Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Sobolev versus H¨older local minimizers and existence of multiple solutions for a singular quasilinear equation

JACQUESGIACOMONI, IANSCHINDLER ANDPETERTAKA´Cˇ

Abstract. We investigate the following quasilinear and singular problem, −pu= λ

u^δ +u^q in ;

u|_∂=0, u>0 in , (P)

whereis an open bounded domain with smooth boundary, 1<p<∞,p−1<

q ≤ p^∗−1,λ > 0, and 0 < δ < 1. As usual, p^∗ = _N^{N p}_−p if 1 < p <

N, p^∗ ∈ (p,∞)is arbitrarily large if p = N, and p^∗ = ∞if p > N. We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) inW₀^1,p(). While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle and a regularity result for solutions to problem (P) inC^1,β()with someβ∈(0,1). Furthermore, we show thatδ <1 is a reasonable sufficient (and likely optimal) condition to obtain solutions of problem (P) inC¹().

Mathematics Subject Classification (2000): 35J65 (primary); 35J20, 35J70 (secondary).

1.

Introduction

In this paper we are interested in the following quasilinear and singular problem:

−pu= λ

u^δ +u^q in ;

u|_∂=0, u>0 in . (P)

Here,is an open bounded domain with smooth boundary,pu=∇·(|∇u|^p−2∇u) denotes the p-Laplace operator, 1 < p < ∞, p−1 < q ≤ p^∗−1,λ > 0, and 0 < δ <1. As usual, p^∗ = _N^{N p}_−p if 1< p < N, p^∗ ∈(p,∞)is arbitrarily large Received December 22, 2005; accepted in revised form January 23, 2007.

if p= N, and p^∗ = ∞if p> N. Such problems arise, for instance, in models of pseudoplastic flows.

Our main concern is the question of existence and multiplicity of weak solu- tions to the Dirichlet boundary value problem (P) in W₀^1,p(). To obtain multiple (at least two distinct, positive) solutions of problem (P), we combine some well- known variational methods (seee.g.Ambrosetti-Brezis-Cerami [2]) with a few new ideas of our own which employ two new results of separate interest: a regularity result for solutions to problem (P) inC^1,β()with someβ ∈(0,1), Theorem 2.2, and a strong comparison principle, Theorem 2.3. Our regularity result is obtained by adapting some ideas from Lieberman [33] for estimates in Campanato spaces.

The strong comparison principle extends a result of Cuesta and Tak´aˇc [14]. More precisely, we look for solutions to problem (P) that are critical points of the energy functionalE_λ: W₀^1,p()→

R

^{defined by}

E_λ(u)= 1 p

|∇u|^pdx

− λ 1−δ

(u⁺)^1−δdx− 1 q+1

(u⁺)^q+1dx

(1.1)

in the Sobolev space W₀^1,p(). As usual,r⁺ = max{r,0}andr⁻ = max{−r,0} for r ∈

R

. Note that E_λ is not of classC¹ onW₀^1,p()because of the singular term(u⁺)^1−δ; consequently, one cannot directly apply classical variational meth- ods, such as the Mountain Pass lemma of Ambrosetti-Rabinowitz [4].

First, we show that the number

^def= inf{λ >0 : (P) has no weak solution} (1.2) satisfies 0< <∞. Then we prove the existence of multiple (at least two distinct, positive) solutions of problem (P) for every λ ∈ (0, ): a local minimizer and a saddle point for the functional E_λ. Indeed, this existence and multiplicity result is a consequence of a competition between the positive and two negative terms in the energy functionalE_λ. Notice thatE_λ(0)=0 and 0<1−δ <1< p<q+1. Let 0< λ < . The first negative term,

− λ 1−δ

(u⁺)^1−δdx, dominates providedu>0 is “small”, the positive term,

1 p

|∇u|^pdx,

becomes dominant foru>0 “mid-sized”, and the second negative term,

− 1 q+1

(u⁺)^q+1dx,

becomes dominant foru > 0 “large”. This intuitive picture clearly suggests two critical points for E_λ: a local minimizer between “small” and “mid-sized”, and a saddle point between “mid-sized” and “large”. Asλ ∈ (0, )approaches, the two critical points merge into a single one forλ=which disappears forλ > ; see definition (1.2).

The local minimizer is obtained first in theC¹topology with the help of our C^1,β regularity result (Theorem 2.2) combined with our strong comparison princi- ple (Theorem 2.3). Then we take advantage of arguments due to Brezis and Niren- berg [12] and Ambrosetti, Brezis, and Cerami [2] in order to show that the local minimizer in theC¹topology is also a local minimizer for E_λin the W₀^1,p topol- ogy. In contrast, the saddle point is obtained by a modification of the Mountain Pass lemma of Ambrosetti-Rabinowitz [4],cf.Ghoussoub and Preiss [25].

Before giving our main results, let us briefly recall the literature concerning re- lated singular problems. When p=2, the following problem has been investigated in quite a large number of papers:





−u= λ κ(x)

u^δ +µ(x)u^q in ; u|_∂=0, u>0 in .

(1.3) The weightsκ, µ: →

R

are assumed to be nonnegative and (essentially) bounded.

Whenµ= 0 (the purely singular problem), Crandall, Rabinowitz, and Tartar [15]

show that, for anyδ > 0, problem (1.3) admits a unique solutionu_λinC²()∩ C(); furthermore, if 0< δ <1 thenu_λis inC²()∩C¹(). Whenµ >0 is small enough, Coclite and Palmieri [13] prove the existence of a solution to problem (1.3) for 0 < λ < , withas in (1.2), 0 < <∞. Assuming 0 < δ <1, Yijing, Shaoping, and Yiming [44] apply variational arguments based on Nehari’s method [35] to show the existence of at least two solutions forq >1 subcritical: q <∞ if N = 1 or 2, andq < 2^∗−1 = ^N_N⁺₋²₂ ifN ≥ 3. The critical caseq = 2^∗−1 and N ≥3 was settled almost simultaneously in Haitao [29] and Hirano, Saccon, and Shioji [31] by two different methods: Perron’s method and Nehari’s method, respectively. To get the existence of at least two solutions, Haitao [29] shows that for any 0< λ < , the solution obtained by Perron’s method is a local minimizer for the energy functional E_λ. His arguments depend on the strong maximum principle (see Brezis and Nirenberg [11, Theorem 3]). In Adimurthi and Giacomoni [1], the existence of at least two solutions in dimension N = 2 is extended to 0< δ < 3 and to critical nonlinearities of Trudinger-Moser type (see Moser [34]). Note that δ <3 is the optimal condition onδ(δ >0) to obtain solutions inW₀^1,2().

In the casep =2 the question of multiplicity of solutions has been investigated for problems with convex and concave nonlinearities of the following kind:

−pu=λu^δ+u^q in ;

u|_∂=0, u>0 in , (1.4)

where 0< δ < p−1< q ≤ p^∗−1. Ambrosetti, Garc´ıa Azorero, and Peral [3]

establish the existence of at least two solutions to problem (1.4) for the subcritical

(q < p^∗−1) and radially symmetric case (= B_R(0)a ball). Their main tools are some uniform a priori estimates (that require radial symmetry) and global bifurca- tion theory. The critical caseq = p^∗−1 is treated in Garc´ıa Azorero and Peral [20]

with additional restrictions on pandλ > 0 small enough. These restrictions are used to prove that the levels of certain Palais-Smale sequences are strictly below the first critical levelS^N/p/N at which the Palais-Smale condition fails. Recall

S= inf

0 =u∈W₀^1,p()

|∇u|^pdx

|u|^p∗dx_p/p∗ .

Note that in this work, the existence of at least two solutions is not obtained for allλ∈ (0, ); only forλ >0 small enough. The restriction thatλ >0 be small was removed in Garc´ıa Azorero, Peral, and Manfredi [21] using the approach of C¹ versus W₀^1,p local minimizers ([2]). Essential elements in their approach are aC^1,β regularity result of DiBenedetto [19] and a strong comparison principle of Guedda and Veron [27]. A similar result for problem (P) with radial symmetry is obtained in Giacomoni and Sreenadh [22] when 0< δ,λ >0 is small enough, and q > p−1 > 0. The radially symmetric setting enables a shooting method to be employed; see also Atkinson and Peletier [8] and Prashanth and Sreenadh [37] for 1< p<N andλ∈(0, ).

The outline of this paper is as follows. Our main results are stated in Section 2.

In Section 3 we prove the existence of a solution that is a local minimizer of E_λin W₀^1,p()for 0< λ < . In the proof we use Theorem 2.2 which follows from the regularity results contained in Appendices A and B. In Section 4, using Ekeland’s principle and minimax arguments, we prove the existence of a second solution and thus finish the proof of Theorem 2.1.

ACKNOWLEDGEMENTS. The authors thank the anonymous referee for suggesting a number of valuable improvements and corrections.

2.

Main results

We look forweaksolutions (solutions, for short) of problem (P), that is, for func- tionsu∈W₀^1,p()satisfying ess inf_Ku>0 over every compact setK ⊂and

|∇u|^p−2∇u· ∇φdx =λ

u^−δφdx+

u^qφdx (2.1) for all φ ∈ C_c^∞(). As usual, C_c^∞() denotes the space of all C^∞ functions φ: →

R

with compact support. We denote by p^∗ = N p/(N − p)the critical Sobolev exponent for 1 < p < N; we take p^∗ ∈ (p,∞) arbitrarily large for

p= N, andp^∗ = ∞for p> N.

We introduce some notation which will be used throughout the paper. Given 1≤ p<∞, the norm inL^p()is denoted by

uL^p()def

= |u|^pdx 1/p

and the norm inW₀^1,p()by u_W1,p

0 () def=

|∇u|^pdx _1/p

.

The normalized positive eigenfunction associated with the principal eigenvalueλ1

of−pis denoted byφ1:

−pφ1=λ1|φ1|^p−2φ1 in ; φ1=0 on ∂, (2.2) φ1∈W₀^1,p()is normalized byφ1>0 inand

φ₁^pdx =1.

The functiond(x)denotes the distance from a pointx ∈ to the boundary

∂, where=∪∂is the closure of⊂

R

^N. This means that d(x)^def= dist(x, ∂)≡ inf

y∈∂|x−y|.

Note that the strong maximum and boundary point principles from V´azquez [43, Theorem 5, page 200] guaranteeφ1 > 0 in and ^∂φ_∂ν¹ < 0 on∂, respectively.

Hence, since φ1 ∈ C¹(), there are constants and L, 0 < < L, such that d(x)≤φ1(x)≤L d(x)for allx ∈.

The open ball inW₀^1,p()of radiusr centered atuis denoted by

B

r(u)^def= {v∈W₀^1,p(): u−v_W^1,p

0 ()<r},

for somer ∈ (0,∞) andu ∈ W₀^1,p(). Ifu = 0, we abbreviate

B

r ≡

B

r(0). Finally, the open ball in

R

^{N of radiusr} centered atxis denoted byB_r(x).

Our main result is the following theorem.

Theorem 2.1. Let the pair(p,q)satisfy either p∈(1,∞)and q ∈(p−1,p^∗−1), or else p∈

2N N+2,2

∪

3N N+3,3

and q = p^∗−1. Then there exists∈(0,∞) with the following properties:

(i) For every0< λ < there exist at least two solutions of problem(P), u_λand v_λ, such that u_λ, v_λ∈C¹()and u_λ

^vλ.

(ii) Forλ=there exists at least one solution of(P)in C¹(). (iii) For everyλ > there is no solution of(P).

To prove Theorem 2.1, we establish aC^1,β(), Theorem B.1 in Appendix B. The- orem B.1 gives the following regularity result for weak solutions to problem (P).

Theorem 2.2. Let0< δ < 1,1< p< ∞, and p−1<q ≤ p^∗−1. Then any weak solution to problem(P)belongs to C^1,β()for someβ∈(0,1).

This regularity result motivates and complements the following new strong comparison principle.

Theorem 2.3. Let u, v ∈C^1,β(), for some0< β <1, satisfy0

^u,0

^vand

−pu−λu^−δ= f, (2.3)

−pv−λv^−δ=g, (2.4)

with u = v = 0on∂, where f,g ∈ C()are such that0 ≤ f < g pointwise everywhere in. Then, the following strong comparison principle holds:

0<u < v in and ∂v

∂ν < ∂u

∂ν <0 on ∂. (2.5) Remark 2.4. Theorem 2.3 holds if we replace thep-Laplacian operator by a more general quasilinear operator; see, for instance, conditions (3)-(7) in Cuesta and Tak´aˇc [14].

Proof of Theorem 2.3. First, note that from the strong maximum of V´azquez (see Theorem 5 in [43]), we infer thatu > 0 in and ^∂_∂ν^u < 0 on∂. Hence, since u ∈ C¹(), there are constants andL, 0 < < L, such thatd(x) ≤u(x) ≤ L d(x)near the boundary∂. Analogous results hold forv. Moreover, f ≤gin guaranteesu≤vin, by the weak comparison principle which can be proved by a standard variational argument. Consequently,

d(x)≤u(x)≤v(x)≤L d(x) (2.6) near the boundary∂. As in the proof of Proposition 2.4 in Cuesta and Tak´aˇc [14]

(see page 729), we define anη-neighborhood_η⊂of the boundary∂, _η^def= {x ∈:d(x) < η}, (2.7) forη >0, and setw^def= v−u, 0≤w∈C^1,β()withw=0 on∂. There exists η >0 small enough, such that in the open set_ηwe have

−div(A(x)∇w)−B(x)w

= − N i,j=1

∂

∂x_i a_{i j}(x)∂w

∂x_j

−λB(x)w=g− f >0. (2.8)

The coefficientsa_{i j}(x)are given by a_{i j}(x)=

₁

0 |(1−t)∇u(x)+t∇v(x)|^p−2

×





δi j+(p−2)

∂

∂x_i((1−t)u+tv) ∂

∂x_j((1−t)u+tv)

|(1−t)∇u(x)+t∇v(x)|²





 dt (2.9)

for x ∈ _η and i, j = 1,2, . . . ,N, where δi j denotes the Kronecker symbol:

δi j =1 ifi = j;δi j =0 ifi = j. The differential operator above induced by the matrix (a_{i j})i,j=1,2,...,N is uniformly elliptic in_η witha_{i j} ∈ C^0,β(_η)provided η >0 is chosen small enough. The coefficientB(x)satisfies

B(x)= −δ ₁

0

dt

((1−t)u(x)+tv(x))^δ+1 <0. (2.10) Inequalities in (2.6) guarantee that B(x)satisfies the conditions of Lemma 2.7 in Hern´andez, Mancebo, and Vega [30]. We conclude that the (classical) strong maxi- mum principle applies to inequality (2.8) in each connected component of the open set_η, thus yielding inequalities (2.5) in_η.

Finally, we will show thatu < v throughout . Let η ∈ (0, η) and˜ ^def= \_η. Employingw >0 in_η, we can findc>0 such thatw≥con∂˜ ⊂_η. Moreover, recalling f,g∈C()with 0≤ f < gpointwise everywhere in, we can choosec>0 small enough, such that also

λ

u^δ − λ

(u+c)^δ ≤g− f holds in.˜ It follows thatu+c≤von∂˜ together with

−p(u+c)− λ

(u+c)^δ ≤ f +(g− f)=g= −pv− λ

v^δ in.˜ Consequently, we may apply the weak comparison principle (see Proposition 2.3 in [14]) in order to conclude thatu+c≤vholds throughout˜. As=_η∪ ˜, we have verifiedu< vthroughout.

3.

Existence of weak solutions

3.1. Existence of a solution for0< λ≤

First, let us consider the following purely singular Dirichlet problem:

−pu=λu^−δ in ;

u|_∂=0, u>0 in . (3.1)

Recall 0 < δ < 1. By requiring “u > 0 in” we actually mean ess inf_Ku > 0 for any compact set K ⊂ . We look for a solutionu ∈ W₀^1,p()that satisfies equation (3.1) in the sense of distributions. More precisely, if u₀ ∈ W₀^1,p()is a distributional solution of problem (3.1), with ess inf_Ku₀ > 0 for any compact set K ⊂, thenu₀∈C¹()by interior regularity due (independently) to DiBenedetto [19, Theorem 2, page 829] and Tolksdorf [42, Theorem 1, page 127].

Lemma 3.1. Assume0 < δ < 1andλ > 0. Then problem (3.1)has a unique weak solution in W₀^1,p()in the sense of distributions. This solution, denoted by u_λ, satisfies u_λ≥_λφ1a.e. in, where_λ>0is a constant.

Proof. First, we observe that an energy functional on W₀^1,p() formally corre- sponding to problem (3.1) can be given by

E˜_λ(u)^def= 1 p

|∇u|^pdx− λ 1−δ

(u⁺)^1−δdx, u∈W₀^1,p().

Owing to the Poincar´e inequality and 0 < 1−δ < 1 < p < ∞, this func- tional is coercive and weakly lower semicontinuous on W₀^1,p(). It follows that E˜_λpossesses a global minimizeru₀ ∈ W₀^1,p(). We haveu₀ ≡0 in, owing to E˜_λ(0)=0>E˜_λ(φ1)for every >0 small enough.

Second, the polar decompositionu =u⁺−u⁻ of any functionu ∈W₀^1,p() gives ∇u = ∇u⁺ − ∇u⁻. Thus, ifu₀ is a global minimizer for E˜_λ, then so is its absolute value |u₀|, by E˜_λ(|u₀|) ≤ ˜E_λ(u₀). The equality E˜_λ(|u₀|) = ˜E_λ(u₀) holds if and only if u⁻₀ = 0 a.e. in , that is, if and only if u₀ ≥ 0 a.e. in . Thus, any global minimizeru₀forE˜_λmust satisfyu₀ ≥0 a.e. in. Equivalently, u∈W₀^1,p()₊where

W₀^1,p()₊ ^def=

u∈W₀^1,p():u≥0 a.e. in stands for the positive cone inW₀^1,p().

Third, we will show that evenu₀ ≥ φ1 holds almost everywhere in with a constant > 0 small enough. To this end, let us first remark that the Gˆateaux derivative E˜_λ(φ1)ofE˜_λatφ1exists and satisfies

E˜_λ(φ1)= −p(φ1)−λ(φ1)^−δ =λ1(φ1)^p−1−λ(φ1)^−δ

=(φ1)^−δ

λ1(φ1)^p−1+δ−λ

≤ −λ

2(φ1)^−δ<0

(3.2)

whenever >0 is small enough, say, 0< ≤_λ.

On the contrary to our claim above, suppose that the (nonnegative) function v=(u₀−_λφ1)⁻=(_λφ1−u₀)⁺does not vanish identically in. Denote

⁺= {x ∈: v(x) >0}.

Let us investigate the functionξ(t) ^def= ˜E_λ(u₀+tv)oft ∈

R

+ = [0,∞). This function is convex thanks to the fact that the restriction of the functional E˜_λto the positive coneW₀^1,p()₊is convex. We haveξ(t)≥ξ(0)for allt ≥0. Furthermore, owing to u₀+tv ≥ max{u₀, t_λφ1} ≥ t_λφ1 fort > 0, the Gˆateaux derivative E˜_λ(u₀+tv)ofE˜_λatu₀+tvexists and yieldsξ(t)= ˜E_λ(u₀+tv), vfort >0.

This derivative is nonnegative and nondecreasing. Consequently, for 0<t <1 we have

0≤ξ(1)−ξ(t)= ˜E_λ(u₀+v)− ˜E_λ(u₀+tv), v

=

⁺

E˜_λ(_λφ1) vdx−ξ(t)

≤ −λ 2

⁺(_λφ1)^−δvdx<0,

(3.3)

by inequality (3.2) andξ(t) ≥ 0, a contradiction. We have verifiedv ≡ 0 in, that is,u₀≥_λφ1a.e. in.

Finally, we have proved that every global minimizer u₀ for E˜_λ onW₀^1,p() must satisfy u₀ ≥ _λφ1 a.e. in . The functional E˜_λ being strictly convex on W₀^1,p()₊, we conclude thatu₀is the only critical point of E˜_λinW₀^1,p()₊ with the property ess inf_Ku₀ > 0 for any compact setK ⊂ . Consequently,u_λ= u₀ provides the unique weak solution to problem (3.1).

Remark 3.2. In our proof of Lemma 3.1 above,v0 = 0 is a critical point for the functional v → ˜E_λ(u₀+v) defined for all v ∈ C_c^∞() only. We have proved that the functional E˜_λrestricted to W₀^1,p()₊ has precisely one critical point that stays away from zero, uniformly on any compact set K ⊂ , namely, the global minimizeru₀.

We obtain the following result regarding.

Lemma 3.3. Let0< δ <1and p−1<q ≤ p^∗−1. Then0< <∞.

Proof. We give the proof only in the critical case,i.e. q = p^∗−1. In the subcritical case,i.e.,q < p^∗−1, the proof is simpler since the energy functional E_λdefined below is weakly lower semicontinuous inW₀^1,p(). Letu_λbe the unique solution to (3.1). Define

f_λ(x,s)^def=

λs^−δ+s^q if s >u_λ(x);

λ(u_λ(x))^−δ+(u_λ(x))^q if s ≤u_λ(x). (3.4)

LetF_λ(x,s)=_s

0 f_λ(x,t)dt. DefineE_λ : W₀^1,p()→

R

^by E_λ(u)^def= 1

p

|∇u|^pdx−

F_λ(x,u)dx. (3.5) From Lemma A.4,E_λisC¹(W₀^1,p(),

R

⁾. We consider the following minimization problem:

I_λ= min

u∈Br

E_λ(u). (3.6)

Clearly, we haveI_λ>−∞. Note that

1

p|∇u|^p− _q+¹₁|u|^q+1

dx >0 for every u ∈∂

B

r providedr > 0 is small enough. Fix suchr >0; the other negative term in E_λ(u)may be made arbitrarily small by takingλ >0 small enough. Therefore we findrandλsuch that

umin∈∂Br

E_λ(u) >0. (3.7)

Moreover, sinceE_λ(tu) <0 fortsmall, we have

I_λ<0. (3.8)

Let{u_n}^∞_n=1 be a minimizing sequence,i.e. u_n ⊂

B

r and E_λ(u_n) → I_λ asn →

∞. From (3.7) and (3.8),{u_n}^∞_n=1 satisfies dist(u_n, ∂

B

r) ≥ η0for some η0 > 0.

Therefore, there exists 0<r₀<r such that

u_n∈

B

^r0. (3.9)

Now, from Ekeland’s variational principle, there existr₀ ≤r₁ <r and a sequence {vn}^∞_n=1⊂

B

r1satisfying

dist(u_n, vn)≤ 1

n, E_λ(u_n)≤E_λ(vn) and E_λ(vn)→0 in W^−1,p()asn→ ∞.

(3.10)

From the first statement of (3.10),{vn}^∞_n=1is a minimizing sequence forI_λand up to a subsequence satisfiesvn u˜_λasn→ ∞withu˜_λ∈

B

^r1. From the last statement of (3.10), we have

−p(vn)− f_λ(x, vn)=o_n(1) in W^−1,p(). (3.11) From (3.11), Theorem 2.1 in Boccardo and Murat [9] with

f_n(x)=(max{vn(x),u_λ(x)})^−δ+o_n(1), g_n(x)=(max{vn(x),u_λ(x)})^q

(note that from Hardy’s inequality and since q = p^∗ −1, {f_n}^∞_n=1 and{g_n}^∞_n=1 satisfy the conditions in Theorem 2.1 in [9]), Remark 2.1 in [9] and from Brezis and Lieb [10], it follows that

vn_W1,p

0 ()= vn− ˜u_λW1,p

0 ()+ ˜u_λW1,p

0 ()+o_n(1) and

vn_Lq+1()= vn− ˜u_λLq+1()+ ˜u_λLq+1()+o_n(1) (3.12) asn→ ∞. From (3.9), (3.10) and (3.12), it follows thatu˜_λ,vn− ˜u_λ∈

B

^{r. Thus,}

1

p|∇vn− ˜u_λ|^p− 1

q+1|vn− ˜u_λ|^q+1

dx >0. (3.13) From (3.12) and (3.13), we get

I_λ= E_λ(vn)+o_n(1)

= E_λ(u˜_λ)+ 1

pvn− ˜u_λ^p

W₀^1,p()− 1

q+1vn− ˜u_λ^q_L⁺q+1¹()+o_n(1)

≥E_λ(u˜_λ)+o_n(1).

Hence,E_λ(u˜_λ)=I_λand

−pu˜_λ= f_λ(x,u˜_λ) in ;

˜

u_λ|_∂=0.

Now, Theorem 2.3 imply thatu˜_λ >u_λin, henceu˜_λis a weak solution to prob- lem (P). Thus >0.

Now, let us show that < ∞. We argue by contradiction: suppose there exists a sequenceλn→ ∞such that problem (P) admits a solutionu_n. There exists λ_∗>0 such that

λ

t^δ +t^q ≥(λ1+)t^p−1for allt >0, ∈(0,1)andλ > λ_∗. Chooseλn > λ_∗. Clearlyu_nis a supersolution of the problem

−pu=(λ1+)u^p−1 in ;

u>0, u|∂=0. (3.14)

for all ∈(0,1). We now use Lemma 3.1 to chooseµ < λ1+small enough so thatµφ1(x) <u_n(x)andµφ1is a subsolution to problem (3.14). By a monotone interation procedure we obtain a solution to (3.14) for any∈(0,1), contradicting the fact thatλ1is an isolated point in the spectrum of−pinW₀^1,p()(see Anane [5]).

We prove now the existence of positive weak solution to (P) for any 0< λ <

. Precisely, we have the following result:

Proposition 3.4. For anyλ ∈ (0, ), there exists u_λ a positive weak solution to (P). Moreover, E_λ(u_λ) <0.

Proof. Fix 0< λ < λ2< .λ2is such that there exist solutions to (P) forλ=λ2. Letu_λbe the solution of (3.1) andu_λ2 is one solution of (P) (whenλ= λ2in the equation of (P)). Clearly, from Theorem B.1 in Appendix B,u_λ,u_λ2are inC^1,β() for some 0 < β < 1 andu_λ≤ u_λ2 in. Indeed, setting ^def= {u_λ− ¯u > 0}and from the equations satisfied byu_λandu¯=u_λ2we have

(+pu¯−pu_λ)(u_λ− ¯u)dx ≤λ

(u^−δ_λ − ¯u^−δ)(u_λ− ¯u)dx≤0 (3.15)

and

(+pu¯−pu_λ)(u_λ− ¯u)dx

≥

(|∇u_λ|^p−2∇u_λ− |∇ ¯u|^p−2∇ ¯u)(∇u_λ− ∇ ¯u)dx

≥







 C_p

u_λ− ¯u>0

|∇(u_λ− ¯u)|²

(|∇u_λ| + |∇ ¯u|)^2−pdx if 1< p<2 C_p

u_λ− ¯u>0

|∇(u_λ− ¯u)|^pdx ifp≥2

≥0

(3.16)

from Lemma 4.1 in Ghoussoub and Yuan [26]. Hence from (3.15) and (3.16), we getu_λ≤ ¯u.

By the strong comparison principle (Theorem 2.3), we obtain u¯ > u_λ in , ^∂_∂ν^u¯ < ^∂_∂ν^uλ on∂. Define

f˜_λ(x,s)=







λ¯u(x)^−δ+ ¯u(x)^q ifs>u¯(x),

λs^−δ+s^q ifu_λ(x)≤s≤ ¯u(x), λu_λ(x)^−δ+u_λ(x)^q ifs<u_λ(x).

LetF˜_λ(x,s)=_s

0 f˜_λ(x,t)dt. Define the functionalE˜_λ:W₀^1,p()→

R

^by E˜_λ(u)= 1

p

|∇u|^pdx−

F˜_λ(x,u)dx.

E˜_λis bounded below in W₀^1,p()and is weakly lower semi-continuous. Hence, E˜_λachieves its global minimum at some u_λ ∈ W₀^1,p(). Moreover, since E˜_λ is

C¹ by Lemma A.4, u_λ solves the equation−pu_λ = ˜f_λ(x,u_λ)in. From the strong maximum principle of V´azquez (see Theorem 5 in [43]) we getu_λ > 0 in . It follows by regularity results (see again Theorem B.1 in Appendix B) that u_λ ∈ C^1,β()for someβ ∈ (0,1). Again, by Therorem 2.3, we conclude that u_λ < u_λ < u¯ in and _∂ν^∂ (u_λ−u_λ) < 0,_∂ν^∂ (u¯ − u_λ) < 0 on ∂. Hence, f˜_λ(x,u_λ) =λu^−δ_λ +u^q_λforx ∈and sou_λis a weak solution to (P). Moreover, we have that

E˜_λ(u_λ)≤ ˜E_λ(u_λ)= E_λ(u_λ) < 1 p

|∇u_λ|^pdx− λ 1−δ

u_λ^1−δdx <0. This completes the proof of Proposition 3.4.

Now, we show the following result.

Proposition 3.5. There exists at least one positive weak solution forλ=to(P).

Proof. Let{λk}^∞_k=1such thatλk ask→ ∞. Then, from Proposition 3.4, there existsu_k =u_λk ≥u_λ

k to a weak positive solution to (P) forλ=λk. Therefore, for anyφ∈C_c^∞(), we have

|∇u_k|^p−2∇u_k∇φdx=λk

(u_k)^−δφdx+

u^q_kφdx. (3.17) Since u_k ∈ W₀^1,p() and u_k ≥ u_λ

k it is easy to see that (3.17) holds also for φ ∈W₀^1,p(). Moreover, from Proposition 3.4

E_λk(u_k) <0. (3.18) From (3.18), it follows that

sup

k

u_kW1,p

0 ()<∞. (3.19)

Hence, there existsu ≥u_λ

k such thatu_k u inW₀^1,p()ask → ∞and then by Sobolev imbedding:

u_k u inL^q() and pointwise a.e. ask → ∞. (3.20) From (3.17), (3.19) and (3.20), we get for anyφ∈W₀^1,p():

|∇u|^p−2∇u∇φdx =λ

(u)^−δφdx+

u^qφdx (3.21) which completes the proof of Proposition 3.5.

From the above propositions, we get the following corollary:

Corollary 3.6. Let1< p<∞, p−1<q ≤ p^∗−1,0< δ <1, and0< λ≤. Then there exists a minimal solution to(P).

Proof. We use here the weak comparison principle (see Proposition 2.3 in Cuesta and Tak´aˇc [14] or Tolksdorf [41]) and the following monotone iterative scheme:





−pu_n− λ

u^δ_n =u^q_n−1 in ; u_n|_∂=0,

(3.22) where u₀ = u_λ, the unique solution to (3.1). Note thatu₀ is a weak subsolution to (P) andu₀ ≤ u where u is the solution to (P) obtained in Proposition 3.5.

Then, from the weak comparison principle, we get easily thatu₀≤u₁and{u_n}^∞_n=1 is nondecreasing. Furthermore, u_n ≤ u and{u_n}^∞_n=1 is uniformly bounded in W₀^1,p(). Hence, it is easy to prove that{u_n}^∞_n=1 converges weakly inW₀^1,p() and pointwise to uˆ_λ, a weak solution to (P). Let us show that uˆ_λis the minimal solution to (P) for anyλ ∈ (0, ]. Letv_λa weak solution to (P) forλ ∈ (0, ].

Then,u₀=u_λ≤v_λ. From the weak comparison principle,u_n ≤v_λfor anyn≥0.

Lettingn→ ∞, we getuˆ_λ≤v_λ. This completes the proof of Corollary 3.6.

3.2. C¹versusW^1,plocal minimizers of the energy

Let u_λ be the solution to (P) given by Proposition 3.4. The main result in this paragraph is

Proposition 3.7. For 0< λ < , u_λis a local minimizer of E_λin W₀^1,p(). Proof. We observe first thatu_λis a local minimizer in theC¹-topology. Indeed, taking advantage of the strong comparison principle shown in Theorem 2.3 and the definition ofu_λ, we have that forν >0 small enough,

u−u_λC¹_()¯ ≤ν ⇒u_λ≤u≤ ¯u. (3.23) whereu¯is defined in the proof of Proposition 3.4. From (3.23), we get that

u−u_λC¹_()¯ ≤ν ⇒ E_λ(u_λ)= ˜E_λ(u_λ)≤ ˜E_λ(u)= E_λ(u).

Now, let us show that u_λ is a local minimizer of E_λ in W₀^1,p(). Suppose not and we derive a contradiction. First, we deal with the subcritical case, i.e. q <

p^∗−1. In this case,E_λis weakly lower semicontinuous onW₀^1,p()and achieves its minimum on bounded subsets ofW₀^1,p(). Hence, ifu_λis not a local minimum for E_λ, for every >0 we obtainvsuch that 0<v_W^1,p

0 () ≤and

E_λ(u_λ+v) < E_λ(u_λ), E_λ(u_λ+v)= inf

v_W1,p

0 ()≤E_λ(u_λ+v). (3.24)