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Advanced Nonlinear Studies

Editor-in-Chief: Ahmad, Shair


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Volume 16, Issue 1

Issues

Minimal Energy Solutions and Infinitely Many Bifurcating Branches for a Class of Saturated Nonlinear Schrödinger Systems

Rainer Mandel
Published Online: 2015-12-02 | DOI: https://doi.org/10.1515/ans-2015-5022

Abstract

We prove a conjecture recently formulated by Maia, Montefusco and Pellacci saying that minimal energy solutions of the saturated nonlinear Schrödinger system

{-Δu+λ1u=αu(αu2+βv2)1+s(αu2+βv2)in n,-Δv+λ2v=βv(αu2+βv2)1+s(αu2+βv2)in n

are necessarily semitrivial whenever α,β,λ1,λ2>0 and 0<s<max{α/λ1,β/λ2} except for the symmetric case λ1=λ2, α=β. Moreover, it is shown that for most parameter samples α,β,λ1,λ2, there are infinitely many branches containing seminodal solutions which bifurcate from a semitrivial solution curve parametrized by s.

Keywords: Nonlinear Schrödinger System with Saturation; Bifurcation; Ground States

MSC 2010: 35J47; 35J50; 35Q55

1 Introduction

In this paper, we intend to continue the study on nonlinear Schrödinger systems for saturated optical materials that was recently initiated by Maia, Montefusco and Pellacci [10]. In their paper, the system of elliptic partial differential equations

{-Δu+λ1u=αu(αu2+βv2)1+s(αu2+βv2)in n,-Δv+λ2v=βv(αu2+βv2)1+s(αu2+βv2)in n(1.1)

was suggested in order to model the interaction of two pulses within the optical material under investigation. Here, the parameters satisfy λ1,λ2,α,β,s>0 and n. One way to find classical fully nontrivial solutions of (1.1) is to use variational methods. The Euler functional Is:H1(n)×H1(n) associated to (1.1) is given by

Is(u,v):=12(uλ12+vλ22-αsu22-βsv22)+12s2nln(1+s(αu2+βz2))=12(uλ12+vλ22)-12s2ng(sZ),(1.2)

where Z(x):=αu(x)2+βv(x)2 and g(z):=z-ln(1+z) for all z0. The symbol 2 denotes the standard norm on L2(n) and the norms λ1, λ2 are defined via

uλ1:=(n|u|2+λ1u2)1/2,vλ2:=(n|v|2+λ2v2)1/2.

Since we are interested in minimal energy solutions (that is, ground states) for (1.1), the ground states us,vs of the scalar problems associated to (1.1) turn out to be of particular importance. These are positive radially symmetric and radially decreasing smooth functions satisfying

{-Δus+λ1us=α2us31+sαus2in n,-Δvs+λ2vs=β2vs31+sβvs2in n.(1.3)

Since we will encounter these solutions many times, let us recall some facts from the literature. The existence of positive finite energy solutions us,vs of (1.3) for parameters 0<s<α/λ1 and 0<s<β/λ2 can be deduced from [19, Theorem 2.2] for n3 or from [5, Theorem 1 (i)] for n2, respectively. In the case n=1, the positive functions us,vs are given by us(x)=us(-x), vs(x)=vs(-x) for all x and

us|[0,+)-1(z)=zus(0)(1λ1x2-s-2g(sαx2))1/2𝑑xfor z(0,us(0)],vs|[0,+)-1(z)=zvs(0)(1λ2x2-s-2g(sβx2))1/2𝑑xfor z(0,vs(0)],

where us(0),vs(0)>0 are uniquely determined by

λ1us(0)2-s-2g(sαus(0)2)=λ2vs(0)2-s-2g(sβvs(0)2)=0.(1.4)

As in the explicit one-dimensional case, it is known also in the higher-dimensional case that us,vs are radially symmetric, see [6, Theorem 2]. Finally, the uniqueness of us,vs follows from [17, Theorem 1] in the case n3 and from [12, Theorem 1] in the case n=2. The uniqueness result for n=1 is a direct consequence of the existence proof we gave above.

In this paper, we strengthen the results obtained by Maia, Montefusco and Pellacci [10] concerning ground state solutions and (component-wise) positive solutions of (1.1), so let us shortly comment on their achievements. In Theorem 3.7 of their paper, they proved the existence of nonnegative radially symmetric and nonincreasing ground state solutions of (1.1) for all n2 and for parameter values 0<s<max{α/λ1,β/λ2}, where the upper bound for s is in fact optimal by Lemma 3.2 in the same paper. It was conjectured that each of these ground states is semitrivial except for the special case α=β, λ1=λ2, where the totality of ground state solutions is known in a somehow explicit way, see [10, Theorem 2.1] or Theorem 1.1 (i) below. In [10], this conjecture was proved for parameters smin{α/λ1,β/λ2}, see Theorem 3.15 and Theorem 3.17 therein. Our first result shows that the full conjecture is true even in the case n=1, which was not considered in [10].

Let n, α,β,λ1,λ2>0 and 0<s<max{α/λ1,β/λ2}. Then, the following holds.

  • (i)

    In the case α=β and λ1=λ2 , all ground states of ( 1.1 ) are given by (cos(θ)us,sin(θ)vs) for θ[0,2π).

  • (ii)

    In the case αβ or λ1λ2 , every ground state solution of ( 1.1 ) is semitrivial.

The proof of this result will we presented in Section 2. Our approach is based on a suitable min-max characterization of the mountain pass level associated to (1.1) involving a fibering map technique as in [11]. This method even allows to give an alternative proof for the existence of a ground state solution of (1.1) which is significantly shorter than the one presented in [10] and which, moreover, incorporates the case n=1, see Proposition 2.1. More importantly, this approach yields the optimal result.

In view of Theorem 1.1, it is natural to ask how the existence of fully nontrivial solutions of (1.1) can be proved. In [10], Maia, Montefusco and Pellacci found necessary conditions and sufficient conditions for the existence of positive solutions of (1.1) which, however, partly contradict each other. For instance, [10, Theorem 3.21] claims that positive solutions exist for parameters α=β, λ1λ2 and s>0 sufficiently small contradicting the nonexistence result from [10, Theorem 3.10]. The error leading to this contradiction is located on [10, p. 338, l. 13], where the number λ2/s must be replaced by λ2s, which makes the results from Theorem 3.19 and Theorem 3.21 in that paper break down. Our approach to finding positive solutions and, more generally, seminodal solutions of (1.1) is to apply bifurcation theory to the semitrivial solution branches

𝒯1:={(0,vs,s):0<s<βλ2},𝒯2:={(us,0,s):0<s<αλ1},

which was motivated by the papers of Ostrovskaya and Kivshar [13] and Champneys and Yang [3]. In the case n=1 and λ1=1, λ2=ω2(0,1), α=β=1, they numerically detected a large number of solution branches emanating from 𝒯2 containing seminodal solutions. Moreover, they conjectured that the bifurcation points on 𝒯2 accumulate near s=1, see [3, p. 2184 ff.]. Our results confirm these observations. For simplicity, we will only discuss the bifurcations from 𝒯2 since the corresponding analysis for 𝒯1 is the same up to interchanging the roles of λ1,λ2 and α,β. Investigating the linearized problems associated to (1.1) near (us,0,s) for parameters close to the boundary of the parameter interval (0,α/λ1), we prove the existence of infinitely many bifurcating branches containing fully nontrivial solutions of a certain nodal pattern. Despite the fact that the question whether fully nontrivial solutions bifurcate from 𝒯1,𝒯2 makes perfect sense for all space dimensions n, our bifurcation result is restricted to n{1,2,3}. Later, we will comment on this issue in more detail, see Remark 3.6. In order to formulate our bifurcation result, let us define the positive numbers μ¯k to be the k-th eigenvalues of the linear compact self-adjoint operators ϕ(-Δ+λ2)-1(αβu02ϕ) mapping Hr1(n) to itself, where u0 denotes the positive ground state solution of the first equation in (1.3) for s=0. By Sturm–Liouville theory, we know that these eigenvalues are simple and that they satisfy

μ¯0>μ¯1>μ¯2>>μ¯k0+as k+.

Deferring some more or less standard notational conventions to a later stage, we come to the statement of our result.

Let n{1,2,3} and let α,β,λ1,λ2>0 and k00 satisfy

λ2λ1<βα𝑎𝑛𝑑μ¯k0<1.

Then, there is an increasing sequence (sk)kk0 of positive numbers converging to α/λ1 such that continua 𝒞k𝒮 containing (0,k)-nodal solutions of (1.1) emanate from 𝒯2 at s=sk for all kk0. In the case k0=0, we necessarily have λ1>λ2 and there is a C>0 such that all positive solutions (u,v,s)𝒞0 with s0 satisfy

uλ1+vλ2<C𝑎𝑛𝑑s<α-βλ1-λ2<αλ1.(1.5)

In the case n{2,3}, we can estimate μ¯0 from above in order to obtain a sufficient condition for the conclusions of Theorem 1.2 to hold for k0=0. This estimate, which leads to Corollary 1.3, is based on the Courant–Fischer min-max principle and Hölder’s inequality. In the one-dimensional case, the values of all eigenvalues μ¯k are explicitly known, which results in Corollary 1.4.

Let n{2,3}. Then, the conclusions of Theorem 1.2 are true for k0=0 if

λ2λ1<βα<(λ2λ1)4-n4.(1.6)

Let n=1. Then, the conclusions of Theorem 1.2 are true in the case

λ2λ1<βα<12(λ2λ1+2k0)(λ2λ1+2k0+1).(1.7)

As we mentioned above, one can find sufficient criteria for the existence of (k,0)-nodal solutions bifurcating from 𝒯1 by reversing the roles of λ1,λ2 and α,β in the statement of Theorem 1.2 as well as in its corollaries.

Theorem 1.2 gives rise to many questions which would be interesting to solve in the future. A list of open problems is provided in Section 5. Before going on with the proof of our results, let us clarify the notation which we used in Theorem 1.2. The set 𝒮X× is the closure of all solutions of (1.1) which do not belong to 𝒯2 and a subset of 𝒮 is called a continuum if it is a maximal connected set within 𝒮. Finally, a fully nontrivial solution (u,v) of (1.1) is called (k,l)-nodal if both component functions are radially symmetric and u has precisely k+1 nodal annuli and v has precisely l+1 nodal annuli. In other words, since double zeros cannot occur, (u,v) is (k,l)-nodal if the radial profiles of u, respectively v, have precisely k, respectively l, zeros.

2 Proof of Theorem 1.1

According to the assumptions of Theorem 1.1, we will assume throughout this section that the numbers λ1,λ2,α,β are positive, that s lies between 0 and max{α/λ1,β/λ2}=:s*, and that the space dimension is an arbitrary natural number. Furthermore, we define the energy levels

cs=inf{Is(u,v):(u,v)H1(n)×H1(n) solves (1.1),(u,v)(0,0)},cs*=inf{Is(u,v):(u,v)H1(n)×H1(n) solves (1.1),u=0,v0 or u0,v=0}.

The first step towards the proof of Theorem 1.1 is a more suitable min-max characterization of the least energy level cs of (1.1) which, as in [11], gives rise to a simple proof for the existence of a ground state. To this end, we introduce the Nehari manifold

c𝒩s:=inf𝒩sIs,𝒩s:={(u,v)H1(n)×H1(n):(u,v)(0,0) and Is(u,v)[(u,v)]=0}.

The value

cs=c𝒩s=inf(u,v)(0,0)supr>0Is(ru,rv).(2.1)

is attained at a radially symmetric and radially nonincreasing ground state of (1.1).

Proof.

From [10, (3.15), (3.52)] we get cs=c𝒩s, so let us prove the second equation in (2.1). For every fixed u,vH1(n) satisfying (u,v)(0,0), we set

β(r):=Is(ru,rv)=r2(uλ12+vλ22)-12s2ng(rsZ),

so that (ru,rv)𝒩s holds for r>0 if and only if β(r)=0. Since β is smooth and strictly concave with β(0)>0, a critical point of β is uniquely determined and it is a maximizer (whenever it exists). Since the supremum of β is +, when there is no maximizer of β we obtain

c𝒩s=inf𝒩sIs=inf(u,v)(0,0)supr>0Is(ru,rv),

which proves (2.1).

Due to 0<s<max{α/λ1,β/λ2}, we can find a semitrivial function (u,v)H1(n)×H1(n) satisfying

uλ12+vλ22<αsu22+βsv22,

which implies that cs<+ according to (2.1). So, let (uk,vk) be a minimizing sequence in H1(n)×H1(n) satisfying supr>0Is(ruk,rvk)cs as k+. Using the classical Polya–Szegő inequality and the extended Hardy–Littlewood inequality

nln(1+rs(αuk2+βvk2))nln(1+rs(αuk*2+βvk*2))for all r>0,

for the spherical rearrangement taken from [1, Theorem 2.2], we may assume uk,vk to be radially symmetric and radially decreasing. Since the function g(z)=z-ln(1+z) strictly increases on (0,+) from 0 to +, we may moreover assume that (uk,vk) are rescaled in such a way that the equality

12s2ng(sZk)=1

holds for Zk:=αuk2+βvk2. The inequality

cs+o(1)=limk+supr>0Is(ruk,rvk)lim supk+Is(uk,vk)=12lim supk+(ukλ12+vkλ22)-1

implies that the sequence (uk,vk) is bounded in H1(n)×H1(n). Using the uniform decay rate and the resulting compactness properties of radially decreasing functions bounded in H1(n)×H1(n) (apply, for instance, [18, Compactness Lemma 2]), we may take a subsequence, again denoted by (uk,vk), such that (uk,vk)(u,v) in H1(n)×H1(n) pointwise always everywhere and

12s2ng(rsZ)=limk+12s2ng(rsZk)for all r>0.

From this we infer that

12s2ng(sZ)=1

and, thus, (u,v)(0,0). Hence, for all r>0, we obtain

cs=limk+supρ>0Is(ρuk,ρvk)lim supk+(r2(ukλ12+vkλ22)-12s2ng(rsZk))r2(uλ12+vλ22)-12s2ng(rsZ)=Is(ru,rv),

so that (u,v) is a nontrivial radially symmetric and radially decreasing minimizer. Taking for r the maximizer of the map rIs(ru,rv), we obtain the ground state solution (u¯,v¯):=(ru,rv) having the properties we claimed to hold. Indeed, the Nehari manifold may be rewritten as

𝒩s={(u,v)H1(n)×H1(n):(u,v)(0,0),H(u,v)=0}

for

H(u,v):=Is(u,v)[(u,v)]=uλ12+vλ22-nZ21+sZ,

so that the Lagrange multiplier rule applies due to

H(u,v)[(u,v)]=2(uλ12+vλ22)-n4Z2+2sZ3(1+sZ)2=-n2Z21+sZ<0

for all (u,v)𝒩s. ∎

Let us note that cs equals c=m𝒩=m𝒫 from [10, Lemma 3.6] and, therefore, corresponds to the mountain pass level of Is. Given Proposition 2.1, we are in position to prove Theorem 1.1.

Proof of Theorem 1.1.

Part (i) was proved in [10, Lemma 3.2], so let us prove (ii). First, we show that the ground state energy level cs equals cs*. Since we have cscs* by definition, we have to show that

supr>0Is(ru,rv)cs*for all u,vH1(n) with u,v0.(2.2)

From (1) we deduce that if uλ12(α/s)u22, then we have Is(ru,rv)Is(0,rv) for all v0 and r>0, which implies (2.2). In the same way, one proves (2.2) in the case vλ22(β/s)v22, so it remains to prove (2.2) for functions (u,v) satisfying

uλ12<αsu22,andvλ22<βsv22.(2.3)

To this end, let r>0 be arbitrary but fixed. From (2.3) we infer that the numbers

t(u,v):=αsu22-uλ12αsu22+βsv22-uλ12-vλ22,r(u,v):=r(αsu22+βsv22-uλ12-vλ22)

satisfy t(u,v)(0,1) and r(u,v)>0 as well as

Is(ru,rv)=-r(u,v)2+12s2nln(1+r(u,v)s(αu2+βv2)αsu22+βsv22-uλ12-vλ22).(2.4)

The concavity of the logarithm yields

nln(1+r(u,v)s(αu2+βv2)αsu22+βsv22-uλ12-vλ22)=nln(t(u,v)(1+r(u,v)sαu2αsu22-uλ12)+(1-t(u,v))(1+r(u,v)sβv2βsv22-vλ22))t(u,v)nln(1+r(u,v)sαu2αsu22-uλ12)+(1-t(u,v))nln(1+r(u,v)sβv2βsv22-vλ22)min{nln(1+r(u,v)sαu2αsu22-uλ12),nln(1+r(u,v)sβv2βsv22-vλ22)}.

Combining this inequality with (2.4) gives

Is(ru,rv)min{-r(u,v)2+nln(1+r(u,v)sαu2αsu22-uλ12),-r(u,v)2+nln(1+r(u,v)sβv2βsv22-vλ22)}.

Taking the supremum with respect to r>0, gives (2.2) and, therefore, cscs*, which is what we had to show.

It remains to prove that every ground state is semitrivial unless λ1=λ2, α=β. To this end, assume that (u,v) is a fully nontrivial ground state solution of (1.1), so that in particular Is(u,v)=cs holds. Then, cs=cs* implies that the inequalities above are equalities for some r>0. In particular, since the logarithm is strictly concave and t(u,v)(0,1), we get

1+r(u,v)sαu2αsu22-uλ12=k(1+r(u,v)sβv2βsv22-vλ22)a.e. on n

for some k>0. This implies that k=1, so that u,v have to be positive multiples of each other. From the Euler–Lagrange equation (1.1) we deduce that λ1=λ2 and α=β, which finishes the proof. ∎

3 Proof of Theorem 1.2

In this section, we assume λ1,λ2,α,β>0 as before but the space dimension n is supposed to be 1,2 or 3. In Remark 3.6, we will comment on the reason for this restriction. Let us first provide the functional analytic framework we will be working in. In the case n2, we set X:=Hr1(n)×Hr1(n) to be the product of the radially symmetric functions in H1(n) and define F:X×(0,+)X by

F(u,v,s):=(u-(-Δ+λ1)-1(αuZ(1+sZ)-1)v-(-Δ+λ2)-1(βvZ(1+sZ)-1)),where Z:=αu2+βv2.(3.1)

Hence, finding solutions of (1.1) is equivalent to finding zeros of F. Using the compactness of the embeddings Hr1(n)Lq(n) for n2 and 2<q<2n/(n-2), one can check that the function F(,s) is a smooth compact perturbation of the identity in X for all s, so that the Krasnosel’skii–Rabinowitz global bifurcation theorem [9, 15] is applicable. In the case n=1, however, this structural property is not satisfied, which motivates a different choice for X. In Appendix A, we show that one can define a suitable Hilbert space X of exponentially decreasing functions such that F(,s):XX is again a smooth compact perturbation of the identity in X. Except for this technical inconvenience, the case n=1 can be treated in a similar way to the case n{2,3}, so we carry out the proofs for the latter case only. Furthermore, we always assume that λ2/λ1<β/α according to the assumption of Theorem 1.2.

The first step in our bifurcation analysis is to investigate the linearized problems associated to the equation F(u,v,s)=0 around the elements of the semitrivial solution branch 𝒯2. While doing this, we make use of a nondegeneracy result for ground states of semilinear problems which is due to Bates and Shi [2]. Amongst other things, it tells us that us is a nondegenerate solution of the first equation in (1.3), that is, we have the following result.

The linear problem

-Δϕ+λ1ϕ=3α2us2+sα3us4(1+sαus2)2ϕ,ϕHr1(n), 0<s<αλ1,

only admits the trivial solution ϕ=0.

Proof.

In order to apply [2, Theorem 5.4 (6)], we set

g(z):=-λ1z+α2z31+sαz2,z,

so that us is the ground state solution of -Δu=g(u) in n which is centered at the origin. In the notation of [2], one can check that g is of class (A). Indeed, the properties (g1), (g2), (g3A), (g4A), (g5A) from [2, p. 258] are satisfied for

b=(λ1α2-αλ1s)1/2,K=1

and the unique positive number θ>b satisfying

(αs-λ1)θ2-1s2ln(1+sαθ2)=0.

Notice that (g4A), (g5A) follow from the fact that Kg(z):=zg(z)/g(z) decreases from 1 to - on the interval (0,b) and that it decreases from + to K=1 on (b,+). Having checked the assumptions of [2, Theorem 5.4 (6)], we obtain that the space of solutions of -Δϕ-g(us)ϕ=0 in n is spanned by 1us,,nus, implying that the linear problem only has the trivial solution in Hr1(n). Due to

g(us)=-λ1+3α2us2+sα3us4(1+sαus2)2,(3.2)

this proves the claim. ∎

Using this preliminary result, we can characterize all possible bifurcation points on 𝒯2 which are, due to the implicit function theorem, the points where the kernel of the linearized operator is nontrivial. For notational purposes, we introduce the linear compact self-adjoint operator L(s):Hr1(n)Hr1(n) for parameters 0<s<α/λ1 by setting

L(s)ϕ:=(-Δ+λ2)-1(Wsϕ),Ws(x):=αβus(x)21+sαus(x)2,0<s<αλ1,

for ϕHr1(n). Denoting by (μk(s))k0 the decreasing null sequence of eigenvalues of L(s), we will observe that finding bifurcation points on 𝒯2 amounts to solving μk(s)=1 for s(0,α/λ1) and k0. In fact, we have the following result.

We have

ker(XF(us,0,s))={0}×ker(Id-L(s))for 0<s<αλ1.

Proof.

For (u,v),(ϕ1,ϕ2)X, we have

XF1(u,v,s)[ϕ1,ϕ2]=ϕ1-(-Δ+λ1)-1(sαZ2+3α2u2+αβv2(1+sZ)2ϕ1+2αβuv(1+sZ)2ϕ2),XF2(u,v,s)[ϕ1,ϕ2]=ϕ2-(-Δ+λ2)-1(sβZ2+3β2v2+αβu2(1+sZ)2ϕ2+2αβuv(1+sZ)2ϕ1).

Plugging in u=us, v=0 and Z=αu2+βv2=αus2 gives

XF1(us,0,s)[ϕ1,ϕ2]=ϕ1-(-Δ+λ1)-1(3α2us2+sα3us4(1+sαus2)2ϕ1),XF2(us,0,s)[ϕ1,ϕ2]=ϕ2-(-Δ+λ2)-1(sβα2us4+αβus2(1+sαus2)2ϕ2)=ϕ2-(-Δ+λ2)-1(αβus21+sαus2ϕ2)=ϕ2-(-Δ+λ2)-1(Wsϕ2)=ϕ2-L(s)ϕ2.

From these formulas and Proposition 3.1, we deduce the claim. ∎

Given this result, our aim is to find sufficient conditions for the equation μk(s)=1 to be solvable. Since there is only few information available for any given s>0, our approach consists of proving the continuity of μk and calculating the limits of μk(s) as s approaches the boundary of (0,α/λ1). It will turn out that the limits at both sides of the interval exist and that they lie on opposite sides of the value 1 provided our sufficient conditions from Theorem 1.2 are satisfied. As a consequence, these conditions and the intermediate value theorem imply the solvability of μk(s)=1 and it remains to add some technical arguments in order to apply the Krasnosel’skii–Rabinowitz global bifurcation theorem to prove Theorem 1.2. Calculating the limits of μk at the ends of (0,α/λ1) requires Proposition 3.3 and Proposition 3.4.

We have

usu0𝑎𝑛𝑑Wsαβu02as s0,

where the convergence is uniform in n.

Proof.

As in Lemma A.1 in Appendix A, one shows that on every interval [0,s0] with 0<s0<α/λ1, there is an exponentially decreasing function which bounds each of the functions us with s[0,s0] from above. In particular, the Arzelà–Ascoli theorem shows that usu0 and Wsαβu02 as s0 locally uniformly in n, so that the uniform exponential decay gives usu0 and Wsαβu02 uniformly in n. ∎

We have

us+𝑎𝑛𝑑Wsβλ1αas sαλ1,

where the convergence is uniform on bounded sets in n.

Proof.

First, we show that

us(0)=maxnus+as ss*:=αλ1.(3.3)

Otherwise, we would observe that us(0)a for some subsequence, where a0. In the case a>0, a combination of elliptic regularity theory for (1.3) and the Arzelà–Ascoli theorem would imply that us converges locally uniformly to a nontrivial radially symmetric function uC1(n) satisfying

-Δu+λ2u=α2u31+s*αu2in n

in the weak sense and u(0)=u=a. As in Lemma A.1, we conclude that the functions us are uniformly exponentially decaying, so that u even lies in Hr1(n). Hence, we may test the differential equation with u and obtain

λ1nu2n|u|2+λ1u2=nα2u41+s*αu2<αs*nu2=λ1nu2,

which is impossible. It therefore remains to exclude the case a=0. In this case, the functions us would converge uniformly in n to the trivial solution, implying that us/us(0) would converge to a nonnegative bounded function ϕC1(n) satisfying -Δϕ+λ1ϕ=0 in n and ϕ(0)=ϕ=1. Hence, ϕ is smooth, so that Liouville’s theorem applied to the function (x,y)ϕ(x)cos(λ1y) defined on n+1 implies that ϕ is constant and, thus, ϕ0, contradicting ϕ(0)=1. This proves (3.3).

Now, set ϕs:=us/us(0). Using

-Δϕs+λ1ϕs=αϕsαus21+sαus2in n

and the fact that αus2/(1+sαus2) remains bounded as ss*, we get that the functions ϕs converge locally uniformly as ss* to some nonnegative radially nonincreasing function ϕC1(n) satisfying ϕ(0)= ϕ=1. In order to prove our claim, it is sufficient to show that ϕ1, since this implies us=us(0)ϕs+ locally uniformly and, in particular, Wsβλ1/α locally uniformly.

First, we show that ϕ>0. If this were not true, then there would exist a number ρ(0,+) such that ϕ|Bρ>0 and ϕ|Br=0 for all r[ρ,+). In Bρ, we have us+ and α2us2/(1+sαus2)λ1 implies -Δϕ+λ1ϕ=λ1ϕ in Bρ and ϕ|Bρ=0, in contradiction to the maximum principle. Hence, we must have ϕ>0 in n. Repeating the above argument, we find -Δϕ+λ1ϕ=λ1ϕ in n and ϕ(0)=ϕ=1, so that Liouville’s theorem implies ϕϕ(0)=1. ∎

The previous propositions enable us to calculate the limits of the eigenvalue functions μk(s) as s approaches the boundary of (0,α/λ1).

For all k0, the functions μk are positive and continuous on (0,α/λ1). Moreover, we have

μk(s)μ¯kas s0,μk(s)βλ1αλ2as sαλ1.

Proof.

As in Proposition 3.3, the uniform exponential decay of the functions us for s[0,s*) for s*:=α/λ1 implies that usus0, WsWs0 uniformly in n whenever s0[0,s*]. Hence, the Courant–Fischer min-max characterization for the eigenvalues μk(s) implies the continuity of μk as well as μk(s)μ¯k as s0.

In order to evaluate μk(s) for ss*, we apply Lemma C.1 from Appendix C. The conditions (i) and (ii) of the lemma are satisfied since we have Ws=Ws(0)βλ1/α and Wsβλ1/α locally uniformly as ss* by Proposition 3.4. From the lemma we get μk(s)βλ1/αλ2 as ss*, which is all we had to show. ∎

The eigenvalue functions μk0,…,μk0+3$\mu_{k_{0}},\ldots,\mu_{k_{0}+3}$ on (0,α/λ1)$(0,\alpha/\lambda_{1})$.
Figure 1

The eigenvalue functions μk0,,μk0+3 on (0,α/λ1).

When n4, the statement of Proposition 3.3 is not meaningful since u0 does not exist in this case by Pohožaev’s identity. So, it is natural to ask how us,Ws and μk behave when s approaches zero and n4. Having found an answer to this question, it might be possible to modify our reasoning in order to prove sufficient conditions for the existence of bifurcation points from 𝒯2 in the case n4.

The above propositions are sufficient for proving the mere existence of the continua 𝒞k from Theorem 1.2. So, it remains to show that positive solutions lie to the left of the threshold value (α-β)/(λ1-λ2) and that they are equibounded in X. The latter result will be proved in Lemma A.1 whereas the first claim follows from the following nonexistence result which slightly improves [10, Theorem 3.10 and Theorem 3.11].

If positive solutions of (1.1) exist, then we either have

(i)λ1=λ2,α=β𝑜𝑟(ii)s<α-βλ1-λ2<min{αλ1,βλ2}.

Proof.

Assume there is a positive solution (u,v) of (1.1). Testing (1.1) with (v,u) leads to

nuv(λ1-λ2-(α-β)Z1+sZ)=0.

Hence, the function λ1-λ2-(α-β)Z/(1+sZ) vanishes identically or it changes sign in n. In the first case, we get (i), so let us assume that the function changes sign. Then, we have λ1λ2 and αβ, so that [10, Theorem 3.11 and Remark 3.18] imply that

0<α-βλ1-λ2<min{αλ1,βλ2}.

Moreover, s(α-β)/(λ1-λ2) would imply that

|λ1-λ2-(α-β)Z1+sZ|>|λ1-λ2|-|α-β|s0in n,

contradicting the assumption that λ1-λ2-(α-β)Z/(1+sZ) changes sign. Hence, we have s<(α-β)/(λ1-λ2), which concludes the proof. ∎

Proof of Theorem 1.2.

The main ingredient of our proof is the Krasnosel’skii–Rabinowitz global bifurcation theorem (cf. [9, 15] or [8, Theorem II.3.3]) which, roughly speaking, says that a change of the Leray–Schauder index along a given solution curve over some parameter interval implies the existence of a bifurcating continuum emanating from the solution curve within this parameter interval. In our application, the solution curve is 𝒯2 and the first task is to identify parameter intervals within (0,α/λ1) where the index changes. For notational purposes, we set s*:=α/λ1.

Step 1. Existence of Solution Continua 𝒞k Bifurcating from 𝒯2. By the assumptions of Theorem 1.2 and Proposition 3.5, we have

lims0μk(s)=μ¯k<1andlimss*μk(s)=βλ1αλ2>1for all kk0.

The continuity of the eigenvalue functions μk on (0,s*) as well as the fact that μk(s)>μk+1(s) for all kk0, s(0,s*), therefore implies that 0<ak0<ak0+1<ak0+2<<α/λ1 for the numbers ak given by

ak:=sup{0<s<αλ1:μk(s)<1},kk0.

By the definition of ak, we can find a¯k<ak<a¯k such that the following inequalities hold:

(i)μk(s)<1<μk-1(a¯k)for all sa¯k,kk0,(ii)μk(s)>1>μk-1(a¯k)for all sa¯k,kk0,(iii)ak-1/k<a¯k<a¯k<a¯k+1for all kk0.

In fact, one first chooses a¯k(ak,ak+1) such that (ii) is satisfied and then a¯k<ak sufficiently close to ak such that (i) and (iii) hold.

Now, let us show that the Leray–Schauder index ind(F(,s),(us,0)) changes sign on each of the mutually disjoint intervals (a¯k,a¯k). The index of F(,s) near (us,0) is computed using the Leray–Schauder formula which involves the algebraic multiplicities of the eigenvalues μ>1 of the compact linear operator Id-XF(us,0,s), see [8, (II.2.11)]. From the formulas appearing in Proposition 3.2 we find that μ>1 is such an eigenvalue if and only if one of the equations

(-Δ+λ1)-1(3α2us2+sα3us4(1+sαus2)2ϕ)=μϕin n,ϕHr1(n),ϕ0,L(s)ψ=(-Δ+λ2)-1(Wsψ)=μψin n,ψHr1(n),ψ0,

is solvable. If s=a¯k, then the second equation is solvable with μ>1 if and only if μ is an eigenvalue of L(a¯k) larger than 1. By (3.4) (i), this is equivalent to μ{μ0(a¯k),,μk-1(a¯k)}. Due to Sturm–Liouville theory, each of these eigenvalues is simple. The first equation is solvable with μ>1 if and only if Δ+g(us) has a negative eigenvalue in Hr1(n), where g is defined as in (3.2). From [2, Theorem 5.4 (4)–(6)] we infer that there is precisely one such eigenvalue μ>1 and μ has algebraic multiplicity one. Denoting the Hr1(n) spectrum with σ, we arrive at the formula

ind(F(,a¯k),(0,va¯k))=(-1)#{μσ(Id-XF(0,va¯k,a¯k)):μ>1}=(-1)k+1=-(-1)k+2=-(-1)#{μσ(Id-XF(0,va¯k,a¯k)):μ>1}=-ind(F(,a¯k),(0,va¯k)).

The Krasnosel’skii–Rabinowitz theorem implies that the interval (a¯k,a¯k) contains at least one bifurcation point (usk,0,sk), so that the maximal component 𝒞k in 𝒮 satisfying (usk,0,sk)𝒞k is nonempty. By Proposition 3.2, this implies μj(sk)=1 for some j0 and (3.4) implies j=k, that is, μk(sk)=1. Indeed, property (ii) gives μk-1(sk)>1 and (i) gives μk+1(sk)<1.

Step 2. sks* as k+. If the claim did not hold, then we would have sks¯ from below for some s¯<s*. From sk(a¯k,a¯k), the inequality a¯k>ak-1/k and the definition of ak, we deduce that μk(t)1 whenever tsk+1/k, kk0, and, thus,

μk(t)1for all t(s¯+s*2,s*) and kk1

for some sufficiently large k1. This contradicts μk(t)0 as k+ for all t(0,s*) and the claim is proved.

Step 3. Existence of Seminodal Solutions within 𝒞k. We briefly show that fully nontrivial solutions of (1.1) belonging to a sufficiently small neighbourhood of (usk,0,sk) are (0,k)-nodal. Indeed, if solutions (um,vm,sm) of (1.1) converge to (usk,0,sk), then vm/vm(0) converges to the eigenfunction ϕ of L(sk) with ϕ(0)=1 which is associated to the eigenvalue 1. Due to the fact that μk(sk)=1 and Sturm–Liouville theory, ϕ has precisely k+1 nodal annuli, so that the same is true for vm and sufficiently large m. On the other hand, the convergence umu implies that um must be positive for large m, which proves the claim.

Step 4. Positive Solutions. The claim concerning positive solutions of (1.1) follows directly from Proposition 3.7 and Lemma A.1 from Appendix A. ∎

4 Proof of Corollary 1.3 and Corollary 1.4

Let ζHr1(n) be the unique positive function which satisfies -Δζ+ζ=ζ3 in n, so that u0,v0 can be rewritten as

u0(x)=λ1α-1ζ(λ1x),v0(x)=λ2β-1ζ(λ2x).

Hence, Corollary 1.3 follows from Theorem 1.2 and the estimate

μ¯0=maxϕ0αβu0ϕ22ϕλ22maxϕ0αβu042ϕ42ϕλ22=αβu042v042v0λ22=αβu042v042=βα(λ1λ2)4-n4.

In the case n=1, we have ζ(x)=2sech(x) and it is known (see, for instance, [4, Lemma 5.1]) that the eigenvalue problem μ(-ϕ′′+ω2ϕ)=ζ2ϕ in admits nontrivial solutions in Hr1() if and only if 2/μ= (ω+2k)(ω+2k+1) for some k0. This implies that

μ¯k=βα2(λ2λ1+2k)(λ2λ1+2k+1),k0,

and Corollary 1.4 follows from Theorem 1.2.

5 Open Problems

Let us finally summarize some open problems concerning (1.1) which we were not able to solve and which we believe provide a better understanding of the equation. Especially the open questions concerning global bifurcation scenarios are supposed to be very difficult from the analytical point of view so that numerical indications would be very helpful, too. The following questions might be of interest.

  • (i)

    As in the author’s work on weakly coupled nonlinear Schrödinger systems [11], one could try to prove the existence of positive solutions by minimizing the Euler functional over the “system Nehari manifold” s consisting of all fully nontrivial functions (u,v)X which satisfy I(u,v)[(u,0)]=I(u,v)[(0,v)]=0. For which parameter values α,β,λ1,λ2,s are there such minimizers and do they belong to 𝒞0?

  • (ii)

    What is the existence theory and the bifurcation scenario when αλ2=βλ1 and αβ, λ1λ2?

  • (iii)

    In the case α=β, λ1=λ2, the points on 𝒯1,𝒯2 are connected by a smooth curve and the same is true for every semitrivial solution. Do these connections break up when the parameters of the equation are perturbed? This is related to the question whether the continuum 𝒞0 contains 𝒯1.

  • (iv)

    It would be interesting to know if the eigenvalue functions μk are strictly monotone. The monotonicity of μk would imply that sk are the only solutions of μk(s)=1 so that the totality of bifurcation points is given by (sk)kk0.

  • (v)

    We expect that 𝒯1,𝒯2 extend to semitrivial solution branches 𝒯~1,𝒯~2 containing also negative parameter values s. A bifurcation analysis for such branches remains open. Let us shortly comment on why we expect an interesting outcome from such a study. In the model case n=1 and β=λ2=1, one obtains from (1.4) the existence of us for all s<0 as well as the a priori information us(0)2(1/(|s|+1),1/|s|). Using this, one successively proves that sus(0)2-1 and s(1+sus(0)2)0 as s-. This implies that Ws(0)=us(0)2/(1+sus(0)2)+ as s-, so that one expects that μk(s)+ as s- for all k0. In view of μ¯k0<1, this leads to the natural conjecture that there are also infinitely many bifurcating branches (𝒞~k)kk0 in the parameter range s<0.

  • (vi)

    Our paper does not contain any existence result for fully nontrivial solutions when n4 and λ1λ2 or αβ. It would be interesting to know whether there is such a nonexistence result.

A A Priori Bounds

In our proof of the a priori bounds for positive solutions (u,v) of (1.1), we will use the notation s*:=min{α/λ1, β/λ2} and u(x)=u^(|x|), v(x)=v^(|x|), so that u^,v^ denote the radial profiles of u,v. Notice that all nonnegative solutions are radially symmetric and radially decreasing by [10, Lemma 3.8]. We want to highlight the fact that the main ideas leading to Lemma A.1 are taken from [7, Section 2].

Let n{1,2,3}. For all ε>0, there are cε,Cε>0 such that all nonnegative solutions (u,v) of (1.1) for λ1,λ2,α,β[ε,ε-1] and s[0,min{α/λ1,β/λ2}-ε] satisfy

uλ1+vλ2<Cε𝑎𝑛𝑑u(x)+v(x)Cεe-cε|x|for all xn.

Proof.

We will break the proof into three steps.

Step 1. Boundedness in L(n)×L(n). Assume that there is an sequence (uk,vk) of nonnegative solutions of (1.1) for parameters (λ1)k,(λ2)k,αk,βk[ε,ε-1] and sk[0,s*-ε] which is unbounded in L(n)×L(n). As always, we write Zk(x):=αkuk(x)2+βkvk(x)2. Passing to a subsequence, we may assume that Zk(0)=maxnZk+ and ((λ1)k,(λ2)k,αk,βk,sk)(λ1,λ2,α,β,s) for some s[0,s*-ε] and λ1,λ2,α,β[ε,ε-1]. Let us distinguish the cases s>0 and s=0 to lead this assumption to a contradiction.

For the case s>0, the functions

ϕk:=ukZk(0)-1/2,ψk:=vkZk(0)-1/2

are bounded in L(n) and satisfy αkϕk(0)2+βkψk(0)2=1 as well as

-Δϕk+(λ1)kϕk=αkϕkZk1+skZkin n,-Δψk+(λ2)kϕk=βkψkZk1+skZkin n.

Using the fact that Zk/(1+skZk)sk-1=s-1+o(1) and De Giorgi–Nash–Moser estimates, we obtain from the Arzelà–Ascoli theorem that there are bounded nonnegative radially symmetric limit functions ϕ,ψC1(n) satisfying αϕ(0)2+βψ(0)2=1 and

-Δϕ+λ1ϕ=αsϕin n,-Δψ+λ2ψ=βsψin n.

From λ1<α/s and λ2<β/s we obtain

ϕ(r)=κ1r2-n2Jn-22((αs-λ1)1/2r)andψ(r)=κ2r2-n2Jn-22((βs-λ2)1/2r)for r0

and for some κ1,κ2. Since the functions ϕ,ψ are nonnegative, this is only possible in the case κ1=κ2=0, which contradicts αϕ(0)2+βψ(0)2=1. Hence, the case s>0 does not occur.

For the case s=0, we first show that skZk0 uniformly on n which, due to the fact that Zk(0)= maxnZk, is equivalent to proving that skZk(0)0. So, let κ be an arbitrary accumulation point of the sequence (skZk(0))k and without loss of generality we assume that skZk(0)κ[0,+], so that we are left to show that κ=0. To this end, set

ϕk(x):=uk(skx)Zk(0)-1/2,ψk(x):=vk(skx)Zk(0)-1/2.

The functions ϕk,ψk satisfy αkϕk(0)2+βkψk(0)2=1 as well as

-Δϕk+sk(λ1)kϕk=αkϕkskZk1+skZk=αkϕkskZk(0)(αkϕk2+βkψk2)1+skZk(0)(αkϕk2+βkψk2)in n,-Δψk+sk(λ2)kψk=βkψkskZk1+skZk=βkψkskZk(0)(αkϕk2+βkψk2)1+skZk(0)(αkϕk2+βkψk2)in n.

The Arzelà–Ascoli theorem implies that a subsequence (ϕk),(ψk) converges locally uniformly to nonnegative functions ϕ,ψC1(n) satisfying αϕ(0)2+βψ(0)2=1 and

-Δϕ=αϕκ(αϕ2+βψ2)1+κ(αϕ2+βψ2)in n,-Δψ=βψκ(αϕ2+βψ2)1+κ(αϕ2+βψ2)in n.

For the case κ=+, we arrive at a contradiction as in the case s>0, so let us assume that κ<+. Then, z:=ϕ+ψ is nonnegative, nontrivial and the inequality αϕ2+βψ2αϕ(0)2+βψ(0)2=1 implies that

-Δz=(αϕ+βψ)κ(αϕ2+βψ2)1+κ(αϕ2+βψ2)min{α,β}(ϕ+ψ)κ1+κ(αϕ2+βψ2)c(κ)(ϕ+ψ)3=c(κ)z3,

where c(κ)=min{α,β}2κ/(2(1+κ)). From [14, Theorem 8.4] we infer that c(κ)=0 and, thus, κ=0. Hence, every accumulation point of the sequence (skZk(0)) is zero, so that skZk converges to the trivial function uniformly on n.

With this result at hand, one can use the classical blow-up technique by considering

ϕ~k(x):=uk(Zk(0)-1/2x)Zk(0)-1/2,ψ~k(x):=vk(Zk(0)-1/2x)Zk(0)-1/2.

These functions satisfy αkϕ~k(0)2+βkψ~k(0)2=1 as well as

-Δϕ~k+Zk(0)-1(λ1)kϕ~k=αkϕ~kZkZk(0)-11+skZkin n,-Δψ~k+Zk(0)-1(λ2)kψ~k=βkψ~kZkZk(0)-11+skZkin n.

Then, we have skZk0 uniformly in n and similar arguments as the ones used above lead to a bounded nonnegative nontrivial solution ϕ,ψ of

-Δϕ=αϕ(αϕ2+βψ2)in n,-Δψ=βψ(αϕ2+βψ2)in n,

which we may lead to a contradiction as above. This finally shows that Zk(0)+ is also impossible in the case s=0, so that the nonnegative solutions (u,v) of (1.1) are pointwise bounded by some constant depending on ε.

Step 2. Uniform Exponential Decay. Let us assume for contradiction that there is a sequence (uk,vk,sk) of positive solutions of (1.1) satisfying

u^k(rk)+v^k(rk)ke-rk/kfor all k and some rk>0.(A.1)

Due to the L-bounds for (uk,vk) which we proved in the first step, we can use De Giorgi–Nash–Moser estimates and the Arzelà–Ascoli theorem to obtain a smooth bounded radially symmetric limit function (u,v) of a suitable subsequence of (uk,vk). As a limit of positive radially decreasing functions, u,v are also nonnegative and radially nonincreasing. In particular, we may define

u:=limr+u^(r)0,v:=limr+v^(r)0.

Our first aim is to show that u=v=0. Since (u^,v^) decreases to some limit at infinity, we have u^(r),v^(r), u^′′(r),v^′′(r)0 as r+, so that (1.1) implies that

λ1u=αuZ1+sZ,λ2v=βvZ1+sZ,where Z=αu2+βv2.(A.2)

Now, define

Ek(r):=u^k(r)2+v^k(r)2-λ1u^k(r)2-λ2v^k(r)2+s-2g(sZk(r)),E(r):=u^(r)2+v^(r)2-λ1u^(r)2-λ2v^(r)2+s-2g(sZ(r)).

The differential equation implies that

Ek(r)=-2(n-1)r(u^k(r)2+v^k(r)2)0,

so that Ek decreases to some limit at infinity. The monotonicity of u^k,v^k and the fact that u^k(r),v^k(r)0 as r+ imply that this limit must be 0. In particular, we obtain that Ek0 and the pointwise convergence EkE implies that E is a nonnegative nonincreasing function. From this we obtain that

0limr+E(r)=-λ1u2-λ2v2+s-2g(sZ)=(A.2)-Z21+sZ+s-2g(sZ)=1s2(sZ1+sZ-ln(1+sZ)).

This equation implies that Z=0 and, hence, u=v=0.

Now, let μ satisfy 0<μ<min{λ1,λ2} and choose δ>0. Due to the fact that u=v=0, we may choose r0>0 such that u^(r0)+v^(r0)<δ/2 holds. From u^k(r0)u^(r0), v^k(r0)v^(r0) and the fact that u^k,v^k are decreasing, we obtain that u^k(r)+v^k(r)δ for all rr0 and all kk0 for some sufficiently large k0. Having chosen δ>0 sufficiently small, the inequality u^k,v^k0 implies that

-(u^k+v^k)′′+μ2(u^k+v^k)0on [r0,+) for all kk0.

Hence, the maximum principle implies that for any given R>r0, the function wR(r):=e-μ(r-r0)+e-μ(R-r) satisfies u^k+v^kwR on (r0,R). Indeed, wR dominates u^k+v^k on the boundary of (r0,R) due to the fact that

wR(r0)=wR(R)1δ(u^k+v^k)(r0)=max{(u^k+v^k)(r0),(u^k+v^k)(R)}.

Sending R to infinity, we obtain that

(u^k+v^k)(r)e-μ(r-r0)for all rr0,

which, together with the a priori bounds from the first step, yields a contradiction to the assumption (A.1). This proves the uniform exponential decay.

Step 3. Conclusion. Given the uniform exponential decay of (u,v), we obtain a uniform bound on uL4(n), vL4(n) which, using the differential equation (1.1), gives a uniform bound on uλ1,vλ2. This finishes the proof. ∎

Let us mention that in view of Proposition 3.4, the a priori bounds from the above lemma cannot be extended to the interval s[0,min{α/λ1,β/λ2}]. Furthermore, positive solutions of (1.1) are not uniformly bounded for parameters s belonging to neighbourhoods of 0 when n4, see Remark 3.6. Notice that the assumption n{1,2,3} in the proof of the above lemma only becomes important when we apply [14, Theorem 8.4].

B A Functional Analytic Setting for n=1

In this section, we show that in the one-dimensional case, the function F(,s):XX given by (3.1) is a compact perturbation of the identity for an appropriately chosen Banach space X such that 𝒯1,𝒯2 are continuous curves in X×(0,+). Let σ(0,1) be fixed and set (X,,X) to be the Hilbert space given by

X:={(u,v)Hr1()×Hr1():(u,v),(u,v)X<+}

with

(u,v),(u~,v~)X:=0+e2σμ1x(uu~+μ12uu~)𝑑x+0+e2σμ2x(vv~+μ22vv~)𝑑x,

where μ1:=λ1 and μ2:=λ2. One may check that (X,,X) is a Hilbert space and the subspace C0,r()×C0,r() consisting of smooth even functions having compact support is dense in X. We will use the formula

((-Δ+μ2)-1f)(x)=μ2e-μ|x-y|f(y)𝑑y=0+μΓ(μx,μy)f(y)𝑑y(B.1)

for all fC0,r() and μ>0, where

Γ(x,y)=12(e-|x-y|+e-|x+y|).

Proof of Well-Definedness. First, let us prove for all (u,v)X the estimate

μ1|u(r)|(u,v)Xe-σμ1randμ2|v(r)|(u,v)Xe-σμ2rfor r0.(B.2)

It suffices to prove these inequalities for u,vC0,r(). For such functions, we have

μ1u(r)22μ1r+|uu|𝑑xe-2σμ1rre2σμ1x(u2+μ12u2)𝑑x(u,v)X2e-2σμ1r,μ2v(r)22μ2r+|vv|𝑑xe-2σμ2rre2σμ2x(v2+μ22v2)𝑑x(u,v)X2e-2σμ2r.

Next, using that u(0)=v(0)=0 and the fact that u,v have compact support, we obtain

0+e2σμ1x(u2+μ12u2)𝑑x=0+(e2σμ1xuu)-2σμ1e2σμ1xuu+e2σμ1xu(-u′′+μ12u)dx=-2σμ10+e2σμ1xuu𝑑x+0+e2σμ1xu(-u′′+μ12u)𝑑xσ0+e2σμ1x(u2+μ12u2)𝑑x+0+e2σμ1xu(-u′′+μ12u)𝑑x.

Then, performing the analogous rearrangements for v, yields for all u,vC0,r() that

(u,v)X211-σ0+e2σμ1xu(-u′′+μ12u)𝑑x+11-σ0+e2σμ2xv(-v′′+μ22v)𝑑x.(B.3)

Applying this inequality to (u,v)=((-Δ+μ12)-1(f)χR,(-Δ+μ22)-1(g)χR) for f,gC0,r() and a suitable family (χR)R>0 of cut-off functions converging to 1, we obtain

((-Δ+μ12)-1(f),(-Δ+μ22)-1(g))X2(B.3)11-σ0+e2σμ1x(-Δ+μ12)-1(f)(x)f(x)𝑑x+11-σ0+e2σμ2x(-Δ+μ22)-1(g)(x)g(x)𝑑x=(B.1)μ11-σ0+0+e2σμ1xΓ(μ1x,μ1y)f(x)f(y)𝑑x𝑑y+μ21-σ0+0+e2σμ2xΓ(μ2x,μ2y)g(x)g(y)𝑑x𝑑yμ11-σ0+0+eσμ1xeσμ1y|f(x)||f(y)|𝑑x𝑑y+μ21-σ0+0+eσμ2xeσμ2y|g(x)||g(y)|𝑑x𝑑y=μ11-σ(0+eσμ1x|f(x)|𝑑x)2+μ21-σ(0+eσμ2x|g(x)|𝑑x)2.

Plugging in

f:=fu,v:=αuZ1+sZαu(αu2+βv2),g:=gu,v:=βvZ1+sZβv(αu2+βv2)

and using the estimate (B.2), we find that there is a positive number C depending on σ,μ1,μ2,α,β but not on u,v such that

((-Δ+μ12)-1(fu,v),(-Δ+μ22)-1(gu,v))XC(u,v)X3.(B.4)

By the density of C0,r()×C0,r() in X, this inequality also holds for (u,v)X. If now (uk,vk) is a sequence in C0,r()×C0,r() converging to (u,v)X, then similar estimates based on (B.2) show that

((-Δ+μ12)-1(fuk,vk-fum,vm),(-Δ+μ22)-1(guk,vk-gum,vm))XC(uk-um,vk-um)X((uk,vk)X+(um,vm)X)2

for some C>0, implying that F:X×(0,+)X is well defined and that (B.4) also holds for (u,v)X.

Proof of Compactness of Id-F. Let now (um,vm) be a bounded sequence in X. Then, without loss of generality, we can assume that (um,vm)(u,v)X and (um,vm)(u,v) pointwise almost everywhere. We set

fm:=αumZm1+sZm,gm:=βvmZm1+sZm,f:=αuZ1+sZ,g:=βvZ1+sZ,

where Zm:=αum2+βvm2 and Z:=αu2+βv2. Then, we have fmf and gmg pointwise almost everywhere and the estimate (B.2) implies that

|fm(r)|+|f(r)|α(|um(r)|Zm(r)+|u(r)|Z(r))C(e-3σμ1r+e-σ(μ1+2μ2)r),(B.5)|gm(r)|+|g(r)|β(|vm(r)|Zm(r)+|v(r)|Z(r))C(e-3σμ2r+e-σ(μ2+2μ1)r)(B.6)

for some positive number C>0. Using the estimate from above, we therefore obtain that

(Id-F)(um,vm)-(Id-F)(u,v)X2=((-Δ+μ12)-1(fm-f),(-Δ+μ22)-1(gm-g))X2μ11-σ(0+eσμ1x|fm(x)-f(x)|𝑑x)2+μ21-σ(0+eσμ2x|gm(x)-g(x)|𝑑x)2.

Using (B.5), (B.6) and the dominated convergence theorem, we finally get that

(Id-F)(um,vm)-(Id-F)(u,v)X0as m+,

which is all we had to show.

C A Spectral Theoretic Result

Finally, we prove a spectral theoretical result which we used in the proof of Proposition 3.5 and for which we could not find a reference in the literature. The key ingredient of this result is the min-max principle for eigenvalues of semibounded self-adjoint Schrödinger operators, see, for instance, [16, Theorem XIII.2]. As in Proposition 3.5, we denote by μk(s), k0, the k-th eigenvalue of the compact self-adjoint operator

Ls:Hr1(n)Hr1(n)withLsϕ:=(-Δ+λ)-1(Wsϕ)(C.1)

for potentials Ws vanishing at infinity, that is, Ws(x)0 as |x|+.

Let n, κ,λ>0, a<b and let (Ws)s(a,b) be a family of radially symmetric potentials Ws:n [0,+) vanishing at infinity and satisfying

(i)lim supsbWs=κ𝑎𝑛𝑑(ii)Wsκ locally uniformly as sb.

Then, we have μk(s)κ/λ as sb for all k0.

Proof.

The min-max principle and (i) imply that

lim supsbμk(s)lim supsbWsλ=κλ.

So, it remains to show the corresponding estimate from below. Given the assumptions Ws0 and (ii), we find that it is sufficient to show that μkεκ/λ as ε0, where μkε denotes the k-th eigenvalue of the compact self-adjoint operator Mε:Hr1(n)Hr1(n) defined by Mεϕ=(-Δ+λ)-1((κ-ε)1B1/εϕ). Here, 1B1/ε denotes the indicator function of the ball in n centered at the origin with radius 1/ε. Since εMε is continuous on (0,+) with respect to the operator norm, the min-max characterization of the eigenvalues implies that the mapping

εωkε

is continuous on (0,+), where

ωkε:=κ-εμkε-λ.

By the definition of μkε,ωkε, the boundary value problem

{-ϕ′′(r)-n-1rϕ(r)=ωkεϕ(r)for 0rε-1,-ϕ′′(r)-n-1rϕ(r)=-λϕ(r)for rε-1

with

ϕ(0)=0andϕ(r)0 as r+

for ϕC1([0,+)) has a nontrivial solution. Testing the differential equation on [0,ε-1] with ϕ, we obtain that ωkε>0. Hence, ϕ is given by

ϕ(r)=α{cr2-n2Jn-22(ωkεr)if rε-1,r2-n2Kn-22(λr)if rε-1

for some α0. Here, K denotes the modified Bessel function of the second kind and J represents the Bessel function of the first kind. From ϕC1([0,+)) we get the conditions

Kn-22(λε-1)=cJn-22(ωkεε-1),λKn-22(λε-1)=ωkεcJn-22(ωkεε-1)

on c and ωkε. Due to the continuity of εωkε on (0,+) and due to the fact that K is positive whereas J has infinitely many zeros going off to infinity, we infer that ωkεε-1 is bounded on (0,+). In particular, this gives that ωkε0 and, thus, μkεκ/λ as ε0, which is all we had to show. ∎

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About the article

Received: 2015-03-31

Accepted: 2015-10-09

Published Online: 2015-12-02

Published in Print: 2016-02-01


Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: MA 6290/2-1

This project was supported by the Deutsche Forschungsgemeinschaft (DFG) through the grant MA 6290/2-1.


Citation Information: Advanced Nonlinear Studies, Volume 16, Issue 1, Pages 95–113, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2015-5022.

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