2005-Oujda International Conference on Nonlinear Analysis.
Electronic Journal of Differential Equations, Conference 14, 2006, pp. 149–153.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
MULTIPLICITY RESULTS FOR NONLINEAR ELLIPTIC
EQUATIONS
SAMIRA BENMOULOUD, MOSTAFA KHIDDI, SIMOHAMMED SBAI
the
Abstract. Let Ω be a bounded domain in RN , N ≥ 3, and p = N2N
−2
limiting Sobolev exponent. We show that for f ∈ H01 (Ω)∗ , satisfying suitable
conditions, the nonlinear elliptic problem
−∆u = |u|p−2 u + f
u=0
in Ω
on ∂Ω
has at least three solutions in H01 (Ω).
1. Introduction
It is well known [6, Theorems 1 and 2] that for f 6= 0 and kf k sufficiently small,
the problem
−∆u = |u|p−2 u + f on Ω
(1.1)
u = 0 on ∂Ω
has at least two distinct solutions u0 and u1 which are critical points of the functional
Z
Z
Z
1
1
I(u) =
|∇u|2 −
|u|p −
f u,
2 Ω
p Ω
Ω
such that I(u1 ) > I(u0 ). In this note we suppose f ≥ 0 and satisfies
α N
(1.2)
kf k < S 4 ,
N
where
N − 2 N +2
1
<α<(
) 4 , and S =
inf
k∇uk22 ,
2
N +2
u∈H01 (Ω)kukp =1
which corresponds to the best constant for the Sobolev embedding H01 (Ω) ,→ Lp (Ω).
We determine a special ωε , from the extremal functions for the Sobolev inequality
in RN , and consider Γ the class of continuous paths joining 0 to ωε .
Proposition 1.1. Let
c = inf sup I(γ(t)).
γ∈Γ t∈[0,1]
2000 Mathematics Subject Classification. 35J20, 35J65.
Key words and phrases. Semilinear elliptic equations; critical Sobolev exponent.
c 2006 Texas State University - San Marcos.
Published September 20, 2006.
149
150
S. BENMOULOUD, M. KHIDDI, M. SBAI
EJDE/CONF/14
Then there is a sequence (uj ) ⊂ H01 (Ω) such that
I(uj ) → c,
0
I (uj ) → 0
in (H01 (Ω))∗ ,
I(u0 ) < I(u1 ) < c.
Let u denotes the weak limit in H01 (Ω) of (a subsequence of) (un ), our principal
result is as follows.
Theorem 1.2. Let f ∈ H01 (Ω)∗ , f ≥ 0 satisfies (1.2). Then either
(1) I(u) = c and Problem (1.1) has at least three solutions. Or
(2) I(u) ≤ c − N1 S N/2 .
Note that the existence results of biharmonic analogue of Problem (1.1) have
been studied in [2], so a result similar to that of Theorem 1.2 may be established
for the bilaplacian operator.
2. The proof of Proposition 1.1
We start with a variant of the mountain pass theorem of Ambrosetti-Rabinowitz
without the Palais-Smale condition
Theorem 2.1. Let E be a real Banach space and I ∈ C 1 (E, R). Suppose there
exists a neighborhood U of 0 in E and a constant ρ > 0 such that
(H1) I(u) ≥ ρ, for all u ∈ ∂U .
(H2) I(0) < ρ and, I(v) < ρ for some v ∈ E \ U .
Let
c = inf max I(γ(t)),
γ∈Γ t∈[0,1]
where
Γ = {γ : [0, 1] → E, is continuous, γ(0) = 0, γ(1) = v}.
Then there is a sequence (un ) in E such that
I(un ) → c,
0
I (un ) → 0
in E ∗ .
On H01 (Ω) we define a variational functional I : H01 (Ω) → R for problem (1.1),
by
Z
1
1
2
p
f u.
I(u) = k∇uk2 − k ukp −
2
p
Ω
Clearly I is C 1 on E and I(0) = 0. We shall verify the assumptions of Theorem 2.1
Verification of (H1). Let r ∈]0, αS N/4 [ and u ∈ H01 (Ω)) be such that k∇uk2 = r.
We have
1
1
I(u) ≥ r2 − rp S −p/2 − kf kr.
2
p
Letting r → αS N/4 , we obtain
1
1
1 2 N/2
I(u) ≥ α2 S N/2 − αp S N/2 −
α S
.
2
p
4N
Set
αp S N/2
ρ=
,
2N
hence I(u) > ρ for all u ∈ ∂B(0, r).
EJDE/CONF/14
MULTIPLICITY RESULTS
151
Verification of (H2). Assume 0 ∈ Ω and let φ ∈ C0∞ (Ω) be a fixed function such
that φ ≡ 1 for x in some neighborhood of 0. For ε > 0, define
u(x)
φ(x)
u (x) =
v (x) =
.
N −2 ,
2
ku
2
kp
( + |x| )
Hence, from [4],
N −2
k∇v k22 = S + O( 2 ).
For every µ 6= 0, [6, Lemma 2.1], gives a real t+ > 0 such that
t+ > (
(2.1)
N −2
1
k∇µv k22
1 N − 2 N −2
p−2 =
(
) 4 k∇v k2 2
p)
(p − 1)kµv kp
µ N +2
(2.2)
N −2
1
k∇v k2 2 .
µ
(2.3)
and
t+ <
Set ω = t+ µv . We have
N
N
N − 2 N −2
N − 2 N −2 N
) 4 k∇v k22 > (
) 4 S 4 > αS 4 > r.
N +2
N +2
On the other hand, from (2.2) and (2.3), we get
1
1
I(ω ) < (t+ )2 k∇ω k22 − (t+ )p
2
p
1
1 1 N − 2 p(N −2)
<
k∇v kN
) 4 k∇v kN
2 − p (
2 .
2
2µ
µ p N +2
k∇ω k2 = t+ µk∇v k2 > (
Using (2.1), we deduce
I(ω ) < (
N −2
1
1 N −2 N −2 N
p S N/2
− p
(
) 2 )(S + O( 2 ))N/2 < 0
,
2
2µ
µ N +2 N +2
2N
for µ large enough. Then c ≥ ρ > I(ω ). Recall that ω ∈ Λ− ([6, Lemma 2.1] with
Λ− = {u ∈ H01 (Ω)/ < I 0 (u), u >= 0, k∇uk22 − (p − 1)kukpp < 0},
and that inf Λ− I is attained by u1 [6, Theorem 2]. We conclude that
c ≥ ρ > I(ω ) ≥ I(u1 ) > I(u0 ).
3. Proof of the Theorem 1.2
Applying Proposition 1.1 we obtain a sequence (uj ) ⊂ H01 (Ω) such that
I(uj ) → c,
0
I (uj ) → 0
in
H01 (Ω)∗ .
(3.1)
(3.2)
This implies that k∇uj k2 is uniformly bounded. Hence for a subsequence of uj ,
still denoted by uj , we can find u ∈ H01 (Ω) such that
uj → u weakly in H01 (Ω),
uj → u strongly in Lq , q < p,
uj → u a.e. on Ω.
From (3.2), we deduce that u is a (weak) solution of Problem (1.1). In particular
u satisfies
Z
2
p
kuk2 − kukp = f u
(3.3)
152
S. BENMOULOUD, M. KHIDDI, M. SBAI
EJDE/CONF/14
Let uj = u + vj , where vj → 0 weakly in H01 (Ω) and vj → 0 a.e on Ω. We have
k∇uj k22 = k∇uk22 + k∇vj k22 + ◦(1).
and by (3.1),
1
1
I(u) + k∇vj k22 − kvj kpp = c + o(1),
2
p
thanks to Brezis-Lieb Lemma [5]. By (3.2) and (3.3), k∇vj k22 − kvj kpp = o(1), which
gives
1
I(u) + k∇vj k22 = c + o(1).
N
Set l = limj→+∞ k∇vj k22 , then limj→+∞ kvj kpp = l. Using Sobolev inequality one
N
see that l ≥ Sl2/p . Then l = 0, or l ≥ S 2 . We get, either
I(u) = c,
and since
I(u) > I(u1 ) > I(u0 ),
u is a solution of Problem (1.1) distinct from uo and u1 , or
I(u) ≤ c −
Remark 3.1. One can show that c <
case
1 N
S2.
N
N
1
2
NS
, consequently I(u) < 0 in the second
4. Semilinear biharmonic equation
In [2], Benmouloud considered the problem
∆2 u = |u|p−2 u + f
in Ω
∆u = u = 0 on ∂Ω
2
where Ω is a bonded domain in RN , N ≥ 5 p = N2N
−4 and ∆ denotes the biharmonic
−1
operator. She proved that for f ∈ H subject to a suitable condition, this problem
has at least two distinct solutions in H 2 (Ω) ∩ H01 (Ω). The existence of on solution
follows from the mountain-pass theorem, with Palais-Smale condition, and a second
is obtained by a constrained minimization (see also [3]).
It follows from this study that an analog result of Theorem 1.2 may be established
by a similar argument with suitable smallness condition on f .
References
[1] A. Ambrosetti, P. Rabinowitz; Dual Variational Methods in Critical Point Theory and Applications, J. Funct. Anal, Vol. 11, 1973, pp. 349-381.
[2] S. Benmouloud, Existence de solutions pour un problme biharmonuque non homogne avec
exposant critique de Sobolev. Bull. Belg. Math. Soc. 8 (2001), 555-565
[3] S. Benmouloud, M. Sbai; A perturbed biharmonic minimization problem with critical exponent
à paraitre dans le volume 7 de Math-Recherche et applications.
[4] H. Brezis, L. Niremberg; Positive Solutions of NonLinear Elliptic Equations Involving Critical
Sobolev Exponents, comm. Pure. Appl. Math 36(1983), 437-477.
[5] H. Brezis, E. Lieb; A relation between pointwise convergence of functions and convergence of
functionals, Proc. Amer. Math. Sco. 88 pp. 486-490 (1983).
[6] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann.
Inst. Henri Poincaré, Vol. 9, no. 3, 1992, p. 281-304.
EJDE/CONF/14
MULTIPLICITY RESULTS
153
Samira Benmouloud
E.G.A.L, Dépt. Maths, Fac. Sciences, Université Ibn Tofail, BP. 133, Kénitra, Maroc
E-mail address: ben.sam@netcourrier.com
Mostafa Khiddi
E.G.A.L, Dépt. Maths, Fac. Sciences, Université Ibn Tofail, BP. 133, Kénitra, Maroc
Simohammed Sbai
E.G.A.L, Dépt. Maths, Fac. Sciences, Université Ibn Tofail, BP. 133, Kénitra, Maroc
E-mail address: sbaisimo@netcourrier.com