Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy–Leray potential

Ann. I. H. Poincaré – AN 31 (2014) 1–22

www.elsevier.com/locate/anihpc

Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy–Leray potential

Susana Merchán

, Luigi Montoro

, Ireneo Peral

, Berardino Sciunzi

aDepartamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain bDipartimento di Matematica, UNICAL, Ponte Pietro Bucci 31 B, 87036 Arcavacata di Rende, Cosenza, Italy

Received 10 September 2012; accepted 18 January 2013 Available online 1 February 2013

Abstract

In this work we deal with the existence and qualitative properties of the solutions to a supercritical problem involving the

−_p(·)operator and the Hardy–Leray potential. Assuming 0∈Ω, we study the regularizing effect due to the addition of a first order nonlinear term, which provides the existence of solutions with a breaking of resonance. Once we have proved the existence of a solution, we study the qualitative properties of the solutions such as regularity, monotonicity and symmetry.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:35J20; 35J25; 35J62; 35J70; 35J92; 46E30; 46E35

Keywords:Quasilinear elliptic equations; Hardy potential; Supercritical problems; Existence and nonexistence; Regularity; Symmetry of solutions

1. Introduction

In this paper we shall study the existence and qualitative properties of weak positive solutions to the supercritical problem

−pu+ |∇u|^p=ϑ u^q

|x|^p +f inΩ, u0 inΩ, u=0 on∂Ω,

(P)

whereΩis a bounded domain inR^Nsuch that 0∈Ω,ϑ >0,p−1< q < p,f 0,f∈L¹(Ω)and 1< p < N. The existence in the semilinear casep=2 has been investigated in the recent work[15]. We start giving the following definition.

✩ S.M., L.M. and I.P. have been partially supported by project MTM2010-18128 of MICINN Spain. L.M and B.S. are partially supported by PRIN-2009-grant:Metodi Variazionali e Topologici nello Studio di Fenomeni non Lineari. B.S. is also supported by ERC-2011-grant:Elliptic PDE’s and symmetry of interfaces and layers for odd nonlinearities.

* Corresponding author.

E-mail addresses:susana.merchan@uam.es(S. Merchán),montoro@mat.unical.it(L. Montoro),ireneo.peral@uam.es(I. Peral), sciunzi@mat.unical.it(B. Sciunzi).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.01.003

Definition 1.We say thatuis a weak solution to

−_pu+ |∇u|^p=ϑ u^q

|x|^p +f inΩ, ifu∈W₀^1,p(Ω)and

Ω

|∇u|^p−2(∇u,∇φ)+

Ω

|∇u|^pφ=ϑ

Ω

u^q

|x|^pφ+

Ω

f φ ∀φ∈W₀^1,p(Ω)∩L^∞(Ω).

The behavior of the supercritical problem with 0∈∂Ωis quite different. See details in[10]and[15]forp=2 and [16]for thep-Laplacian case 1< p < N.

In this work we consider 0∈Ω and, because of the regularizing effect due to the presence of the gradient term

|∇u|^pon the left-hand side of problem (P), we are able to prove the existence of a weak solutionu(see Definition1) to problem (P), remarkably for anyϑ >0 and for eachf ∈L¹(Ω), f 0. As nowadays well understood, the solution obtained is calledsolutions obtained as limits of approximations, or simply SOLA, see[5]. By using the results in[6]

in this case SOLA is equivalent toentropy solution, see[1], orrenormalized solution.

We have the following result:

Theorem 1.Consider problem(P)with1< p < N,p−1< q < pand assume thatf∈L¹(Ω)is a positive function.

Then for allϑ >0there exists a weak solutionu∈W₀^1,p(Ω)to(P).

This result emphasizes the fact that the term |∇u|^p on the left-hand side of (P) is enough to get a resonance breaking result. The scheme of the proof is the following:

(i) We prove the existence of a solution to the truncated problem

−_pu_k+ |∇u_k|^p=ϑ T_k u^q_k

|x|^p

+T_k(f ) inΩ, u_k∈W₀^1,p(Ω).

whereTk(s)=max{min{k, s},−k},k >0. This is done by solving the regularized problem (3) below and passing to the limit inW₀^1,p(Ω).

(ii) We show that the sequence of solutions to the truncated problem converges weakly inW₀^1,p(Ω)and then we deduce the a.e. convergence of the gradients. Finally we exploit it to deduce strong convergence inW₀^1,p(Ω).

(iii) We pass to the limit in the truncated problem and we obtain the existence of a solution to (P).

Let us remark that, because of the presence of the gradient term (which causes the existence of solutions), to pass to the limit in the truncated problem it is necessary to deduce the convergence ofuk(solutions of the truncated problem) inW₀^1,p(Ω). A convergence inW₀^1,q(Ω)withq < p, in the spirit of[3], would not be sufficient to pass to the limit and get a weak formulation of the problem.

In the second part of this paper we deal with the study of the qualitative properties of weak solutions to (P). First we point out some regularity properties of the solutions and then we prove the following result:

Theorem 2.Letu∈C¹(Ω\{0})be a weak solution to(P). Consider the domainΩstrictly convex w.r.t. theν-direction (ν∈S^N−1)and symmetric w.r.t.T₀^ν, where

T₀^ν=

x∈R^N: x·ν=0 .

Moreover, assumef ∈C¹(Ω\ {0})to be non-decreasing w.r.t. theν-direction in the set Ω₀^ν= {x∈Ω: x·ν <0}

and even w.r.t.T₀^ν. Thenuis symmetric w.r.t.T₀^νand non-decreasing w.r.t. theν-direction inΩ₀^ν. Moreover, ifΩ is a ball, thenuis radially symmetric with ^∂u_∂r(r) <0forr=0.

Remark 1.Notice that the extra regularity hypothesis onf is sufficient to have the corresponding regularity of a solution. See[20]for details on regularity.

We point out that Theorem2will be a consequence of a more general result, see Proposition3below, which states a monotonicity property of the solutions in general domains near strictly convex parts of the boundary. This can be useful for example in blow-up analysis.

Also, it will be clear from the proof, that the same technique could be applied to study the case of more general nonlinearities. In particular, we note that the nonlinearity in problem (P) is in general locally Lipschitz continuous only in(0,∞).

The main ingredient in the proof of the symmetry result is the well knownMoving Plane Method[21], that was used in a clever way in the celebrated paper[12]for the semilinear nondegenerate case. Actually our proof is more similar to the one of[2]and is based on the weak comparison principle in small domains. TheMoving Plane Method was extended to the case ofp-Laplace equations firstly in[7]for the case 1< p <2 and later in[9]for the casep2.

In the casep2 it is required the nonlinearity to be positive and as can be seen in some examples, this assumption is in general necessary.

The first crucial step is the proof of a weak comparison principle in small domains that we carry out in Proposition2.

This is based on some regularity results in the spirit of[9]. These results hold only away from the origin due to the presence of the Hardy potential in our problem. This will require more attention in the application of the moving plane procedure. Moreover, the presence of the gradient term|∇u|^p, leads to a proof of the weak comparison principle in small domains which makes use of the right choice of test functions.

Notation.Generic fixed numerical constants will be denoted byC(with subscript in some case) and will be allowed to vary within a single line or formula. Moreoverf⁺andf⁻will stand for the positive and negative part of a function, i.e.f⁺=max{f,0}andf⁻=min{f,0}.We also denote|A|the Lebesgue measure of the setA.

2. Existence of an energy solution to the problem (P)

It will be useful to refer to the following result.

Lemma 1(Hardy–Sobolev inequality). Suppose1< p < N andu∈W^1,p(R^N). Then we have

R^N

|u|^p

|x|^p CN,p

R^N

|∇u|^p,

withC_N,p=(_N^p_−p)^poptimal and not achieved constant.

2.1. Existence of a solution to the truncated problem

First, we are going to study the existence of a solution to the truncated problem

−_pu_k+ |∇u_k|^p=ϑ T_k u^q_k

|x|^p

+T_k(f ) inΩ, u_k∈W₀^1,p(Ω), (1) whereT_k(s)=max{min{k, s},−k},k >0.

Theorem 3.There exists a positive solution to problem(1).

Notice thatφ≡0 is a subsolution to problem (1). Considerψthe solution to −pψ=ϑ·k+Tk(f ) inΩ,

ψ=0 on∂Ω. (2)

In factψturns to be a supersolution to (1).

To prove Theorem3we will consider a sequence of approximated problems that we solve by iteration and by using some convenient comparison argument. We take as starting pointw₀=0 and consider iteratively the problem,

⎧⎨

⎩−_pw_n+ |∇w_n|^p

1+¹_n|∇w_n|^p =ϑ T_k w_n^q₋₁

|x|^p

+T_k(f ) inΩ,

w_n=0 on∂Ω.

(3) Notice that the subsolution φ≡0 and the supersolutionψ to problem (1) are subsolution and supersolution to the problem (3).

Next proposition follows using a comparison argument from[4].

Proposition 1.There existswn∈W₀^1,p(Ω)∩L^∞(Ω)solution to(3).

Moreover,0w_nψ∀n∈N.

Proof of Theorem3. We proceed in two steps.

Step 1: Weak convergence ofwninW₀^1,p(Ω).By simplicity let us set H_n(∇w_n)= |∇w_n|^p

1+¹_n|∇w_n|^p. (4)

Takingw_nas a test function in the approximated problems (3), we obtain

Ω

|∇w_n|^pdx+

Ω

H_n(∇w_n)w_ndx=ϑ

Ω

T_k w^q_n−1

|x|^p

w_ndx+

Ω

T_k(f )w_ndx

ϑ

Ω

kw_ndx+

Ω

f w_ndx.

Sincew_n∈W₀^1,p(Ω)∩L^∞(Ω)andf∈L¹(Ω), there exists a positive constantC(k, f, ψ, ϑ, Ω)such that

Ω

|∇wn|^pdx+

Ω

Hn(∇wn)wndxC(k, f, ψ, ϑ, Ω).

Moreover, since

ΩH_n(∇w_n)w_ndx0, we have

Ω

|∇w_n|^pdxC(k, f, ψ, ϑ, Ω). (5)

Therefore, up to a subsequence,wn uk weakly inW₀^1,p(Ω)andwn uk weakly-* inL^∞(Ω), giving uk∈W₀^1,p(Ω)∩L^∞(Ω).

Step 2: Strong convergence ofwninW₀^1,p(Ω)and passing to the limit in (1).To get the strong convergence in W₀^1,p(Ω)first of all we notice that

wn−uk_W^1,p

0 (Ω)(wn−uk)⁺

W₀^1,p(Ω)+(wn−uk)⁻

W₀^1,p(Ω). (6)

Thus, we proceed estimating each term on the right-hand side of (6).

Asymptotic behaviour of(wn−uk)⁺_W^1,p

0 (Ω).Choosing(w_n−u_k)⁺as a test function in (3) we obtain

Ω

|∇w_n|^p−2

∇w_n,∇(w_n−u_k)⁺ dx+

Ω

H_n(∇w_n)(w_n−u_k)⁺dx

=ϑ

Ω

T_k w^q_n−1

|x|^p

(w_n−u_k)⁺dx+

Ω

T_k(f )(w_n−u_k)⁺dx. (7)

Since w_n u_k in W₀^1,p(Ω), one has w_n →u_k a.e. in Ω and thus (w_n −u_k)⁺ →0 a.e. in Ω together with (w_n−u_k)⁺ 0 inW₀^1,p(Ω)as well. Therefore, the right-hand side of (7) goes to zero whenngoes to infinity.

Then, since

ΩH_n(∇w_n)(w_n−u_k)⁺dx0, (7) becomes

Ω

|∇w_n|^p−2

∇w_n,∇(w_n−u_k)⁺

dx=o(1) asn→ +∞. (8)

Since

Ω

|∇u_k|^p−2

∇u_k,∇(w_n−u_k)⁺

=o(1) asn→ +∞, it follows

Ω

|∇wn|^p−2∇wn− |∇uk|^p−2∇uk,∇(wn−uk)⁺

dx=o(1). (9)

Then, from (9) we have o(1)=

C₁(p) ^{|∇(wn−uk)+|2}

(|∇wn|+|∇u_k|)^2−p if 1< p <2,

C₁(p)|∇(w_n−u_k)⁺|^p ifp2, (10)

withC1(p)a positive constant depending onp. In any case, since for 1< p <2 using Hölder’s inequality one has

Ω

∇(w_n−u_k)^+p

Ω

|∇(w_n−u_k)⁺|² (|∇wn| + |∇uk|)^(2−p)

^p

2

Ω

|∇w_n| + |∇u_k|p^2−p

2

, (11)

we obtain

(w_n−u_k)⁺

W₀^1,p(Ω)→0 asn→ +∞. (12)

Asymptotic behaviour of(wn−uk)⁻_W^1,p

0 (Ω).Let us considere^−wn[(w_n−u_k)⁻]as a test function in (3),

Ω

e^−wn|∇wn|^p−2

∇wn,∇(wn−uk)⁻ dx+

Ω

e^−wn

|∇w_n|^p

1+¹_n|∇w_n|^p − |∇wn|^p

(wn−uk)⁻dx

=ϑ

Ω

e^−wnT_k w^q_n−1

|x|^p

(w_n−u_k)⁻dx+

Ω

e^−wnT_k(f )(w_n−u_k)⁻dx. (13) We point out that using this test function it follows

Ω

e^−wn

|∇wn|^p

1+¹_n|∇wn|^p − |∇w_n|^p

(w_n−u_k)⁻0. (14)

As above, since(w_n−u_k)⁻→0 a.e. inΩ, the right-hand side of (13) tends to zero as n goes to infinity. Being w_nψ(see Proposition1), one hase^−wnγ >0 uniformly onn. Then Eq. (13) states as

γ

Ω

|∇w_n|^p−2

∇w_n,∇(w_n−u_k)⁻

dx=o(1). (15)

Arguing in the same way as we have done from Eq. (8) to (12), we obtain (w_n−u_k)⁻

W₀^1,p(Ω)→0 asn→ +∞. (16)

From Eq. (6), by using (12) and (16) we get (w_n−u_k)

W₀^1,p(Ω)→0 asn→ +∞

and consequently∇w_n→ ∇u_ka.e. inΩ. Then, by (4) followsH_n(∇w_n)→ |∇u_k|^pa.e. inΩand by Vitali’s Lemma, Hn(∇wn)→ |∇uk|^p inL¹(Ω).

Hence,uk∈W₀^1,p(Ω)∩L^∞(Ω)satisfies the problem in the following sense

Ω

|∇u_k|^p−2(∇u_k,∇φ)+

Ω

|∇u_k|^pφ=ϑ

Ω

T_k u^p_k

|x|^p

φ+

Ω

T_k(f )φ ∀φ∈W₀^1,p(Ω)∩L^∞(Ω), (17) concluding the proof. 2

2.2. Passing to the limit and convergence to a solutionuof(P)

We want to show thatu_k→ustrongly inW₀^1,p(Ω)in order to prove the existence of a solutionuto problem (P).

Proof of Theorem1. We perform the proof in different steps.

Step 1: Weak convergence ofukinW₀^1,p(Ω).We start takingT_n(u_k)as a test function in the truncated problem (1), obtaining

Ω

∇T_n(u_k)^pdx+

Ω

|∇u_k|^pT_n(u_k) dx=ϑ

Ω

T_k u^q_k

|x|^p

Tn(u_k) dx+

Ω

T_k(f )T_n(u_k) dx.

Notice that, defining Ψn(s)=

s 0

Tn(t)^1pdt, (18)

one has

Ω

∇T_n(u_k)^pdx+

Ω

∇Ψ_n(u_k)^pdx=ϑ

Ω

T_k u^q_k

|x|^p

T_n(u_k) dx+

Ω

T_k(f )T_n(u_k) dx

ϑ

Ω

u^q_k

|x|^pT_n(u_k)+nf_L¹_(Ω). (19) By a straightforward calculation it is easy to check that for fixedq∈ [p−1, p),∀ε >0 and∀n >0,there exists C_εsuch that

s^qT_n(s)εΨ_n^p(s)+C_ε s0. (20)

Thanks to Lemma1and (20), Eq. (19) states as

Ω

∇T_n(u_k)^pdx+

Ω

∇Ψ_n(u_k)^pdxε ϑ C_N,p

Ω

∇Ψ_n(u_k)^p+ϑ C_ε

Ω

1

|x|^p +nf_L¹_(Ω). Then choosingε >0 such that 0< ε_C^ϑ

N,p <1, for some positiveCwe get

Ω

∇T_n(u_k)^pdx+C

Ω

∇Ψ_n(u_k)^pdxϑ C_ε

Ω

1

|x|^p +nf_L¹_(Ω)C(ϑ, ε, f, p, n, Ω). (21) Fixedl1, by definition (18) ofΨ_land Eq. (21), one has

Ω

|∇u_k|^pdx

Ω

∇T_l(u_k)^pdx+

Ω∩{u_kl}

|∇u_k|^pdx

Ω

∇T_l(u_k)^p+1 l

Ω

∇Ψ_l(u_k)^pC, (22) uniformly onk. Therefore, up to a subsequence it followsu_k uweakly inW₀^1,p(Ω)and a.e. inΩ.

Step 2: Strong convergence inL¹(Ω)of the singular term.By Hölder inequality we have

Ω

T_k u^q_k

|x|^p

Ω

u^q_k

|x|^p

Ω

u^p_k

|x|^pdx ^q

p

Ω

1

|x|^pdx ^p−q

p C

Ω

|∇u_k|^{p q}

pC, (23)

withC a positive constant that does not depend onk. It follows thatT_k(^u

q

|xk|^p)is bounded inL¹(Ω)and converges almost everywhere to _|^u_x|^qp. In particular Fatou’s Lemma implies_|^u_x^q_|p ∈L¹(Ω). Moreover, letE⊂Ωbe a measurable set, by Fatou’s Lemma we have

E

Tk

u^q_k

|x|^p

E

u^q_k

|x|^p lim

n→+∞

E

w_n^q

|x|^p

E

ψ^q

|x|^p δ

|E| ,

uniformly in kwhere lims→0δ(s)=0, w_n is as in the proof of Theorem3andψ as in Proposition1. Thus, from Vitali’s Theorem it follows

T_k u^q_k

|x|^p

→ u^q

|x|^p inL¹(Ω). (24)

Step 3: Strong convergence of|∇uk|^p→ |∇u|^p inL¹(Ω).To show the strong convergence of the gradients we need some preliminary results. We have the following

Lemma 2.Letu_kbe defined by(1). Then

nlim→∞

{ukn}

|∇u_k|^p=0 (25)

uniformly ink.

Proof. Let us consider the functions

Gn(s)=s−Tn(s), and ψn−1(s)=T1

Gn−1(s) .

Notice that,ψ_n−1(u_k)|∇u_k|^p|∇u_k|^p_χ{ukn}. Usingψ_n−1(u_k)as a test function in (1) we get

{ukn}

|∇u_k|^p

Ω

∇ψ_n−1(u_k)^p+

Ω

|∇u_k|^pψ_n−1(u_k)

=

Ω

ϑ T_k u^q_k

|x|^p

ψ_n−1(u_k)+

Ω

T_k(f )ψ_n−1(u_k). (26)

Since {uk}is uniformly bounded in W₀^1,p(Ω), then up to a subsequence, {uk} strongly converges in L^p(Ω) for 1p < p^∗=_N^Np_−p and a.e. inΩ. Thus we obtain that

x∈Ω: n−1< u_k(x) < n→0 ifn→ ∞, x∈Ω: u_k(x) > n→0 ifn→ ∞,

uniformly onk. Then, from (24) and (26) we have uniformly ink

nlim→∞

{ukn}

|∇u_k|^p=0. 2 (27)

Next lemma shows the strong convergence of the truncated terms.

Lemma 3.Consideru_k uas above. Then one has uniformly inm, T_m(u_k)→T_m(u) inW₀^1,p(Ω)fork→ +∞.

Proof. Notice that T_m(u_k)−T_m(u)

W₀^1,p(Ω)T_m(u_k)−T_m(u)₊

W₀^1,p(Ω)+T_m(u_k)−T_m(u)₋

W₀^1,p(Ω). (28) We are going to estimate each term on the right-hand side of (28).

Asymptotic behaviour of (Tm(uk)−Tm(u))⁺_W^1,p

0 (Ω). We take (T_m(u_k)−T_m(u))⁺ as a test function in (1), obtaining

Ω

|∇u_k|^p−2∇u_k,∇

T_m(u_k)−T_m(u)₊ dx+

Ω

|∇u_k|^p

T_m(u_k)−T_m(u)₊ dx

=

Ω

ϑ Tk

u^q_k

|x|^p

+Tk(f )

Tm(uk)−Tm(u)₊

dx. (29)

SinceT_m(u_k) T_m(u)andT_m(u_k)→T_m(u)a.e. inΩ, we have(T_m(u_k)−T_m(u))⁺ 0 inW₀^1,p(Ω)and(T_m(u_k)− T_m(u))⁺→0 a.e. inΩ. Thus, the right-hand side of (29), by dominated convergence, tends to zero ask goes to infinity. From (29) we have

Ω

|∇u_k|^p−2∇u_k,∇

T_m(u_k)−T_m(u)₊

dx=o(1). (30)

We estimate the left-hand side of (30) as

Ω

|∇u_k|^p−2∇u_k,∇

T_m(u_k)−T_m(u)₊ dx

=

Ω∩{|u_k|m}

∇Tm(uk)^p−2∇Tm(uk)−∇Tm(u)^p−2∇Tm(u),∇

Tm(uk)−Tm(u)₊ dx

+

Ω∩{|uk|m}

∇T_m(u)^p−2∇T_m(u),∇

T_m(u_k)−T_m(u)₊ dx

+

Ω∩{|uk|>m}

|∇u_k|^p−2∇u_k,∇

T_m(u_k)−T_m(u)₊

dx. (31)

Since(T_m(u_k)−T_m(u))⁺ 0 weakly inW₀^1,p(Ω), denotingχ_mthe characteristic function of the set{x∈Ω: |u_k|>

m}, the second term on the right-hand side of (31) becomes

Ω∩{|uk|m}

∇T_m(u)^p−2∇T_m(u),∇

T_m(u_k)−T_m(u)₊ dx

Ω

∇T_m(u)^p−2∇T_m(u),∇

T_m(u_k)−T_m(u)₊ dx

+

Ω∩{|uk|>m}

∇T_m(u)^p−2∇T_m(u),∇

T_m(u_k)−T_m(u)₊ dx

o(1)+Cu^p−1

W₀^1,p(Ω)χ_m∇T_m(u)

L^p(Ω)→0 ask→ +∞,

since, by dominate convergence again,χ_m∇T_m(u)→0 strongly in(L^p(Ω))^N. As above, the last term in (31) can be estimated as

Ω

|∇u_k|^p−2∇u_k, χ_m∇T_m(u) dx

Cu_k^p−1

W₀^1,p(Ω)χ_m∇T_m(u)

L^p(Ω)→0, (32)

ask→ +∞.

Considering that, by the dominated convergence theorem, we have

Ω∩{|uk|>m}

∇T_m(u_k)^p−2∇T_m(u_k)−∇T_m(u)^p−2∇T_m(u),∇

T_m(u_k)−T_m(u)₊ dx

Ω

χ_m∇T_m(u)^p→0 ask→ +∞, Eq. (30) becomes

Ω

|∇u_k|^p−2∇u_k,∇

T_m(u_k)−T_m(u)₊ dx

=

Ω

∇T_m(u_k)^p−2∇T_m(u_k)−∇T_m(u)^p−2∇T_m(u),∇

T_m(u_k)−T_m(u)₊

dx+o(1).

Finally we obtain

Ω

|∇u_k|^p−2∇u_k,∇

T_m(u_k)−T_m(u)₊ dx

C1(p) ^{|∇(Tm(uk)−Tm(u))+|2}

(|∇Tm(uk)|+|∇Tm(u)|)^2−p+o(1) if 1< p <2, C₁(p)|∇(T_m(u_k)−T_m(u))⁺|^p+o(1) ifp2,

(33) withC₁(p)a positive constant depending onp, which implies (together with (11))

T_m(u_k)−T_m(u)₊

W₀^1,p(Ω)→0 ask→ +∞. (34)

Asymptotic behaviour of(Tm(uk)−Tm(u))⁻_W^1,p

0 (Ω).We usee^−Tm(uk)(T_m(u_k)−T_m(u))⁻as a test function in (1) (see Section2.1) obtaining

Ω

e^−Tm(uk)

|∇uk|^p−2∇uk,∇

Tm(uk)−Tm(u)₋ dx

−

Ω

e^−Tm(uk)

T_m(u_k)−T_m(u)₋

|∇u_k|^p−2∇u_k,∇T_m(u_k) dx

+

Ω

|∇u_k|^pe^−Tm(uk)

T_m(u_k)−T_m(u)₋ dx

=

Ω

ϑ Tk

u^q_k

|x|^p

+Tk(f )

e^−Tm(uk)

Tm(uk)−Tm(u)₋

dx. (35)

In this case as well, since(T_m(u_k)−T_m(u))⁻ 0 weakly inW₀^1,p(Ω)and(T_m(u_k)−T_m(u))⁻→0 a.e. inΩ, the right-hand side of (35) tends to zero askgoes to infinity.

The first term on the left-hand side of (35), being(∇T_m(u_k))χ_m=0, states as

Ω

e^−Tm(uk)

|∇uk|^p−2∇uk,∇

Tm(uk)−Tm(u)₋ dx

+

Ω∩{|u_k|>m}

|∇uk|^pe^−Tm(uk)

Tm(uk)−Tm(u)₋

dx=o(1). (36)

We point out that

T_m(u_k)−T_m(u)₋ χ_m=0, hence (36) becomes

Ω

|∇u_k|^p−2∇u_k,∇

T_m(u_k)−T_m(u)₋

=C_mo(1) ask→ +∞, (37)

withC_ma positive constant depending onm.

The choice and use ofe^−Tm(uk)(Tm(uk)−Tm(u))⁻as a test function allows to simplify conveniently Eq. (35) in order to obtain the desired result. In fact, we proceed writing the left-hand side of (37) as

Ω

|∇u_k|^p−2∇u_k,∇

T_m(u_k)−T_m(u)₋ dx

=

Ω∩{|u_k|m}

∇T_m(u_k)^p−2∇T_m(u_k)−∇T_m(u)^p−2∇T_m(u),∇

T_m(u_k)−T_m(u)₋ dx

+

Ω∩{|uk|m}

∇Tm(u)^p−2

∇Tm(u),∇

Tm(uk)−Tm(u)₋ dx

+

Ω∩{|uk|>m}

|∇u_k|^p−2∇u_k,∇

T_m(u_k)−T_m(u)₋

dx. (38)

The second term on the right-hand side of (38) can be estimated as follows

Ω∩{|u_k|m}

∇Tm(u)^p−2

∇Tm(u),∇

Tm(uk)−Tm(u)₋ dx

=

Ω

∇T_m(u)^p−2

∇T_m(u),∇

T_m(u_k)−T_m(u)₋ dx

−

Ω∩{|uk|>m}

∇T_m(u)^p−2

∇T_m(u),∇

T_m(u_k)−T_m(u)₋ o(1)+Cu^p−1

W₀^1,p(Ω)χ_m∇T_m(u)

L^p(Ω)→0 ask→ +∞, (39)

since by weak convergence the first term on the right-hand side of (39) goes to zero, while the second one goes to zero using (22) and the fact that, for dominated convergence,χ_m∇T_m(u)→0 strongly inL^p(Ω). Moreover, we observe that the last term in (38) is zero since(T_m(u_k)−T_m(u))⁻χ_m=0. Finally as above, Eq. (38) becomes

o(1)=

Ω

|∇u_k|^p−2∇u_k,∇

T_m(u_k)−T_m(u)₋ dx

C₁(p) ^{|∇(Tm(uk)−Tm(u))−|2}

(|∇T_m(u_k)|+|∇T_m(u)|)^2−p+o(1) if 1< p <2, C₁(p)|∇(T_m(u_k)−T_m(u))⁻|^p+o(1) ifp2,

(40) withC₁(p)a positive constant depending onp. By (37) and (40) (using (11) again) we get

T_m(u_k)−T_m(u)₋

W₀^1,p(Ω)→0 ask→ +∞. (41)