 1 µ n 1 ρ n (x) v(x) p dx = 1 ν 1 In particular 1 inf { ρ n (x)v(x)dx} 1 ν 1 v S 2 λ n. 2 Since ρ n (x) b, λ 2 (, b) = λ 2 (, b) λ n 2 (, ρ n ) = λ n 2 b (ii) It is sufficient to notice that p u n 2 = λ n 2 u n 2 p 2 u n 2 a.e. on 2 and by using the Sobolev-Poincaré inequality, we have p u n 2 p M(λ n 2 ) p u n 2 p L p () Mνp 2 + β p ν 1 1. (iii) Using p u n 2 = λ n 2 a n u n 2 p 2 u n 2 a.e. on n and with the same argument as above, one gets p u n 2 p M(a 1 ν 1 ) p. n Remark 2.1. (1) By the lemma 2.2 we can choose the sequence (ρ) n N such that (λ n 2 ) n N converges to the positive limit λ 2. For all p > N, u n 2 converges weakly in W 1,p 0 () and strongly in C() to u 2 W 1,p 0 (). (2) ρ n (x) b for all x. 1.

4 158 A. R. EL AMROUSS, Z. EL ALLALI, N. TSOULI EJDE/CONF/14 (3) u 2 0 in L p (). Lemma 2.3. With the above notation, λ 2 = ν 1 Proof. To prove this lemma, we proceed in three steps: First step: We show that p u 2 = λ 2 a u 2 p 2 u 2 a.e. on 1 (2.4) Indeed; for m 1 and by the lemma 2.2, we have p u n 2 p L p ( m) pu n 2 p L p ( n) M(a 1ν 1 ) p n m hence p u n 2 p (W 1,p ( m)) M(a 1 ν 1 ) p n m. It follows that there exists a subsequence, still denoted by ( p u n 2 ), such that p u n 2 T m weakly in the sapce (W 1,p ( m )). By remark 2.1, u n 2 u 2 weakly in W 1,p ( m ). Since p u n 2 = λ n 2 a n u n 2 p 2 u n 2 a.e. on m for all n m, we have lim pu n 2, u n 2 m = T m, u 2 m n + where, m is the duality bracket between W 1,p ( m ) and its dual (W 1,p ( m )). However, p is an operator of type (M), consequently T m = p u 2 is in the space (W 1,p ( m )). We deduce that hence (2.4). Similarly, we prove that Second step: We show that p u 2 = λ 2 a u 2 p 2 u 2 a.e. on m m 1 p u 2 = λ 2 u 2 p 2 u 2 a.e. on 2 (2.5) u 2/ i = 0 in L p ( i ) for i = 1, 2. Indeed, it follows from (2.2) and since n is of C 1 and ( 2 ) n, we have u n 2 / 2 L p ( 2) c n d n u n 2 σ W 1,p ( n ) un 2 1 σ L p ( n ), (2.6) where c n is the constant of the immersion W σ,p ( n) L p ( n); σ = 1 p and d n is the constant of the interpolation of the inequality u W σ,p ( n ) d n u σ W 1,p ( n ) u 1 σ L p ( n ) for all u W σ,p ( n). The two norms of the second member in (2.6) can be estimated as follows: u n 2 p W 1,p ( n ) un 2 p L p ( n ) + un 2 p L p ( n ) M + 1, (2.7) where M is the constant of the Sobolev-Poincaré of lemma 2.2. Moreover, since (u n 2, λ n 2 ) is a solution of P (, ρ n ), by (2.3) and lemma 2.2, we get r n u n 2 p dx u n 2 p dx + a n u n 2 p dx. 2 n n Since b 1, we deduce that u n 2 p dx b r n n u n 2 p dx b r n M. (2.8)

5 EJDE/CONF/14 NODAL LINE OF THE P-LAPLACIAN 159 Thus, by (2.6), (2.7) and (2.8), we have However, lim n + u n 2 / 2 Lp ( 2) c n d n (M + 1) σ p ( bm r n ) 1 σ p. c nd n r (1 σ)/p n = 0 and u 2 W 1,p 0 (), consequently u 2/ 2 = 0 in L p ( 2 ). Similarly, we have u 2/ 1 = 0 in L p ( 1 ) because ( 1 ) n. Third step: We establish that λ 2 = ν 1 = η 1. Indeed, since u 2 = 0 in L p ( 1 ) and L p ( 2 ) with u 2 W 1,p 0 (), we have u 2 W 1,p 0 ( 1 ) and u 2 W 1,p 0 ( 2 ). Moreover, if we use (2.4), (2.5) and the remark 2.1, then (u 2/1, λ 2 ) and (u 2/2, λ 2 ) are respectively solutions of problems (1.1) with = 1 and with = 2. We have by lemma 2.2, λ 2 = lim n + λn 2 ν 1, where ν 1 is the first eigenvalue of (1.1) with = 2. We conclude that λ 2 = ν 1 = η 1. Lemma 2.4. The sequence (f n ) n N admits a subsequence which converges weakly in W 1,p 0 () to v 1, where with (u n 2 ) + = max{0, u n 2 }. Proof. It is known that (u n 2 ) + f n = ( a p )1/p (u n. 2 )+ L p () (u n 2 ) + p dx = λ n 2 If we multiply by ( ( a p )1/p (u n 2 ) + p p, L ()) then p f n p dx = λ n 2 ρ n f n p dx λ n 2 b ρ n (u n 2 ) + p dx. f n p dx = bλ n 2 Thus, by lemmas 2.2 and 2.3, we have f n p dx p a bλ 2 < + n N. p a. (2.9) So, for a subsequence of the sequence (f n ) n N, still denoted (f n ) n N, we have f n f weakly in W 1,p 0 (), then f n f strongly in C() with p > N. Since f n L p () = ( p a )1/p then f L p () 0. Hence f 0 on, since u 2/2 = βw 1 < 0, f = 0 on 2 (2.10) a fortiori f = 0 on 2 and f = 0 on 1. It results that f W 1,p 0 ( 1 ). According to (2.9), we have ) f n p dx = λ n 2 (a n f n p dx r n f n p dx + f n p dx n 2 ) λ n 2 (a n f n p dx + f n p dx. n 2

6 160 A. R. EL AMROUSS, Z. EL ALLALI, N. TSOULI EJDE/CONF/14 Hence, lim inf n + that Thus f n p ( dx λ 2 a f p dx+ f p dx ). From (2.10), we deduce 1 2 lim inf f n n + p dx λ 2 a f p dx. f p dx λ 2 a 1 f p dx = λ 2 p. We have λ 2 = η 1 being the first eigenvalue of (1.1) with = 1 and ρ = a; consequently λ 2 = 1 f p dx and f = v 1. p Proof of the main result. (i) From lemma 2.3, u 2 is an eigenfunction associated to ν 1 (resp. η 1 ). So, there exists α, β R n such that u 2 = αv 1 + βw 1 with α p + β p > 0. (ii) We distinguish two possible cases: First case: If α 0 and β 0, we can assume that α > 0 and β < 0 (the other cases will be treated in the same way ). As u 2 = αv 1 on 1 and v 1 is a positive eigenfunction of class C 1 on 1, then x 0 U such that min{u 2 (x) : x U} = u 2 (x 0 ) > 0. By lemma 2.2, u n 2 converges uniformly (p > N) to u 2 in, consequently for ɛ = u 2 (x 0 ) > 0 there exists n 0 (U) N such that for all n n 0 (U), we have u n 2 (x) > ɛ 2 x U i.e. (U Z(u n 2 )) = for all n n 0 (U). It is the same for (V Z(u n 2 )) = for all n n 0 (V ). We announce here that according to the lemma 2.2 the case where αβ > 0 does not intervene. Second case: If α = 0 or β = 0. We consider now the case where α = 0 and β < 0. The other cases will be treated in the same way. By lemma 2.4, there exists a subsequence, still denoted (f n ) n N, which converges uniformly to f = v 1 in. Moreover v 1 > 0 in U, and there exists x 0 U such that f(x 0 ) = min{f(x) = v 1 (x) : x U} > 0. Thus, for ɛ = f(x 0 ) > 0, there exists n 0 (U) N such that for all n n 0 (U) we have f n (x) > ɛ 2 x U. i.e for all n n 0 (U), U Z(u n 2 ) =. Therefore, since β < 0, it is the same for V Z(u n 2 ) = for all n n 0 (V ). We remark here that by (i) of the lemma 2.2 the case where α = β = 0 does not intervene.

7 EJDE/CONF/14 NODAL LINE OF THE P-LAPLACIAN Consequences of the main result. Corollary 2.5. If Γ is a surface of class C 1 in which subdivides in two connected components, then for all neighborhood [Γ] ɛ of Γ, there exists a weight ρ ɛ L () and an eigenfunction u associated to the second eigenvalue of P (, ρ ɛ ) such that Z(u) [Γ] ɛ ; where [Γ] ɛ = {x : d(x, Γ) ɛ}. Proof. We distinguish two cases First case: Γ =. Let ε > 0, we consider U = 1 \ ([ ] ɛ [Γ] ɛ ) and V = 2 \ ([ ] ɛ [Γ] ɛ ), where [ ] ɛ = {x : d(x, ) ɛ}. Since Γ =, we can choose ɛ enough small, so that [ ] ɛ [Γ] ɛ =. By theorem 2.1, there exists n N such that Z(u n 2 ) [ ] ɛ [Γ] ɛ. We assume that Z(u n 2 ) [ ] ɛ then there exists a nodal component D ɛ of u n 2 included in [ ] ɛ. Thus (u n 2 /Dɛ, λ n 2 ) is a solution of the problem P (D ɛ, ρ n/dɛ ) with λ n 2 its first eigenvalue [1, 7]. By the remark 2.1, we have λ n 2 = λ 1 (D ɛ, ρ n/dɛ ) λ 1 (D ɛ, b) we have meas(d ɛ ) 0 when ɛ 0, consequently λ n 2 = λ 1 (D ɛ, ρ n ) + when ɛ 0 which is absurd with lemma 2.2. So Z(u n 2 ) [Γ] ɛ. Second case: Γ. Let ɛ > 0, there exists a surface Γ ɛ of C 1 which subdivide in two connected components such that Γ ɛ [Γ] ɛ and Γ ɛ = Let η > 0 (enough small) so that [Γ ɛ] η [Γ] ε and [ ] η [Γ ɛ] η =, finally we conclude the result by applying the proof of the first case with Γ ɛ. Remark 2.2. The result of the corollary 2.5 remains true even if Γ is not of class C 1, only it is enough to approach Γ by a surface Γ of class C 1 which located in [Γ] ɛ. Corollary 2.6. For all L > 0, there exists ρ L () and an eigenfunction u associated to the second eigenvalue of P (, ρ) such that the length of Z(u) is larger than L. Proof. Let L > 0, there exists a surface Γ is of class C 1 in which subdivide in two connected components such that Γ = and meas(γ) > L + 1. For ɛ > 0 (enough small) we consider [Γ] ɛ and [ ] ɛ two neighborhood of Γ and respectively such that [ ] ɛ [Γ] ɛ = Denote by U = 1 \([ ] ɛ [Γ] ɛ ) and V = 2 \([ ] ɛ [Γ] ɛ ) two open. In virtue of the Theorem 2.1 and of the Corollary 2.5, n N such that U Z(u n 2 ) = V Z(u n 2 ) = and Z(u n 2 ) [Γ] ɛ. Let us suppose that for an infinity of ɛ > 0, u n 2 admits a nodal component D ɛ included in [Γ] ɛ. So λ n 2 = λ 1 (D ɛ, ρ n/dɛ ) λ 1 (D ɛ, b). Since lim ɛ 0 meas(d ɛ ) = 0, it follows that λ n 2 lim ɛ 0 λ 1 (D ɛ, b) = + which is absurd with the lemma 2.2. Thus, for ɛ enough small, there exists n N such that Z(u n 2 ) is a closed surface in [Γ] ɛ with Z(u n 2 ) = W where W is an open containing ɛ i which is an open included in i such that ɛ i [Γ] ɛ. So if meas(γ) > L + 1, then ɛ > 0 (enough small) and n N such that meas(z(u n 2 ) > L for i = 1, 2.

8 162 A. R. EL AMROUSS, Z. EL ALLALI, N. TSOULI EJDE/CONF/14 References [1] A. Anane, Simplicité et isolation de la première valeur propre du p-laplacien avec poids, C. R. Acad Sci. Paris t.305 Série I (1987), [2] A. Anane and N.Tsouli, On the second eigenvalue of the p-laplacian, Pitman Research Notes in Mathematics Series, 343, 1-9. [3] M. Cuesta D. de Figueiredo and J.P. Gossez; A nodal domain property for the p-laplacian, C. R. Acad. Sci. Paris, t. 330, Serie I (2000), [4] D. G. De Figuerido, The Dirichlet problem for non linear elliptic equation: a Hilbert space approch, in lecture note in math, No 446, Springer, Berlin (1975), [5] T. Kappeler and B. Ruf, On the nodal line of the second eigenfunction of elliptic operators in two dimension, Journal Rvine und. Ange Maths. (396), [6] M. Otani, A remark on certain nonlinear elliptic equation, Proc. Fac. Sci. Tokai Univ. No. 19 (1984), [7] N. Tsouli, Etude de l ensemble nodal des fonctions propres et de la non-résonance pour l opérateur p-laplacien, Thèse d état, Université Mohamed I, Oujda (1996). Abdel R. El Amrouss Département de Mathématiques et Informatique Faculté des Sciences, Université Mohamed 1er, Oujda, Maroc address: amrouss@sciences.univ-oujda.ac.ma Zakaria El Allali Département de Mathématiques et Informatique Faculté des Sciences, Université Mohamed 1er, Oujda, Maroc address: zakaria@sciences.univ-oujda.ac.ma Najib Tsouli Département de Mathématiques et Informatique Faculté des Sciences, Université Mohamed 1er, Oujda, Maroc address: tsouli@sciences.univ-oujda.ac.ma