# Multiplicity Results for a Perturbed Elliptic Neumann Problem

Authors: Gabriele Bonanno; Giuseppina D'Aguì
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

The existence of three solutions for elliptic Neumann problems with a perturbed nonlinear term depending on two real parameters is investigated. Our approach is based on variational methods.


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## Body

## 1. Introduction

Here and in the sequel,Ω⊂ℝn is a bounded open set, with a boundary of class C1, q∈L∞(Ω) with essinfΩq>0, p>n; f:Ω×ℝ→ℝ and g:Ω×ℝ→ℝ are L1-Carathéodory functions.The aim of this paper is to study the following perturbed boundary value problem with Neumann conditions:(Pλ,μ)-Δpu+q(x)|u|p-2u=λf(x,u)+μg(x,u),inΩ,∂u∂ν=0,on∂Ω,
where Δp=div(|∇u|p-2∇u) is the p-Laplacian, ν is the outer unit normal to ∂Ω, λ and μ are positive real parameters.Nonlinear boundary value problems involving thep-Laplacian operator Δp (with p≠2) arise from a variety of physical problems. They are used in non-Newtonian fluids, reaction-diffusion problems, flow through porous media, and petroleum extraction (see, e.g., [1, 2]).In the last years, several researchers have studied nonlinear problems of this type through different approaches. In [1], the authors have obtained results on the existence of a solution for the problem (1.1)-Δpu+g(x,u)=f(u),inΩ,∂u∂ν=0,on∂Ω,
by using the perturbation result on sums of ranges of nonlinear accretive operators. Subsequently, Wei and Agarwal, in [2], have studied the same problem by developing some new techniques in the wake of [1]. Problem (Pλ,μ), when q=0, λ=1 and g does not depend on u, has been studied in [3]. In this paper, the authors have obtained the existence of at least three solutions for small μ, by using Implicit Function Theorem and Morse Theory. By using variational methods and in particular critical point results given by Ricceri in [4], Faraci, in her nice paper [5], has dealt with a Neumann Problem involving the p-Laplacian (for any p) of type (1.2)-Δpu+q(x)|u|p-2u=f(x,u)+μg(x,u),inΩ,∂u∂ν=0,on∂Ω.
In particular, [5, Theorems 8, 9] assure the existence of three solutions for the problem given above.In the present paper, we establish some results (Theorems3.1, 3.2), which assure the existence of at least three weak solutions for the problem (Pλ,μ). In particular the following result is a consequence of Theorem 3.2.Theorem 1.1.
Letf:ℝ→ℝ be a nonnegative continuous function such that
(1.3)limt→0+f(t)t=0,∫06f(t)dt<6∫01f(t)dt.
Then, for every λ∈]3/4∫01f(t)dt,9/2∫06f(t)dt[ and for every positive continuous function g:[0,1]×ℝ→ℝ there exists δ⋆>0 such that, for each μ∈]0,δ*[, the problem
(1.4)-u′′+u=λf(u)+μg(x,u)in]0,1[,u'(0)=u′(1)
has at least three nonzero classical solutions.With respect to [3, 5], we stress that our results hold under different assumptions (see Remarks 3.4 and 3.5). In particular, in Theorem 1.1, no asymptotic condition at infinity is required on the nonlinear term. We also point out that in Theorems 3.1 and 3.2, precise estimates of parameters λ and μ are given.


## 2. Preliminaries and Basic Notations

Our main tools are three critical point theorems that we recall here in a convenient form. The first has been obtained in [6], and it is a more precise version of Theorem 3.2 of [7]. The second has been established in [7].Theorem 2.1 (see [6, Theorem 2.6]).
LetX be a reflexive real Banach space, Φ:X→ℝ a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X*, Ψ:X→ℝ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that
(2.1)Φ(0)=Ψ(0)=0.
Assume that there exist r>0 and x¯∈X, with r<Φ(x¯), such that:  (a1)
supΦ(x)≤rΨ(x)/r<Ψ(x¯)/Φ(x¯);(a2)
for eachλ∈Λr:=]Φ(x¯)/Ψ(x¯),r/supΦ(x)≤rΨ(x)[ the functional Φ-λΨ is coercive.Then, for each λ∈Λr, the functional Φ-λΨ has at least three distinct critical points in X.Theorem 2.2 (see [7, Corollary 3.1]).
LetX be a reflexive real Banach space, Φ:X→ℝ a convex, coercive and continuously Gâteaux differentiable functional whose Gâteaux derivative admits a continuous inverse on X*, Ψ:X→ℝ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that
(2.2)infXΦ=Φ(0)=Ψ(0)=0.
Assume that there exist two positive constants r1,r2, and x¯∈X, with 2r1<Φ(x¯)<r2/2, such that  (b1)
supx∈Φ-1(]-∞,r1[)Ψ(x)/r1<(2/3)(Ψ(x¯)/Φ(x¯));(b2)
supx∈Φ-1(]-∞,r2[)Ψ(x)/r2<(1/3)(Ψ(x¯)/Φ(x¯));(b3)
for eachλ∈Λr1,r2′:=](3/2)(Φ(x¯)/Ψ(x¯)), min{r1/supx∈Φ-1(]-∞,r1[)Ψ(x), r2/2supx∈Φ-1(]-∞,r2[)Ψ(x)}[ and for every x1,x2∈X, which are local minima for the functional Φ-λΨ, and such that Ψ(x1)≥0 and Ψ(x2)≥0 one has inft∈[0,1]Ψ(tx1+(1-t)x2)≥0.Then, for each λ∈Λr1,r2′ the functional Φ-λΨ admits three critical points which lie in Φ-1(]-∞,r2[).Now we recall some basic definitions and notations.A functionh:Ω×ℝ→ℝ is called an L1-Carathéodory function if x→h(x,t) is measurable for all t∈ℝ, t→h(x,t) is continuous for almost every x∈Ω, for all M>0 one has sup|t|≤M|h(x,t)|∈L1(Ω). Clearly, if h is continuous in Ω¯×ℝ, then it is L1-Carathéodory.We also recall that a weak solution of the problem (Pλ,μ) is any u∈W1,p(Ω), such that (2.3)∫Ω(|∇u(x)|p-2∇u(x)∇v(x))dx+∫Ω(q(x)|u(x)|p-2u(x)v(x))dx-∫Ω[λf(x,u(x))+μg(x,u(x))]v(x)dx=0,∀v∈W1,p(Ω).Put(2.4)k=supu∈W1,p(Ω)∖{0}supx∈Ω|u(x)|(∫Ω|∇u(x)|pdx+∫Ωq(x)|u(x)|pdx)1/p.IfΩ is convex, an explicit upper bound for the constant k is(2.5)k≤2(p-1)/pmax{(1∫Ωq(x)dx)1/p,diam(Ω)n1/p(p-1p-nm(Ω))(p-1)/p∥q∥∞∫Ωq(x)dx}
(see, e.g., [8, Remark 1]).PutF(x,ξ)=∫0ξf(x,t)dt for all (x,ξ)∈Ω×ℝ and G(x,ξ)=∫0ξg(x,t)dt for all (x,ξ)∈Ω×ℝ.Moreover, setGc:=∫Ωmax|ξ|≤cG(x,ξ)dx for all c>0 and Gd:=infΩ×[0,d]G for all d>0. Clearly, Gc≥0 and Gd≤0.


## 3. Main Results

In this section, we present our main results on the existence of at least three weak solutions for the problem (Pλ,μ).In order to introduce our first result, fixingc,d>0 such that ∥q∥1dp/∫ΩF(x,d)dx<cp/kp∫Ωmax|ξ|≤cF(x,ξ)dx and picking λ∈Λ:=]∥q∥1dp/p∫Ω F(x,d)dx,  cp/pkp∫Ω max|ξ|≤cF(x,ξ)dx[, put(3.1)δ:=min{cp-λpkp∫Ωmax|ξ|≤cF(x,ξ)dxpkpGc,dp∥q∥1-pλ∫ΩF(x,d)dx|Ω|pGd},(3.2)δ¯:=min{δ,1max{0,p|Ω|kplim sup|ξ|→+∞[(supx∈ΩG(x,ξ))/ξp]}},
where we read r/0=+∞ so that, for instance, δ¯=+∞ when lim sup|ξ|→+∞ (supx∈ΩG(x,ξ)/ξp)≤0 and Gd=Gc=0.Theorem 3.1.
Letf:Ω×ℝ→ℝ be an L1-Carathéodory function, and put F(x,ξ)=∫0ξf(x,t)dt for all (x,ξ)∈Ω×ℝ. Assume that there exist two positive constants c,d, with c<(k∥q∥11/p)d, such that  (i)
∫Ωmax|ξ|≤cF(x,ξ)dx/cp<1kp∥q∥1∫ΩF(x,d)dxdp, where k is given by (2.4);(ii)
lim sup|ξ|→+∞[(supx∈ΩF(x,ξ))/ξp]≤0.Then, for every λ∈Λ:=]∥q∥1dp/p∫ΩF(x,d)dx,cp/pkp∫Ωmax|ξ|<cF(x,ξ)dx[ and for every L1(Ω)-Carathédory function g:Ω×ℝ→ℝ such that  (iii)
lim sup|ξ|→+∞[(supx∈ΩG(x,ξ))/ξp]<+∞, where G(x,ξ)=∫0ξg(x,t)dt for all (x,ξ)∈Ω×ℝ,there exists δ¯>0 given by (3.2) such that, for each μ∈[0,δ¯[, Problem (Pλ,μ) has at least three weak solutions.Proof.
Fixλ, g, and μ as in the conclusion. Take X=W1,p(Ω) endowed with the norm
(3.3)∥u∥=(∫Ω|∇u(x)|pdx+∫Ωq(x)|u(x)|pdx)1/p.
On the space C0(Ω¯), we consider the norm ∥u∥∞:=supx∈Ω¯|u(x)|. Since p>n, X is compactly embedded in C0(Ω¯), we have
(3.4)∥u∥∞≤k∥u∥.
Put, for each u∈X,
(3.5)Φ(u)=1p∥u∥p,Ψ(u)=∫Ω[F(x,u(x))+μλG(x,u(x))]dx.
Since the critical points of the functional Φ-λΨ on X are weak solutions of problem (Pλ,μ), our aim is to apply Theorem 2.1 to Φ and Ψ. To this end, taking into account that the regularity assumptions of Theorem 2.1 on Φ and Ψ are satisfied, we will verify (a1) and (a2).
Putr=1/p(c/k)p taking into account (3.4), one has(3.6)supΦ(u)≤rΨ(u)=supΦ(u)≤r∫Ω[F(x,u(x))+μλG(x,u(x))]dx≤∫Ωmax|ξ|≤cF(x,ξ)dx+μλGc.
Now, fix u¯=d. Clearly u¯∈X and moreover Φ(u¯)>r. One has
(3.7)Ψ(u¯)=∫ΩF(x,d)dx+μλ∫ΩG(x,d)dx,Φ(u¯)=1p∥u¯∥p=1pdp∥q∥1.
Hence,
(3.8)Ψ(u¯)Φ(u¯)≥p∥q∥1∫ΩF(x,d)dxdp+p|Ω|∥q∥1μλGddp,supΦ(u)≤rΨ(u)r≤pkp∫Ωmax|ξ|≤cF(x,ξ)dxcp+pkpμλGccp.
Since μ<δ, one has
(3.9)μ<cp-λpkp∫Ωmax|ξ|<cF(x,ξ)dxpkpGc,thismeanspkp∫Ωmax|ξ|≤cF(x,ξ)dxcp+pkpμλGccp<1λ.
Furthermore, μ<(dp∥q∥1-pλ∫ΩF(x,d)dx)/p|Ω|Gd; this means that (p/∥q∥1)(∫ΩF(x,d)dx/dp)+(p|Ω|/∥q∥1)(μ/λ)(Gd/dp)>1/λ.
Then,(3.10)pkp∫Ωmax|ξ|≤cF(x,ξ)dxcp+pkpμλGccp<1λ<p∥q∥1∫ΩF(x,d)dxdp+|Ω|p∥q∥1μλGddp.
Hence, from (3.8) and (3.10), condition (a1) of Theorem 2.1 is verified.
Finally, sinceμ<δ¯, we can fix l>0 such that lim sup|ξ|→+∞[(supx∈ΩG(x,ξ))/ξp]<l and μl<1/|Ω|pkp. Therefore, there exists a function h∈L1(Ω) such that(3.11)G(x,ξ)≤lξp+h(x)
for each (x,ξ)∈Ω×ℝ.
Now, fix0<ɛ<(1/p|Ω|kpλ)-(μl/λ); from (ii) there is a function hɛ∈L1(Ω) such that(3.12)F(x,ξ)≤ɛξp+hɛ(x)
for each (x,ξ)∈Ω×ℝ. It follows that, for each u∈X,
(3.13)Φ(u)-λΨ(u)≥(1p-λ|Ω|kpɛ-μ|Ω|kpl)∥u∥p-λ∥hɛ∥1-μ∥h∥1.
This leads to the coercivity of Φ-λΨ, and condition (a2) of Theorem 2.1 is verified. Since, from (3.8) and (3.10),
(3.14)λ∈]Φ(u¯)Ψ(u¯),rsupΦ(u)≤rΨ(u)[,
Theorem 2.1 assures the existence of three critical points for the functional Φ-λΨ, and the proof is complete.Now, we state a variant of Theorem3.1. Here no asymptotic condition on g is requested; on the other hand, the functions f, g are supposed to be nonnegative.Fixc1,d,c2>0 such that 3∥q∥1dp/2∫ΩF(x,d)dx<1/kpmin{c1p/∫ΩF(x,c1)dx,  c2p/2∫ΩF(x,c2)dx} and picking (3.15)λ∈Λ:=]3∥q∥1dp2p∫ΩF(x,d)dx,1pkpmin{c1p∫ΩF(x,c1)dx,c2p2∫ΩF(x,c2)dx}[,
put (3.16)δ*:=min{c1p-λpkp∫ΩF(x,c1)dxpkpGc1,c2p-2λpkp∫ΩF(x,c2)dx2pkpGc2}.Theorem 3.2.
Assume that there exist three positive constantsc1,c2,d, with 21/pc1<k∥q∥11/pd<2-1/pc2, such that  (j)
f(x,ξ)≥0 for each (x,ξ)∈Ω×[0,c2];(jj)
∫ΩF(x,c1)dx/c1p<(2/3kp∥q∥1)(∫ΩF(x,d)dx/dp);(jjj)
∫ΩF(x,c2)dx/c2p<(1/3kp∥q∥1)(∫ΩF(x,d)dx/dp). Then, for every
(3.17)λ∈Λ:=]3∥q∥1dp2p∫ΩF(x,d)dx,1pkpmin{c1p∫ΩF(x,c1)dx,c2p2∫ΩF(x,c2)dx}[
and for every nonnegative L1-Carathédory function g:Ω×ℝ→ℝ, there exists δ*>0 given by (3.16) such that, for each μ∈]0,δ*[, the problem (Pλ,μ) has at least three weak solutions ui, i=1,2,3, such that
(3.18)0≤ui(x)<c2,∀x∈Ω,i=1,2,3.Proof.
Without loss of generality, we can assumef(x,t)≥0 for all (x,t)∈Ω×ℝ. Fix λ, g, and μ as in the conclusion and take X, Φ and Ψ as in the proof of Theorem 3.1.
We observe that the regularity assumptions of Theorem2.2 on Φ and Ψ are satisfied.
Then, our aim is to verify(b1) and (b2). To this end, put u¯ as in Theorem 3.1, r1=(1/p)(c1/k)p, and r2=(1/p)(c2/k)p. Therefore, one has 2r1<Φ(u¯)<r2/2 and, since μ<δ* and Gd=0, one has(3.19)supΦ(u)<r1Ψ(u)r1≤pkp∫ΩF(x,c1)dxc1p+pkpμλGc1c1p<1λ<2p3∥q∥1∫ΩF(x,d)dxdp+μλ2|Ω|p3∥q∥1Gddp≤23Ψ(u¯)Φ(u¯),2supΦ(u)<r2Ψ(u)r2≤2pkp∫ΩF(x,c2)dxc2p+2pkpμλGc2c2p<1λ<2p3∥q∥1∫ΩF(x,d)dxdp+μλ2|Ω|p3∥q∥1Gddp≤23Ψ(u¯)Φ(u¯).
Therefore, (b1) and (b2) of Theorem 2.2 are verified.
Finally, we verify thatΦ-λΨ satisfies assumption (b3) of Theorem 2.2.
Letu1 and u2 be two local minima for Φ-λΨ. Then u1 and u2 are critical points for Φ-λΨ, and so, they are weak solutions for the problem (Pλ,μ). For every positive parameter λ,μ, and for every (x,t)∈Ω×[0,+∞[ one has (λf+μg)(x,t)≥0, hence, owing to the Weak Maximum Principle (see for instance [9]) we obtain u1(x)≥0,u2(x)≥0,forallx∈Ω. Then, it follows that su1+(1-s)u2≥0,foralls∈[0,1], and that (λf+μg)(x,su1+(1-s)u2)≥0, and, hence, Ψ(su1+(1-s)u2)≥0,foralls∈[0,1].
From Theorem2.2, the functional Φ-λΨ has at least three distinct critical points which are weak solutions of (Pλ,μ) and the conclusion is achieved.Example 3.3.
Consider the following problem(3.20)-Δ3u+|u|u=320|x|2u31+u8+μ(|x|2+1)|u|3,inΩ,∂u∂ν=0,on∂Ω,
where Ω={x∈ℝ2:|x|<1}. Then, owing to Theorem 3.2, for each μ∈]0,10-3[, the problem given above has at least three weak solutions. It is enough to choose, for instance, c1=10-6,c2=102 and d=1.Remark 3.4.
We observe that [5, Theorem 8] cannot be applied to the problem of Example 3.3 since the assumption(N6′)
thereexists>1,α>0,β∈L1(Ω)withs<psuch that|G(x,u)|≤α|u|s+β(x) does not hold. Moreover, also [5, Theorem 9] cannot be applied since condition(N7′′)
lim supu→0+(infx∈Ω∫0ug(x,t)dt/|u|p)=+∞ is not verified.
Finally, we observe that [5, Theorem 7] and [5, Theorem 9] ensure only two solutions and two nonzero solutions, respectively.
Furthermore, contrary to theorems in [5], owing to our results, we have  precise values of μ for which the problem admits solutions.
On the other hand, we observe that in [5], the case p≤n is investigated too.Remark 3.5.
It is easy to verify that our results and theorems in [3] are mutually independent. In particular, we remark that none of theorems in [3] can be applied to the problem in Example 3.3. We also observe that in [3] no estimate of small μ is given.Proof of Theorem1.1.
Our aim is to apply Theorem3.2 by choosing n=1,p=2,andq=1. Put d=1 and c2=6. Therefore, taking into account that k=2, one has
(3.21)3∥q∥1dp2pF(d)=34∫01f(t)dt,1pkpc2p2F(c2)=92∫06f(t)dt.
Moreover, since limt→0+(F(t)/t2)=0, there is a positive constant c1<1, such that (F(c1)/c12)<(1/3)∫01f(t)dt and c12/F(c1)>18/∫06f(t)dt. Hence, a simple computation shows that all assumptions of Theorem 3.2 are satisfied, and the conclusion follows.Remark 3.6.
We explicitly observe that in Theorem3.2, as well as in Theorem 1.1, no asymptotic condition at infinity on the nonlinear term is requested.Example 3.7.
The functionf¯:ℝ→ℝ(3.22)f¯(x)={0,t≤12,2t-1,12<t≤1,33t-32t,1<t≤263256,18,263256<t≤6,et-e6+18,t>6,
satisfies all assumptions of Theorem 1.1.Remark 3.8.
We explicitly observe that the very nice Theorem1 of [10] cannot be applied to the function of Example 3.7 since limt→+∞(f¯(t)/t)=+∞.



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