THE EIGENVALUE PROBLEM FOR A SINGULAR QUASILINEAR ELLIPTIC EQUATION

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1 Electronic Journal of Differential Equations, Vol. 2004(2004), o. 16, ISS: URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu ft ejde.math.txstate.edu (login: ft) THE EIGEVALUE PROBLEM FOR A SIGULAR QUASILIEAR ELLIPTIC EQUATIO BEJI XUA Abstract. We show that many results about the eigenvalues and eigenfunctions of a quasilinear ellitic equation in the non-singular case can be extended to the singular case. Among these results, we have the first eigenvalue is associated to a C 1,α () eigenfunction which is ositive and unique (u to a multilicative constant), that is, the first eigenvalue is simle. Moreover the first eigenvalue is isolated and is the unique ositive eigenvalue associated to a non-negative eigenfunction. We also rove some variational roerties of the second eigenvalue. 1. Introduction In this aer, we shall study the eigenvalue roblem of the singular quasilinear ellitic equation div( x a Du 2 Du) = λ x (a+1)+c u 2 u, u = 0, on, in (1.1) where R n is an oen bounded domain with C 1 boundary, 0, 1 < < n, 0 a < (n )/, and c > 0. For a = 0, c =, there are many results about the eigenvalues and eigenfunctions of roblem (1.1), such as λ 1 is associated to a C 1,α () eigenfunction which is ositive in and unique (u to a multilicative constant), that is, λ 1 is simle. Moreover λ 1 is isolated, and is the unique ositive eigenvalue associated to a nonnegative eigenfunction (cf. [11, 1, 6] and references therein). In this aer, we will show that many results about the eigenvalues and eigenfunctions in the case where a = 0, c = can be extended to the more general case where 0 a < (n )/, c > 0. The starting oint of the variational aroach to these roblems is the following weighted Sobolev-Hardy inequality due to Caffarelli, Kohn and irenberg [3], which is called the Caffarelli-Kohn-irenberg inequality Mathematics Subject Classification. 35J60. Key words and hrases. Singular quasilinear ellitic equation, eigenvalue roblem, Caffarelli-Kohn-irenberg inequality. c 2004 Texas State University - San Marcos. Submitted August 15, Published February 6, Suorted by grants and from the ational atural Science Foundation of China. 1

2 2 BEJI XUA EJDE-2004/16 Let 1 < < n. For all u C0 (R n ), there is a constant C a,b > 0 such that ( ) /q x bq u q dx Ca,b x a Du dx, (1.2) R n R n where < a < n, a b a + 1, q = (a, b) = n (1.3) n d, d = 1 + a b. Let R n be an oen bounded domain with C 1 boundary and 0, and let () be the comletion of C0 (R n ), with resect to the norm defined by D 1, a ( u = x a Du dx) 1/. From the boundedness of and the standard aroximation argument, it is easy to see that (1.2) holds for any u Da 1, () in the sense ( ) /r x α u r dx C x a Du dx, (1.4) for 1 r n n, α (1 + a)r + n(1 r ); that is, the imbedding D1, a () L r (, x α ) is continuous, where L r (, x α ) is the weighted L r sace with norm ( u r,α := u Lr (, x α ) = x α u r dx) 1/r. In fact, we have the following comact imbedding result which is an extension of the classical Rellich-Kondrachov comactness theorem (cf. [4] for = 2 and [15] for the general case). For the convenience of the reader, we include its roof here. Theorem 1.1 (Comact imbedding theorem). Let R n be an oen and bounded domain with C 1 boundary and 0, 1 < < n, < a < n. The imbedding Da 1, () L r (, x α ) is comact if 1 r < n n and α < (1 + a)r + n(1 r ). Proof. The continuity of the imbedding is a direct consequence of the Caffarelli- Kohn-irenberg inequality (1.2) or (1.4). To rove the comactness, let {u m } be a bounded sequence in Da 1, (). For any ρ > 0 with B ρ (0) is a ball centered at the origin with radius ρ, it follows that {u m } W 1, (\B ρ (0)). Then the classical Rellich-Kondrachov comactness theorem guarantees the existence of a convergent subsequence of {u m } in L r ( \ B ρ (0)). By taking a diagonal sequence, we can assume without loss of generality that {u m } converges in L r ( \ B ρ (0)) for any ρ > 0. On the other hand, for any 1 r < n n, there exists a b (a, a + 1] such that r < q = (a, b) = n n d, d = 1 + a b [0, 1). From the Caffarelli-Kohn- irenberg inequality (1.2) or (1.4), {u m } is also bounded in L q (, x bq ). By

3 EJDE-2004/16 THE EIGEVALUE PROBLEM 3 Hölder inequality, for any δ > 0, it follows that x α u m u j r dx x <δ ( x (α br) q x <δ ( δ C 0 = Cδ n (α br) q ) 1 r ( q q r dx ) r n 1 (α br) q 1 r q q r dr q r, ) r/q x br u m u j r dx (1.5) where C > 0 is a constant indeendent of m. Since α < (1 + a)r + n(1 r ), it follows that n (α br) q q r > 0. Therefore, for a given ε > 0, we first fix δ > 0 such that x α u m u j r dx ε, m, j. 2 x <δ Then we choose such that x α u m u j r dx C α \B δ (0) \B δ (0) u m u j r dx ε, m, j, 2 where C α = δ α if α 0 and C α = (diam()) α if α < 0. Thus x α u m u j r dx ε, m, j, that is, {u m } is a Cauchy sequence in L q (, x bq ). For studying the eigenvalue roblem (1.1), we introduce the following two functionals on Da 1, (): Φ(u) := x a Du dx, J(u) := x (a+1)+c u dx. For c > 0, J is well-defined. Furthermore, Φ, J C 1 (Da 1, (), R), and a real value λ is an eigenvalue of roblem (1.1) if and only if there exists u Da 1, () \ {0} such that Φ (u) = λj (u). At this oint, let us introduce the set M := {u D 1, a () : J(u) = 1}. Then M and M is a C 1 manifold in Da 1, (). It follows from the standard Lagrange multilier argument that eigenvalues of (1.1) corresond to critical values of Φ M. From Theorem 1.1, Φ satisfies the (PS) condition on M. Thus a sequence of critical values of Φ M comes from the Ljusternik-Schnirelman critical oint theory on C 1 manifolds. Let γ(a) denote the Krasnoselski genus on Da 1, () and for any k, set Then the values Γ k := {A M : A is comact, symmetric and γ(a) k}. λ k := inf max A Γ k u A Φ(u) (1.6) are critical values and hence are eigenvalues of roblem (1.1). Moreover, λ 1 λ 2 λ k +.

4 4 BEJI XUA EJDE-2004/16 One can also define another sequence of critical values minimaxing Φ along a smaller family of symmetric subsets of M. Let us denote by S k the unit shere of R k+1 and O(S k, M) := {h C(S k, M) : h is odd}. Then for any k, the value µ k := inf h O(S k 1,M) max Φ(h(t)) (1.7) t S k 1 is an eigenvalue of (1.1). Moreover λ k µ k. This new sequence of eigenvalues was first introduced by [9] and later used in [7, 6] for a = 0, c =. From the Caffarelli-Kohn-irenberg inequality (1.2) or (1.4), it is easy to see that λ 1 = µ 1 = inf{φ(u) : u D 1, a (), J(u) = 1} > 0, and the corresonding eigenfunction e 1 0. To obtain the roerties of the eigenvalues of roblem (1.1), first we need some boundedness and regularity results of the eigenfunctions of roblem (1.1). In section 2, based on the Moser s iteration technique, we shall deduce the L boundedness and C 1,α ( \ {0}) regularity results. In section 3, we shall obtain the simlicity of the first eigenvalue λ 1. In section 4, we shall rove that the first eigenvalue λ 1 is isolated. Section 5 is concerned with the roerties of the second eigenvalue λ Regularity results In this section, we will rove the L boundedness and C 1,α ( \ {0}) regularity results of the weak solution to roblem (1.1) (cf. [4, 5] for the case = 2). Theorem 2.1. Assume that 1 < < n, 0 a < n, c > 0, and u D1, a () is a solution of (1.1). Then u L (, x α ) and u C 1,α ( \ {0}) for some α 0. Proof. By the standard ellitic regularity theory (e.g. [13]), it suffices to show the L boundedness of u. To do this, we aly the Moser s iteration as in [10] and [5]. For k > 0, q 1, we define two C 1 functions on R, h and H by { sign(t) t q, if t k, h(t) = sign(t){qk q 1 t + (1 q)k q (2.1) }, if t > k, and H(t) = t 0 (h (s)) ds. Thus, it is easy to see that h (t) 0 for all t R and H(u(x)) Da 1, () if u Da 1, (). In fact, a simle calculation shows that { h q t q 1, if t k, (t) = qk q 1 (2.2), if t > k and q H(t) = (q 1) + 1 t (q 1)+1 sign(t), if t k, q ( 1 (q 1) + 1 k(q 1)+1 + k (q 1) ( t k) ) sign(t), if t > k. It is trivial to verify that (2.3) H(t) q h(t) (h (t)) 1, H(t) t 1 q h(t), (2.4)

5 EJDE-2004/16 THE EIGEVALUE PROBLEM 5 for all t R. In fact, for all t k, q 1, we see that H(t) = q (q 1) + 1 t (q 1)+1 q h(t) (h (t)) 1 = q t (q 1)+1 and H(t) t 1 q = (q 1) + 1 t q q h(t) = q t q. For t > k, q 1, a direct calculation shows that and H(t) q h(t) (h (t)) 1 = q ( 1 ( (q 1) + 1 1)k(q 1)+1 + (1 q)k (q 1) ( t k) ) 0 H(t) t 1 q h(t) = q ( 1 ( (q 1) + 1 1)k(q 1)+1 t 1 q k (q 1) ( t k) + k (q 1) ( t k ) ) 0 Let ψ(x) = η H(u(x)) be a test function defined in, where η is a non-negative smooth function in to be secified later. Then from (1.1), it follows that x a Du 2 Du Dψ dx = λ x (a+1)+c u 2 uψ dx. (2.5) From the definitions of h, H, ψ, (2.4) imlies that x a Du 2 Du Dψ dx = x a η Du 2 Du DH(u) dx + x a η 1 H(u) Du 2 Du Dη dx x a η Dh(u) dx q x a η 1 Du 1 Dη h(u) (h (u)) 1 dx. By the Hölder inequality, it follows that q x a η 1 Du 1 Dη h(u) (h (u)) 1 dx 1 x a η Dh(u) dx + Cq x a h(u) Dη dx, 2 (2.6) (2.7) where and hereafter C is a universal ositive constant indeendent of k, q. Inserting (2.7) into (2.6), we see that x a Du 2 Du Dψ dx 1 (2.8) x a η Dh(u) dx Cq x a h(u) Dη dx. 2 Equation (2.4) also imlies λ x (a+1)+c u 2 uψ dx = λ x (a+1)+c u 2 uη H(u) dx λq x (a+1)+c η h(u) dx. (2.9)

6 6 BEJI XUA EJDE-2004/16 n nr For any r (, ), let α = n + (a + 1)r (ar, (a + 1)r), from the n Caffarelli-Kohn-irenberg inequality (1.4), it follows that ( x α ηh(u) r dx ) /r C x a D(ηh(u)) dx. (2.10) Thus, substituting (2.8) (2.10) into (2.5), it is easy to show that ( x α ηh(u) r dx ) /r q C { x a h(u) Dη dx + x (a+1)+c η h(u) dx } (2.11). For each x 0, let η C0 (B 2R (x 0 )), R < 1, such that 0 η 1, η 1 in B R (x 0 ), Dη < 2/R. Then (2.11) imlies that ( x α h(u) r dx ) /r BR(x0) q C { x a B 2R(x 0) R h(u) dx + x (a+1)+c h(u) dx }. B 2R (x 0) Letting k in (2.11), the Hölder inequality imlies ( x α u qr dx ) /r BR(x0) q C { x a B 2R(x 0) R u q dx + x (a+1)+c u q dx } B 2R (x 0) q C ( x α u qs dx ) 1/s, B 2R(x 0) (2.12) (2.13) where s ( max { n α 1, n (a + 1) + c, n α } r ),. n a A simle covering argument yields that ( x α u qr dx ) /r q C ( x α u qs dx ) 1/s, (2.14) that is, u L qr (, x α ) (Cq) 1/q u L qs (, x α ), which is a reversed Hölder inequality and imlies that u L qr (, x α ) for all q > 1. Then letting q = χ m, m = 0, 1, 2,, χ = r s > 1, the Moser s iteration technique (cf. [12]) imlies 1 u L sχ (, x α ) (Cχ m ) χ m u L s (, x α ) m=0 C σ χ τ u L s (, x α ) C u L s (, x α ), where σ = 1 m=0 χ m, τ = 1 m=0 mχ m. Letting, we therefore obtain u L (, x α ) <.

7 EJDE-2004/16 THE EIGEVALUE PROBLEM 7 Based on the above regularity result, the strong maximum rincile due to Vazquez [14] imlies the following ositivity of nonnegative eigenfunction. Corollary 2.2. Suose that 1 < < n, 0 a < n, c > 0, u 0 is an eigenfunction corresonding to λ > 0. Then u > 0 in \ {0}. 3. Simlicity of λ 1 In this section, we rove the simlicity of λ 1, that is, any two eigenfunctions both corresonding to λ 1 are roortional. Theorem 3.1. Suose that 1 < < n, 0 a < n c > 0, u 0 and v 0 are eigenfunctions both corresonding to λ 1. Then u and v are roortional. Proof. From Theorem 2.1, u, v are bounded. We use the modified test-functions as in [11]: η = (u + ε) (v + ε) (v + ε) (u + ε) (u + ε) 1 and (v + ε) 1 (3.1) in the corresonding equations for u and v, resectively, where ε is a ositive arameter. Direct calculation imlies { Dη = 1 + ( 1) ( v + ε ) }Du ( v + ε ) 1Dv, (3.2) u + ε u + ε and, by symmetry, the gradient of the test-function in the corresonding equations for v has a similar exression with u and v interchanged. Set u ε = u + ε, v ε = v + ε. Inserting the chosen test-functions into their resective equations and adding these, it follows that λ 1 x (a+1)+c[ u 1 u 1 ε v 1 ] (u vε 1 ε vε) dx ) } Du ε dx u ε = x a{ 1 + ( 1) ( v ε + x a{ 1 + ( 1) ( u ε ) } Dv ε dx v ε x a ( v ε ) 1 Duε 2 Du ε Dv ε dx u ε x a ( u ε ) 1 Dvε 2 Dv ε Du ε dx v ε = x a (u ε vε)( D log u ε D log v ε ) dx x a vε D log u ε 2 D log u ε (D log v ε D log u ε ) dx x a u ε D log v ε 2 D log v ε (D log u ε D log v ε ) dx 0, (3.3) where the last inequality is a consequence of the following simle calculus inequality (cf. [11]): w 2 > w 1 + w 1 2 w 1 (w 2 w 1 ) (3.4)

8 8 BEJI XUA EJDE-2004/16 for oints in R n, w 1 w 2, > 1. By the Lebesgue s Dominated Convergence Theorem, it follows that lim x (a+1)+c[ u 1 ε 0 + u 1 v 1 ] (u ε vε 1 ε vε) dx = 0. (3.5) The same argument as in [11] imlies that vdu = udv a.e. in, which imlies that u and v are roortional. 4. Isolation of λ 1 In this section, we rove the isolation of λ 1. First, we show that only the first eigenfunctions are non-negative. Theorem 4.1. Suose that 1 < < n, 0 a < n, c > 0. If v 0 is any eigenfunction corresonding to the eigenvalue λ, then λ = λ 1. Proof. Let u 0 denote a first eigenfunction, then the same rocedure as in Section 3 yields x (a+1)+c[ λ 1 u 1 u 1 ε λ v 1 ] (u vε 1 ε vε) dx 0, (4.1) and arguing as before, it follows that (λ 1 λ) x (a+1)+c (u v ) dx 0. (4.2) This leads to a contradiction, if λ > λ 1, since u can be relaced by any of the functions 2u, 3u, 4u,. Thus λ = λ 1. From Theorem 4.1, for any eigenvalue λ > λ 1, the corresonding eigenfunction v must change sign. ext, we need an estimate of the measure of the nodal domains of an eigenfunction v. We recall that a nodal domain of v is a connected comonent of \ {x : u = 0}. Theorem 4.2. Suose that 1 < < n, 0 a < n, c > 0. If v is any eigenfunction corresonding to the eigenvalue λ > λ 1 > 0 and is a nodal domain of v, then for some ositive constant C > 0, where σ = 1 r 1 s c n n r, s ( r r, nr n nr cr n ) if 0 < c < n r. (Cλ) 1/σ (4.3), r (, n n ), s > r r if Proof. Assume that v > 0 in, the case v < 0 being comletely analogous. Since v Da 1, (), then v Da 1, ( ). Hence the function w(x) = v(x) if x and w(x) = 0 if \ belongs to Da 1, (). Using w as a test function in the weak equation satisfied by v yields x a Dv dx = λ x (a+1)+c v dx. (4.4)

9 EJDE-2004/16 THE EIGEVALUE PROBLEM 9 For r (, n n ), let α = (1 + a)r + n(1 r ). Then the Hölder inequality imlies that x (a+1)+c v dx σ( ) α 1/s ( ( (a+1)+c+ x r )s dx x α v r) /r C σ( x α v r) /r, (4.5) since the choice of s imlies that ( (a + 1) + c + α r )s > n. On the other hand, the Caffarelli-Kohn-irenberg inequality (1.2) or (1.4) imlies that ( x α v r) /r C x a Dv dx, (4.6) where C = C(a, α). Thus (4.4) (4.6) imly (4.3). Corollary 4.3. Each eigenfunction has a finite number of nodal domains. Proof. Let j be a nodal domain of an eigenfunction associated to some ositive eigenvalue λ. It follows from (4.3) that and the claim follows. j j (Cλ) 1/σ j Theorem 4.4. Suose that 1 < < n, 0 a < n, c > 0. λ 1 is isolated. Proof. Suose, on the contrary, there exists a sequence of eigenvalues {ν m } such that ν m λ 1 and ν m λ 1 as m. Let u m be an eigenfunction associated to ν m such that u m D 1, a () = 1. Thus, u to a subsequence, {u m} converge weakly in Da 1, () and strongly in L (, x (a+1)+c ) to a function u Da 1, (). Furthermore, the limit function u is an eigenfunction associated to the first eigenvalue λ 1. Without loss of generality, assume that u 0. Then for any δ > 0, by the Egorov theorem, u m converges uniformly to u on a subset δ, with \ δ < δ. Let m be a nodal domain of u m such that u m < 0 in m, then m 0 as m, which contradicts (4.3). 5. Variational roerty of the second eigenvalue Since λ 1 is isolated in the sectrum and there exist eigenvalues different from λ 1, it makes sense to define the second eigenvalue of (1.1) as λ 2 := inf{λ R : λ is eigenvalue and λ > λ 1 } > λ 1. It follows from the closure of the set of eigenvalues of (1.1) that λ 2 is a different eigenvalue of (1.1) from λ 1. Theorem 5.1. λ 2 = λ 2, where λ 2 is defined by (1.6). Proof. It is trivial that λ 2 λ 2. It suffices to show that λ 2 λ 2. Suose that v is the eigenfunction associated to λ 2, then from Theorem 4.1 and Corollary 4.3, let 1,, r, r 2 denote the nodal domains of v. For i = 1,, r, set v(x) [ v i (x) = i x (a+1)+c v dx ], if x 1/ i 0, if x \ i. 1

10 10 BEJI XUA EJDE-2004/16 It is easy to see that v i Da 1, (). Let F r denote the subsace of Da 1, () sanned by {v 1,, v r } and A r = {u F r : J(u) = 1}. For each u F r, u = r α iv i, it follows that r r J(u) = α i J(v i ) = α i. Thus the set A r can also be reresented as A r = { r r α i v i : α i = 1 }. It is easy to see that A r is comact, symmetric and γ(a r ) = r 2, that is, A r Γ 2. On the other hand, inserting v i into the corresonding equation of v yields Dv i dx = λ 2 x (a+1)+c v i dx. (5.1) i i Then for any u = r α iv i A r, it follows that r r Φ(u) = α i Φ(v i ) = λ 2 α i J(v i ) = λ 2. (5.2) Thus, it follows that λ 2 := inf which imlies the conclusion. max A Γ 2 u A Φ(u) max u A r Φ(u) = λ 2, (5.3) The above argument imlies the following further variation characterization of the second eigenvalue (cf. [9, 8, 2] for a = 0, c = ). Theorem 5.2. λ 2 = λ 2 = µ 2 = inf h athcalf max u h([ 1,1]) Φ(u), where F := {h C([ 1, 1], M) : h(±1) = ±e 1 } and e 1 M is the ositive eigenfunction associated to λ 1. References [1] A. Anane and. Tsouli, On the second eigenvalue of the -Lalacian, onlinear PDE, Ed. A. Benkirane and J.-P. Gossez, Pitman Research otes in Mathematics, Vol. 343(1996), 1-9. [2] M. Arias, J. Camos, M. Cuesta & J.P. Gossez, Assymmetric eigenvalue roblems with weights, Prerint, See also Sur certains roblémes ellitiques asymétriques avec oids indéfinis, C.R.A.S., t.332, Série I, 1-4, [3] L. Caffarrelli, R. Kohn and L. irenberg, First order interolation inequalities with weights, Comositio Mathematica, Vol. 53(1984), [4] K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc., Vol. 2(1993), [5] K.-S. Chou and D. Geng, On the critical dimension of a semilinear degenerate ellitic equation involving critical Sobolev-Hardy exonent, onlinear Anal., TMA, Vol. 26(1996), [6] M. Cuesta, Eigenvalue roblems for the -Lalacian with indefinite weights, Electron. J. Diff. Eqns., Vol. 2001(2001), o. 33, 1-9. [7] M. Cuesta, On the Fučik sectrum of the Lalacian and -Lalacian, Proceedings of 2000 Seminar in Differential Equations, Kvilda (Czech Reublic), to aear. [8] M. Cuesta, D. DeFigueiredo and J. P. Gossez, The beginning of the Fučik sectrum for the -Lalacian, J. of Diff. Eqns., Vol. 159(1999), [9] P. Drabek and S. Robinson, Resonance roblems for the -Lalacian, J. of Funct. Anal., Vol. 169(1999), [10] M. Guedda and L. Veron, Quasilinear ellitic equations involving critical Sobolev exonents, onlinear Anal., TMA, Vol. 13(1989),

11 EJDE-2004/16 THE EIGEVALUE PROBLEM 11 [11] P. Lindqvist, On the equation div ( u 2 u) + λ u 2 u = 0, Proc. Amer. Math. Soc., Vol. 109(1990), PP Addendum in Proc. Amer. Math. Soc., Vol. 116(1992), [12] J. Moser, A new roof of De Giorgi s theorem concerning the regularity roblem for ellitic differential equations, Comm. Pure Al. Math, Vol. 13(1960), [13] P. Tolksdorff, Regularity for a more general class of quasilinear ellitic equations, J. Diff. Eqns, 51(1984), [14] J. L. Vazquez, A strong maximum rincile for some quasilinear ellitic equations, Al. Math. Otim, Vol. 12(1984), [15] B.-J. Xuan, The solvability of Brezis-irenberg tye roblems of singular quasilinear ellitic equation, submitted to onliear Anal., Series A. Benjin Xuan Deartment of Mathematics, University of Science and Technology of China Deartment of mathematics, Universidad acional, Bogota Colombia address: wenyuanxbj@yahoo.com