Existence of positive solutions for a Kirchhoff type problem involving sign-changing weight functions

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problem involving sign-changing weight functions

Tsung-fang Wu

Department of Applied Mathematics National University of Kaohsiung, Taiwan (Joint work with Ching-yu Chen and Yueh-cheng Kuo )

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1 Introduction

2 Motivation and Problems

3 Nehari manifold

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Kirchhoff type problems:      −M Z Ω |∇u|2dx ∆u = h(x, u) in Ω, u = 0 in ∂Ω, (1)

where Ω is a smooth bounded domain in RN with h ∈ Ω × R → R

being continuous, M (s) = as + b and a, b > 0.

Such problems are often referred to as being nonlocal because of the presence of the integral over the entire domain Ω. It is degenerate if b = 0 and non-degenerate otherwise.

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Problem (1) is analogous to the stationary case of equations that arise in the study of string or membrane vibrations, namely,

%utt− E 2L Z L 0 ∂u ∂x 2 dx +P0 A ! uxx = h(x, u), (2)

where u denotes the displacement, h(x, u) the external force, ρ is the mass density, E is the Young’s modulus of the material, L is the length of the string, P0 the initial tension and A is the area of the cross section.

Equations of this type were first proposed by Kirchhoff in 1883 to describe the transversal oscillations of a stretched string, particularly, taking into account the subsequent change in string length caused by oscillations.

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Existence of positive solutions of Problem (1) Ma, Mu˜noz Rivera [Appl. Math. Lett., 2003] Corrˆea [Nonlinear Anal., 2004]

Alves, Corrˆea, Ma [Comput. Math Appl. 2005] Perera, Zhang [JDE, 2006]

Perera, Zhang [J. Math. Anal. Appl., 2006] Bensedki, Bouchekif [Math. Comp. Model., 2009] ..

.

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When a = 0, b = 1 and h(x, u) = λ |u|q−2u + |u|p−2u.

Ambrosetti-Brezis-Cerami [J. Funct. Anal., 1994], consider the semilinear elliptic equation involving concave and convex nonlinearities

−∆u = λ |u|q−2u + |u|p−2u in Ω,

u = 0 on ∂Ω, (Eλ) where (i) λ > 0, (ii) 1 < q < 2 < p ≤ 2∗ 2∗= _{N −2}2N , if N ≥ 3; 2∗ = ∞, if N = 2 , (iii) Ω is a smooth bounded domain in RN.

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There exists λ0 > 0 such that

– Eq. (Eλ) has at least two positive solutions for λ < λ0,

– Eq. (Eλ) has a positive solution for λ = λ0,

– Eq. (Eλ) does not admit any positive solution for λ > λ0.

If Ω is a starshaped domain and p = 2∗, then for any sequence {λn} ⊂ R+ with λn& 0 as n → ∞, there exists a sequence {uλn} of

positive solutions of Eq. (Eλn) such that

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When Ω = BN(0; 1) =_{x ∈ R}N : |x| < 1 . There exists λ0 > 0 such that

– Eq. (Eλ) has exactly two positive solutions for λ < λ0,

– Eq. (Eλ) has exactly one positive solution for λ = λ0,

– Eq. (Eλ) does not admit any positive solution for λ > λ0.

Adimurthi-Pacella-Yadava [Diff. Int, Eqns, 1997],

Damascelli-Grossi-Pacella [Annls Inst. H. Poincar´e Analyse Non lineair´e, 1999],

Ouyang and Shi [J. Diff. Eqns., 1999],

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0 k k

u

l

0 / Fig. 1 : 2 < p < 2∗

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Extended Equations

The equations involving sign-changing weight functions

−∆u = λa (x) |u|q−2u + b (x) |u|p−2u in Ω,

u = 0 in ∂Ω, (Ea,b)

where (i) λ > 0,

(ii) 1 < q < 2 < p ≤ 2∗,

(iii) Ω is a smooth bounded domain in RN, (iv) a ∈ Lrq_{(Ω) and a}+_{= max {a, 0} 6≡ 0 where r}

q = _r−qr for some

r ∈ (q, 2∗) ;

(v) b ∈ Lsp_{(Ω) and b}+ _{= max {b, 0} 6≡ 0 where s}

p = _s−ps for some

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If p < 2∗, then there exists λ∗ > 0 such that Eq. (Ea,b) has at least two

positive solutions for λ < λ∗.

de Figueiredo-Gossez-Ubilla [J. Funct. Anal., 2003] Wu [J. Math. Anal. Appl., 2006](Palais-Smale method) Brown-Wu [Elect. J. Diff. Eqns., 2007] (Fibering method)

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0 k k

u

l

* /

l

*

?

Fig. 2 : 2 < p < 2∗

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Motivated by these findings, we now extend the analysis to the Kirchoff type equation with h(x, u) = λ f (x) |u|q−2u + g(x) |u|p−2u. Our

intension here is to illustrate the difference in the solution behaviour which arises from the consideration of the nonlocal effect. The problem we consider is thus      −MR Ω|∇u| 2 dx

∆u = λf (x) |u|q−2u + g (x) |u|p−2u in Ω,

u = 0 in ∂Ω,

(Eλ,M)

where Ω is a smooth bounded domain in RN with

1 ≤ q < 2 < p < 2∗, M (s) = as + b and the parameters a, b, λ > 0. The weight functions f, g ∈ C Ω satisfy the following conditions:

(D1) f+= max {f, 0} 6= 0,

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0 = Z Ω −M Z Ω |∇u|2dx

∆u − λf |u|q−2u − g |u|p−2u ϕdx = Z Ω M Z Ω |∇u|2dx

∇u∇ϕ − λf |u|q−2uϕ − g |u|p−2uϕdx

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It is well known that the solution u ∈ H₀1(Ω) of problem (Eλ,M) is a

critical point of the energy functional Jλ,M ∈ C1(H, R) defined by

Jλ,M(u) = 1 2Mc kuk2_H1 −λ q Z Ω f |u|qdx −1 p Z Ω g |u|pdx, (3)

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For ϕ ∈ C₀∞(Ω) , J_λ,M0 (u) ϕ = lim t→0 Jλ,M(u + tϕ) − Jλ,M(u) t = Z Ω M ( Z Ω

|∇u|2dx)∇u∇ϕ − λf |u|q−2uϕ − g |u|p−2uϕdx

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Since Jλ,M(u) = 1 2Mc kuk2_H1 −λ q Z Ω f |u|qdx − 1 p Z Ω g |u|pdx ≥ a 4kuk 4−p H1 − S− p 2 p kg+k∞ p ! kukp_H1 + b 2kuk 2−q H1 − λS− q 2 q kf+k∞ q ! kukq_H1,

for all u ∈ H₀1(Ω) \ {0} , and so Jλ,M is bounded below on H01(Ω) when

p < 4. In the case when p > 4, however, it is clear that lim

t→∞Jλ,M(tu) = −∞

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In order to obtain existence results, we introduce the Nehari manifold Nλ,M =u ∈ H01(Ω) \ {0} | Jλ,M0 (u) , u = 0

where h , i denote the usual duality. Thus, u ∈ Nλ,M if and only if

Mkuk2_H1 kuk2_H1− λ Z Ω f |u|qdx − Z Ω g |u|pdx = 0. Note that Nλ,M contains every nonzero solution of equation (Eλ,M) .

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The Nehari manifold Nλ,M is closely linked to the behavior of functions of

the form Ku,M : t → Jλ,M(tu) for t > 0. Such maps are known as fibering

maps which were introduced by Dr´abek-Pohozaev , Brown-Zhang and

Brown-Wu. If u ∈ H1 0(Ω) , we have Ku,M(t) = 1 2Mc t2kuk2_H1 − λt q q Z Ω f |u|qdx −t p p Z Ω g |u|pdx; K_u,M0 (t) = tMt2kuk2_H1 kuk2_H1− λtq−1 Z Ω f |u|qdx − tp−1 Z Ω g |u|pdx; K_u,M00 (t) = M t2kuk2_H1 kuk2_H1+ 2t2M0 t2kuk2_H1 kuk4_H1 −λ (q − 1) tq−2 Z Ω f |u|qdx − (p − 1) tp−2 Z Ω g |u|pdx.

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Clearly, tK_u,M0 (t) = Mktuk2_H1 ktuk2_H1− λ Z Ω f |tu|qdx − Z Ω g |tu|pdx and so, for u ∈ H₀1(Ω) \ {0} and t > 0, K_u,M0 (t) = 0 if and only if tu ∈ Nλ,M, i.e., positive critical points of Ku,M correspond to points on

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In particular, K_u,M0 (1) = 0 if and only if u ∈ Nλ,M. It is natural therefore

to split Nλ,M into three parts corresponding to local minima, local

maxima and points of inflection. Accordingly, we define N+_λ,M = u ∈ Nλ,M | Ku,M00 (1) > 0 ;

N0_λ,M = u ∈ Nλ,M | Ku,M00 (1) = 0 ;

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Definition.(Natural constrains) Let E be a Hilbert space and

J ∈ C1(E, R). A manifold M is called natural constraint for J if there exists a functional I ∈ C1(E, R) with the property that every constrained critical point of I on M is indeed a stationary point of J .

Lemma. Suppose that u0 is a local minimizer for Jλ,M on Nλ,M and that

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Proposition. We have the following results.

(i) If p > 4 and 0 < λ < λ0(a), then Nλ,M = N+λ,M ∪ N − λ,M and

N±_λ,M 6= ∅ for all a > 0.

(ii) If p = 4, a < _Λ1 and 0 < λ < bλ0(a), then Nλ,M = N+_λ,M ∪ N−_λ,M and

N±_λ,M 6= ∅.

(iii) If p = 4 and a ≥ _Λ1, then N+_λ,M = Nλ,M 6= ∅ for all λ > 0.

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Theorem

Suppose that N = 1, 2, 3 and p > 4. Then for each a > 0 and

0 < λ < λ0(a) , equation (Eλ,M) has two positive solutions u+λ,M ∈ N + λ,M

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8 6 4 2 0 8 6 4 2 0 8 6 4 2 0 0 2 4 6 8 10 0 5 10 15 0 20 40 60 80 (a) (b) (c) u∞ 2 3 10 a_{= ×} − 3 3 10 a_{= ×} − _a_{= ×}_{1 10}−1 λ λ λ Fig. 3:f (x) = 1 = g(x), b = 1 and Ω = (0, 1) .

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Remark: We illustrate the finding of Theorem 1 graphically in Figure 3 (a)-(c) with increasing values of a. The pictures show the existence of two branches of solutions representing the two positive solutions with the upper and lower branch corresponding to u−_λ,M and u+_λ,M respectively. These curves are drawn using a continuation method and computed numerically with Matlab. The starting point for the continuation method is derived using the fixed point iteration method developed in

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Set Λ = inf kuk4_H1 | u ∈ H₀1(Ω) , Z Ω g |u|4dx = 1 . Theorem

Suppose that N = 1, 2, 3 and p = 4. Then

(i) for each a ≥ _Λ1 and λ > 0, N+_λ,M = Nλ,M and equation (Eλ,M) has

at least one positive solution;

(ii) for each a < _Λ1 and 0 < λ < 1₂bλ₀(a), equation (E_λ,M) has two positive solutions u+_λ,M ∈ N+_λ,M and u−_λ,M ∈ N−_λ,M, and

lim a→1 Λ − inf u∈N−_λ,M Jλ,M(u) = ∞.

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0 100 200 300 0 5 10 15 20 0 2 4 6 8 10 (a) (b) (c) 14 12 10 8 6 4 2 0 14 12 10 8 6 4 2 0 14 12 10 8 6 4 2 0 u∞ 2 3 10 a_{= ×} − 2 1.5 10 a₌ _× − 3 3 10 a_{= ×} − λ λ λ Fig. 4:f (x) = 1 = g(x), b = 1 and Ω = (0, 1) .

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Define L (θ) = θC∗q f+ ∞+ C p ∗ g+ ∞ |Ω| . Theorem

Suppose that p < 4. Then

(i) for each a, λ > 0, equation (Eλ,M) has a positive solution ua,λ.

Furthermore, for each a > bA0 and λ > 0, ua,λ∈ N+_λ,M = Nλ,M;

(ii) if g ≥ 0, then for each θ > 0 and 0 < a < _pAb(p−2)

0L(θ) there exists

e

λ0∈ (0, θ] such that for 0 < λ < eλ0, equation (Eλ,M) has two

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Theorem

Suppose that p < 4 and f, g > 0. Then for each θ > 0 and 0 < a <n_pAb(p−2)

0L(θ), A∗

o

there exists a positive number eλ∗ ≤ min

n θ, bΛo such that for 0 < λ < eλ∗, equation (Eλ,M) has three positive solutions

u(1),+_λ,M , u(2),+_λ,M and u−_λ,M such that u(i),+_λ,M ∈ N+

λ,M and u

− λ,M ∈ N

− λ,M

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35 30 25 20 15 10 5 0 35 30 25 20 15 10 5 0 35 30 25 20 15 10 5 0₀ ₅ ₁₀ ₁₅ ₂₀ ₀ ₁₀ ₂₀ ₃₀ ₄₀ ₀ ₂₀ ₄₀ ₆₀ ₈₀ λ λ λ (a) (b) (c) u∞ 3 3 10 a_{= ×} − _a₌_{3.6 10}_× −3 _a_{= ×}_{5 10}−3 Fig. 5:f (x) = 1 = g(x), b = 1 and Ω = (0, 1) .

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