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

Abstract. Let Ω be a bounded domain in ℝ^{N}, N ≥ 3, and p = \frac{2N}{N−2} the limiting Sobolev exponent. We show that for f ∈ H_{0}^{1}(Ω)^{∗}, satisfying suitable conditions, the nonlinear elliptic problem

−Δu = |u|^{p−2}u + f in Ω

u = 0 on ∂Ω

has at least three solutions in H_{0}^{1}(Ω).

1. Introduction

It is well known [6, Theorems 1 and 2] that for f ≠ 0 and ∥f∥ sufficiently small, the problem

\begin{matrix}−Δu = |u|^{p−2}u + f on Ω \\ u = 0 on ∂Ω \end{matrix}  (1.1)

has at least two distinct solutions u_{0} and u_{1} which are critical points of the functional

I(u) = \frac{1}{2} ∫_{Ω} |∇u|^{2} − \frac{1}{p} ∫_{Ω} |u|^{p} − ∫_{Ω} fu,

such that I(u_{1}) > I(u_{0}). In this note we suppose f ≥ 0 and satisfies

∥f∥ < \frac{α}{N} S^{\frac{N}{4}} ,  (1.2)

where

\frac{1}{2} < α < (\frac{N − 2}{N + 2})^{\frac{N+2}{4}} , and S = inf_{u∈H_{0}^{1}(Ω)∥u∥_{p}=1} ∥∇u∥_{2}^{2},

which corresponds to the best constant for the Sobolev embedding H_{0}^{1}(Ω) ↪ L^{p}(Ω). We determine a special ω_{ε}, from the extremal functions for the Sobolev inequality in ℝ^{N}, and consider Γ the class of continuous paths joining 0 to ω_{ε}.

Proposition 1.1. Let

c = inf_{γ∈Γ} sup_{t∈[0,1]} I(γ(t)).

2000 Mathematics Subject Classification. 35J20, 35J65.

Key words and phrases. Semilinear elliptic equations; critical Sobolev exponent.

\begin{matrix}c \\ ◯ \end{matrix}2006 Texas State University - San Marcos.

Published September 20, 2006.

Then there is a sequence (u_{j}) ⊂ H_{0}^{1}(Ω) such that

I(u_{j}) → c,
I′(u_{j}) → 0 in (H_{0}^{1}(Ω))^{∗},
I(u_{0}) < I(u_{1}) < c.

Let u denotes the weak limit in H_{0}^{1}(Ω) of (a subsequence of) (u_{n}), our principal result is as follows.

Theorem 1.2. Let f ∈ H_{0}^{1}(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 − \frac{1}{N} 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_{γ∈Γ} max_{t∈[0,1]} I(γ(t)),

where

Γ = {γ : [0, 1] → E, is continuous, γ(0) = 0, γ(1) = v}.

Then there is a sequence (u_{n}) in E such that

I(u_{n}) → c,
I′(u_{n}) → 0 in E^{∗}.

On H_{0}^{1}(Ω) we define a variational functional I : H_{0}^{1}(Ω) → ℝ for problem (1.1), by

I(u) = \frac{1}{2}∥∇u∥_{2}^{2} − \frac{1}{p}∥ u∥_{p}^{p} − ∫_{Ω} fu.

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 ∈ H_{0}^{1}(Ω)) be such that ∥∇u∥_{2} = r. We have

I(u) ≥ \frac{1}{2}r^{2} − \frac{1}{p}r^{p}S^{−p/2} − ∥f∥r.

Letting r → αS^{N/4}, we obtain

I(u) ≥ \frac{1}{2}α^{2}S^{N/2} − \frac{1}{p}α^{p}S^{N/2} − \frac{1}{4N} α^{2}S^{N/2}.

Set

ρ = \frac{α^{p}S^{N/2}}{2N} ,

hence I(u) > ρ for all u ∈ ∂B(0, r).

Verification of (H2). Assume 0 ∈ Ω and let ϕ ∈ C_{0}^{∞}(Ω) be a fixed function such that ϕ ≡ 1 for x in some neighborhood of 0. For ε > 0, define

u_{ϵ}(x) = \frac{ϕ(x)}{(ϵ + |x|^{2})^{\frac{N−2}{2}}} , v_{ϵ}(x) = \frac{u_{ϵ(x)}}{∥u_{ϵ}∥_{p}} .

Hence, from [4],

∥∇v_{ϵ}∥_{2}^{2} = S + O(ϵ^{\frac{N−2}{2}} ).  (2.1)

For every μ ≠ 0, [6, Lemma 2.1], gives a real t^{+} > 0 such that

t^{+} > (\frac{∥∇μv_{ϵ}∥_{2}^{2}}{(p − 1)∥μv_{ϵ}∥_{p}^{p}})^{\frac{1}{p−2}} = \frac{1}{μ}(\frac{N − 2}{N + 2})^{\frac{N−2}{4}} ∥∇v_{ϵ}∥2^{\frac{N−2}{2}}  (2.2)

and

t^{+} < \frac{1}{μ}∥∇v_{ϵ}∥2^{\frac{N−2}{2}} .  (2.3)

Set ω_{ϵ} = t^{+}μv_{ϵ}. We have

∥∇ω_{ϵ}∥_{2} = t^{+}μ∥∇v_{ϵ}∥_{2} > (\frac{N − 2}{N + 2})^{\frac{N−2}{4}} ∥∇v_{ϵ}∥2^{\frac{N}{2}} > (\frac{N − 2}{N + 2})^{\frac{N−2}{4}} S^{\frac{N}{4}} > αS^{\frac{N}{4}} > r.

On the other hand, from (2.2) and (2.3), we get

I(ω_{ϵ}) < \frac{1}{2}(t^{+})^{2}∥∇ω_{ϵ}∥_{2}^{2} − \frac{1}{p}(t^{+})^{p}

< \frac{1}{2μ^{2}} ∥∇v_{ϵ}∥_{2}^{N} − \frac{1}{μ^{p}} \frac{1}{p}(\frac{N − 2}{N + 2})^{\frac{p(N−2)}{4}} ∥∇v_{ϵ}∥_{2}^{N} .

Using (2.1), we deduce

I(ω_{ϵ}) < (\frac{1}{2μ^{2}} − \frac{1}{μ^{p}} \frac{N − 2}{N + 2}(\frac{N − 2}{N + 2})^{\frac{N}{2}} )(S + O(ϵ^{\frac{N−2}{2}} ))^{N/2} < \frac{ϵ_{0}^{p}S^{N/2}}{2N} ,

for μ large enough. Then c ≥ ρ > I(ω_{ϵ}). Recall that ω_{ϵ} ∈ Λ^{−} ([6, Lemma 2.1] with

Λ^{−} = {u ∈ H_{0}^{1}(Ω)/ < I′(u), u >= 0, ∥∇u∥_{2}^{2} − (p − 1)∥u∥_{p}^{p} < 0},

and that inf_{Λ^{−}} I is attained by u_{1} [6, Theorem 2]. We conclude that

c ≥ ρ > I(ω_{ϵ}) ≥ I(u_{1}) > I(u_{0}).

3. Proof of the Theorem 1.2

Applying Proposition 1.1 we obtain a sequence (u_{j}) ⊂ H_{0}^{1}(Ω) such that

I(u_{j}) → c,  (3.1)
I′(u_{j}) → 0 in H_{0}^{1}(Ω)^{∗}.  (3.2)

This implies that ∥∇u_{j}∥_{2} is uniformly bounded. Hence for a subsequence of u_{j}, still denoted by u_{j}, we can find u ∈ H_{0}^{1}(Ω) such that

u_{j} → u weakly in H_{0}^{1}(Ω),
u_{j} → u strongly in L^{q}, q < p,
u_{j} → u a.e. on Ω.

From (3.2), we deduce that u is a (weak) solution of Problem (1.1). In particular u satisfies

∥u∥_{2}^{2} − ∥u∥_{p}^{p} = ∫ fu  (3.3)

Let u_{j} = u + v_{j}, where v_{j} → 0 weakly in H_{0}^{1}(Ω) and v_{j} → 0 a.e on Ω. We have

∥∇u_{j}∥_{2}^{2} = ∥∇u∥_{2}^{2} + ∥∇v_{j}∥_{2}^{2} + ∘(1).

and by (3.1),

I(u) + \frac{1}{2}∥∇v_{j}∥_{2}^{2} − \frac{1}{p}∥v_{j}∥_{p}^{p} = c + o(1),

thanks to Brezis-Lieb Lemma [5]. By (3.2) and (3.3), ∥∇v_{j}∥_{2}^{2} −∥v_{j}∥_{p}^{p} = o(1), which gives

I(u) + \frac{1}{N} ∥∇v_{j}∥_{2}^{2} = c + o(1).

Set l = lim_{j→+∞} ∥∇v_{j}∥_{2}^{2}, then lim_{j→+∞} ∥v_{j}∥_{p}^{p} = l. Using Sobolev inequality one see that l ≥ Sl^{2/p}. Then l = 0, or l ≥ S^{\frac{N}{2}} . We get, either

I(u) = c,

and since

I(u) > I(u_{1}) > I(u_{0}),

u is a solution of Problem (1.1) distinct from u_{o} and u_{1}, or

I(u) ≤ c − \frac{1}{N} S^{\frac{N}{2}} .

Remark 3.1. One can show that c < \frac{1}{N} S^{\frac{N}{2}} , consequently I(u) < 0 in the second case

4. Semilinear biharmonic equation

In [2], Benmouloud considered the problem

Δ^{2}u = |u|^{p−2}u + f in Ω
Δu = u = 0 on ∂Ω

where Ω is a bonded domain in ℝ^{N}, N ≥ 5 p = \frac{2N}{N−4} and Δ^{2} denotes the biharmonic operator. She proved that for f ∈ H^{−1} subject to a suitable condition, this problem has at least two distinct solutions in H^{2}(Ω) ∩ H_{0}^{1}(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.

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