Among these problems, we have the obstacle problem, the seepage surface problem, and the Elenbaas equation

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

A DISCONTINUOUS PROBLEM INVOLVING THE

P-LAPLACIAN OPERATOR AND CRITICAL EXPONENT IN R^N

CLAUDIANOR OLIVEIRA ALVES & ANA MARIA BERTONE

Abstract. Using convex analysis, we establish the existence of at least two nonnegative solutions for the quasilinear problem

−∆pu=H(u−a)u^p∗−1+λh(x) inR^N

where ∆puis thep-Laplacian operator,His the Heaviside function,p^∗is the Sobolev critical exponent, andhis a positive function.

1. Introduction

The interest in the study of nonlinear partial differential equations with discon- tinuous nonlinearities has increased because many free boundary problems arising in mathematical physics may be stated in this from. Among these problems, we have the obstacle problem, the seepage surface problem, and the Elenbaas equation;

see for example [9, 10, 11].

Among the typical examples, we have chosen the model for the heat conductivity in electrical media. This model has a discontinuity in its constitutive laws. In fact, considering a domain Ω⊂R³(which in particular could be taken as the whole space R³[4]) with electrical media, the thermal and electrical conductivity are denoted by K(x, t) and σ(x, t), respectively. Herexis in Ω andt represents the temperature.

Since we are considering an electrical media, the functionσmay have discontinuities int, and the distribution of the temperature is unknown. The differential equation describing this distribution is

−

n

X

i=1

∂

∂xi

K(x, u(x))∂u(x)

∂xi

=σ(x, u(x)). (1.1)

Note that this equation is related to a free boundary problem in which the jump surface of the electrical conductivity is unknown. We describe this surface as being the set

Γα(u) ={x∈Ω, u(x) =α, σis discontinuous at α}. (1.2)

2000Mathematics Subject Classification. 35A15, 35J60, 35H30.

Key words and phrases. Variational methods, discontinuous nonlinearities, critical exponents.

c

2003 Southwest Texas State University.

Submitted September 23, 2002. Published April 16, 2003.

Partially supported by PRONEX-MCT/Brazil and Millennium Institute for the Global Advancement of Brazilian Mathematics - IM-AGIMB.

1

When the thermal conductivityKis constant and the electrical conductivityσhas a single jump and a critical growth, the model becomes

−∆u=H(u−a)u^2∗−1+λh(x) in Ω. (1.3) Here H is the Heaviside function (i.e. H(t) = 0 if t ≤0 and H(t) = 1 if t >0), 2^∗ ≡ 2N/(N −2) is the well known Sobolev critical exponent for N > 2, λis a positive parameter, andhis a measurable function defined in Ω.

Note that in this model the jump surface of the solution (1.2) is represented by the set

Γa(u) ={x∈R^N, u(x) =a}. (1.4) Related to problem (1.3) for the special case of a= 0, i.e., without jump discon- tinuities, we cite the works of Tarantello [17] when p=2, and Alves [2], Cao, Li &

Zhou [8] and Gon¸calves & Alves [13] for the casep≥2. In the casea >0, we cite the work of Alves, Bertone & Gon¸calves [3].

In this paper we employ variational techniques to study existence and multiplicity of nonnegative solutions of a family of elliptic equations of type (1.3) in the whole spaceR^N. More precisely, we shall study the quasilinear problem

−∆pu=H(u−a)u^p∗−1+λh inR^N, (1.5) where here p^∗ is the critical Sobolev exponent defined by _N−p^pN with N > p. We considera >0 andλ >0 real parameters,h:R^N →(0,∞) a positive measurable function with

h∈L^θ(R^N)∩L¹(R^N), 1 θ+ 1

p^∗ = 1. (1.6)

As a solution of (1.5) we understand a functionu∈ D^1,pverifying

−∆_pu(x)−λh(x)∈fb(u(x)) a.e inR^N, (1.7) wherefbis the multi-valued function

fb(s) =

({f(s)}, ifs6=a [f(a− −), f(a+)], ifs=a,

withf(t) =H(t−a)t^p∗−1,f(t+ 0) = lim_δ→0+f(t+δ),f(t−0) = lim_δ→0+f(t−δ).

We recall that the solutions of (1.5) are exactly the critical points of the func- tionalIλ,a:D^1,p→Rgiven by

Iλ,a(u) = 1 pkuk^p−

Z

R^N

F(u)dx−λ Z

R^N

h(x)udx. (1.8)

where F(u) = Ru

0 f(t)dt and D^1,p is the closure of C₀^∞(R^N) with respect to the norm

kφk^p= Z

R^N

|∇φ(x)|^pdx.

The set Γ_a(u) has a relevant role when its Lebesgue measure is zero, since the solutions would satisfy (1.5) in the “strong” sense, i.e.,

−∆pu(x) =H(u(x)−a)u(x)^p∗−1+λh(u(x)), (1.9) almost everywhere (a.e. for short) inR^N.

Another important remark is that we are considering only nontrivial solutions which means that the functionsu 6≡0 and verify meas{x∈ R^N, u(x)> a} >0.

We observe that there exists a functionwλwhich satisfies

−∆_pu=λh(x), u(x)>0 inR^N (1.10) and|wλ|_∞≤a, then it is a solution of (1.5) whenλis small. Furthermore, we will denote byw=wλthe unique solution of (1.10).

Our main result is the following.

Theorem 1.1. Assume thathsatisfies(1.6). Then, there existsλ∗>0anda∗>0 such that ifλ∈(0, λ_∗)anda∈(0, a_∗), problem (1.5)has two nonnegative solutions u_i,i= 1,2with the following properties:

(i) ∆_pu_i∈L^θ(R^N);

(ii) meas{x∈R^N, ui(x)> a}>0,i= 1,2;

(iii) meas Γ_a(u_i) = 0;

(iv) I_λ,a(u₂)<0< I_λ,a(u₁).

The proof of theorem (1.1) relies on some results of Convex Analysis since the functionalIλ,a is locally Lipschitz. To get critical points forIλ,a, we use a version of the Mountain Pass for locally Lipschitz functional and the Ekeland Variational Principle. However, the arguments involved are not standard ones: First of all because we are working with the p-Laplacian operator, which is not linear, the growth of the nonlinear part is critical, and the domain is the whole spaceR^N. The second reason is that the arguments used whena= 0 ( the classical case ) cannot be used immediately in our context and because of that a new estimates appear, for instance, to prove that the energy functional verifies the Palais-Smale condition at some levels.

To finish this introduction, we would like to say that the our main result complete the results obtained in [1], [2] and [3], in the following sense, in [1] and [2] was considered the casea= 0 and in [3] was considered the situation where the operator is the Laplacian and the Heaviside function is multiplying the term involving the functionh.

2. Basic results from convex analysis

Throughout this paperX is a Banach space, Φ∈ Lip_loc(X,R) means that the functional isLocally LipschitzianonX. Thegeneralized directional derivative ofΦ inu∈X is the function denoted by Φ⁰(u;·) and defined by the formula

Φ⁰(u;v) = lim sup

h→0, λ↓0

Φ(u+h+λv)−Φ(u+h)

λ .

Since Φ⁰(u;·) is continuous and convex it makes sense to consider the subdifferential of Φ⁰(u;·), which is, by definition,

∂Φ⁰(u;z) ={µ∈X^∗:hµ, v−ziX^∗,X ≤Φ⁰(u;v)−Φ⁰(u;z)∀v∈X}.

We define asgeneralized gradientof Φin uthe set

∂Φ(u) ={µ∈X^∗ | hµ, viX^∗,X ≤Φ⁰(u;v)∀v∈X}, and we shall denote it by∂Φ(u). Since Φ⁰(u; 0) = 0 we have

∂Φ(u) =∂Φ⁰(u; 0).

An important property of the generalized gradient is the following: Ifu∈X then

∂Φ(u) is a nonempty convex set and it isw^∗−compact. In particular, there exists ˆ

ω∈∂Φ(u) such thatkωkˆ X^∗ = min_{ω∈∂Φ(u)}kωkX^∗.

We say that {un} verifiesthe Palais Smale Condition for the functional Φand the valuec (denoted by (P S)_c) if{un}verifies

Φ(un)→c and kωnk= min

ω∈∂Φ(un)

kωkX^∗ →0, (2.1)

then it implies that there is a subsequence ofun which converges inD^1,p.

Next we shall enunciate two crucial results that will be used throughout this work. One is the well known Mountain Pass theorem, in a locally Lipchitzian version. The other is a characterization of the elements of the generalized gradient of a determined functional. The proof of these results can be found in [3].

Theorem 2.1. LetΦ∈Lip_loc(X,R). Suppose thatΦ(0) = 0and there isη, r₁>0, e∈X with kek> r₁ such that

Φ(u)≥η ifkuk=r1,Φ(e)≤0. (2.2) If c≡infγ∈Γmax0≤t≤1Φ(γ(t))and

Γ≡ {γ∈ C([0,1], X)|γ(0) = 0, γ(1) =e}, thenc >0and there exists a sequence{un} ⊂X satisfying (2.1).

Proposition 2.2. LetΦ(u) =R

R^NF(u)dxbe the functional defined in (1.8). Then, Φ∈Liploc(L^p∗(R^N);R),∂Φ(u)⊂(L^p∗(R^N))⁰ and ifω∈∂Φ(u), it satisfies

ω(x)∈fb(u(x)), a.e. x∈R^N. (2.3) 3. Preliminary Results

Hereafter we shall useL^s, s >1 to represent the Lebesgue spaceL^s(R^N) and|·|s

its usual norm. Besides, ifg is a Lebesgue integrable function, we shall write R g forR

R^NgdxandSdenotes the best Sobolev constant of the imbeddingD^1,p,→L^p∗, that is,

S= min

u∈D^1,p, u6=0

R|∇u|^p R|u|^p∗_p^p∗

Our first Lemma is a version for vectorial functions in R^N of a result due to Brezis & Lieb ( see [6] ). Its proof can be found in [1].

Lemma 3.1. Let η_n : R^N → R^K (K ≥ 1) with η_n ∈ L^p(R^N)×. . .×L^p(R^N) (p ≥ 2), ηn(x) → 0 a.e in R^N and A(y) = |y|^p−2y, for all y ∈ R^K. Then, if

|ηn|_Lp(R^N)≤C, for alln∈Nwe have Z

R^N

|A(ηn+w)−A(ηn)−A(w)|^p−1p =on(1), for eachw∈L^p(R^N)×. . .×L^p(R^N) fixed.

The next lemma is standard and its proof use similar arguments to those in [3].

It shows that the functionalIλ,a verifies the mountain pass geometry.

Lemma 3.2. There is λ₀ >0 such that forλ∈(0, λ₀) the functionalI_λ,a verifies the mountain pass geometry (2.2), for alla >0.

Using the lemma above, we conclude by Theorem 2.1 that there exists {un} in D^1,psuch that

Iλ,a(un)→c andkwnk= min

w_n∈∂Iλ,a(u_n)kwk →0 Lemma 3.3. The functional I_λ,a satisfies the condition(P S)_c, for

c∈(−∞, 1

NS^Np −c₁λ^p−1p ),

wherec₁=c₁(N, S, θ,|h|θ)is a positive constant that verifies the following inequal- ity

1 Nt^p−λ

θ|h|_θt≥ −c₁λ^p−1p , for allt≥0.

Proof. Supposeu_n satisfies (2.1). One hasu_n bounded and there existsu₀∈ D^1,p such thatu_n converges weakly inD^1,pand a.e. inR^N tou₀. Letv_n=u_n−u₀and suppose thatkvnk^p→l >0. Thus,

hwn, vni= Z

|∇un|^p−2∇un∇vn−λ Z

h(x)vn− hρn, vni whereρn∈∂Φ(un). Using Proposition 2.1, we have

0≤ρ_n(x)≤u^p_n^∗−1(x) a.e inR^N

and repeating similar arguments explored in [13], it is possible to show the existence of a set Γ⊂R^N empty or finite such that {un} is strongly convergent inL^p∗(K) for all K ⊂ (R^N \Γ) compact set. The above information imply that, up to subsequence, we can assume

ρ_n(x)→ρ₀(x) a.e inR^N.

The above properties involving the sequences {un} and {ρn} together with the arguments explored in [13] are sufficient to show

∇un(x)→ ∇u0(x) a.e inR^N. From Lemma 3.1 we have

hwn, vni= Z

|∇vn|^p+ Z

|∇u0|^p−2∇u0∇vn− hρn, vni+on(1).

Hence,hwn, vni=l− hρn, vni+on(1), which implies

n→∞limhρn, vni=l. (3.1)

Moreover, by recalling that hρn, v_ni ≤

Z

f(u_n+ 0)v_n++ Z

f(u_n−0)(−v_n−), we get

hρn, vni ≤ Z

u^p_n^∗−1vn+= Z

un>u0

u^p_n^∗−1(un−u0).

Consequently, hρn, vni ≤

Z u^p₀^∗+

Z

|vn|^p∗− Z

un≤u0

u^p_n^∗− Z

u^p_n^∗−1u0+ Z

un≤u0

u^p_n^∗−1u0+on(1).

Therefore,

hρn, vni ≤ Z

|vn|^p∗ +on(1).

The last inequality implies that

n→∞limhρn, vni ≤ lim

n→∞

Z

|vn|^p∗. (3.2)

Now, from (3.1) and (3.2), we obtain thatSl^pp∗ ≤l, which infers

S^Np ≤l. (3.3)

On the other hand, we have

Iλ,a(un) +on(1) =Iλ,a(un)− 1

p^∗hwn, uni that is,

I_λ,a(u_n) +o_n(1)≥ 1

Nkunk^p−λ θ Z

h(x)u_n. Thus,

Iλ,a(un) +on(1)≥ 1

Nkvnk^p−λ^p−1p c1+on(1), wherec1=c1(N, S, θ,|h|θ) is the constant stated in the Lemma.

From the last inequality and (3.3), we get c≥ S^Np

N −λ^p−1p c₁, which contradicts that c∈(−∞,^S

N p

N −λ^p−1p c1). Therefore, we should havel = 0 and consequentlyun→u0 inD^1,p. This finished the proof of the lemma.

Lemma 3.4. There existsλ1>0,a_∗ >0, and e∈ D^1,p such that, forλ∈(0, λ1) anda∈(0, a∗), we havee∈B^c_ρ(0)with Iλ,a(e)<0, and

0< r≤c= inf

γ∈Γ max

0≤t≤1I(γ(t))< 1

NS^Np −c1λ^p−1p , (3.4) withΓ ={γ∈C([0,1],D^1,p), γ(0) = 0, γ(1) =e}.

Proof. Letλ₂>0 such that ^S

N p

N −λ

p p−1

2 c₁>0 ∀λ∈(0, λ₂).It is known by Talenti in [16] that the family of functions

wε(x) =

hN ε ^N−p_p−1p−1i^N−p_p2

(ε+|x|^p−1p )^N−pp

ε >0 satisfies

kwεk^p=|wε|^p_p^∗∗ =S^Np. Note that, there ist₀>0 such that fort≤t₀, we have

Iλ,a(twε)≤ 1

NS^Np −c1λ^p−1p ∀λ∈(0, λ2).

Moreover, ift≥t₀

Ω_a ={t0w_ε> a} ⊂ {twε> a},

thus,

I_λ,a(tw_ε)≤ t^p

pS^Np −λt Z

h(x)w_ε− Z

Ωa

F(tw_ε)

= t^p

pS^Np −λt Z

h(x)wε−t^p∗ p^∗

Z

Ωa

w^p_ε^∗+a^p∗ p^∗ |Ωa|.

Therefore, the function P(t) = t^p

pS^Np −λt Z

h(x)w_ε−t^p∗ p^∗

Z

Ω_a

w^p_ε^∗+a^p∗ p^∗ |Ω_a|, has a maximum atγ1>0 and the functiong(t) =^t_p^p−^tp

∗

p^∗ attains its maximum in t= 1. As a consequence we get

Iλ,a(twε)≤ 1

NS^Np −λt0

Z

h(x)wε+γ^p₁^∗ p^∗

Z

Ω^c_a

w_ε^p∗+a^p∗ p^∗ |Ωa|.

Now, noticing that

|Ωa| ≤ ω_NK_εt

N N−p

0

a^N−pN , whereKεis a constant that dependents of ε, one obtains

a^p∗|Ωa| →0 asa→0.

Then, by takingλ₃>0 such that λt0

Z

h(x)wε> λ^p−1p c1,

for allλ∈(0, λ3), we choosea^∗=a(λ3) satisfying

−λt0

Z

h(x)w_ε+γ^p₁^∗ p^∗

Z

Ω^c_a

w_ε^p∗+a^p∗

p^∗ |Ωa|<−λ^p−1p c₁ ∀a∈(0, a^∗).

Finally, fora∈(0, a_∗) andλ∈(0, λ₁), withλ₁= min{λ2, λ₃}, we have I_λ,a(tw_ε)≤ 1

NS^Np −c₁λ^p−1p ∀λ∈(0, λ₂) ∀t≥t₀,

and the proof is complete

4. Proof of Theorem 1.1

4.1. First solution (Mountain Pass). Letλ_∗ = min{λ0, λ1}, whereλ0 and λ1

were given by Lemmas 3.2 and 3.4. By Theorem 2.1 there exists a sequence (P S)c, forcdefined in (3.4). Therefore we obtainρn∈∂Φ(un) such that

w_n=Q⁰(u_n)−Ψ⁰(u_n)−ρ_n, (4.1) where herewn was defined in (2.1), and

Q(u) = 1 p

Z

|∇u|^pdx, Ψ(u) =λ Z

h(x)u(x)dx.

Using straightforward arguments, we find that{un}is bounded inD^1,p. Moreover, using the fact that

hwn, u_n−i=o_n(1)

we have ku_n−k → 0, where un− is the negative part of un. Then there exists a nonnegativeu1 ∈ D^1,p such thatun * u1, un(x)→u1(x), a.e. x∈R^N. Besides, there existsρ0∈L^θ such that ρn * ρ0 inL^θ. Now, sinceρn ∈∂Φ(un), repeating the same arguments explored in the proof of Lemma 3.3 we have

ρn(x)∈fb(un(x)), a.e. x∈R^N, ρ0(x)∈fb(u1(x)), a.e. x∈R^N, and forϕ∈ D^1,p,

Z

|∇u₁|^p−2∇u₁∇ϕ−λ Z

h(x)ϕ− Z

ρ₀ϕ= 0. (4.2)

Proof of i): ∆pu1∈L^θ. In this subsection, we shall adapt for our problem some arguments that could be found in [5]. From (4.2), we have

−∆pu1=J1+J2 in (D^1,p)⁰, whereJ₁, J₂:D^1,p→Rare linear functionals:

J1(v) =λ Z

h(x)v and J2(v) = Z

ρ0(x)v.

Note thatJ1, J2∈(L^p∗)⁰⊂(D^1,p)⁰. Thus, by Riesz’s Theorem,J1, J2∈L^θand so

∆pu1∈L^θ. Since (4.2) holds, then

−∆pu1=λh+ρ0 a.eR^N and, from this equality, we get

−∆pu1(x)−λh(x)∈fb(u(x)), a.e. x∈R^N. This has proved thatu1 is a solution of (1.5).

Proof of ii): meas{x ∈ R^N;u1 > a} > 0. Now, we shall prove that u1 is a nontrivial solution. By Lemmas 3.3 and 3.4 we getun→u1andI(u1)>0, so that u16≡0. Suppose, by contradiction, thatu1≤ainR^N. Then,u1would verify

ku1k^p=λ Z

h(x)u₁, inR^N, and as a consequence

I(u1) = −λ(p−1) p

Z

h(x)u1<0.

This contradicts the fact thatI(u1)>0.

4.2. Proof of iii): meas(Γa(u1)) = 0. Assume by contradiction that meas(Γa(ui))>

0. By using the Morrey-Stampacchia’s Theorem (see [14] and [15]), we have that

−∆pu(x) = 0 a.e. x∈Γa(u). Hence,

−λh(x)∈[0, a^p∗],

which is a contradiction. Thus meas(Γ_a(u_i)) = 0,i= 1,2.

4.3. Second solution (Local Minimization). To prove the existence ofu2, we observe that, fixed a positive functionψ∈C₀^∞(R^N), we have

t→0limIλ,a(tψ)<0.

Consequently

ec= inf

Bρ

Iλ,a<0, fora∈(0, a∗), and −∞ < ec < 0. Now, considering Iλ,a|_B

ρ, we apply the Ekeland variational principle (see [12]) to obtainu_ε∈B_ρ such that

Iλ,a(uε)<inf

Bρ

Iλ,a+ε, (4.3)

and

I_a(u_ε)< I_a(u) +εku−u_εk, u6=u_ε. (4.4) Letεbe a positive number defined by

0< ε < inf

∂B_ρIλ,a−inf

Bρ

Iλ,a.

For this choice ofε, one has

Iλ,a(uε)≤inf

Bρ

Iλ,a+ε < inf

∂B_ρIλ,a,

which implies thatu_ε∈B_ρ. Let γ >0 be small enough thatu_γ =u_ε+γv ∈B_ρ, andv∈ D^1,p. From (4.4) we get

I_λ,a(u_ε+γv)−I_λ,a(u_ε) +γεkvk ≥0.

Thus we have

−εkvk ≤lim sup

γ↓0

Iλ,a(uε+γv)−Iλ,a(uε)

γ ≤I_λ,a⁰ (uε;v).

Now, since the equality below

I_λ,a⁰ (u;v) = max

µ∈∂Iλ,a(u)hµ, vi, u, v∈ D^1,p, holds, it follows that

−εkvk ≤I_λ,a⁰ (uε, v) = max

ω∈∂I_λ,a(u_ε)hω, vi, for allv∈ D^1,p. Interchangingv and−vwe obtain

−εkvk ≤ max

ω∈∂Iλ,a(u_ε)hω,−vi=− min

ω∈∂Iλ,a(u_ε)hω, vi, v∈ D^1,p Therefore,

min

ω∈∂Iλ,a(u_ε)hω, vi ≤εkvk, v∈ D^1,p, concluding that

sup

kvk=1

min

ω∈∂Iλ,a(u_ε)hω, vi ≤ε.

Finally, by Ky Fan’s Min-max theorem ([7, Proposition 1.8]), we get min

ω∈∂Iλ,a(uε)

sup

v∈B1

hω, vi ≤ε,

which along with (4.3) yields the existence ofun ∈Bρ such thatIλ,a(un)→c, and min_{ω∈∂Iλ,a}(u_n)kωk → 0. Therefore, by Lemma 3.3, there exists u2 ∈ D^1,p and a subsequence un_i ofun such that un_i →u2 in D^1,pandIλ,a(u2) =c= inf_B

ρI <0.

Moreover, we have that u2 > a in a open ω ∈ R^N because, otherwise, we would haveu(x)≤ainR^N. This implies thatu2is a solution of (1.10) and by uniqueness we would haveu2=u2(a), for alla∈(0, a^∗). On the other hand,

S|u2|^p_p∗ ≤ ku2k^p < λ Z

h(x)u₂(x)≤λ|h|1a,

which implies that u₂ goes to zero in D^1,p as a goes to zero, hence ii) and iv) hold foru2. The arguments to proof that u2 also verifiesi) and iii) are the same explored in the section 4.1. This conclude the proof of Theorem 1.1.

Acknowledgments. The authors would like to thank the anonymous referee for his/her suggestions and valuable comments.

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Claudianor Oliveira Alves

Universidade Federal de Campina Grande, Departamento de Matem´atica, 58109-970 Campina Grande-PB, Brazil

E-mail address:[email protected]

Ana Maria Bertone

Universidade Federal da Para´ıba, Departamento de Matem´atica, 58059-900 Jo˜ao Pessoa- PB, Brazil

E-mail address:[email protected]