Monotonicity constraints and supercritical Neumann problems

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Monotonicity constraints and supercritical Neumann problems

Enrico Serra

¹

, Paolo Tilli

^∗

Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy Received 22 April 2010; received in revised form 24 September 2010; accepted 24 September 2010

Available online 20 October 2010

Abstract

We prove the existence of a positive and radially increasing solution for a semilinear Neumann problem on a ball. No growth conditions are imposed on the nonlinearity. The method introduces monotonicity constraints which simplify the existence of a minimizer for the associated functional. Special care must be employed to establish the validity of the Euler equation.

Résumé

On démontre l’existence d’une solution positive et radialement croissante pour un problème de Neumann semilinéaire sur une boule. Aucune restriction de croissance n’est imposée sur la nonlinéarité. La méthode indroduit des contraintes de monotonie qui simplifient la preuve de l’existence d’un minimum pour la fonctionnelle associée à l’équation. Une attention particulière est consacrée à la preuve de la validité de l’équation d’Euler.

MSC:35J60; 58E30

Keywords:Constrained variational problems; Supercritical elliptic equations; Neumann problem

1. Introduction and statement of the main result In this paper we consider the Neumann problem

⎧⎪

⎪⎨

⎪⎪

⎩

−u+u=a

|x|

f (u) inB_R,

u >0 inB_R,

∂u

∂ν =0 on∂B_R,

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whereB_Ris the ball of radiusRcentered at zero inR^N, withN2.

The functionf is assumed to behave superlinearly, and our principal scope is to prove, under qualitative assumptions ona, an existence result without any restriction on the growth off. This generality usually prevents the use of

* Corresponding author.

E-mail addresses:enrico.serra@polito.it (E. Serra), paolo.tilli@polito.it (P. Tilli).

1 The author was supported by MIUR Project “Variational Methods and Nonlinear Differential Equations”.

doi:10.1016/j.anihpc.2010.10.003

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variational methods; nevertheless we will show that the problem can still be treated variationally, and, more precisely, that a solution can be obtained by a variational principle on the set of radiallyincreasingfunctions.

In trying to prove existence results for nonlinear elliptic problems, it is common wisdom to exploit the characteristic features of the equation under study to gain useful properties. For example, if a problem (equation and boundary conditions) is invariant under rotation, it is quite natural, as a first try, to work in the space of radially symmetric functions. Then the problem becomes essentially one-dimensional and, for instance, compactness properties are not only much easier to establish, but often hold under less restrictive assumptions. In this case everything works fine in the variational approach because of the Principle of Symmetric Criticality by Palais [8]. This means, loosely speaking, that constraining the problem to the space of radial functions does not prevent one to establish the validity of the Euler equation. One also says that this type of constraint is anatural constraint: constrained critical points (of the energy functional) are indeed free critical points.

This sort of arguments work generally well when the problem is invariant under the action of some group of symmetries, but may fail to be applicable in more general situations. For example, in some cases it is natural to expect a solution enjoying a certain property, but one cannot work directly in the set of functions satisfying that property. The reason for this is that the set to which one constrains the problem may not contain enough functions to prove that the differential of the energy functional vanishes at a supposedly “critical” point.

Problems like (1) have been studied extensively in the literature, and particularly in the casea≡1 andf (u)= u^N^N−2⁺², see for example [5,6,1,2,7,10] and the rich list of references provided by these papers. In contrast, nonau- tonomous problems, wherea is not constant, and even with power nonlinearities, do not seem to have been deeply investigated in the supercritical case (with the exception of singularly perturbed problems, see for example [4] and the references therein). The only nonperturbative result that we are aware of is that of [3], which is concerned with the Neumann problem for the Hénon equation

−u+u= |x|^αu^p.

In [3] the authors prove, by a shooting method, that the problem has a positive and radiallyincreasingsolution for everyp >1 andα >0. The proof of the existence result in [3] makes great use of the fact that the functionr→r^α vanishes atr=0. Also the particularity that all the functions (ofrandu) involved are powers plays a relevant role.

If one neglects the technical aspects, the proof in [3] suggests that if the function|x|^α is replaced by somea(|x|), increasingin|x|, and if the power nonlinearity is substituted by some more generalf (u), the qualitative structure of the problem should not be affected too much. In particular under some reasonable assumptions of this kind, one is led to conjecture that problem (1) should admit an increasing solution.

The central point addressed by the present paper is the following. In absence of growth conditions on f, the variational approach seems useless, since the functional associated to the problem is not well defined on H¹(B_R), and not even on H_rad¹ (B_R). But we are looking for radially increasing solutions. If we could confine ourselves to radially increasing functions only, then the action functional would turn out to be well defined onH_rad¹ (B_R), because nonnegative radially increasingH¹functions are bounded. In this case it could be possible to recast the problem into a variational framework, which is so useful in dealing with subcritical equations. However, a major obstacle has to be faced: are there enough radially increasing functions to prove that the differential of the action functional, say at a constrained minimum point, vanishes? At first sight one is tempted to answer this question in the negative. On the contrary, we shall see that under suitable conditions, a variational approach in the set of radially increasing functions is possible, and does indeed lead to a solution of (1).

The precise setting is the following. Throughout the paper we make the following assumptions:

(H1) a∈L¹(0, R)is increasing, not constant anda(r) >0 a.e. in[0, R];

(H2) f ∈C¹([0,+∞)),f (0)=0 andf(0)=0;

(H3) f(t )t−f (t ) >0 for allt >0;

(H4) there existsμ >2 such thatf (t )tμF (t ):=μ_t

0f (s) dsfor allt∈ [0,+∞).

Assumptions (H2)–(H4) are quite standard when one deals with elliptic problems from a variational point of view (though they can still be slightly relaxed). The point here is the absence of any upper bound on the growth off.

Our main result is the following.

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Theorem 1.1.Assume that(H1)–(H4)hold. Then problem(1)admits at least one radially increasing solution.

Remark 1.2.We have stated the result for nonlinearities of the forma(|x|)f (u)mainly because the assumptions can be expressed in a very simple form. It is not difficult to show that the same result holds for nonlinearities likef (|x|, u), provided (H1)–(H4) are suitably adapted.

Power nonlinearities satisfy (H2)–(H4). Therefore we obtain directly from the preceding theorem the following corollary.

Corollary 1.3.Assume thata∈L¹(0, R)is increasing, not constant and satisfiesa(r) >0 a.e. in[0, R]. Then for everyp >1the problem

⎧⎪

⎨

⎪⎩

−u+u=a(|x|)u^p inBR,

u >0 inBR,

∂u

∂ν =0 on∂B_R,

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admits at least one radially increasing solution.

We point out that, apart from the existence result, we believe that themethodby which the result is proved is quite interesting, and should have a wider range of applicability than that shown here. In particular, variational methods on spaces of monotone functions do not seem to be a standard tool: the only example we are aware of, though in a different context, is the paper [9]. Some extensions to other types of equations will be the object of further research.

Notation.Open balls of center zero and radiusrinR^Nare denoted byBr. For any radial functionu, we write freely u(x), withx∈BR, oru(r), withr∈ [0, R]. The symbol u denotes the standardH¹(BR)norm, while u p, with p∈ [1,+∞], stands for theL^pnorm onBR. Finally, the scalar product inH¹(BR)is denoted byu, v.

2. Minimizing over monotone functions

We begin by listing, for further reference, some properties off andF that we will use frequently throughout the paper, and that can be deduced directly from (H2)–(H4).

The functionsf andF are strictly increasing on[0,+∞)and positive on(0,+∞). Ast→0⁺,f (t )=o(t )and F (t )=o(t²). Finally, there exists a constantC >0 such that

f (t )tCt^μ and F (t )Ct^μ ∀t1. (3)

Our ambient set is that of radially increasing functions. We define

M= u∈H_rad¹ (B_R)u(r)0, u(r)u(s)for everyr, s∈ [0, R], rs .

Remark 2.1.Since we are working with radial functions inH¹(BR), we can assume that any functionuis continuous on(0, R], and sinceuis increasing and nonnegative, continuity can be guaranteed also at zero by definingu(0)= lim_r_→₀+u(r), and we will tacitly use this fact throughout the paper.

The main advantage of working in the setM is that concentration phenomena are prevented, as is pointed out in the next statement.

Lemma 2.2.There exists a positive constantC, depending only onRand on the dimension, such that

u _∞C u ∀u∈M. (4)

Proof. Fixρ∈(0, R). Since anyu∈Mis nonnegative and increasing, we have u _∞= u _L_∞_(B

R\Bρ)C u _H1(B_R\Bρ)C u ,

by the continuity of the embedding ofH_rad¹ (B_R\B_ρ)intoL^∞(B_R\B_ρ). 2

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We now define a functionalI:M→Rby I (u)=1

2

B_R

|∇u|²dx+1 2

B_R

u²dx−

B_R

a

|x| F (u) dx

=1 2 u ²−

B_R

a

|x|

F (u) dx.

Note that the last integral is well defined onMbecauseM⊂L^∞(B_R). Clearly, working onMallows one to defineI without growth conditions onf; on the other hand, it is by no means clear why constrained minimizer should solve the Euler equation (1).

In general, the functionalI will not be bounded from below onM. To overcome this difficulty we use a standard procedure in Critical Point Theory, that consists in restrictingI to a set where it becomes bounded from below. The set we use is the following version of “Nehari manifold”: we define

N =

u∈Mu≡0, u ²=

BR

a

|x|

f (u)u dx

.

This set has been used quite frequently in the literature. Under standard assumptions it can be proved that it is a manifold diffeomorphic to the unit sphere ofH¹. In the present case some of these standard assumptions are missing, and are substituted by others. Moreover, since we are working onM, the setN does not look like the sphere ofH_rad¹ (BR).

Hence, for completeness, we prove all the properties that we need.

Lemma 2.3.The setN is not empty. In particular, for everyu∈M,u≡0, there existst >0such thatt u∈N. Proof. Letu∈M,u≡0. We show that there existst >0 such thatt u∈N, namely such that

t² u ²−

BR

a

|x|

f (t u)t u dx=0. (5)

Letγ : [0,+∞)→Rbe the function oft defined by the left-hand side of (5). We have to prove thatγ vanishes at some positivet.

Nowf (t )=o(t )ast→0⁺, andu∈L^∞(B_R). Therefore, for everyε >0, there existsδ >0 such that|t u(x)|δ implies 0f (t u(x))εt u(x).

Hence, for every 0t _u^δ_∞, 0

BR

a

|x|

f (t u)t u dxεt²

BR

a

|x|

u²dxεt² u ²_∞ a ₁=Cεt².

Therefore,

γ (t )t² u ²−Cεt²>0

fortandεpositive and small. We have proved thatγ is positive in a right neighborhood of zero.

Next, sinceu0 andu≡0, there existα >0 and a setE⊂BRof positive measure such thatu(x)αfor all x∈E. By (3), fort1/α,

BR

a

|x|

f (t u)t u dxCt^μ

E

a

|x| u^μdx,

which is finite becauseu∈L^∞(B_R). Then, γ (t )t² u ²−Ct^μ

E

a

|x|

u^μdx→ −∞

ast→ +∞, becauseμ >2.

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The functionγ, being continuous, must then have a positive zerot₀. This means thatt₀u∈N. 2 We now define

c= inf

u∈NI (u),

and we prove thatcis a positive number. Note that onN, the functionalI reads I (u)=1

2

B_R

a

|x|

f (u)u dx−

B_R

a

|x|

F (u) dx.

Lemma 2.4.There resultsc >0.

Proof. We first prove that

uinf∈N u ²>0. (6)

To see this, we assume that there exists a sequenceu_nof elements ofN such that u_n →0 asn→ ∞, and we seek a contradiction.

LettingCbe the constant defined in (4), we have from Lemma 2.2 u_n ²=

B_R

a

|x|

f (u_n)u_ndxf u_n _∞

u_n _∞

B_R

a

|x| dx

f

C un

C un a 1,

and hence, sincef (t )=o(t )ast→0, from u_n →0 we obtain a contradiction. Therefore u_n ²is bounded away from zero.

Now we turn to the levelc. For everyu∈N we have, using (H4), I (u)=1

2 u ²−

B_R

a

|x| F (u) dx

=1

2 u ²− 1

μ u ²+ 1 μ u ²−

BR

a

|x| F (u) dx

= 1

2 −1 μ

u ²+1 μ

B_R

a

|x|

f (u)u−μF (u) dx

1

2 −1 μ

u ². (7)

By (6), this concludes the proof. 2

We can now show thatI has a minimizer onN.

Lemma 2.5.The levelcis attained:there existsu∈N such thatI (u)=c.

Proof. Letu_nbe a minimizing sequence forI onN. By (7), we see thatI is coercive onN, and hence the sequence u_nis bounded inH¹(B_R). Then, up to subsequences, we can assume that for a suitableu∈H¹(B_R)

• u_n uinH¹(B_R),

• u_n→upointwise almost everywhere inB_R,

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and sinceu_n∈M, we clearly have that alsou∈M. Moreover, by Lemma 2.2, we see that the norms u_n _∞ are uniformly bounded. Hence, by dominated convergence, asn→ ∞,

c+o(1)=I (u_n)=1 2

B_R

a

|x|

f (u_n)u_ndx−

B_R

a

|x|

F (u_n) dx

=1 2

B_R

a

|x|

f (u)u dx−

B_R

a

|x|

F (u) dx+o(1). (8)

Sincec >0, this shows thatu≡0, so thatu∈M\ {0}.

Now ifu∈N, then

B_R

a

|x|

f (u)u dx= u ²,

and the proof is complete, since the last equation shows thatI (u)=c.

If, on the other hand,u /∈N, then weak lower semicontinuity shows that necessarily u ²<

BR

a

|x|

f (u)u dx. (9)

We now prove that this cannot hold, so that u does indeed belong to N. We consider again the function γ: [0,+∞)→Rdefined by

γ (t )=t² u ²−

BR

a

|x|

f (t u)t u dx.

Since u≡0, we certainly haveγ (t ) >0 fort positive and small, as in Lemma 2.3. Moreover, by (9), we see that γ (1) <0, so thatγ must have a zerot₀∈(0,1).

Then the functiont₀uis inN. By (H3), the function t→1

2f (t )t−F (t )

is strictly increasing fort0. Therefore, sincet₀<1, cI (t₀u)=

BR

a

|x|1

2f (t₀u)t₀u−F (t₀u)

dx

<

B_R

a

|x|1

2f (u)u−F (u)

dx=c,

by (8). This shows that (9) cannot hold, and thereforeu∈N and is the required minimizer. 2 3. The Euler equation

In the preceding section we have found a minimizer for I onN. The fact that this minimizer solves the Euler equation is not obvious at all, since a priori there are not enough “test” functions to show that the first variation ofI vanishes on H_rad¹ (B_R). This happens because we are working with increasing functions only. To be more precise, let u∈N be the minimizer found above. If u satisfies an estimate like u(r)δ >0 for all r in some interval (a, b)⊂(0, R), then for every ϕ ∈C^∞₀ (a, b), the functions u+εϕ (after radial projection on N) are admissible variations whenεis small enough, and this is enough to show thatuweakly solves the Euler equation. However, we do not have for the moment any lower bound onu, and in principleumight even be constant on certain subintervals of(0, R). On these intervals, there would be little hope to prove that the Euler equation is satisfied.

We now show how these problems can be overcome. We begin with a necessary condition for minimizers.

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Lemma 3.1.Letube a minimizer forIonN. Assume thatv∈H_rad¹ (B_R)(not necessarily inM)is such thatu+sv∈ Mfor everysin some interval[0, ε). Then

BR

∇u∇v dx+

BR

uv dx

BR

a

|x|

f (u)v dx. (10)

Proof. Sinceudoes not vanish identically, the same holds foru+svfor allssmall enough, and we can then assume that this happens for everys∈ [0, ε).

Each functionu+svis inM, and hence, by Lemma 2.3, for everys∈ [0, ε)there existst=t (s)such that t (s)(u+sv)∈N.

We now study some properties oft (s).

To this aim we define a functionG: [0, ε)×R→Rby G(s, t )=t² u+sv ²−

BR

a

|x| f

t (u+sv)

t (u+sv) dx.

We have

G(0,1)= u ²−

B_R

a

|x|

f (u)u dx=0

becauseu∈N. Next,Gis of classC¹and a simple computation, together with the fact thatu∈N, yields

∂G

∂t (0,1)=2 u ²−

B_R

a

|x|

f(u)u²dx−

B_R

a

|x|

f (u)u dx

=

B_R

a

|x|

f (u)u dx−

B_R

a

|x|

f(u)u²dx

=

B_R

a

|x| u

f (u)−f(u)u dx <0 because of (H3).

Then, a simple variant of the Implicit Function Theorem shows that there existδ >0 and a functiont: [0, δ)→R of classC¹such that

G s, t (s)

=0 ∀s∈ [0, δ), andt (0)=1. This means that

t (s)(u+sv)∈N ∀s∈ [0, δ).

Now defineH: [0, δ)→Ras H (s)=I

t (s)(u+sv)

=1

2t (s)² u+sv ²−

BR

a

|x| F

t (s)(u+sv) dx.

By construction,hisC¹and has a local minimum at 0, so thatH(0)0.

Hence,

0H(0)=t(0) u ²+ u, v −

BR

a

|x|

f (u)t(0)u dx−

BR

a

|x|

f (u)v dx

=t(0)

u ²−

BR

a

|x|

f (u)u dx

+ u, v −

BR

a

|x|

f (u)v dx.

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Sinceu∈N, the first term in the last line vanishes, so that u, v

BR

a

|x|

f (u)v dx,

and (10) is proved. 2

Our scope is to show that (10) holds for every v∈H_rad¹ (B_R). The first step consists in gathering some more information about the minimizeru.

To this aim, consider the problem −w+w=0 inB_R,

w=u on∂B_R. (11)

It is well known that this problem has a unique solutionϕ, which is radial, strictly positive onB_R, and strictly radially increasing. The solutionϕcan be found for instance by solving

min w ²w∈H¹(B_R), w_|_∂B_R=u_|_∂B_R

and its properties follow by a standard application of the strong maximum principle. Actually, problem (11) can be explicitly solved in terms of modified Bessel functions.

The functionϕserves as a lower bound for the minimizeru.

Lemma 3.2.Letube a minimizer forI onN, and letϕbe the solution of (11). Then

u(x)ϕ(x) ∀x∈BR. (12)

In particular,uis strictly positive inBR.

Proof. We first note that for everys∈ [0,1], the function u+s(ϕ−u)⁺

is inM. Indeed,u(x)+s(ϕ(x)−u(x))⁺=max(u(x), sϕ(x)+(1−s)u(x)), and all the functions involved are radially increasing.

By Lemma 3.1,

B_R

∇u∇(ϕ−u)⁺dx+

B_R

u(ϕ−u)⁺dx

B_R

a

|x|

f (u)(ϕ−u)⁺dx0. (13)

Next, the definition ofϕshows that(ϕ−u)⁺∈H₀¹(B_R), and sinceϕsolves problem (11),

BR

∇ϕ∇(ϕ−u)⁺dx+

BR

ϕ(ϕ−u)⁺dx=0.

Subtracting this from (13) we obtain 0

BR

∇(u−ϕ)∇(ϕ−u)⁺dx+

BR

(u−ϕ)(ϕ−u)⁺dx= −(ϕ−u)⁺²,

which means thatuϕonB_R. 2

The next result is crucial in order to prove thatusolves the Euler equation. In its statement and proof, we recall that we write freelyu(x), withx∈B_R, oru(r), withr∈ [0, R], depending on the convenience of the moment, and similarly for all radial functions. The context will always rule out any ambiguity.

Proposition 3.3.Letube a minimizer forI onN. Then

u(r) >0 a.e. in(0, R). (14)

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Remark 3.4.The point here is thestrictinequality. The functionu, being increasing, obviously satisfies u(r)0 whereveru(r)is defined.

Proof. Fix ρ ∈(0, R)such that ρ is a Lebesgue point for u(r)r^N⁻¹ and let λ be apositive real number to be determined later. Finally, letδbe a small positive number that will eventually tend to zero.

Now consider the piecewise linear functionv_δ: [0, R] →Rdefined as v_δ(r)=

⎧⎨

⎩

−1 ifr∈ [0, ρ],

−1+¹⁺_δ^λ(r−ρ) ifr∈ [ρ, ρ+δ], λ ifr∈ [ρ+δ, R].

Clearly, the functionx→vδ(|x|), which we call simplyvδ, is inH_rad¹ (BR).

The functionv_δis increasing, but is not inM, since it is not positive. However, for every nonnegative and smalls, the functionu+sv_δbelongs toM, because of Lemma 3.2.

Then, by Lemma 3.1,

BR

∇u∇v_δdx+

BR

uv_δdx

BR

a

|x|

f (u)v_δdx. (15)

Whenδ→0⁺, the functionsv_δtend to v₀(r)=

−1 ifr∈ [0, ρ], λ ifr∈(ρ, R],

pointwise and in everyL^p(BR)withpfinite.

Hence, asδ→0⁺,

B_R

uvδdx→ −

B_ρ

u dx+λ

B_R\B_ρ

u dx

and, similarly,

B_R

a

|x|

f (u)vδdx→ −

B_ρ

a

|x|

f (u) dx+λ

B_R\B_ρ

a

|x|

f (u) dx.

Furthermore, denoting byσ the area of the unit sphere inR^N,

BR

∇u∇v_δdx=σ R 0

u(r)v_δ(r)r^N⁻¹dr=σ1+λ δ

ρ+δ

ρ

u(r)r^N⁻¹dr.

Sinceρis a Lebesgue point foru(r)r^N⁻¹, asδ→0⁺we have

B_R

∇u∇vδdx→σ (1+λ)ρ^N⁻¹u(ρ).

We can now letδ→0⁺in (15), to conclude that for almost everyρ∈(0, R), σ (1+λ)ρ^N⁻¹u(ρ)−

B_ρ

u dx+λ

B_R\B_ρ

u dx−

B_ρ

a

|x|

f (u) dx+λ

B_R\B_ρ

a

|x|

f (u) dx,

which we write as

(1+λ)ρ^N⁻¹u(ρ) ρ 0

ur^N⁻¹dr−λ R ρ

ur^N⁻¹dr

− ρ 0

a(r)f (u)r^N⁻¹dr+λ R ρ

a(r)f (u)r^N⁻¹dr. (16)

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Now the functions a and u are increasing, and the function t→ ^{f (t )}_t is strictly increasing by (H3). All these functions are nonnegative. Therefore

a(r)f u(r)

=a(r)f (u(r))

u(r) u(r)a(ρ)f (u(ρ))

u(ρ) u(r) for a.e.r∈ [0, ρ], while

a(r)f u(r)

=a(r)f (u(r))

u(r) u(r)a(ρ)f (u(ρ))

u(ρ) u(r) for a.e.r∈ [ρ, R]. Hence we obtain

− ρ 0

a(r)f (u)r^N⁻¹dr−a(ρ)f (u(ρ)) u(ρ)

ρ 0

u(r)r^N⁻¹dr (17)

and λ

R ρ

a(r)f (u)r^N⁻¹drλa(ρ)f (u(ρ)) u(ρ)

R ρ

u(r)r^N⁻¹dr. (18)

The key remark now is that since the functionais not constant, at least one of the two preceding inequalities must be strict. Indeed, if they were both equalities, then we should have

a(r)f (u(r))

u(r) =a(ρ)f (u(ρ))

u(ρ) for almost everyr∈ [0, R].

Ifuis not constant, this is impossible becausef (t )/t is strictly increasing. Ifuis constant, thenamust be constant too, against assumption (H1).

Inserting (17) and (18) into (16), and keeping in mind the preceding remark we see that for almost everyρ∈(0, R), (1+λ)ρ^N⁻¹u(ρ) >

ρ 0

ur^N⁻¹dr−λ R ρ

ur^N⁻¹dr+a(ρ)f (u(ρ)) u(ρ)

− ρ 0

ur^N⁻¹dr+λ R ρ

ur^N⁻¹dr

. (19) We now chooseλ >0 such that

ρ 0

ur^N⁻¹dr=λ R ρ

ur^N⁻¹dr,

which is possible since both integrals are positive, by Lemma 3.2. Then (19) becomes (1+λ)ρ^N⁻¹u(ρ) >0 for almost everyρ∈(0, R).

Sinceλis positive, this is the required inequality. 2

We now show that the estimate provided by the preceding proposition is enough to show that the “first variation”

of the functionalI vanishes on a dense subset ofH_rad¹ (BR).

Proposition 3.5.Letube a minimizer forI onN. Then

BR

∇u∇v dx+

BR

uv dx=

BR

a

|x|

f (u)v dx for everyv∈C_rad¹ (B_R).

Proof. Let Ek=

r∈(0, R)u(r)exists andu(r) >1 k

, k=1,2, . . .

(11)

and note that

r∈(0, R)u(r)exists andu(r) >0

=

k

E_k, meas

[0, R] \

k

E_k

=0 (20)

by the preceding proposition.

Denote byχ_E_k be the characteristic function of the setE_k.

Now take anyv∈C_rad¹ (B_R), and define, for everyk=1,2, . . ., a functionv_k: [0, R] →Rby v_k(r)=v(0)+

r 0

v(s)χ_E_k(s) ds.

Of course,v_k∈W^1,^∞(0, R)and v_k(r)=v(r)χE_k(r) for almost everyr∈ [0, R].

Now for a fixedk, we letε∈Rbe such that

|ε|<min

u(0)

1+ v _∞, 1 k(1+ v _∞)

(21) and for suchε’s we claim that

u+εv_k∈M\ {0}.

To see this, we use (21): we first note that for everyr∈ [0, R], u(r)+εv_k(r)u(0)− |ε| v _∞> u(0)− v _∞

1+ v _∞u(0) >0.

Next, for almost everyr∈(0, R), we have (u+εvk)(r)=u(r)+εv(r)χE_k(r).

Ifr∈E_k, then

(u+εv_k)(r)=u(r)+εv(r)χ_E_k(r)=u(r)+εv(r) >1

k− |ε| v _∞>1

k− v _∞

k(1+ v _∞)>0.

Ifr /∈E_k, then

(u+εvk)(r)=u(r),

and we know from Proposition 3.3, thatu(r) >0 almost everywhere.

It follows that(u+εv_k)(r) >0 for almost everyr∈(0, R), and then the functionu+εv_k is also increasing.

Hence, by Lemma 3.1, R

0

uv_kr^N⁻¹dr+ R 0

uv_kr^N⁻¹dr R 0

a(r)f (u)v_kr^N⁻¹dr, ∀k=1,2, . . . . (22) We now letk→ ∞. To this aim we note that

v_k(r)v(0)+ r 0

v(s)χ_E_k(s)dsv(0)+Rv

∞

and v_k _∞ v _∞. By the Ascoli–Arzelà Theorem,vk→v uniformly on[0, R]. Moreover, sinceEk⊂Ek+1for everyk, and by (20),

v_k(r)→v(r) a.e. in(0, R).

(12)

By dominated convergence, we can letk→ ∞in (22) to obtain R

0

uvr^N⁻¹dr+ R 0

uvr^N⁻¹dr R

0

a(r)f (u)vr^N⁻¹dr.

Sincevis arbitrary, this means

B_R

∇u∇v dx+

B_R

uv dx=

B_R

a

|x|

f (u)v dx for everyv∈C_rad¹ (B_R), and the proof is complete. 2

End of the proof of Theorem 1.1. By the previous result one easily deduces, with a density argument, that

B_R

∇u∇v dx+

B_R

uv dx=

B_R

a

|x|

f (u)v dx for everyv∈H_rad¹ (BR), namely thatu∈Mweakly solves problem (1). 2

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