In this note, we study the variational characterization and some properties of the second smallest eigenvalue of the Hardy-Sobolev operator Lµ:=−∆p−|x|µp with respect to an indefinite weightV(x)

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

ON THE SECOND EIGENVALUE OF A HARDY-SOBOLEV OPERATOR

K. SREENADH

Abstract. In this note, we study the variational characterization and some properties of the second smallest eigenvalue of the Hardy-Sobolev operator Lµ:=−∆p−_|x|^µ_p with respect to an indefinite weightV(x).

1. Introduction

Let Ω be a domain inR^N containing 0. We recall the classical Hardy-Sobolev inequality which states that, for 1< p < N,

Z

Ω

|∇u|^pdx≥N−p p

pZ

Ω

|u|^p

|x|^pdx, ∀u∈C_c^∞(Ω). (1.1) LetD₀^1,p(Ω) be the closure ofC_c^∞(Ω) with respect to the normkuk1,p=k∇uk_Lp(Ω). The Hardy-Sobolev operatorLµonD^1,p₀ (Ω) is defined as

Lµu:=−∆pu− µ

|x|^p|u|^p−2u, 0< µ <N−p p

p

, where ∆pu= div(|∇u|^p−2∇u), is thep-Laplacian.

We are interested in the variational characterization and some properties of the second smallest eigenvalue of the problem

L_µu=λV(x)|u|^p−2u in Ω

u= 0 on∂Ω. (1.2)

On the weight onV(x), we assume the following:

(H1) V ∈L¹_loc(Ω), V⁺ =V1+V2 6≡0 with V1 ∈ L^N/p(Ω) and V2 is such that lim_{x→y, x∈Ω}|x−y|^pV₂(x) = 0 for all y ∈ Ω, lim_{|x|→∞,x∈Ω}|x|^pV₂(x) = 0, whereV⁺(x) = max{V(x),0}.

(H2) There exists r > N/p and a closed subset S of measure zero in R^N such that Ω\S is connected andV ∈L^r_loc(Ω\S).

Here we note that there is no global integrability condition assumed onV⁻. This work is motivated by the work in [8]. The eigenvalue problem with indefinite weights has been studied for the case µ = 0 by Szulkin-Willem [8]. However, some important properties, of the smallest eigenvalue λ1, such as simplicity and

2000Mathematics Subject Classification. 35J20, 35J70, 35P05, 35P30.

Key words and phrases. p-Laplacian, Hardy-Sobolev operator, unbounded domain.

c

2004 Texas State University - San Marcos.

Submitted August 12, 2003. Published Janaury 22, 2004.

1

being isolated were shown only for p= 2. Recently the author in [6] proved the simplicity ofλ1and sign changing nature of eigenfunctions corresponding to other eigenvalues when Ω is bounded. Infact in [6] the author studied these properties for L_µ. Following the same arguments, one can prove these results in the present case.

However, showing that λ₁ is isolated and characterization of the second smallest eigenvalue, were open questions. To prove these properties, we follow the ideas in [5] and in [3]. Here we should mention that our results are new even for the case µ= 0. We use the following results in later sections.

Propositioin 1.1 (Boccardo-Murat [1]). Let Ω be a bounded domain in R^N and letu_n∈W^1,p(Ω)satisfy

−∆pu_n=f_n+g_n in D⁰(Ω) and

(i) un→uweakly inW^1,p(Ω) (ii) un→uinL^p(Ω)

(iii) fn →f inW^−1,p

0

(iv) gn is a bounded sequence of Radon measures.

Then there exists a subsequence {un} of{un}such that ∇un→ ∇ua.e. inΩ.

Propositioin 1.2 (Brezis-Lieb [2]). Let f_n → f a.e in Ω as n → ∞ and f_n be bounded inL^p(Ω), for somep >1. Then

n→∞lim{kfnkp− kfn−fkp}=kfkp.

LetX be a Banach space and letM ={u∈X g(u) = 0} with g ∈C¹. Also letf :X →Rbe aC¹ functional and let ˜f be the restriction off to M. Then we have the following form of the Mountain pass Theorem [7].

Propositioin 1.3. Let u, v∈M with u6≡v and suppose that c:= inf

h∈Γ max

w∈h(t)f(w)>max{f(u), f(v)}

where

Γ :={h∈C([−1,+1], M)h(−1) =u and h(1) =v} 6=∅

Also suppose thatf˜satisfies Pailse-Smale (PS) condition onM. Thencis a critical value off˜.

We define the norm

kf˜⁰k∗= inf{kf⁰(u)−tg⁰(u)kX^∗ :t∈R}.

The variational characterization of the smallest eigenvalue is given by λ1= inf

06≡u∈W₀^1,p(Ω)

R

Ω|∇u|^pdx−R

Ω

|u|^p

|x|^pdx R

Ω|u|^pV(x)dx

and the corresponding eigenfunction is denoted by φ₁, which is unique under the conditionR

Ω|φ|^pV(x)dx= 1 (see [6]). We will prove the following property.

Theorem 1.4. The eigenvalue λ₁ is isolated in the spectrum ofL_µ.

We will establish the following variational characterisation of the second smallest eigenvalue:

λ2= inf

γ∈Γsup

u∈γ

R

Ω|∇u|^pdx−R

Ω

|u|^p

|x|^pdx R

Ω|u|^pV(x)dx ,

where Γ = {γ∈C([−1,1] :M) γ(−1) =−φ1, γ(1) =φ1} and M is defined as in the next section. We show also the following property ofλ2.

Theorem 1.5. If Va≤Vb, then λ2(Va)≥λ2(Vb).

2. Proofs of results

In this section we show thatλ1is siolated and give a variational characterization for second smallest eigenvalue ofLµ.

Lemma 2.1. The mappingu7−→R

ΩV⁺|u|^pdxis weakly continuous.

The proof of this lemma follows from (1.1) and (H1). We refer the reader to [8]

for more details.

Now, we consider the set M =n

u∈D^1,p₀ (Ω) Z

Ω

|u|^pV(x) = 1o .

SinceM is not a manifold inD^1,p₀ (Ω), we defineX ={u∈D^1,p₀ (Ω) kukX <∞}, where

kuk^p_X :=

Z

Ω

|∇u|^pdx+ Z

Ω

|u|^pV⁻dx.

ThenM is aC¹-manifold as a subset of the spaceX. On this space, we define the functional

Jµ(u) = R

Ω|∇u|^pdx−R

Ω µ

|x|^p|u|^pdx R

Ω|u|^pV dx . Let ˜Jµ denote the restriction ofJµ toM. and let kuk^p_Lp(V)=R

Ω|u|^pV(x)dx.

Lemma 2.2. The functionalJ˜µ satisfies the Palais-Smale condition at any positive level.

Proof. Let{un}be a sequence inM such that Jµ(un)→λ >0 and hJ_µ(u_n), φi −J_µ(u_n)

Z

Ω

|u_n|^p−2u_nφV dx=o(1). (2.1) Using Hardy-Sobolev inequality and u_n ∈M, it follows thatu_n is bounded inX which gives the existence of a subsequence {u_n} of{u_n}and usuch thatu_n →u weakly inD^1,p₀ (Ω). Sinceλ >0 we may assume that J_µ(u_n)≥0. Using Lemma 2.1 and (2.1), we get

hJ_µ⁰(un)−J_µ⁰(u), un−ui+Jµ(un) Z

Ω

|un|^p−2un− |u|^p−2u

(un−u)V⁻dx=o(1).

By Fatou’s Lemma, 0 = Z

Ω n→∞lim

|un|^p−2un− |u|^p−2u

[un−u]V⁻

≤lim inf

n→∞

Z

Ω

|un|^p−2un− |u|^p−2u

[un−u]V⁻dx.

Also,un satisfies

−∆pun− µ

|x|^p|un|^p−2un−Jµ(un)|un|^p−2unV(x) =o(1) in D⁰(Ωm), where Ωm is a bounded domain such that Ω = ∪^∞_m=1Ωm. By Proposition 1.1, noting that _|x|^µp|un|^p−2un+Jµ(un)|un|^p−2unV⁻ is a bounded sequence of Radon measures, there exists a subsequence{u^m_n}of{u_n}anusuch that∇u^m_n → ∇ua.e., in Ωm. By the process of diagonalization we can choose a subsequence{un} such that∇un → ∇ua.e. in Ω. By Proposition 1.2, we have

kun−uk^p_1,p=kunk^p_1,p− kuk^p_1,p+o(1) (2.2) kun−u

|x| k^p_Lp(1)=kun

|x|k^p_Lp(1)− k u

|x|k^p_Lp(1)+o(1). (2.3) We also have, by Fatau’s lemma,

Z

Ω

V⁻(|un|^p+|u|^p− |un|^p−2unu− |u|^p−2uun)dx

≥ Z

Ω

V⁻(|un|^p+|u|^p)−Z

Ω

V⁻|un|^p(p−1)/pZ

Ω

V⁻|u|^p1/p

−Z

Ω

V⁻|u|^p(p−1)/pZ

Ω

V⁻|un|^p1/p

=hZ

Ω

V⁻|un|^p(p−1)/p

−Z

Ω

V⁻|u|^p(p−1)/pi

×hZ

Ω

V⁻|un|^p1/p

−Z

Ω

V⁻|u|^p1_pi

≥0. Now using (2.2) and (2.3),

o(1) =hJµ(un)−Jµ(u),(un−u)i+Jµ(un) Z

Ω

[|un|^p−2un− |u|^p−2u](un−u)V⁻dx

≥ Z

Ω

|∇un− ∇u|^p− Z

Ω

µ

|x|^p|un−u|^p+o(1)

≥ 1− µ λN

ku_n−uk_1,p+o(1).

i.e.,u_n→uinD^1,p₀ (Ω). Notice that o(1) =hJµ(u_n)−J_µ(u), u_n−ui

= Z

Ω

V⁻ |un|^p−2un− |u|^p−2u

(un−u)dx+o(1)≥0. Therefore,R

ΩV⁻|u_n|^pdx→R

ΩV⁻|u|^pdxand henceku_nk_X→ kuk_X. Observe that ˜Jµ(u)≥ λ1 and ˜Jµ(±φ1) = λ1. So +φ1 and −φ1 are two global minima of ˜J_µ. Now consider

Γ ={γ∈C([−1,1];M) γ(−1) =−φ1, γ(1) =φ1}.

By Proposition 1.3, there existsu∈X such that ˜J_µ⁰(u) = 0 andJµ(u) =C, where C= inf

γ∈Γsup

u∈γ

J˜µ(u). (2.4)

Lemma 2.3. (i) M is locally arc wise connected

(ii) Any connected open subset B of M is arcwise connected (iii) ifB⁰ is a component of an open setA, then∂B⁰∩B is empty.

The proof of this lemma follows from the fact thatM is a Banach Manifold. For a proof we refer the reader to [3]. DefineO={u∈M J˜µ(u)< r}

Lemma 2.4. Each component of Ocontains a critical point of J˜µ.

Proof. LetO1 be a component of O and let d= inf{J˜µ(u), u∈ O1}, where O1 is X-closure ofO. Suppose this infimum is achieved byv∈ O1. Then by Lemma 2.3 this cannot be in∂O1and hencev is inO1 and is a critical point of ˜Jµ.

Now we show that dis achieved. Let un ∈ O1 be a minimizing sequence with J˜_µ(u_n)≤d+_n¹₂. By Ekeland Variational Principle, we get v_n∈ O₁ such that

J˜_µ(v_n)≤J˜_µ(u_n), (2.5) kv_n−u_nk_X≤ 1

n, (2.6)

J˜_µ(v_n)≤J˜_µ(v) +1

nkv_n−vk_X, ∀v∈ O₁. (2.7) From (2.5) it follows that ˜Jµ(vn) is bounded. Now we claim thatkJ˜µ

0(vn)k∗ →0.

We fixnand choosew∈X tangent toM atvn, i.e.,R

Ω|vn|^p−2vnwV = 0. Now we consider the path

ut= vn+tw kvn+twk_Lp(V)

.

Since ˜J_µ(v_n) ≤d+_n¹ < r forn large, we have v_n ∈ O₁ and by Lemma 2.3 (iii), v_n ∈/ ∂O₁. Sou_t∈ O₁for|t|small. Takingv=u_tin (2.7) we obtain

J˜µ(vn)−J˜µ(vn+tw) t

≤ 1

ntkvn( 1

r(t)−1)kX+1

nkwk+1 t

1

r(t)^p −1J˜µ(vn+tw),

(2.8)

wherer(t) =kvn+twkL^p(V). The last term in (2.8) involves ^r(t)_t^p−1 which can be calculated as

d dtr(s)^p

_s=0= lim

t→0

r(t)^p−1 t . On the other hand sincewis tangent toM atvn,

d dtr(s)^p

_s=0=p Z

Ω

|vn|^p−2vnwV(x)dx= 0.

Therefore, we have ^r(t)_t^p−1 → 0 as t → 0 and that the second term goes to 0.

Similarly, the first term also goes to zero as t→0. Taking limitt→0 in (2.8) we get

hJ_µ⁰(v_n), wi ≤ 1

nkwk_X, for allw∈X tangent toM atv_n.

Now ifw is arbitrary inX. We choose αn so that (w−αnvn) is tangent toM at vn. i.e.,αn=R

Ω|vn|^p−2vnwV(x)dx. So (2.8) gives,

|hJ_µ⁰(vn), wi − hJ_µ⁰(vn), vni Z

Ω

|vn|^p−2vnw| ≤ 1

nkw−αnvnkX

SincekαnvnkX ≤C.kwkX, we have

|hJ_µ⁰(v_n), wi −t_n Z

Ω

|v_n|^p−2v_nwV(x)dx| ≤_nkwk_X

wheret_n =hJ_µ⁰(v_n), v_niand_n→0. Therefore,kJ˜_µ⁰(v_n)k_∗→0 andv_nis a Palais- Smale sequence. Hence by Lemma 2.2, {vn} has a convergent subsequence with

limit, say,v. Thendis achieved atv.

Lemma 2.5. The number C defined by (2.4) is the second smallest eigenvalue of L_µ

Proof. We follow the proof in [3]. Assume by contradiction that there exists an eigenvalue δ such thatλ1 < δ <C. In other words, ˜Jµ has a critical valueδ with λ₁< δ <C. We will construct a path in Γ on which ˜J_µ remains≤δ, which yields a contradiction with the definition ofC. Letu∈M satisfies the equation

−∆pu− µ

|x|^p|u|^p−2u=δV(x)|u|^p−2u in D⁰(Ω), anduchanges sign in Ω. Takingu⁺ andu⁻ as test function we get

Z

Ω

|∇u⁺|^pdx− Z

Ω

µ

|x|^p|u⁺|^pdx=δ Z

Ω

(u⁺)^pV(x)dx Z

Ω

|∇u⁻|^pdx− Z

Ω

µ

|x|^p(u⁻)^pdx=δ Z

Ω

(u⁻)^pV(x)dx . Consequently

J˜µ(u) = ˜Jµ( u⁺ ku⁺k_Lp(V)

) = ˜Jµ( −u⁻ ku⁻k_Lp(V)

) = ˜Jµ( u⁻ ku₋k_Lp(V)

) =δ.

We will consider the following three paths inM, which go respectively from uto

u⁺

ku⁺kLp(V), from _ku+^ukLp(V⁺ ) to _ku−^ukLp(V⁻ ) and _ku−^−ukLp(V⁻ ) tou:

u₁(t) = tu+ (1−t)u⁺ ktu+ (1−t)u⁺kL^p(V)

,

u2(t) = tu⁺+ (1−t)u⁻ ktu+ (1−t)u⁻k_Lp(V)

,

u₃(t) = −tu⁻+ (1−t)u k −tu⁻+ (1−t)uk_Lp(V)

. Also we have

J˜µ(u1(t)) = ˜Jµ(u2(t)) = ˜Jµ(u3(t)) =δ.

By joining the paths u₁(t) and u₂(t) we get a new path which connects u and

u⁻

ku⁻kLp(V) and stays at levels≤δ. Call this path asu4(t). Now we defineO={v∈ M J˜µ(v)< δ}. Clearly φ1,−φ1 ∈O. Since _ku−^uk⁻Lp(V) does not change sign and vanishes on a set of positive measure it is not a critical point of ˜Jµ. So _ku−^uk⁻_Lp(V)

is a regular value of ˜Jµ, and consequently there exists a C¹ pathη : [−, ] →M withη(0) = _ku−^ukLp(V⁻ ) and _dt^d( ˜Jµ(η(t))

_t=06= 0. choose a pointv∈O on this path (this is possible because ˜Jµ

0(η(t))

_t=0 6= 0) we can thus move from _ku−^uk⁻_Lp(V) to v through this path which lies at levels < δ. Taking the component of O which

contains v and applying Lemma 2.3 together with Lemma 2.4, we can connect v to +φ1 (or to −φ1) with a path in M at levels < δ. Let us assume that this is +φ₁which is reached in this way. Now call this path connecting _ku−^uk⁻_Lp(V) andφ₁ asu5(t), and consider the symmetric path−u5(t), which goes from −_ku−^ukLp(V⁻ ) to

−φ1. We evaluate the functional ˜J_µ along−u5(t). Since ˜J_µ is even, J˜µ(−u5(t)) = ˜Jµ(u5(t))≤δ.

Finally withu₃(t) we can connect−_ku−^uk⁻_Lp(V) withuby a path which stays at level δ . Putting every thing together we get a path connecting−φ₁ and φ₁ staying at

levels≤δ. This concludes the proof.

Note that Theorem 1.4 is an immediate consequence of Lemma 2.5. So we have the following characterization ofλ2, the second smallest eigenvalue ofLµ,

λ2= inf

γ∈Γsup

u∈γ

Z

Ω

|∇u|^pdx− Z

Ω

|u|^p

|x|^pdx.

Now letµ_k be the sequence of eigenvalues obtained in [6] which are characterized as

µ_k = inf

A∈Fsup

u∈A

Z

Ω

|∇u|^p− Z

Ω

µ

|x|^p|u|^p, whereF ={A ⊂M the genus ofA ≥k}.

Corollary 2.6. With the notation above,µ2=λ2.

Proof. Letγbe a curve in Γ. By joining this with its symmetric path−γ(t) we can get a set of genus ≥2 where J_µ does not increase its values. Therefore,λ₂≥µ₂. But by Theorem 1.4, there is no eigenvalue betweenλ₁andλ₂. Henceλ₂=µ₂. Lemma 2.7. Let u∈X be a solution of (1.2) and let O be a component of{x∈ Ω u(x)>0}. Then u

O ∈D₀^1,p(O)

Proof. Letun ∈Cc(Ω)∩D^1,p₀ (Ω) such thatun→uin D^1,p₀ (Ω). Thenu⁺_n →u⁺ in D₀^1,p(Ω). Letvn = min(un, u) and let φ:R→Rbe aC¹ function such that

φ(t) =

(0 fort≤1/2 1 fort≥1 and|φ⁰| ≤1. Letψr(x) =φ d(x, S)/r

whered(x, S) = dist(x, S). Then ψ_r(x)

(0 ford(x, S)≤r/2 1 ford(x, S)≥r

and |∇ψr(x)| ≤ C/r for some constant C. Now we define wn,r(x) = ψrvn(x) _O. Since ψrvn ∈ C(Ω), we have wn,r ∈ C(O) and vanishes on the boundary ∂O.

Indeed forx∈∂O ∩S thenψr(x) = 0 and sown,r(x) = 0. Ifx∈∂O ∩Ω andx /∈S then u(x) = 0(since u is continuous except at 0) and so v_n(x) = 0 . If x ∈ ∂Ω then u_n(x) = 0 and hencev_n(x) = 0. So in all the casesw_n,r(x) = 0 forx∈∂O.

Therefore,wn,r∈D^1,p₀ (O).

Z

Ω

|∇(wn,r)− ∇(ψru)|^p= Z

O

|(∇ψr)vn+ψr∇vn−(∇ψr)u−ψr∇u|^pdx

≤ k∇ψrvn− ∇ψruk^p_Lp(O)+kψr∇vn−ψr∇uk^p_Lp(O)

which goes to 0 asn→ ∞. i.e.,wn,r→ψru

_O inD^1,p₀ (O). Now Z

O

|∇ψru+ψr∇u−u|^p≤ Z

O

|ψr∇u− ∇u|^p+ Z

O∩{r/2<|x|<r}

|∇ψr|^pu

→0 asr→0 by (1.1). Therefore,u

O ∈D^1,p₀ (O).

Proof of Theorem 1.5. We denote ˜J_µ corresponding to V_b with ˜J_µ,b. Let u_a be a solution to

−∆pu− µ

|x|^p|u|^p−2u=λ2Va(x)|u|^p−2u in D⁰(Ω).

Assuming that the claim below is true, we have J˜µ,b

v⁺ kv⁺k_Lp(V_b)

< λ2(Va),J˜µ,b

v⁻ kv⁻k_Lp(V_b)

< λ2(Va),J˜µ,b(v)< λ2(Va).

Define Ob = {u ∈ X,R

Ω|u|^pVb = 1,J˜µ,b(v) < λ2(Va).} Now we proceed as in Lemma 2.5, to define the pathsvi(t), i= 1, ...,5 on which ˜Jµ,b < λ2(Va). We join these paths in a way described in Lemma 2.5 to obtain a path γ(t) in Γb (the family of paths corresponds to Vb) such that ˜Jµ,b(γ(t))< λ2(Va). This completes the proof.

Claim: There exists v∈X, which changes sign and R

Ω|∇v⁺|^pdx−R

Ω µ

|x|^p|v⁺|^pdx R

Ω(v⁺)^pV_bdx < λ2(Va), R

Ω|∇v⁻|^pdx−_|x|^µp|v⁻|^pdx R

Ω(v⁻)^pVbdx < λ2(Va).

(2.9)

Proof of Claim: Since ua is an eigenfunction corresponding to λ2 > λ1, it has to change sign in Ω (see [6]). LetO₁and O₂ be positive and negative nodal domains ofu_a respectively such that

Z

O1

Va (u⁺_a)^pdx <

Z

O1

Vb(u⁺_a)^pdx and Z

O2

Va (u⁻_a)^pdx≤ Z

O2

Vb(u⁻_a)^pdx.

By Lemma 2.7,u_{a O}

1 ∈D₀^1,p(O₁) and also inL^p(O₁, V⁻). We have λ1(O1, Vb)≤

R

O1|∇u_a|^p−_|x|^µ_p|u_a|^p R

O1|u_a|^pV_b < λ2(Va).

Therefore, λ1(O1, Vb)< λ2(Va). Simillarilyλ1(O2, Vb)≤λ2(Va). Now we modify O1 and O2 to get ˜O1 and ˜O2 with empty intersection and λ1( ˜O1, Vb) < λ2(Va) and λ1( ˜O2, Vb) < λ2. For η > 0, letO1(η) = {x ∈ O1 dist(x, O₁^c) > η}. Then λ1(O1(η), Vb) ≥λ1(O1, Vb) and λ1(O1(η), Vb)→ λ1(O1, Vb) as η → 0. Therefore, there existsη₀>0 such thatλ₁(O₁(η), V_b)< λ₂(V_a) for 0< η < η₀. Letx∈∂O₂∩Ω and 0 < η < min{η0,dist(x₀,Ω^c)}. Now define ˜O₂ = O₂∪B(x₀, η/2). Then O˜₂∩O₁(η) =∅, λ₁( ˜O₂, V_b)< λ₁(O₂, V_b)< λ₂(V_a). Now we consider the function

v =v1−v2, where vi are the extensions by zero outside ˜Oi of the eigenfunctions associated toλ1( ˜Oi, Vb). Thenv satisfies (2.9).

Acknowledgements. The Author was supported by grant 40/1/2002-R&D-II/165 from the National Board for Higher Mathematics(NBHM), DAE, Govt. of India.

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Konijeti Sreenadh

T.I.F.R. Centre, Post Box No.1234, Bangalore-560012, India E-mail address:[email protected]