While Br´ezis and Kamin use the method of sub and super solutions, we employ variational arguments for the existence of solutions

C. O. ALVES^∗, J. V. CONCALVES^∗ AND L. A. MAIA† Abstract. In this note we give a result for the operatorp-Laplacian com- plementing a theorem by Br´ezis and Kamin concerning a necessary and suffi- cient condition for the equation−∆u=h(x)u^q in IR^N,where 0< q <1,to have a bounded positive solution. While Br´ezis and Kamin use the method of sub and super solutions, we employ variational arguments for the existence of solutions.

1. Introduction

We are concerned with existence of solutions for the quasilinear elliptic problem

(∗)





−∆_pu=h(x)u^q in IR^N, u≥0, u≡0, |∇u|^p <∞,

where ∆_pu=div(|∇u|^p−2∇u), 1< p < N, 0≤q < p^∗−1, p^∗ = _N−p^Np , and h:IR^N →IR is a measurable function withh≡0.

In [3] Br´ezis and Kamin studied (∗) in the casep= 2 and obtained necessary and sufficient conditions for it to have bounded solutions. More precisely it was shown in [3], for p= 2 and by using a priori estimates and the method of sub and super solutions, that (∗) has a bounded solution iff both

(H1) h∈L^∞_loc and h≥0 a.e. in Ω and the linear problem

−∆u=h(x)in IR^N have a bounded solution.

1991Mathematics Subject Classification. 35J20, 35J25.

Key words and phrases. Quasilinear ellipticequation,p-Laplacian, variational method.

* Supported in part by CNPq/Brasil.

†Partially supported by FAP-DF/Brasil.

Received: September 15, 1996.

c

1996 Mancorp Publishing, Inc.

407

The study of problem (∗) in this case (p= 2) is related to the study of the asymptotic behaviour of solutionsU(x, t) (as t→ ∞) of

h(x)U_t+ ∆U^1q = 0 in IR^N ×(0,∞).

Actually,

U(x, t)≡Av(x) 1 (t+B)^1−qq ,

whereB >0 is any constant andA >0 is an appropriate number, solves the evolution equation above if v(x)≥0 is a solution of the equation −∆v^1q = h(x)v in IR^N,i.e., u≡v^1q solves (∗). Eidus [7] treats a situation in which h(x)→0 at ∞.

We point out that (∗) does not always have a solution, see again [3] and also Gidas and Spruck [8] for an important non-existence result.

There is by now an extensive literature on this kind of problem. We refer the reader also to Br´ezis and Nirenberg [4], Noussair and Swanson [9], Tshinanga [13], Alves-Goncalves and Maia [1], Rabinowitz [11], Costa and Miyagaki [5]

and their references.

Our aim in this work is to use variational methods to prove the following result.

Theorem 1.Assume that

(H2) h∈L^∞_loc andh(x)≤0 a.e. inΩ^c for some bounded domainΩ⊂IR^N and

(H₃) inf_ω h >0

for some domain ω ⊂ Ω. Then (∗) has a solution u ∈ D^1,p provided that either

(i) 0≤q < p−1

or

(ii) p−1< q < p^∗−1.

Moreover, u∈W_loc^1,p when 1< p < N and u∈W_loc^p,s for s > N when p= 2.

Our approach to prove Theorem 1 will involve the consideration of the family of problems in the ball of radiusR > 0 centered at the origin IR^N, B_R ≡ BR(0), namely

(∗)_R





−∆pu=h(x)u^q in BR, u∈W_R≡W_o^1,p(B_R), u≥0in BR, u≡0,

and the finding of critical points of the associated energy functional I_R : WR→IR given by

IR(u) = 1 p

BR

|∇u|^p− 1 q+ 1

BR

hu^q+1₊ .

Actually, under appropriate assumptions such as the ones we have stated above,I_R∈ C¹(W_R, IR) and its derivativeI_R (u) is given by

I_R (u), φ=

BR

|∇u|^p−2∇u∇φ−

BR

hu^q₊φ, φ∈W_R.

A distributional solution of (∗)will be found by estimating and passing to the limit as R→ ∞.

One reason for working out this reduction procedure is that the Euler- Lagrange functional of (∗) is not defined over eitherW^1,p(IR^N) orD^1,p(IR^N)

since the integral

hu^q+1₊ may not be defined.

2. Preliminaries

Let W_n ≡ W_o^1,p(B_n). Under the conditions of Section 1 we have I_n ∈ C¹(W_n, IR). We state below some technical lemmas and remarks which will be useful in the next section.

Lemma 1. Assume (H₂),(H₃) and 0≤q < p−1. Then

(i) In(u)→ ∞ asuWn → ∞.

(ii) In(tφ)<0, 0< t < tn for some tn>0 and φ∈C₀^∞. We remark that by Lemma 1 there is some u_n∈W_nsuch that

I_n(u_n) = inf

WnI_n<0

and

I_n(u_n), φ = 0, φ∈W_n

so that in particularu_n is a weak solution of (∗)_n for each n >1.

Lemma 2. There are constants c < 0, and M > 0 independent of n such that

(i) I_n(u_n)≤c, n > 1

(ii) u_nD^1,p ≤M, n >1.

Lemma 3. Assume p−1 < q < p^∗ −1 and (H2),(H3). Then there are constants ρ, r>0 independent of n ande∈C_o^∞(Ω),e_Wn > ρ such that

(i) I_n(u)≥r for u_Wn =ρ

(ii) In(e)≤0, n >1.

Moreover, In satisfies the (PS) condition.

Remark 1.By the Ambrosetti-Rabinowitz Mountain Pass theorem we find a critical pointun∈Wn ofIn such that

I_n(u_n) =c_n≥r >0

where

cn= inf

γ∈Γ max

0≤t≤1In(γ(t)) and

Γ ={γ ∈C([0,1], Wn) |γ(0) = 0, γ(1) =e}

and using the fact thatWn⊂Wn+1 we actually have c₁≥c₂≥ ... c_n≥ ... ≥r >0 so that in particular

In(un) =cn→c for somec≥r >0.

Lemma 4.The sequenceu_n∈W_n⊂D^1,pgiven above bythe Mountain Pass theorem is bounded inD^1,p.

Lemma 5. Under either conditions (i) or (ii) of Theorem 1 we have

(i) un# u in D^1,p,

(ii) un→u a.e. inIR^N,

(iii) ∇u_n→ ∇u a.e. inIR^N

for someu∈D^1,p.

3. Proofs

We now give the proofs of Theorem 1 and Lemmas 1-5.

Proof of Lemma 1.

(i). Since h ∈L^∞_loc, we have h ∈ L^θ(B_n). Thus, by H¨older’s inequality and Sobolev’s embedding theorem,

Bn

hu^q+1₊ ≤C|h|_θ,Bnu^q+1_Wn, whereθ≡ _p∗−(q+1)^p∗ .

Hence

I_n(u)≥ 1

pu^p_Wn−C|h|_θ,Bn

q+ 1 u^q+1_Wn, which shows thatI_n is coercive alongW_n.

(ii). Letting φ∈C_o^∞ withφ≥0, φ≡ 0 andsupt(φ)⊂ω,we get I_n(tφ)<0 for 0< t < t_n

for somet_n>0.

Proof of Lemma 2.We have

In(un) = _p¹_Bn|∇un|^p− _q+1¹ _Bnhu^q+1_n+

≥ _p¹_Bn|∇u_n|^p− _q+1¹ _Ωhu^q+1_n+

≥ _p¹u_n^p_Wn−^|h|_q+1^θ,Ω|u_n|^q+1_p∗,Ω

≥ _p¹u_n^p_Wn−C^|h|_q+1^θ,Ωu_n^q+1_Wn.

Now, it is easy to see that I_n(u_n)≤I_n(t^∗φ) =t^∗p

ω|∇φ|^p−(t^∗)^q+1 q+ 1

ωhφ^q+1≡c < 0 for some t^∗ >0. Thus, we get

u_n^p_Wn =u_n^p_D1,p ≤M for someM >0.

Proof of Lemma 3. We remark first that

In(u) ≥ ¹_pBn|∇u|^p−_q+1¹ _Ωhu^q+1₊

≥ ¹_pu^p_Wn−^|h|_q+1^θ,Ω|u|^q+1_p∗,Ω

≥ ¹_pu^p_Wn−S^{−q+12 |h|}_q+1^θ,Ωu^q+1_Wn,

where S is the best constant for the embedding D^1,p → L^p∗. So, there are r , ρ >0 such that

In(u)≥r foru=ρ.

Note that by taking the extension by zero ofu∈W_ntoIR^N we haveu^p_Wn = u^p_D1,p. ˜So r , ρare independent ofn.

On the other hand, as above, there is some t_o >0 such that I_n(t_oφ) =t^p_o

ω|∇φ|^p− t^q+1_o q+ 1

ωhφ^q+1 ≤0 and, taking e≡toφ,we have In(e)≤0.

Now, letting I_n =I, W_n =W and B_n =B, assume u_k ∈W is a sequence such that

I(u_k)→0 with I(u_k) bounded.

We have

I(u_k)− 1 q+ 1

I(u_k), u_k= 1

p− 1 q+ 1

u_k^p_W, which shows that

u_kW is bounded, and thus,

u_k# u in W.

In addition,

u_k→u in L^r(B), p≤r < p^∗ and u_k →ua.e. in B.

Hence from

I(u_k), u_k=o_k(1) we have

u_k^p_W →

Bhu^q+1₊ .

On the other hand, choosing ψ_j →u in W with ψ_j ∈C_o^∞,we have

B|∇u|^p−2∇u∇ψj →

B|∇u|^p.

Since

B|∇u|^p−2∇u∇ψj=

Bhu^q₊ψ_j

and

Bhu^q₊ψ_j →

Bhu^q+1₊ we infer that

u_kW → uW, which shows that u_k→u in W.

Proof of Lemma 4. We shall use thatW_n⊂D^1,p. By Remark 1, I_n(u_n) =c+o_n(1)

and since

I_n(un), un = 0, forn >1, and recalling that

In(un)− 1 q+ 1

I_n(un), un =c+on(1),

we get

1 p − 1

q+ 1 Bn

|∇u_n|^p =c+o_n(1).

Now, since

Bn|∇un|^p =

|∇un|^p we find someM >0 such that

un_D^1,p ≤M for all n >1.

Next we present the proof of Theorem 1 and we leave the proof of Lemma 5 for a later step.

Proof of Theorem 1. At first we remark that from 0 =I_n(un), un− =

Bn|∇un−|^p

it follows thatu_n≥0 and we have already shown thatu_n≡ 0. On the other

hand,

Bn|∇un|^p−2∇un∇ψ−

Bnhu^q_nψ= 0, ψ ∈C_o^∞. Now, using Lemma 5 and passing to the limit we get

|∇u|^p−2∇u∇ψ−

hu^qψ= 0, ψ∈C_o^∞, which shows that u is a distributional solution of (∗), that is,

−∆_pu=h(x)u^q in D(IR^N).

Next, we show that u ≡ 0. Indeed, assuming that p−1 < q < p^∗−1, we have

In(un) =c+on(1) and 1 q+ 1

I_n(un), un = 0.

Therefore,

c+o_n(1) = 1

p− 1

q+ 1 hu^q+1_n ≤ 1

p − 1

q+ 1 Ωhu^q+1_n . Passing to the limit, we infer that

0< r≤c≤ 1

p − 1

q+ 1 Ωhu^q+1

which shows thatu≡ 0. A similar argument works for the case 0≤q < p−1, recalling thatI_n(u_n)≤c < 0.

Now, u ∈ W_loc^1,p when 1 < p < N, by Sobolev embeddings, and u ∈ W_loc^p,s fors > N when p = 2 by elliptic regularity theory (see DiBenedetto [6] for p= 2.)

The proof of Lemma 5 is adapted from arguments by Noussair-Swanson and Jianfu [10] (see also Alves and Goncalves [2] and their references).

Proof of Lemma 5.We shall only show that

(iii) ∇u_n→ ∇u a.e. in IR^N.

since (i)(ii) are more standard.

So, let us consider the cut-off function

η∈C_o^∞, 0≤η≤1, η= 1 on B₁, η= 0 on B₂^c.

Letρ >0 and η_ρ(x) =η(^x_ρ) and letn >1 such thatB_2ρ⊂B_n. We have I_n(un), φ= 0, φ∈Wn

and sinceu_n−u∈D^1,p, we get (u_n−u)η_ρ∈W_n, so that we also have

Bn

|∇u_n|^p−2∇u_n∇[(u_n−u)η_ρ] =

Bn

hu^q_n(u_n−u)η_ρ.

Hence

Bn

|∇un|^p−2∇un(∇un− ∇u)ηρ=

Bnhu^q_n(un−u)ηρ−

Bn|∇un|^p−2(un−u)∇un∇ηρ. We claim that

(A)

Bn

|∇u_n|^p−2(u_n−u)∇u_n∇η_ρ=o_n(1)

(B)

Bn

hu^q_n(u_n−u)η_ρ=o_n(1) Assume that (A) and (B) hold. Then

Bn

(|∇u_n|^p−2∇u_n− |∇u|^p−2∇u)(∇u_n− ∇u)η_ρ=

−

Bn

|∇u|^p−2∇u(∇u_n− ∇u)η_ρ+o_n(1).

Recalling that

U_n≡(|∇u_n|^p−2∇u_n− |∇u|^p−2∇u)(∇u_n− ∇u)≥0 a.e. inIR^N,

we get

Bρ

U_n≤

Bn

U_nη_ρ=−

Bn

|∇u|^p−2∇u(∇u_n− ∇u)η_ρ+o_n(1).

We claim that (C)

Bn

|∇u|^p−2∇u(∇un− ∇u)ηρ=on(1).

Let us assume that (C) holds. Then we have

Bρ

(|∇u_n|^p−2∇u_n− |∇u|^p−2∇u)(∇u_n− ∇u)η_ρ≤o_n(1).

Now, from Tolksdorff [12, Lemma 1], we have

Bρ(1 +|∇un|+|∇u|)^p−2|∇un− ∇u|²≤on(1), 1< p <2

and

Bρ

|∇u_n− ∇u|^p ≤o_n(1), 2≤p <∞ which shows that

∇un→ ∇u a.e. in Bρ.

Taking a sequence ρ_n→ ∞ and using a diagonal argument, we infer that

∇u_n→ ∇u a.e. in IR^N. Verification of (A). We recall that

D^1,p→W^1,p(B_2ρ)*→L^r(B_2ρ), p≤r < p^∗ so that

u_n→u in L^r(B_2ρ) and hence, using H¨older’s inequality,

Bn||∇u_n|^p−2(u_n−u)∇u_n∇η_ρ| = _B2ρ||∇u_n|^p−2(u_n−u)∇u_n∇η_ρ|

≤ C_ρB2ρ|∇u_n|^p−1|u_n−u|

≤ C_ρ||∇u_n||^p−1_Lp(B2ρ)|u_n−u|_L^p_(B2ρ).

Verification of (B). We have

Bn|hu^q_n(u_n−u)η_ρ| = _B2ρ|hu^q_n||u_n−u|η_ρ

≤ C_h,ρB2ρ|u^q_n||u_n−u|

≤ C_h,ρ|u_n|^q_Lq+1(B2ρ)|u_n−u|_L^q+1_(B2ρ) Verification of (C). Letting

F, w ≡

B2ρ

|∇u|^p−2∇u∇wη_ρ, w ∈W^1,p(B_2ρ) and using H¨older’s inequality we infer that

F ∈(W^1,p(B_2ρ))

and, consequently,

F, u_n−u=

B2ρ

|∇u|^p−2∇u(∇u_n− ∇u)η_ρ→0 This proves lemma 5.

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C. O. Alves

Departamento de Matem´atica Universidade Federal da Paraiba

58100-240 - Campina Grande-(PB), BRASIL E-mail address: [email protected]

J. V. Goncalves and L. A. Maia Departamento de Matem´atica Universidade de Bras´ilia 70.910-900 Brasilia-DF, BRASIL

E-mail addresses: [email protected], [email protected]