1, Jannelli [7] obtained the following results: If 0≤µ≤µ−1, then (1.1) has at least one solutionu∈H01(Ω) for all 0&lt

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

AN EXISTENCE RESULT FOR ELLIPTIC PROBLEMS WITH SINGULAR CRITICAL GROWTH

YASMINA NASRI

Abstract. We prove the existence of nontrivial solutions for the singular critical problem

−∆u−µ u

|x|² =λf(x)u+u^2∗−1

with Dirichlet boundary conditions. Here the domain is a smooth bounded subset ofR^N,N≥3, and 2^∗= _N−2^2N which is the critical Sobolev exponent.

1. Introduction This paper concerns the semilinear elliptic problem

−∆u−µ u

|x|² =λf(x)u+u^2∗−1 in Ω u >0 in Ω

u= 0 on∂Ω,

(1.1)

where Ω is a smooth bounded domain in R^N, N ≥ 3 with 0 ∈ Ω; λ and µ are positive parameters with 0≤µ < µ:= (^N₂⁻²)²,µis the best constant in the Hardy inequality, 2^∗=_N^2N₋₂ is the critical Sobolev exponent andf is a positive measurable function which will be specified later.

In recent years, many people have paid much attention to the existence of non- trivial solutions for singular problems we cite [4, 5, 7, 8] and the references cited therein.

Forf(x) = 1, Jannelli [7] obtained the following results:

If 0≤µ≤µ−1, then (1.1) has at least one solutionu∈H₀¹(Ω) for all 0< λ <

λ₁(µ) whereλ₁(µ) is the first eigenvalue of the operator (−4 −_|x|^µ₂) inH₀¹(Ω).

Ifµ−1< µ < µ, then (1.1) has at least one solutionu∈H₀¹(Ω) for allµ^∗< λ <

λ1(µ) where

µ^∗= min

ϕ∈H¹₀(Ω)

R

Ω

|∇ϕ(x)|²

|x|^2σ dx R

Ω

|ϕ(x)|²

|x|^2σ dx andσ=√

µ+√ µ−µ.

2000Mathematics Subject Classification. 35J20, 35J60.

Key words and phrases. Palais-Smale condition; singular potential; Sobolev exponent;

mountain-pass theorem.

c

2007 Texas State University - San Marcos.

Submitted February 6, 2007. Published June 6, 2007.

1

Ifµ−1< µ < µand Ω =B(0, R) then (1.1) has no solution forλ≤µ^∗. If λ ≤ 0 and Ω is star shaped then (1.1) has no nontrivial solutions using Pohozaev-type identity.

For the quasi-linear form of (1.1) the problem has been studied by [5] forµ= 0 and f(x) = _|x|¹_q where 0≤q < p. The purpose of the present paper is to extend (partially) the results obtained by [7] to the case wheref can be singular.

This paper is organized as follows. In section 2, we recall some preliminaries results. In section 3, we give the proof of our theorem using mountain pass Theorem.

2. Notation and Preliminaries We make use the following notation:

L^p(Ω), 1≤p≤ ∞, denote Lebesgue spaces, the normL^p is denoted byk · kpfor 1≤p≤ ∞;

D^1,2(R^N) denotes the closure space ofC₀^∞(R^N) with respect the normk·k_D1,2(R^N):=

R

R^N|∇u|²dx1/2

;

B_r(0) is the ball centred at 0 with radiusr;

C,C₁,C₂ will denote various positive constants;

OnH₀¹(Ω) we use the norm kukµ=Z

Ω

(|∇u|²−µ u²

|x|²)dx1/2

.

By Hardy’s inequality [6], this norm is equivalent to the usual norm ofH₀¹(Ω). Let F=

f : Ω→R⁺: lim

|x|→0|x|²f(x) = 0 withf ∈L^∞_loc(Ω\{0}) ; for 0≤β <2, we set

F2,β =

f ∈ F: 0< lim

|x|→0|x|^βf(x)<∞ . Now, we recall the following results.

Lemma 2.1 ([4]). Let 0≤µ < µ= (^N−2₂ )²,λ∈R⁺,f ∈ F. Then the eigenvalue problem

−∆u−µ u

|x|² =λf(x)u inΩ u= 0 on ∂Ω

admits a nontrivial weak solutions inH₀¹(Ω)corresponding toλ∈(λ^k_µ(f))^∞_k=1where 0< λ¹_µ(f)< λ²_µ(f)≤λ³_µ(f)≤ · · · →+∞.

Lemma 2.2 ([4]). Let Ω be a bounded domain in R^N and f ∈ F. Then the embedding H₀¹(Ω),→L²(Ω, f dx)is compact.

Lemma 2.3 ([4]). Let 2^∗_β = ^2(N_N−2^−β), if f ∈ F_2,β, 0≤β <2; then the embedding H₀¹(Ω),→L^q(Ω, f dx)is (i) continuous for all2≤q≤2^∗_β, (ii) compact for2≤q <

2^∗_β.

Now, we give some examples of function f ∈ F having lower order singularity than|x|⁻² at the origin:

(a) Any bounded function.

(b) In a small neighbourhood of 0,f is|x|^−β for 0< β <2.

(c) f(x) =|x|^−β/|log|x||in a small neighbourhood of 0.

Definition 2.4. Letc∈R,Ebe a Banach space andI∈C¹(E,R). We say thatI satisfies the Palais-Smale condition at the levelc, for short (P S)c, if every sequence (un)n inE such thatI(un)→candI⁰(un)→0 asn→+∞in E⁰ (dual ofE), has a convergent subsequence inE.

Definition 2.5. A functionuinH₀¹(Ω) is said to be a weak solution of (1.1) ifu satisfies

Z

Ω

∇u∇v−µ uv

|x|² −λf(x)uvdx−u^2∗−1v

dx= 0 for allv∈H₀¹(Ω).

It is well known that the nontrivial solutions of (1.1) are equivalent to the non zero critical points of the energy functional

Jλ,µ(u) =1 2

Z

Ω

|∇u|²dx−µ 2 Z

Ω

u²

|x|²dx−λ 2 Z

Ω

f(x)u²dx− 1 2^∗

Z

Ω

|u|^2∗dx.

Define the constant

Sµ= inf

u∈D^1,2(R^N)\{0}

R

R^N|∇u|²dx−µR

R^N u²

|x|²dx R

R^N|u|^2∗2/2^∗ . It is known thatSµ is achieved by the family of functions

u^∗_ε= C_ε

(ε|x|^σ0/√µ+|x|^σ/√µ)^√µ where Cε= (4εN(µ−µ)/(N−2))

õ

2 , σ=√ µ+√

µ−µand σ⁰ =√ µ−√

µ−µ, see [8] for the details.

Note thatu^∗_ε satisfies

−∆u−µ u

|x|² =|u|^2∗−2u foru∈D^1,2(R^N)\{0}.

Hence, we have

ku^∗_εk²_µ=ku^∗_εk²₂^∗∗ = (S_µ)^N/2. Let 0≤φ(x)≤1 be a function inC₀^∞(Ω) defined as

φ(x) =

(1 if|x| ≤R 0 if|x| ≥2R, whereB_2R(0)⊂Ω. Set

uε=φ(x)u^∗_ε and vε= uε

ku_εk₂^∗, (2.1)

so thatkvεk²₂^∗∗ = 1.

In the present paper we prove the following result.

Theorem 2.6. Let f ∈ F2,β and 0 ≤ β < 2. If 0 ≤ µ ≤ µ−(^2−β₂ )² and 0< λ < λ¹_µ(f), then (1.1)has at least one positive solution.

3. Proof of the main theorem First, we establish some lemmas.

Lemma 3.1. Assume thatf ∈ F2,β and0< λ < λ¹_µ(f). ThenJ_λ,µ satisfies(P S)_c for allc <(Sµ)^N/2/N.

Proof. Let (un)n be a sequence such that

J_λ,µ(u_n)→c and J_λ,µ⁰ (u_n)→0 in [H₀¹(Ω)]⁰ asn→+∞. (3.1) We remark that

2Jλ,µ(un)−

J_λ,µ⁰ (un), un

= (1− 2

2^∗)kunk²₂^∗∗ ≤2c+o(1), (3.2) combining (3.1) and (3.2) we show that (u_n) is bounded in H₀¹(Ω).

From Lemmas 2.2 and 2.3, and the reflexivity ofH₀¹(Ω) we extract a subsequence, still denotedu_n such that

u_n →u weakly inH₀¹(Ω) u_n→u inL^r(Ω) if 1< r <2^∗,

un→u almost everywhere, u_n

x → u

x weakly inL²(Ω), un→u strongly inL²(Ω, f dx).

(3.3)

From (3.3) we deduce that

hJ_λ,µ⁰ (u), ϕi= 0 for allϕ∈H₀¹(Ω), (3.4) henceuis a solution of (1.1).

Denotevn :=un−u, then the Brezis-Lieb lemma [2] implies k∇unk²₂=k∇uk²₂+k∇vnk²₂+o(1);

kunk²₂^∗∗ =kuk²₂^∗∗+kvnk²₂^∗∗+o(1);

Z

Ω

u²_n

|x|²dx= Z

Ω

u²

|x|²dx+ Z

Ω

v²_n

|x|²dx+o(1).

(3.5)

Using (3.1), (3.5) and lemma 2.2, we obtain J_λ,µ(u) +1

2kvnk²_µ− 1

2^∗kvnk²₂^∗∗ =c+o(1), (3.6) and

kuk²_µ=kuk²₂^∗∗ +λ Z

Ω

f(x)u²dx− kvnk²_µ+kvnk²₂^∗∗+o(1).

From (3.4) it follows that

kvnk²_µ− kvnk²₂^∗∗ =o(1).

We may therefore assume that

kvnk²_µ→a and kvnk²₂^∗∗ →a, by the definition ofS_µ, we have

S_µkv_nk²₂∗ ≤ kv_nk²_µ, in the limit we have

S_µa^2/2∗ ≤a,

it follows that eithera= 0 ora≥(Sµ)^N/2.

Ifa≥(Sµ)^N/2 passing in the limit in (3.6) we obtain Jλ,µ(u) + 1

Na=c using the assumptionc < _N¹(Sµ)^N/2, we find

Jλ,µ(u)<0. (3.7)

On the other hand, from (3.4) we obtain Jλ,µ(u) = 1

Nkuk²₂^∗∗ ≥0,

which is a contradiction with (3.7). Thenun →ustrongly inH₀¹(Ω).

Lemma 3.2. Assume that f ∈ F2,β then 1/ There exist α, δ > 0 such that J_λ,µ(u) ≥ α for all u ∈ H₀¹(Ω) such that kuk_µ = δ for all 0 < λ < λ¹_µ(f).

2/Jλ,µ(v)<0 for allv∈H₀¹(Ω)such that kvkµ> δ.

Proof. Using the definition ofSµ and the fact that 0< λ < λ¹_µ(f), we obtain J_λ,µ(u)≥ 1

2 1− λ λ¹_µ(f)

kuk²_µ− 1

2^∗(Sµ)^2∗/2kuk²_µ^∗. So forδ >0 sufficiently small there existsα >0 such that

Jλ,µ(u)≥α forkukµ=δ.

Fort >0,

Jλ,µ(tu) = t²

2(kuk²_µ− Z

Ω

f(x)u²dx)−t^2∗

2^∗kuk²₂^∗∗dx,

as t → +∞ we have Jλ,µ(tu) → −∞. Then there exists v ∈ H₀¹(Ω) such that

Jλ,µ(v)<0 for kvkµ> δ.

Lemma 3.3. Assume that0< λ < λ¹_µ(f)and0≤µ≤µ−(^2−β₂ )². Then sup

0≤t<∞

Jλ,µ(tvε)< 1

N(Sµ)^N/2 providedε >0 is a small enough.

Proof. Consider the functions g(t) :=Jλ,µ(tvε) =t²

2(kvεk²_µ−λ Z

Ω

f(x)v²_εdx)−t^2∗ 2^∗,

wherev_εis the extremal function defined in (2.1). Note that lim_t→+∞g(t) =−∞

andg(t)>0 when tis close to 0. So that sup_t≥0g(t) is attained for somet_ε>0.

From

0 =g⁰(tε) =tε kvεk²_µ−λ Z

Ω

f(x)v²_εdx

−t²_ε^∗−1kvεk²₂^∗∗, we have

t_ε=h

kvεk²_µ−λ Z

Ω

f(x)v²_εdxi2∗ −2¹

. Thus,

g(tε) = 1 N

kvεk²_µ−λ Z

Ω

f(x)v_ε²dx ²

∗ 2∗ −2

.

Then as in [7] (see also [3]), we have the following estimates:

Z

Ω

|∇vε|²dx−µv_ε²

|x|²

dx=Sµ^N2 +Cε^N−22 ;

since f ∈ F2,β, there exist r > 0 and C1, C2 > 0 such that K1|x|^−β ≤ f(x) ≤ K₂|x|^−β onB_R(0). Thus

C₁ε

√¯µ 2√

¯ µ−µ(2−β)

≤ Z

Ω

f(x)v_ε²dx≤C₂ε

√µ¯ 2√

¯ µ−µ(2−β)

ifµ < µ−(2−β 2 )²; C₁ε^N−22 |logε| ≤

Z

Ω

f(x)v_ε²dx≤C₂ε^N−22 |logε| ifµ=µ−(2−β 2 )². Consequently,

g(t_ε)≤ (₁

NS

N

µ2 +Cε^N−22 −C1ε^N−22 |logε| ifµ=µ−(^2−β₂ )²,

1

NSµ^N2 +Cε^N−22 −C1ε

√µ¯ 2√

¯ µ−µ(2−β)

ifµ < µ−(^2−β₂ )². Therefore, forε >0 sufficiently small andµ≤µ−(^2−β₂ )² we get

sup

t≥0

Jλ,µ(tvε)< 1 NS_µ^N/2.

Proof of Theorem 2.6. From Lemmas 3.1, 3.2 and 3.3,Jλ,µsatisfies all assumptions of mountain pass Theorem [1], thenc is a critical value i.e. there existsu∈H₀¹(Ω) such that J_λ,µ⁰ (u) = 0 and Jλ,µ(u) = c > 0. Since Jλ,µ(u) = Jλ,µ(|u|) = c, thus

problem (1.1) admits a positive solution.

References

[1] A. Ambrosetti, P. Rabinowitz;Dual variational methods in critical point theory and applica- tions, J. Funct. Anal. 14 (1973), 349-381.

[2] H. Br´ezis, E. Lieb;A relation between pointwise convergence of functions and convergence of functionals, Proc. AMS 88 (1983), 486-490.

[3] J. Chen;Existence of solutions for nonlinear PDE with an inverse square potential, J. Diff.

Eq. 195 (2003), 497-519.

[4] N. Chaudhuri, M. Ramaswamy; Existence of positive solutions of some semilinear elliptic equations with singular coefficients, J. Proc. Soc. Ed 131 (2001), 1275-1295.

[5] J. P. Garcia Azorero, I. Peral Alonso;Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Eq. 144 (1998), 441-476.

[6] G. Hardy, J. E. Littlewood and G. Polya;Inequality, Cambridge univ. Press, Cambridge, UK, 1934.

[7] E. Jannelli;The role played by space dimension in elliptic critical problem, J. Diff. Eq. 156 (1999), 407-426.

[8] S. Terracini,On positive solutions to a class equations with singular coefficient and critical exponent, Adv. Diff. Eq. 2 (1996),241-264.

Yasmina Nasri

Universit´e de Tlemcen, d´epartement de math´ematiques, BP 119 Tlemcen 13000, Alg´erie E-mail address:y [email protected]