A CLASS OF DEGENERATE NONLINEAR ELLIPTIC EQUATIONS

A CLASS OF DEGENERATE NONLINEAR ELLIPTIC EQUATIONS

MIHAI MIH ˘AILESCU

Received 11 January 2005; Revised 4 July 2005; Accepted 17 July 2005

The goal of this paper is to study the existence and the multiplicity of non-trivial weak solutions for some degenerate nonlinear elliptic equations on the whole space R^N. The solutions will be obtained in a subspace of the Sobolev spaceW^1,p(R^N). The proofs rely essentially on the Mountain Pass theorem and on Ekeland’s Variational principle.

Copyright © 2006 Mihai Mih˘ailescu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The goal of this paper is to study a nonlinear elliptic equation in which the divergence form operator−div(a(x,∇u)) is involved. Such operators appear in many nonlinear dif- fusion problems, in particular in the mathematical modeling of non-Newtonian fluids (see [5] for a discussion of some physical background). Particularly, thep-Laplacian op- erator−div(|∇u|^p−2∇u) is a special case of the operator−div(a(x,∇u)). Problems in- volving thep-Laplacian operator have been intensively studied in the last decades. We just remember the work on that topic of Jo˜ao Marcos B. do ´O [7], Pfl¨uger [12], R˘adulescu and Smets [14] and the references therein. In the case of more general types of operators we point out the papers of Jo˜ao Marcos B. do ´O [6] and N´apoli and Mariani [4]. On the other hand, when the operator−div(a(x,_∇u)) is of degenerate type we refer to Cˆırstea and R˘adulescu [15] and Motreanu and R˘adulescu [11].

In this paper we study the existence and multiplicity of non-trivial weak solutions to equations of the type

−diva(x,∇u)=Ᏺ(x,u), x∈R^N, (1.1) where the operator div(a(x,∇u)) is nonlinear (and can be also degenerate),N≥3 and functionᏲ(x,u) satisfies several hypotheses. Our goal is to show how variational tech- niques based on the Mountain Pass theorem (see Ambrosetti and Rabinowitz [2]) and Ekeland’s Variational principle (see Ekeland [8]) can be used in order to get existence of

Hindawi Publishing Corporation Boundary Value Problems

Volume 2006, Article ID 41295, Pages1–17 DOI10.1155/BVP/2006/41295

one or two solutions for equations of type (1.1). Results regarding the multiplicity of so- lutions have been originally proven by Tarantello [16], but in the case of linear equations and in a different framework. More precisely, Tarantello proved that the equation

−Δu= |u|^4/(N−2)u+Γ(x) (1.2) has at least two distinct solutions, in a bounded domain of R^N (N≥3), provided that Γ≡0 is sufficiently “small” in a suitable sense.

2. Main results

The starting point of our discussion is the equation

−Δv+b(x)v=f(x,v) x∈R^N (2.1) studied by Rabinowitz in [13]. Assuming that function f(x,v) is subcritical and satisfies a condition of the Ambrosetti-Rabinowitz type (see [2]) and functionb(x) is sufficiently smooth and unbounded at infinity, it is showed in [13] that problem (2.1) has a nontrivial weak solution in the classical Sobolev spaceW^1,2(R^N).

In the case whenb(x) is continuous and nonnegative and f(x,v)=h(x)v^α+v^βis such thath: R^N→R is some integrable function and 1< α <2< β <(N+ 2)/(N−2),N≥3, Gonc¸alves and Miyagaki proved in [9] that problem (2.1) has at least two nonnegative solutions in a subspace ofW^1,2(R^N). In a similar framework, when f(x,v)=λv^α+v²⁻¹ with 0< α <1 and 2=(2N)/(N−2),N≥3 it is shown in [1] that problem (2.1) has a nonnegative solution forλ positive and small enough. Furthermore, in [1] it is also proved that in the caseN≥4 andα=1 problem (2.1) has a nonnegative solution pro- vided thatλis positive and small enough. For more information and connections on (2.1) the reader may consult the references in [9].

In this paper our aim is to study the problem

−diva(x,∇u)+b(x)u^p−2u= f(x,u), x∈R^N, (2.2) whereN≥3 and 2≤p < N.

We point out the fact that in the case whena(x,∇u)= |x|^α∇u,α∈(0, 2) and p=2 problem (2.2) was studied by Mih˘ailescu and R˘adulescu in [10]. In that paper the authors present the connections between such equations and some Schr¨odinger equations with Hardy potential and show that (2.2) has a nontrivial weak solution. A discussion of some physical applications for equations of type (2.2) and a list of papers devoted with the study of such problems is also included in [10].

In the following we describe the framework in which we will study (2.2).

Considera: R^N×R^N→R^N,a=a(x,ξ), is the continuous derivative with respect toξ of the continuous functionA: R^N×R^N→R,A=A(x,ξ), that is,a(x,ξ)=(d/dξ)A(x,ξ).

Suppose thataandAsatisfy the hypotheses below:

(A1)A(x, 0)=0 for allx∈R^N;

(A2)|a(x,ξ)| ≤c1(θ(x) +|ξ|^p−1), for all x,ξ ∈R^N, with c1 a positive constant and θ: R^N→R is a function such that θ(x)≥0 for allx∈R^N andθ∈L^∞(R^N)∩ L^p/(p−1)(R^N);

(A3) there existsk >0 such that A

x,ξ+ψ

2

≤1

2A(x,ξ) +1

2A(x,ψ)−k|ξ−ψ|^p (2.3) for allx,ξ,ψ∈R^N, that is,A(x,·) isp-uniformly convex;

(A4) 0≤a(x,ξ)·ξ≤pA(x,ξ), for allx,ξ∈R^N; (A5) there exists a constantΛ >0 such that

A(x,ξ)≥Λ|ξ|^p, (2.4)

for allx,ξ∈R^N.

Examples. (1)A(x,ξ)=(1/ p)|ξ|^p,a(x,ξ)=|ξ|^p−2ξ, withp≥2 and we get thep-Laplacian operator

div|∇u|^p−2∇u. (2.5)

(2)A(x,ξ)=(1/ p)|ξ|^p+θ(x)[(1 +|ξ|²)^1/2−1],a(x,ξ)= |ξ|^p−2ξ+θ(x)(ξ/(1 +|ξ|²)^1/2), withp≥2 andθa function which verifies the conditions from (A2). We get the operator

div|∇u|^p−2∇u+ div

⎛

⎝θ(x) ^∇u 1 +|∇u|²1/2

⎞

⎠ (2.6)

which can be regarded as the sum between the p-Laplacian operator and a degenerate form of the mean curvature operator.

(3) A(x,ξ) = (1/ p)[(θ(x)^2/(p−1) +|ξ|²)^p/2 −θ(x)^p/(p−1)], a(x,ξ) = (θ(x)^2/(p−1) +

|ξ|²)^(p−2)/2ξ, with p≥2 and θa function which verifies the conditions from (A2). We get the operator

divθ(x)^2/(p−1)+|∇u|²(p−2)/2

∇u (2.7)

which is a variant of the generalized mean curvature operator, div((1 +|∇u|²)^(p−2)/2∇u).

Assume that functionb: R^N→R is continuous and verifies the hypotheses:

(B) There exists a positive constantb0>0 such that

b(x)≥b0>0, (2.8)

for allx∈R^N.

In a first instance we assume that function f : R^N×R→R satisfies the hypotheses:

(F1) f ∈C¹(R^N×R, R), f =f(x,z) and f(x, 0)=0 for allx∈R^N;

(F2) there exist two functionsτ1,τ2: R^N→R,τ1(x),τ2(x)≥0 for a.e.x∈R^Nand two constantsr,s∈(p−1, (N p−N+p)/(N−p)) such that

fz(x,z)≤τ1(x)|z|^r−1+τ2(x)|z|^s−1, (2.9) for all x∈R^N and all z∈R, where τ1∈L^r0(R^N)∩L^∞(R^N), τ2∈L^s0(R^N)∩ L^∞(R^N), withr0=N p/(N p−(r+1)(N−p)) ands0=N p/(N p−(s+1)(N−p));

(F3) there exists a constantμ > psuch that 0< μF(x,z) :=μ

_z

0 f(x,t)dt≤z f(x,z), (2.10) for allx∈R^Nand allz∈R\ {0}.

Next, we study the problem

−diva(x,∇u)+b(x)|u|^p−2u=h(x)|u|^q−1u+g(x)|u|^s−1u, x∈R^N (2.11) with 1< q < p−1< s <(N p−N+p)/(N−p) andN≥3.

Our basic assumptions on functionshandg: R^N→R are the following:

(H)h(x)≥0 for allx∈R^N andh∈L^q0(R^N)∩L^∞(R^N), whereq0=N p/(N p−(q+ 1)(N−p));

(G)g(x)≥0 for all x∈R^N and g∈L^s0(R^N)∩L^∞(R^N), wheres0=N p/(N p−(s+ 1)(N−p)).

LetW^1,p(R^N) be the usual Sobolev space under the norm u1=

R^N

|∇u|^p+|u|^p dx

1/ p

(2.12) and consider the subspace ofW^1,p(R^N)

E=

u∈W^1,p(R^N);

R^N

|∇u|^p+b(x)|u|^p dx <∞

. (2.13)

The Banach spaceEcan be endowed with the norm u^p=

R^N

|∇u|^p+b(x)|u|^p

dx. (2.14)

Moreover,

u ≥m^{1/ p}0 u1, (2.15)

withm0=min{1,b0}. Thus the continuous embeddings

E W^1,pR^N LⁱR^N, p_≤i_≤p, p_=N^{N p}_−p (2.16) hold true.

We say thatu∈Eis a weak solution for problem (2.2) if

R^Na(x,∇u)· ∇ϕ dx+

R^Nb(x)|u|^p−2uϕ dx−

R^N f(x,u)ϕ dx=0, (2.17) for allϕ∈E.

Similarly, we say thatu∈Eis a weak solution for problem (2.11) if

R^Na(x,∇u)· ∇ϕ dx+

R^Nb(x)|u|^p−2uϕ dx

−

R^Nh(x)|u|^q−1uϕ dx−

R^Ng(x)|u|^s−1uϕ dx=0,

(2.18)

for allϕ∈E.

Our main results are given by the following two theorems.

Theorem 2.1. Assuming hypotheses (A1)–(A5), (B) and (F1)–(F3) are fulfilled then prob- lem (2.2) has at least one non-trivial weak solution.

Theorem 2.2. Assume 1< q < p−1< s <(N p−N+p)/(N−p) and conditions (A1)–

(A5), (B), (H) and (G) are fulfilled. Then problem (2.11) has at least two non-trivial weak solutions provided that the producth^(s+1_L^q0(R^−Np)/(s) ^−q)· g^(p_L^s0−(R^q−N1)/(s) ^−q)is small enough.

3. Auxiliary results

In this section we study certain properties of functionalT:E→R defined by T(u)=

R^NA(x,∇u)dx+ 1 p

R^Nb(x)|u|^pdx, (3.1) for allu∈E. It is easy to remark thatT∈C¹(E, R) and

T(u),v=

R^Na(x,∇u)· ∇v dx+

R^Nb(x)|u|^p−2uv dx, (3.2) for allu,v∈E.

Proposition 3.1. FunctionalTis weakly lower semicontinuous.

Proof. Letu∈Eand>0 be fixed. Using the properties of lower semicontinuous func- tions (see [3, Section I.3]) is enough to prove that there existsδ >0 such that

T(v)≥T(u)−, ∀v∈E withu−v< δ. (3.3) We remember Clarkson’s inequality (see [3, page 59])

α+β 2

^p+α−β 2

^p≤1 2

|α|^p+|β|^p

, ∀α,β∈R. (3.4)

Thus we deduce that

R^Nb(x)u+v 2

^pdx+

R^Nb(x)u−v 2

^pdx

≤1 2

R^Nb(x)|u|^pdx+1 2

R^Nb(x)|v|^pdx, ∀u,v∈E.

(3.5)

The above inequality and condition (A3) imply that there exists a positive constantk1>0 such that

T u+v

2

≤1

2T(u) +1

2T(v)−k1u−v^p, ∀u,v∈E, (3.6) that is,Tisp-uniformly convex.

SinceTis convex we have

T(v)≥T(u) +T(u),v−u, ∀v∈E. (3.7) Using condition (A2) and H¨older’s inequality we deduce that there exists a positive con- stantC >0 such that

T(v)≥T(u)−

R^N

a(x,∇u)· |∇v− ∇u|dx−

R^Nb(x)|u|^p−1|u−v|dx

≥T(u)−

R^Nc1

θ(x) +|∇u|^p−1

|∇v− ∇u|dx

−

R^Nb(x)^{(p−1)/ p}|u|^p−1b(x)^{1/ p}|u−v|dx

≥T(u)−c1·

θL^p/(p−1)(R^N)+∇u_L^p−p(R^1N)

·

R^N|∇v− ∇u|^pdx 1/ p

−

R^Nb(x)|u|^pdx (p−1)/ p

·

R^Nb(x)|v−u|^pdx 1/ p

≥T(u)−Cu−v, ∀v∈E.

(3.8)

It is clear that takingδ=/C relation (3.3) holds true for allv∈Ewithv−u< δ.

Thus we have proved thatT is strongly lower semicontinuous. Taking into account the fact that T is convex then by [3, Corollary III.8] we conclude thatT is weakly lower semicontinuous and the proof ofProposition 3.1is complete.

Proposition 3.2. Assume{u_n}is a subsequence fromEwhich is weakly convergent tou∈E and

lim sup

n→∞

Tun

,un−u≤0. (3.9)

Then{un}converges strongly touinE.

Proof. Since{un}is weakly convergent touinEit follows that{un}is bounded inE.

By conditions (A2) and (A3) we have 0≤A(x,ξ)=

₁

0

d

dtA(x,tξ)dt= ₁

0a(x,tξ)·ξ dt

≤c1

₁

0

θ(x) +|ξ|^p−1t^p−1dt

≤c1

θ(x)|ξ|+1 p^|ξ|^p

, ∀x,ξ∈R^N.

(3.10)

Thus, there exists a constantc2>0 such that A(x,ξ)≤c2

θ(x)|ξ|+|ξ|^p

, ∀x,ξ∈R^N. (3.11)

Relation (3.11) and H¨older’s inequality imply

R^NAx,∇u_ndx≤c2

R^Nθ(x)∇u_ndx+

R^N

∇u_n^pdx

≤c2·

θL^p/(p−1)(R^N)·un+un^p .

(3.12)

The above inequality and the fact that{un}is bounded inEshow that there existsM1>0 such thatT(un)≤M1for alln. Then we may assume thatT(un)→γ. UsingProposition 3.1we find

T(u)≤lim inf

n→∞ Tun

=γ. (3.13)

SinceTis convex the following inequality holds true

T(u)≥Tu_n+Tu_n,u_n−u, ∀n. (3.14) Relation (3.9) and the above inequality implyT(u)≥γand thusT(u)=γ.

We also have (un+u)/2 converges weakly touinE. Using againProposition 3.1we deduce

γ=T(u)≤lim inf

n→∞ T un+u

2

. (3.15)

If we assume by contradiction thatun−udoes not converge to 0 then there exists>0 such that passing to a subsequence{unm}we haveunm−u ≥. That fact and relation (3.6) imply

1

2T(u) +1 2Tunm

−T

u+u_nm 2

≥k1u−unm^p≥k1^p. (3.16) Lettingm→ ∞we find

lim sup

m→∞ T

u+unm

2

≤γ−k1^p (3.17)

and that is a contradiction with (3.15). Thus we have

un−u−→0. (3.18)

The proof ofProposition 3.2is complete.

4. Proof ofTheorem 2.1

In order to proveTheorem 2.1we define the functional J(u)=

R^NA(x,∇u)dx+1 p

R^Nb(x)|u|^pdx−

R^NF(x,u)dx. (4.1) J:E→R is well defined and of classC¹with the derivative given by

J(u),ϕ=

R^Na(x,∇u)· ∇ϕ dx+

R^Nb(x)|u|^p−2uϕ dx−

R^N f(x,u)ϕ dx, (4.2) for allu,ϕ∈E. We have denoted by,the duality pairing betweenEandE, whereE is the dual ofE.

We remark that the critical points of the functionalJcorrespond to the weak solutions of (2.2). Thus, our idea is to apply the Mountain Pass theorem (see [2]) in order to obtain a non-trivial critical point and thus a non-trivial weak solution.

First, we prove a lemma which shows that functionalJhas a mountain-pass geometry.

Lemma 4.1. (1) There existρ >0 andρ>0 such that

J(u)≥ρ>0, ∀u∈E withu =ρ. (4.3) (2) There existsu0∈Esuch that

limt→∞Jtu0

= −∞. (4.4)

Proof. (1) By (F2) there existA1,A2>0 two constants such that

0≤F(x,z)≤A1|z|^r+1+A2|z|^s+1. (4.5) Then we deduce that

|limz|→0

F(x,z)

|z|^p =0, lim

|z|→∞

F(x,z)

|z|^p =0. (4.6)

Then, for a>0 there exist two constantsδ1andδ2such that F(x,z)<|z|^p ∀z with|z|< δ1,

F(x,z)<|z|^p ∀z with|z|> δ2. (4.7) Relation (4.5) implies that for allzwith|z| ∈[δ1,δ2] there exists a positive constantC >0 such that

F(x,z)< C. (4.8)

We obtain that for all>0 there existsC>0 such that

F(x,z)≤|z|^p+C|z|^p. (4.9) Relation (4.9), conditions (A5) and (b1) and the Sobolev embedding imply

J(u)=

R^NA(x,∇u)dx+1 p

R^Nb(x)|u|^pdx−

R^NF(x,u)dx

≥Λ

R^N|∇u|^pdx+1 p

R^Nb(x)|u|^pdx−

R^N|u|^pdx−C

R^N|u|^pdx

≥min Λ,1

p

· u^p− b0

R^Nb(x)|u|^pdx−C

R^N|u|^pdx

≥ u^p_·

min Λ,1

p

− b0

−C_· u^p−p

.

(4.10)

Letting∈(0, min{Λ, 1/ p} ·b0) be fixed, we obtain that the first part ofLemma 4.1holds true.

(2) To prove the second part of the lemma, first, we remark that by condition (F3) we have

F(x,z)_≥λ_|z_|^μ, ∀|z_{| ≥}η,x_∈R^N, (4.11) whereλandηare two positive constants.

On the other hand we claim that

A(x,zξ)≤A(x,ξ)z^p, ∀z≥1,x,ξ∈R^N. (4.12) Indeed, if we putα(t)=A(x,tξ) then by (A1) and (A4) we have

α(t)=a(x,tξ)·ξ=1

ta(x,tξ)·(tξ)≤ p

tA(x,tξ)= p

tα(t). (4.13) Hence

α(t) α(t) ^≤

p

t (4.14)

or

logα(t)−logα(1)≤plog(t). (4.15) We deduce thatα(t)/α(1)≤t^pand thus (4.12) holds true.

Let nowu0∈Ebe such that meas({x∈R^N; |u0(x)| ≥η})>0. Using relations (4.11) and (4.12) we obtain

Jtu0

=

R^N

Ax,t∇u0

+1

pb(x)t^pu0^p dx−

R^NFx,tu0

dx

≤t^p

R^N

Ax,∇u0

+ 1

pb(x)u0^p dx−

{x∈R^N;|u0(x)|≥η}Fx,tu0

dx

−

{x∈R^N;|u0(x)|≤η}Fx,tu0

dx

≤t^p

R^N

Ax,∇u0

+ 1

pb(x)u0^p dx−t^μλ

{x∈R^N;_|u0(x)|≥η}

u0^μdx.

(4.16)

Sinceμ > pthe right-hand side of the above inequality converges to−∞ast→ ∞.

The lemma is completely proved.

Proof ofTheorem 2.1. UsingLemma 4.1we may apply the Mountain Pass theorem (see [2]) to functionalJ. We obtain that there exists a sequence{un}inEsuch that

Jun

−→c >0, Jun

−→0 inE. (4.17)

We prove that{u_n}is bounded inE. We assume by contradiction thatu_n → ∞asn→

∞. Then, using relation (4.17) and conditions (A4), (A5) and (F3) we deduce that forn large enough the following inequalities hold

c+ 1 +un≥Jun

−1 μ

Jun

,un

=

R^N

Ax,∇u_n−1

μax,∇u_n· ∇u_n

dx +

R^N

1

pb(x)un^p−1

μb(x)un^p dx +

R^N

1

μfx,u_nu_n−Fx,u_ndx

≥

1−p μ

R^NAx,∇un

dx+ 1

p⁻ 1 μ

R^Nb(x)un^pdx

≥

1−p μ

Λ

R^N

∇un^pdx+ 1

p⁻ 1 μ

R^Nb(x)un^pdx

≥min

1−p μ

Λ,1 p⁻

1 μ

·un^p.

(4.18)

Dividing byu_nand lettingn→ ∞we obtain a contradiction. Therefore{u_n}is bounded inEby a positive constant denoted byM. It follows that there existsu∈Esuch that, pass- ing to a subsequence still denoted by{un}, it converges weakly touinEandun(x)→u(x) a.e.x∈R^N. SinceEis continuously embedded inL^p(R^N) by [17, Theorem 10.36] we de- duce thatunconverges weakly touinL^p(R^N). Then it is clear that|un|^r−1unconverges weakly to|u|^r−1uinL^p/r(R^N).

Define the operatorU:L^p/r(R^N)→R by U,w =

R^Nτ1(x)uw dx. (4.19)

We remark thatUis linear and continuous provided thatτ1∈L^r0(R^N),u∈L^p(R^N) and 1/ p+r/ p+ 1/r0=1. All the above pieces of information imply

U,u_n^r−1u_n−→

U,|u|^r−1u, (4.20) that is,

nlim→∞

R^Nτ1(x)u_n^r−1u_nu dx=

R^Nτ1(x)|u|^r+1dx. (4.21) With the same arguments we can show that

nlim→∞

R^Nτ2(x)un^s−1unu dx=

R^Nτ2(x)|u|^s+1dx, (4.22)

nlim→∞

R^Nτ1(x)un^r+1dx=

R^Nτ1(x)|u|^r+1dx, (4.23)

nlim→∞

R^Nτ2(x)un^s+1dx=

R^Nτ2(x)|u|^s+1dx. (4.24) Relations (4.21), (4.23) and the fact that

R^Nτ1(x)u_n^r−1u_nu_n−udx=

R^Nτ1(x)u_n^r+1dx−

R^Nτ1(x)|u|^r+1dx +

R^Nτ1(x)|u|^r+1dx−

R^Nτ1(x)u_n^q−1u_nu dx (4.25)

yield

nlim→∞

R^Nτ1(x)un^r−1un

un−udx=0. (4.26)

Similarly we obtain

nlim→∞

R^Nτ2(x)un^s−1un

un−udx=0. (4.27)

By (4.26), (4.27) and condition (F2) we get

nlim→∞

R^N fx,u_nu_n−udx=0. (4.28) On the other hand we have

R^Nax,∇u_n· ∇u_ndx+

R^Nb(x)u_n^p−2u_nu_n−udx

=

Ju_n,u_n−u+

R^N fx,u_nu_n−udx.

(4.29)

Relations (4.28) and (4.29) imply

nlim→∞

R^Nax,∇un

· ∇

un−udx+

R^Nb(x)un^p−2

un−udx

=0, (4.30) that is,

nlim→∞

Tun

,un−u=0, (4.31)

whereTis the functional defined in the above section. Then applyingProposition 3.2we deduce that{un}converges strongly touinE. SinceJ∈C¹(E, R) by (4.17) we deduce that J(u),ϕ =0 for allϕ∈E, that is,uis a weak solution of problem (2.2). Relation (4.17) also implies thatJ(u)=c >0 and that shows thatuis non-trivial.

The proof ofTheorem 2.1is complete.

5. Proof ofTheorem 2.2

We remark that the weak solutions of (2.11) correspond to the critical points of the energy functionalI:E→R defined as follows

I(u)=

R^NA(x,∇u)dx+1 p

R^Nb(x)|u|^pdx− 1 q+ 1

R^Nh(x)|u|^q+1dx

− 1 s+ 1

R^Ng(x)|u|^s+1dx, ∀u∈E.

(5.1)

A simple calculation shows thatIis well defined onEandI∈C¹(E, R) with I(u),ϕ=

R^Na(x,∇u)· ∇ϕ dx+

R^Nb(x)|u|^p−2uϕ dx

−

R^Nh(x)|u|^q−1uϕ dx−

R^Ng(x)|u|^s−1uϕ dx,

(5.2)

for alluandϕ∈E.

Lemma 5.1. The following assertions hold.

(i) There existρ >0 andρ>0 such that

I(u)≥ρ>0, ∀u∈E withu =ρ. (5.3) (ii) There existsψ∈Esuch that

limt→∞I(tψ)= −∞. (5.4)

(iii) There existsϕ∈Esuch thatϕ≥0,ϕ=0 and

I(tϕ)<0 (5.5)

fort >0 small enough.