# Global Bifurcation of Positive Solutions of Asymptotically Linear Elliptic Problems

Authors: Ruyun Ma; Yanqiong Lu; Ruipeng Chen
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We are concerned with determining values ofλ, for which there exist positive solutions of the nonlinear elliptic problem -Δu=λa(x)f(u)inΩ, ∂u/∂n+b(x)g(u)=0on∂Ω. The proof of our main results is based upon unilateral global bifurcation theorem of López-Gómez.


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## Body

## 1. Introduction

LetΩ be a bounded domain of Euclidean space RN, N≥2, with smooth boundary ∂Ω. In this paper, we consider the nonlinear elliptic boundary value problem
(1)-Δu=λa(x)f(u)inΩ,∂u∂n+b(x)g(u)=0on⁡∂Ω,
where Δ=∑j=1N‍(∂2/∂xj2), λ>0 is a parameter, and n is the unit exterior normal to ∂Ω.We make the following assumptions.(H0)
a∈Cθ(Ω-) with a(x)>0 in x∈Ω-; b∈C1+θ(∂Ω) with b≥0 and b≢0 on ∂Ω.(H1)
f∈C1(R,R) is an odd function with f(s)>0 for s>0 and there exist constants f0,f∞∈(0,∞) and functions ξ,h∈C1(R,R), such that
(2)f(s)=f0s+ξ(s),ξ(s)=o(|s|)ass⟶0,(3)f(s)=f∞s+h(s),h(s)=o(|s|)ass⟶+∞.(H2)
g∈C1(R,R) is an odd function with g(s)>0 for s>0 and there exist constants g0,g∞∈(0,∞) and functions ζ,k∈C1(R,R), such that
(4)g(s)=g0s+ζ(s),ζ(s)=o(|s|)ass⟶0,g(s)=g∞s+k(s),k(s)=o(|s|)ass⟶+∞.A solutionu∈C2(Ω-) of (1) is said to bepositive if u>0 on Ω-. The purpose of this paper is to study the global bifurcation of positive solutions for theasymptotically linear elliptic eigenvalue problems (1).LetX=C(Ω-) be the space of continuous functions on Ω-. Then, it is a Banach space with the norm
(5)∥u∥=max⁡{|u(x)|∣x∈Ω-}.
Let
(6)P≔{u∈X∣u(x)≥0,x∈Ω-}.
Then P is a cone which is normal and has a nonempty interior and X=P-P¯. Moreover,
(7)int⁡⁡P={u∈P:u(x)>0forx∈Ω}.By a constantλ1∞ we denote the first eigenvalue of the eigenvalue problem
(8)-Δφ=λa(x)f∞φinΩ,∂φ∂n+b(x)g∞φ=0on⁡∂Ω.
By a constant λ10 we denote the first eigenvalue of the eigenvalue problem
(9)-Δφ=λa(x)f0φinΩ,∂φ∂n+b(x)g0φ=0on⁡∂Ω.
It is well known (cf. Krasnosel’skii [1]) that, for ν∈{0,∞}, λ1ν is positive and simple and that it is a unique eigenvalue with positive eigenfunctions φ1ν∈C2+θ(Ω-). In what follows, the positive eigenfunction φ1ν is normalized as ∥φ1ν∥=1.LetS be the closure of the set {(λ,u)∈(0,∞)×C(Ω-)∣(λ,u) is a positive solution of (1) in R×C(Ω-)}.Theorem 1.
Let (H0)–(H2) hold. Then, there exists an unbounded, closed, and connected componentC+⊂(0,∞)×C(Ω-) in S, which joins (λ10,0) with (λ1∞,∞).Corollary 2.
Let (H0)–(H2) hold. Assume that either(10)λ10<λ<λ1∞,
or
(11)λ1∞<λ<λ10.
Then (1) has at least one positive solution.Remark 3.
Ambrosetti et al. [2] and Umezu [3, 4] only studied the bifurcation from infinity for nonlinear elliptic eigenvalue problems. Nonlinear eigenvalue problems of ordinary differential equations have been extensively studied by many authors via fixed point theorem in cones and bifurcations techniques; see Henderson and Wang [5] and Ma [6, 7] and the references therein. Ma and Thompson [7] considered the two-point boundary value problem
(12)u′′(t)+ra~(t)f~(u)=0,u(0)=u(1)=0.
By using the well-known Rabinowitz global bifurcation theorem [8], they proved the following.Theorem A (see [7, Theorem 1.1]).
Assume that(A1)
a~:[0,1]→[0,∞) is continuous and a~(t)≢0 on any subinterval of [0,1];(A2)
f~∈C(R,R) with sf~(s)>0 for s≠0;(A3)
there existf~0,f~∞∈(0,∞) such that f~0=lim⁡|s|→0(f~(s)/s) and f~∞=lim⁡|s|→∞(f~(s)/s);(A4)
λ~k is the kth-eigenvalue of u′′(t)+λa~(t)u=0, u(0)=u(1)=0.If either λ~k/f~∞<r<λ~k/f~0 or λ~k/f~0<r<λ~k/f~∞, then (12) has two solutions uk+ and uk- such that uk+ has exactly k-1 zero in (0,1) and is positive near 0, and uk- has exactly k-1 zero in (0,1) and is negative near 0.Obviously, Corollary2 is a higher dimensional analogue of Ma and Thompson [7, Theorem 1.1] with k=1.Remark 4.
Shi [9] studied the exact number of all nontrivial solutions for
(13)-Δu=λf(u)inΩ,u=0on⁡∂Ω,
for λ in certain parameter range. He proved the existence of global smooth branches of positive solutions by using the implicit function theorem under some further restrictions on f.Remark 5.
Nonlinear elliptic eigenvalue problems have been studied in [4, 10] via topological degree and global bifurcation techniques. The positone case f(0)≡0 is considered in [10], which is extended to the semipositone case f(0)<0 in [4]. An emphasis is, in Theorem 1 and Corollary 2, no assumption imposed on the boundedness of the function h in (3).Remark 6.
Precup [11] applied the Moser-Harnack inequality for nonnegative superharmonic functions to produce a suitable cone and developed fixed point theorem in cones of Krasnoselskii-type to discuss the existence and multiplicity of positive solutions to elliptic boundary value problems
(14)Δu+f(u)=0inΩ,u(x)>0inΩ,u=0on⁡∂Ω.
The constant A in [11, (3.1)] and the constant B in [11, (3.2)] are not optimal so that [11, Theorem 3.1] is not sharp. However, (10) and (11) in Corollary 2 are optimal. In fact, for the function
(15)f*(s)=λ10s+arctan(s21+s2),
which satisfies f0*=f∞*=λ10, the elliptic problem
(16)Δu+f*(u)=0inΩ,∂u∂n+u=0on⁡∂Ω,
has no positive solution.The rest of this paper is organized as follows. The proof of our main results is based upon the unilateral global bifurcation theorem of López-Gómez, which is different from the topological degree arguments used in [2–4, 10]. So, in Section 2, we state a preliminary result based upon unilateral global bifurcation theorem of López-Gómez. In Section 3, we reduce (1) into a compact operator equation. Section 4 is devoted to the proof of Theorem 1.


## 2. Unilateral Global Bifurcation Theorem of López-Gómez

LetU be a Banach space with the normal ∥·∥. Let L(U) stand for the space of linear continuous operators in U. Let J=(a,b)⊂R. Let F:R×U→U be a nonlinear operator of the form
(17)F(λ,u)=IU-λK+N(λ,u),
where N:J×U→U is a continuous operator compact on bounded sets such that
(18)N(λ,u)=∘(∥u∥),
as u→0 uniformly in any compact interval of J, K∈L(U) is a linear compact operator, and r0≠0 is a simple characteristic value of K; that is,
(19)ker⁡⁡[IU-r0K]=span⁡[φ0],
for some φ0∈U∖{0} satisfying
(20)φ0∉R[IU-r0K].LetS be the closure of the set
(21){(λ,u)∣(λ,u)∈J×U,F(λ,u)=0,u≠0}.
Let C+ (resp., C-) be the component of S that meets (r0,0) and around (r0,0) lies in S∖Qϵ,η- (resp., S∖Qϵ,η+); see [12, Section 6.4] for the details.Let(22)Σ≔{λ∈J:dim⁡ker⁡[IU-λK]≥1}.
Then we present the unilateral global bifurcation theorem of López-Gómez; see [12, Theorem 6.4.3].Lemma 7 (see [12], unilateral global bifurcation of López-Gómez).
AssumeΣ is discrete, r0∈Σ satisfies (19), and the index Ind(0,N(λ)) changes sign as λ crosses r0. Then, for ν∈{+,-}, the component Cν satisfies one of the following:(i)
Cν is unbounded in R×U;(ii)
there existsr1∈Σ∖{r0} such that (r1,0)∈Cν;(iii)
Cν contains a point
(23)(λ,y)∈R×(Y∖{0}),
where Y is the complement of ker⁡[IU-r0K] in U.


## 3. Reduction to a Compact Operator Equation

To establish Theorem1 we begin with the reduction of (1) to a suitable equation for compact operators. According to Gilbarg and Trudinger [13], let K∞:Cθ(Ω-)→C2+θ(Ω-) be the resolvent of the linear boundary value problem
(24)-Δu=ϕinΩ,∂u∂n+b(x)g∞u=0on⁡∂Ω.
By Amann [14, Theorem 4.2], K∞ is uniquely extended to a linear mapping of C(Ω-) compactly into C1(Ω-) and it is strongly positive, meaning that K∞ϕ>0 on Ω- for any ϕ∈C(Ω-) with the condition that ϕ≥0 and ϕ≢0 on Ω-.LetR∞:C1+θ(∂Ω)→C2+θ(Ω-) be the resolvent of the linear boundary value problem
(25)-Δu=0inΩ,∂u∂n+b(x)g∞u=ψon⁡∂Ω.
According to Amann [15, Section 4], R∞ is uniquely extended to a linear mapping of C(∂Ω) compactly into C(Ω-). By the standard regularity argument, problem (1) is equivalent to the operator equation
(26)u=λK∞[af(u)]+R∞[bτ(-k(u))]inC(Ω-).
Here τ:C(Ω-)→C(∂Ω) is the usual trace operator.Similarly, letK0:Cθ(Ω-)→C2+θ(Ω-) be the resolvent of the linear boundary value problem
(27)-Δu=ϕinΩ,∂u∂n+b(x)g0u=0on⁡∂Ω.
Then K0 is uniquely extended to a linear mapping of C(Ω-) compactly into C1(Ω-) and it is strongly positive.LetR0:C1+θ(∂Ω)→C2+θ(Ω-) be the resolvent of the linear boundary value problem
(28)-Δu=0inΩ,∂u∂n+b(x)g0u=ψon⁡∂Ω.
Then R0 is uniquely extended to a linear mapping of C(∂Ω) compactly into C(Ω-). Furthermore, (1) is equivalent to the operator equation
(29)u=λK0[af(u)]+R0[bτ(-ζ(u))]inC(Ω-).


## 4. The Proof of Main Results

Obviously, (H1) and (H2) imply that(30)lim⁡|s|→0ξ(s)|s|=0,lim⁡|s|→0ζ(s)|s|=0,(31)lim⁡|s|→∞h(s)|s|=0,lim⁡|s|→∞k(s)|s|=0.Let(32)h~(r)=max⁡{|h(s)|∣0≤s≤r},k~(r)=max⁡{|k(s)|∣0≤s≤r}.
Then h~ and k~ are nondecreasing and there exist rn↗∞ such that
(33)lim⁡n→∞h~(rn)rn=lim⁡n→∞k~(rn)rn=0.
Indeed, for any r>0, there exists s≤r such that h~(r)/r=|h(s)|/r. Additionally, if we assume s≤C for some C>0, then it follows that h~(r)/r≤C/r→0 as r→∞. On the other hand, we assume that there exists rn such that sn→∞; then, it follows from (31) that h~(rn)/rn≤|h(sn)|/sn→0 as j→∞, as desired.We consider(34)u=λK0[a(f0u+ξ(u))]+R0[bτ(-ζ(u))]inC(Ω-)
as a bifurcation problem from the trivial solution u≡0.DefineK:X→X(35)Ku(t)≔K0[af0u](t);
then, K is a strongly positive linear operator on X. It is easy to verify that K:P→P is completely continuous. From [14, Theorem 3.2], it follows that
(36)r(K)=[λ10]-1.
Define N:[0,∞)×X→X by
(37)N(λ,u)≔λK0[aξ(u)]+R0[bτ(-ζ(u))]inC(Ω-);
then, we have from (30) that
(38)∥N(λ,u)∥=∘(∥u∥),
locally and uniformly in λ.It is very easy to check that (34) enjoys the structural requirements for applying the unilateral global bifurcation theory of [12, Sections 6.4, 6.5] (by a counter example of Dancer [16], the global unilateral theorem of Rabinowitz [8] is false as stated. So, it cannot be used). As the theorem of Crandall and Rabinowitz [17] is applied to get the local bifurcation to positive solutions from (λ10,0), the algebraic multiplicity of Esquinas and López-Gómez [18] (see [19, Chapter 4]) equals 1 and, therefore, by [12, Theorem 5.6.2] or [19, Proposition 12.3.1], the local index of 0 as a fixed point of I-λK0 changes sign as λ crosses λ10. Therefore, it follows from Lemma 7 that there exists the component C+ that satisfies one of the following:(i)
C+ is unbounded in R×U;(ii)
there existsλ*∈(0,∞) with λ*≠λ10 and ψ∈C(Ω-) which changes its sign on Ω, such that
(39)[I-λ*K0]ψ=0,(λ*,0)∈C+;(iii)
C+ contains a point
(40)(λ,y)∈R×(Y∖{0}),
where Y is the complement of ker⁡[IX-λ10K0(a(·))] in X.In what follows, we will show that the above Case (ii) and Case (iii) do not occur.In fact, if(μ,y)∈C+ is a nontrivial solution of (34), then y satisfies the problem
(41)-Δy=μa(x)f(y)inΩ,∂y∂n+b(x)g(y)=0on⁡∂Ω,
that is to say, y satisfies the linear problem
(42)-Δy=μa(x)f^(x)yinΩ,∂y∂n+b(x)g^(x)y=0on⁡∂Ω,
where
(43)g^(x)={g(y(x))y(x),asy(x)≠0,g0,asy(x)=0,f^(x)={f(y(x))y(x),asy(x)≠0,f0,asy(x)=0.We claim that(44)y∈C+:y≠0⟹y∈int⁡P∪int⁡(-P).Suppose, on the contrary, that(μ,y)∈C+ with y≠0 and y∈∂P. Then, there exists a sequence {(μn,un)}⊂C+∩P with un>0 for n∈N, such that
(45)(μn,un)⟶(μ,y),asn⟶∞,
and consequently
(46)y(x)≥0,x∈Ω-.
Combining this with the fact that (μ,y) is a nontrivial solution of (34) and using the strong maximum principle [20, Theorem 2.4] and (42), it concludes that
(47)y(x)>0,x∈Ω-,
contracting y∈∂P.Similarly, for(η,y)∈C+ with y≠0 and y∈∂(-P), we get the desired contradiction.Therefore, the claim (44) is true.Sincef and g are odd in R, (λ,-u) is a solution of (34) if and only if (λ,u) is a solution of (34). Combining this and (44) and using the fact that the eigenfunctions corresponding to the eigenvalue λj of the operator K with j≥2 have to change its sign in Ω, it concludes that Case (ii) and Case (iii) cannot occur. Therefore, there exists an unbounded connected subset C+ of the set
(48){(λ,u)∈(0,∞)×P:u=A(λ,u),u∈int⁡⁡P}∪{(λ10,0)}
such that (λ10,0)∈C+.Proof of Theorem1.
It is clear that any solution of (34) of the form (λ,u) yields a solution u of (1). We will show that C+ joins (λ10,0) to (λ1∞,∞).
Let(μn,yn)∈C+ satisfy
(49)μn+∥yn∥⟶∞.
We note that μn>0, for all n∈N since u=0, is the only solution of (34) (i.e., (26)) for λ=0.
In fact, suppose on the contrary thatu is a nontrivial solution of the problem
(50)-Δu=0inΩ;∂u∂n+b(x)g(u)=0on⁡∂Ω;
then, u satisfies the linear problem
(51)-Δu=0inΩ,∂u∂n+b~(x)u=0on⁡∂Ω,
where b~(x)=b(x)(g(u)/u). Together with (H2) and the results of Krasnosel’skii [1], it follows that u≡0 in Ω-, which is a contradiction. Therefore, (34) (i.e., (26)) with λ=0 has only trivial solution.
Case  1
(
λ
1
∞
<
λ
<
λ
1
0
). In this case, we show that
(52)(λ1∞,λ10)⊆(λ∈R∣∃(λ,u)∈C+).
We divide the proof into two steps.
Step  1. We show that if there exists a constant number M>0 such that
(53)μn⊂(0,M],
then C+ joins (λ10,0) to (λ1∞,∞).
From (53), we have ∥yn∥→∞. We divide the equation
(54)yn=μnK∞[a(f∞yn+h(yn))]+R∞[bτ(-k(yn))]iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiinC(Ω-),
by ∥yn∥ and set vn=yn/∥yn∥. Since vn is bounded in X, choosing a subsequence and relabeling if necessary, we see that vn→v- for some v-∈X with ∥v-∥=1. Moreover, from (31) and the fact that h~ and k~ are nondecreasing together with the assertion ∥yn∥→∞, there exists some rnj≤∥yn∥ such that h~(rnj)≥h~∥yn∥ and rnj→∞. It follows that h~(∥yn∥)/∥yn∥≤h~(rnj)/rnj→0 as n→∞. Subsequently, we have
(55)lim⁡n→∞|h(yn(t))|∥yn∥=lim⁡n→∞|k(yn(t))|∥yn∥=0,
since |h(yn(t))|/∥yn∥≤h~(|yn(t)|)/∥yn∥≤h~(∥yn∥)/∥yn∥ and |k(yn(t))|/∥yn∥≤k~(|yn(t)|)/∥yn∥≤k~(∥yn∥)/∥yn∥. Therefore,
(56)v-=μ-K∞[af∞v-]inC(Ω-),
where μ-≔lim⁡n→∞μn, again choosing a subsequence and relabeling if necessary.
Thus(57)-Δv-=μ-a(x)f∞v-inΩ,∂v-∂n+b(x)g∞v-=0on⁡∂Ω.
Since ∥v-∥=1 and v-≥0, the strong positivity of K∞ ensures that
(58)v->0on⁡Ω-.
Thus μ-=λ1∞, and accordingly, C joins (λ10,0) to (λ1∞,∞).
Step  2. We show that there exists a constant M such that μn∈(0,M], for all n.
From (H1) and (H2), there exist constantsα,β∈(0,∞) such that
(59)f(u)≥αu,u∈[0,∞),g(u)≤βu,u∈[0,∞).
By the same method to defineK∞ and R∞ in Section 3, we may define K* and R* as follows.
LetK*:Cθ(Ω-)→C2+θ(Ω-) be the resolvent of the linear boundary value problem
(60)-Δu=ϕinΩ,∂u∂n+b(x)βu=0on⁡∂Ω.
Then, K* is uniquely extended to a linear mapping of C(Ω-) compactly into C1(Ω-) and it is strongly positive. Let R*:C1+θ(∂Ω)→C2+θ(Ω-) be the resolvent of the linear boundary value problem
(61)-Δu=0inΩ,∂u∂n+b(x)βu=ψon⁡∂Ω.
Let λ1* be the eigenvalue of the linear problem
(62)-Δu=λa(x)αuinΩ,∂u∂n+b(x)βu=0on⁡∂Ω,
and let ψ*∈P be the corresponding eigenfunction. Then
(63)-Δψ*=λ1*a(x)αψ*inΩ,∂ψ*∂n+b(x)βψ*=0on⁡∂Ω.
Since (μn,yn)∈C+ satisfies (1) and (λ1*,ψ*) satisfies (63), it follows from (59) and Green’s formula that
(64)(μn-λ1*)∫Ω‍a(x)αψ*(x)un(x)dx≤μn∫Ω‍a(x)f(un(x))ψ*(x)dx-λ1*∫Ω‍a(x)αψ*(x)un(x)dx=∫Ω‍Δψ*(x)un(x)dx-∫Ω‍Δun(x)ψ*(x)dx=∫∂Ω‍un∂ψ*∂ndσ-∫∂Ω‍ψ*∂un∂ndσ=-∫∂Ω‍b(x)βψ*undσ+∫∂Ω‍b(x)g(un)ψ*dσ≤∫∂Ω‍b(x)[βunψ*-βψ*un]dσ=0,
and here dσ is the surface element of ∂Ω. Subsequently, μn≤λ1*. Therefore, the component C+ joins (λ10,0) to (λ1∞,∞).
Case  2
(
λ
1
0
<
λ
<
λ
1
∞
). In this case, if (μn,yn)∈C+ is such that
(65)lim⁡n→∞(μn+∥yn∥)=∞,lim⁡n→∞μn=∞,
then
(66)(λ10,λ1∞)⊆{λ∈(0,∞)∣(λ,u)∈C+},
and moreover,
(67)({λ}×X)∩C+≠∅.
Assume that there exists M>0, such that, for all n∈N,
(68)μn∈(0,M].
Applying a similar argument to that used in Step 1 of Case 1, after taking a subsequence and relabeling if necessary, it follows that
(69)(μn,yn)⟶(λ1∞,∞),n⟶∞.
Again C+ joins (λ10,0) to (λ1∞,∞) and the result follows.Proof of Corollary2.
It is a directly desired consequence of Theorem1.



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