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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Multiple solutions for an elliptic system with indefinite Robin boundary conditions

Pablo Amster
  • Corresponding author
  • Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires; and IMAS – CONICET, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina
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Published Online: 2017-07-08 | DOI: https://doi.org/10.1515/anona-2017-0034

Abstract

Multiplicity of solutions is proved for an elliptic system with an indefinite Robin boundary condition under an assumption that links the linearisation at 0 and the eigenvalues of the associated linear scalar operator. Our result is based on a precise calculation of the topological degree of a suitable fixed point operator over large and small balls.

Keywords: Nonlinear elliptic systems; indefinite Robin condition; multiplicity of solutions; topological degree

MSC 2010: 35J57; 35J60

1 Introduction

This work is devoted to the study of existence and multiplicity of vector solutions u(x)=(u1(x),,uN(x)) of the problem

Lu(x)=g0(x,u(x))+p(x)xΩ,(1.1)

with the boundary condition

uν(x)=γ(x)u(x)xΩ.(1.2)

Here L denotes the vector Laplacian given by Lu:=(Δu1,,ΔuN), Ωn is a smooth bounded domain and g0:Ω¯×NN is a smooth function. Without loss of generality, it shall be assumed throughout the paper that g0(x,0)=0. We shall also assume that pC(Ω¯,N) and γC1(Ω¯). It is remarked that, unlike in the most standard Robin condition, we do not assume that γ0.

The problem arises in many different applications; for instance (see [9] and the references therein) for some particular g0, it can be seen as the steady state of the population density of some species, governed by a parabolic problem. The boundary condition corresponds to the law of the population flux, given on the boundary by γ. Thus, the inflow/outflow of population to the region occurs when γ(x)>0 or γ(x)<0, respectively. Our direct motivation for the present work can be found in [4], where a Painlevé II (ordinary) scalar equation was considered, namely

u′′(x)=u(x)32+δ(x)u(x)+p

under the radiation boundary conditions u(0)=a0u(0), u(1)=a1u(1) with a0,a1>0. It was shown that a1>0 is compensated by the effect of the superlinear term u32 in such a way that the associated functional is still coercive, giving rise to a unique solution when a0a1 and exactly three solutions when a0<a1, provided that the positive parameter p is small. It is worth noticing that, in this model, δ is a linear function depending on a0 and a1; thus, the uniqueness/multiplicity of solutions can be understood as a consequence of the interaction between the nonlinear term of the equation and the Robin coefficients. The previous results have been extended in [3] and [2] for the equation u′′(x)=g0(x,u(x))+p(x), where g0 is a general superlinear function, that is,

lim|u|g0(x,u)u=+

uniformly on x. Indeed, it was shown that the interaction between g,a0 and a1 can be expressed in a very precise way in terms of the spectrum of the associated linear operator -u′′ under the Robin conditions: if

g0u(x,u)>-λ1(1.3)

for all x and all u, then the problem has a unique solution, and if

-λkg0u(,0)-λk+1(1.4)

for some odd k, then the problem has at least three solutions when p is small. Furthermore, if (1.4) holds with k even, then the problem has at least five solutions for small p, provided that λ1<0.

The main purpose of this paper consists in obtaining suitable extensions of the results mentioned above in two directions: in the first place, we shall consider a system instead of a scalar equation and in the second place, we shall not be longer dealing with ODEs but with a PDE system. We shall study system (1.1) in terms of the eigenvalues λ1<λ2λ3+ of the scalar problem -Δφ=λφ associated to the Robin condition φν|Ω=γφ and λ0:=-. At first sight, it seems reasonable to assume that g0 is superlinear, that is,

lim|u|g0(x,u),u|u|2=+

uniformly on x, where , and || denote the usual inner product and norm of N, respectively. However, we shall employ a weaker condition, namely, that there exists a constant μ-λ1 depending only on Ω and γ such that, if

lim inf|u|g0(x,u),u|u|2>μ(1.5)

uniformly on x, then the problem has a solution (see Theorem 3.3). This constant can be estimated: for example, it will be shown that it always suffices to take

μ:=-λ1+φ1φ12,

where φ1>0 is the (unique) principal eigenfunction such that φ1L2=1. For the ODE case n=1 or when g grows like |u|q for some qn+2n-2, it can be seen that the condition μ=-λ1 is optimal for our purposes.

Extra solutions will be obtained when p is small, under an appropriate condition on the interaction between Dug0(x,0) and the spectrum of the linear associated operator. In more precise terms, if for simplicity we assume that g0=uG0 for some smooth function G0:Ω¯×N, then our main hypothesis is that the Hessian matrix at u=0 satisfies

A-HuG0(x,0)B,

where A,BN×N are symmetric matrices with eigenvalues a1aN and b1bN such that

λνk<akbk<λνk+1(1.6)

for some νk0 and all k=1,,N. Here, the ordering over the set of symmetric functions is understood in the usual sense that YZ if and only if Z-Y is positive semi-definite or equivalently, Yu,uZu,u for all uN.

It shall be seen that if ν1++νN is an odd number, then the problem has at least two solutions (and generically three), provided that pL2 is small enough. In fact, the result holds for a general function g0, with the role of the Hessian matrix now played by the Jacobian matrix Dug0(x,0), provided it is symmetric. In contrast with the case g0=uG0, the general situation does not have variational structure and is presented in our main result through the Leray–Schauder degree method (see Theorem 5.2 below). To our knowledge, there are no previous similar results in the literature concerning the multiplicity of solutions for an elliptic system with indefinite Robin conditions.

The paper is organised as follows. In the next section, we recall some basic facts concerning the spectrum of the associated linear operator. In Section 3, we prove the existence of at least one solution by means of an appropriate space-dependent Hartman condition. In Section 4, we obtain H1 bounds for all solutions, provided that μ-λ1. Furthermore, enlarging μ if necessary, we also obtain L bounds, which shall be the key for proving our multiplicity result. This task is accomplished in Section 5, where we firstly prove a lemma that helps to compute the degree of certain linear operators and then apply the lemma for the computation of the degree of I-K, where K is an appropriate compact fixed point operator. Roughly speaking, for p=0 we shall prove that the degree is equal to 1 over a large ball and equal to -1 over a small neighbourhood of the origin. This guarantees, when p is small, the existence of a solution close to the origin and typically two ‘large’ solutions.

2 Preliminaries

For convenience, throughout the paper we shall employ the notation g(x,u):=g0(x,u)+p(x). It is clear that if g0 satisfies (1.5), then so does g for arbitrary pC(Ω¯). Sometimes we shall express the condition in the following equivalent form: there exist Cμ0 and ε>0 such that

g(x,u),u>(μ+ε)|u|2-CμxΩ¯,uN.(2.1)

Before establishing our main results, we need to recall some basic facts concerning the spectrum of the associated linear operator. Let us observe, in the first place, that the scalar operator -Δ is symmetric for the Robin boundary condition. Thus, by standard arguments (see e.g. [5]) one can deduce the existence of a sequence of eigenvalues λ1λ2λ3+ and an associated orthonormal basis of L2(Ω) consisting of eigenfunctions {φj}j. For convenience, we shall denote λ0:=-.

It is verified (see [1]) that λ1 is simple and the associated eigenfunctions do not vanish on Ω¯. From now on, we shall always assume that φ1 is such that φ1>0. Moreover, φ1 can be obtained as a global minimiser of the functional

(φ):=Ω|φ(x)|22𝑑x-Ωγ(x)φ(x)22𝑑S

subject to the restriction φL2=1. It follows that

λ1φL22-ΩΔφ(x)φ(x)𝑑x(2.2)

for any smooth function φ satisfying the Robin condition. In particular, adding (η-λ1)φL22 to both sides for some η>0, the Cauchy–Schwarz inequality yields the following estimate:

φL21ηΔφ+(λ1-η)φL2.(2.3)

Remark 2.1.

Using the standard norm of L2(Ω,N) still denoted by uL2, namely

uL2:=(i=1NuiL22)1/2,

we observe that (2.2) and (2.3) can be extended in an obvious way for vector functions, replacing Δ by L. In particular, the latter inequality implies, as an immediate consequence, the uniqueness of solutions, provided that g(x,u)+λ1u is (strictly) monotone increasing, that is,

g(x,u)-g(x,v)+λ1(u-v),u-v>0

for all x and uv. Clearly, this extends condition (1.3) for N>1.

In order to understand the dependence of φ1 with respect to γ, observe that if ~ is the functional associated to some γ~ and if λ~1,φ~1 are the respective eigenvalue and eigenfunction, then, by virtue of the trace embedding H1(Ω)L2(Ω), we deduce that

|(φ)-~(φ)|cγ-γ~φH12.

In particular,

λ~1-λ1~(φ1)-λ1|(φ1)-~(φ1)|cγ-γ~

and

λ1-λ~1(φ~1)-λ~1|(φ~1)-~(φ~1)|cγ-γ~.

We conclude that if γ~γ uniformly, then λ~1λ1. Furthermore, since -Δφ~1=λ~1φ~1 we deduce that the family {φ~1} is uniformly bounded in H2(Ω), and by a bootstrapping argument, we conclude that it is uniformly bounded in C2(Ω¯). We claim that φ~1φ1 for the C1 norm. Indeed, otherwise we may suppose that φ~1 converges for the C1 norm to some φφ1. Observe that φL2=1 and φ0; moreover, since Δφ~1-λ1φ1, it is readily seen that Δφ=-λ1φ and hence φ=φ1, a contradiction.

3 A space-dependent Hartman condition

Our general existence result is based on the fact that g satisfies the following Hartman type condition (the original condition was formulated in [6]):

  • (H)

    There exists a smooth function R:Ω¯(0,+) such that

    g(x,u),uR(x)ΔR(x)+|R(x)|2(3.1)

    for all xΩ and all uN such that |u|=R(x).

Lemma 3.1.

Assume that there exists R:Ω¯(0,+) as before such that (3.1) holds and

Rν(x)R(x)γ(x)xΩ.(3.2)

Then problem (1.1)–(1.2) has at least one solution u such that |u(x)|R(x) for all xΩ.

Proof.

Let P(x,u):=min{R(x),|u|}u|u|. For fixed ε,θ>0, a straightforward application of Schauder’s theorem shows that the truncated problem

{Lu(x)-εu(x)=g(x,P(x,u(x)))-εP(x,u(x)),xΩ,uν(x)+θu(x)=(γ(x)+θ)P(x,u(x)),xΩ

has at least one solution u. We claim that, in view of (3.1) and (3.2) it can be deduced that |u(x)|R(x) for all x and, consequently, u is a solution of the original problem (1.1)–(1.2). Indeed, otherwise the function ϕ(x):=|u(x)|-R(x) achieves a strictly positive maximum value at some point x^. If x^Ω, then

0=jϕ(x^)=u(x^)|u(x^)|,ju(x^)-jR(x^)

for all j and Δϕ(x^)0, that is,

0-1|u(x^)|3ju(x^),ju(x^)2+1|u(x^)|(|u(x^)|2+u(x^),Lu(x^))-ΔR(x^),

where we employed the notation |u|2:=i=1N|ui|2. Then

u(x^)|u(x^)|,Lu(x^)ΔR(x^)+|R(x^)|2|u(x^)|ΔR(x^)+|R(x^)|2R(x^).

Moreover, observe that

Lu(x^),u(x^)=g(x^,P(x^,u(x^))),u(x^)+εu(x^)-P(x^,u(x^))),u(x^)>g(x^,P(x^,u(x^))),u(x^)

and consequently, using (3.1), we obtain

ΔR(x^)+|R(x^)|2R(x^)>u(x^)|u(x^)|,g(x^,P(x^,u(x^)))=1R(x^)P(x^,u(x^)),g(x^,P(x^,u(x^)))ΔR(x^)+|R(x^)|2R(x^),

a contradiction.

Now suppose x^Ω, then

0ϕν(x^)=u(x^)|u(x^)|,uν(x^)-Rν(x^).

This implies that

Rν(x^)<γ(x^)R(x^),

which contradicts (3.2). ∎

Remark 3.2.

Observe that if (3.2) is strict, then the previous proof is still valid for θ=0. Furthermore, if (3.1) is strict, then |u(x)|<R(x) for all x.

In order to establish the existence of solutions, let us observe that, without loss of generality, we may assume that Ω=F-1(-,0) for some C2 mapping F:Ω¯ such that 0 is a regular value of F. Thus, the outer normal at xΩ is simply computed as ν(x)=F(x)|F(x)|. For convenience, let us fix the following constants:

a>maxxΩγ(x)|F(x)|,C:=aΔF+2a2|F|2.(3.3)

Theorem 3.3.

Let C be defined by (3.3) and assume that (1.5) holds with μC. Then problem (1.1)–(1.2) admits at least one solution for arbitrary p.

Proof.

Define R(x):=eaF(x)+b with a as before and some b>0 to be specified. Direct computation shows that

R(x)=aR(x)F(x)

and

Rν(x)=aR(x)Fν(x)aR(x)>γ(x)R(x)

for all xΩ. On the other hand,

ΔR(x)=aΔF(x)R(x)+a2|F(x)|2R(x),

and hence

R(x)ΔR(x)+|R(x)|2CR(x)2

with C given by (3.3). Then we may fix R0 such that

g(x,u),u|u|2>C

for |u|R0 and b0 such that R(x)R0 for all x. Hence, for |u|=R(x) it is seen that

g(x,u),u>CR(x)2R(x)ΔR(x)+|R(x)|2

and the previous lemma applies. ∎

4 A priori bounds for the solutions

4.1 H1 bounds

This section is devoted to obtain, when (1.5) is satisfied, suitable a priori H1 bounds for the solutions of (1.1)–(1.2) depending only on μ and the constant Cμ in condition (2.1).

To this end, let us firstly observe that necessarily μ-λ1: indeed, for μ<-λ1 we may consider

g(x,u):=-λ1u+p(x),

for which the problem has an unbounded set of solutions if all the coordinates of p are orthogonal (in the L2 sense) to the eigenfunction associated to λ1, and no solutions otherwise.

Remark 4.1.

In view of Theorem 3.3, the same choice of g leads to the conclusion that if C is as in (3.3) then C-λ1.

We claim that the condition μ-λ1 is also sufficient for getting appropriate H1 bounds. Indeed, using the extension of (2.2) for vector functions (see Remark 2.1), we can multiply the equation by u and integrate to obtain

Ωg(x,u(x)),u(x)𝑑x=ΩLu(x),u(x)𝑑x-λ1uL22.

From inequality (2.1) and using the fact that μ-λ1 it is verified that g(x,u),u)(ε-λ1)|u|2-Cμ, which in turn yields

λ1uL22(λ1-ε)uL22+Cμ|Ω|.

Hence uL2Cμε and, moreover,

Ω|u(x)|2𝑑x(λ1-ε)uL22+Cμ|Ω|+Ωγ(x)|u(x)|2𝑑S.

Next, extend the outer normal to a smooth vector field ν:Ω¯N and define Φ(x):=γ(x)|u(x)|2ν(x). Then

Ωγ(x)|u(x)|2𝑑S=ΩΦ(x),ν(x)𝑑S=ΩdivΦ(x)𝑑x.

Now let D:=maxj=1,,nγνj and assume, without loss of generality, that ε<1D. Since

divΦ(x)=j[j(γνj)(x)|u(x)|2+2γ(x)νj(x)u(x),ju(x)]j[(j(γνj)(x)+γ(x)νj(x)ε)|u(x)|2+γ(x)νj(x)ε|ju(x)|2],

it follows that

Ω|u(x)|2𝑑xC(ε)uL22+DεuL22,

where C(ε):=j(j(γνj)+γνjε)+[(λ1ε-1)+|Ω|]Cμ and the desired bound is obtained.

Thus we have proved the following lemma.

Lemma 4.2.

Fix ε>0 small enough, μ-λ1 and Cμ0. Then there exists an H1 bound for the solutions of (1.1)–(1.2) for any g satisfying (2.1).

Remark 4.3.

The preceding argument shows that, if g(x,u)=uG(x,u), then the associated functional is coercive. However, the functional is not defined over H1(Ω) unless a growth condition is assumed for G. As we shall see, by enlarging μ in an appropriate way, we can still apply the variational method because g may be replaced by a suitable truncation.

4.2 L bounds

In this section we shall prove that if (1.5) holds with μC, where C is defined as in (3.3), then the solutions of (1.1)–(1.2) also admit a priori bounds for the L norm.

With this aim, assume that (2.1) holds with μC and set R(x)=eaF(x)+b as before such that (3.2) holds strictly. Fix R0 as in the proof of Theorem 3.3 and set b large enough such that R(x)>R0 for all x. From the previous computations it is deduced that if u is a solution and |u(x)|R(x) for some x, then the absolute maximum of the function ϕ(x)=|u(x)|-R(x) is achieved at the boundary. Indeed, with the previous notation observe that if the (nonnegative) maximum value of ϕ is achieved at some x^Ω, then it is verified as before that

u(x^),Lu(x^)C|u(x^)|2<g(x^,u(x^)),u(x^),

a contradiction. Let u be a solution and suppose that |u(x)|R(x) for some x. The function ψ(b):=maxϕ(x) is continuous with respect to b, so (since ψ(b)- as b+) enlarging b, if necessary we may assume that ψ(b)=0, that is, |u(x)|<R(x) for xΩ and |u(x^)|=R(x^) for some x^Ω. This implies that

γ(x^)R(x^)<Rν(x^)γ(x^)|u(x^)|=γ(x^)R(x^),

a contradiction.

In summary, there exists a constant b such that if u is a solution of the problem, then |u(x)|<R(x). Furthermore, observe that the choice of R depends only on the fact that g(u),u>C|u|2 for |u|>R0 and the constant C itself is independent of g; thus, replacing g by the function

g^(x,u):={g(x,u),if |u|R(x),|u|R(x)g(x,R(x)|u|u),if |u|>R(x),

we see that the resulting problem has the same solutions. Hence, we may assume that g grows linearly with respect to u. In particular, if g=uG, then the existence of at least one solution is also deduced by variational methods (see Remark 4.3).

Remark 4.4.

An alternative choice of R in the Hartman condition is

R(x)=:cφ1(x)

for some c sufficiently large, provided that

g(x,u),uR(x)ΔR(x)+|R(x)|2=c2(-λ1φ1(x)2+|φ1(x)|2)

for |u|=c|φ1(x)|, that is,

g(x,u),u|u|2(-λ1+|φ1(x)|2φ1(x)2).

Thus, it suffices to consider

μ:=-λ1+φ1φ12.(4.1)

Obviously, this R does not satisfy (3.2) strictly, so the previous argument for the L bound fails. However, we may overcome this difficulty by taking, instead, γ~:=γ+η for some η sufficiently small such that μ~<μ+ε with ε as in (2.1) and R~:=cφ~1, which satisfies the Hartman condition and

R~ν=γ~R~>γR~.

5 Multiplicity

5.1 A useful lemma

The key for a proof of our multiplicity result shall be the computation of the degree of an associated linear operator. For η>0 and a fixed continuous matrix function MC(Ω¯,N×N) let us define the compact linear operator KM:L2(Ω,N)L2(Ω,N) given by KMv:=u, the unique solution of the problem

{Lu(x)+(λ1-η)u(x)=M(x)v(x)+(λ1-η)v(x),xΩ,uν(x)=γ(x)u(x),xΩ.

Then we have the following lemma.

Lemma 5.1.

Let MC(Ω¯,RN×N) be a symmetric matrix function such that A-MB, where the symmetric matrices A,BRN×N satisfy (1.6). Then ker(I-KM)={0}. Furthermore, if VL2(Ω,RN) is a bounded open neighbourhood of 0, then

deg(I-KM,V,0)=(-1)ν1++νN.

Proof.

Following the ideas in [7], we can verify that the problem

Lu(x)=M(x)u(x)

has no nontrivial solutions satisfying the boundary condition (1.2). This implies that I-KM vanishes only at u=0 and, consequently, deg(I-KM,V,0) is well defined.

Next, observe that KMv can be computed as

KMv(x)=-ΩGη(x,y)Mη(y)v(y)𝑑y,

where Gη is the Green function associated to the scalar operator -Δ-(λ1-η)I under the Robin boundary conditions and

Mη:=M+(λ1-η)IN,

with the identity matrix INN×N. Recall that Gη can be written as

Gη(x,y)=j=1φjη(x)φjη(y)λjη,

where {φjη} is an orthonormal basis of eigenfunctions associated to the eigenvalues λjη. It is clear that λjη=λj-λ1+η and we may take φjη=φj. Thus,

(I-KM)u=u+j=1φjλjηΩφj(y)Mη(y)u(y)𝑑y.

Moreover, it is seen that the degree, regarded as a function of M, is continuous, that is, locally constant. Hence, using the fact that the subset of N×N defined by {M symmetric:A-MB} is connected (because it is convex), we may assume that M is a constant matrix, say M=-A.

Writing uj:=Ωφj(x)u(x)𝑑x, the coordinate of the vector function u in the basis {φj}, we obtain

(I-KM)u=u+j=1φjλjηMηuj=j=1φjMjηuj,

where

Mjη:=IN+Mηλjη=λjIN+Mλj-λ1+η.

Observe also that

j=q+1φjλjηujL2uL2|λq+1η|0

uniformly for uL2ρ. This implies that, for some large enough q, it suffices to compute the degree restricted to the subspace spanned by {φj}jq or, equivalently, the degree of the mapping

(u1,,uq)(M1ηu1,,Mqηuq).

In other words, it suffices to compute the sign of the determinant of the block matrix

(M1ηM2ηMqη).

Let wk be the eigenvector of the matrix M=A associated to the eigenvalue ak, then

(λjIN+M)wk=(λj+ak)wk

and, since λjη>0, it follows that

sgn[det(Mjη)]=sgn(k=1N(λj+ak)).

Thus, if q>νk for all k, it is seen that

deg(I-KM,V,0)=sgn(k=1Nj=1q(λj+ak))

and the result is deduced from the fact that

λνk+ak<0<λνk+1+ak.

5.2 Multiple solutions for p small

Theorem 5.2.

Assume that Dug0(x,0) is symmetric and that (1.5) holds with C as in (3.3). Furthermore, assume that there exist symmetric matrices A and B with respective eigenvalues a1aN and b1bN satisfying

λνk<akbk<λνk+1

for some νkN0, and all k=1,,N such that

A-Dug0(x,0)B

for all x. If ν1++νN is odd, then there exists r>0 such that the problem has at least two solutions for all p such that pL2<r.

Proof.

Assume firstly that p=0. According to Section 4.1, we may fix a constant depending only on μ and Cμ such that uL2< for any solution u. Moreover, as mentioned in Section 4.2, we may also assume that g0 has linear growth; thus, for fixed η>0, the operator K:L2(Ω,N)L2(Ω,N) given by Kv:=u, the unique solution of the problem

{Lu(x)+(λ1-η)u(x)=g0(x,v(x))+(λ1-η)v(x),xΩ,uν(x)=γ(x)u(x),xΩ

is well defined and clearly compact. Let KM be defined as in Lemma 5.1, with M=(μ+η)IN, then the operator sK+(1-s)KM has no fixed points on B(0) for 0s1, because gs(x,u):=sg0(x,u)+(1-s)Mu satisfies (2.1). Since μ-λ1, the assumptions of Lemma 5.1 hold with A=B=M and νk=0 for all k, then

deg(I-K,B(0),0)=deg(I-KM,B(0),0)=1.

Next we shall prove that if ρ>0 is small enough then Kuu on Bρ(0) and

deg(I-K,Bρ(0),0)=-1.

To this end, set M(x):=Dug0(x,0) and observe that if vBρ(0), then by letting Lη:=L+(λ1-η)I, we deduce from estimate (2.3) that

Kv-KMvL21ηLη(Kv-KMv)L2=o(ρ).

Due to the linearity and compactness of KM, it is easy to verify that there exists a constant θ>0 such that

v-KMvL2θρ

for all vBρ(0). It follows, for ρ>0 sufficiently small, that the operator defined by sK+(1-s)KM has no fixed points on Bρ(0) for s[0,1], because

v-sKv-(1-s)KMvL2v-KMvL2-KMv-KvL2θρ-o(ρ)>0

for vBρ(0). This implies, for small ρ, that the degree of I-K is well defined and, according to Lemma 5.1,

deg(I-K,Bρ(0),0)=deg(I-KM,Bρ(0),0)=(-1)ν1++νN=-1,

which proves the claim. Moreover, by the excision property of the degree,

deg(I-K,B(0)Bρ(0),0)=2.

Since the degree is locally constant with respect to the third coordinate, it is deduced that, for any PL2(Ω,N) sufficiently close to 0, the degrees deg(I-K,Bρ(0),P) and deg(I-K,B(0)Bρ(0),P) are well defined and equal to -1 and 2, respectively.

In order to complete the proof, for each pL2(Ω,N) we define P:=Θ(p) as the unique solution of the linear problem

LηP=p,Pν|Ω=γP.

The previous estimate (2.3) yields PL21ηpL2; that is, if p is close to 0 then so is P and the problem u-Ku=P has at least one solution in Bρ(0) and another one in B(0)Bρ(0). The result is thus deduced from the fact that if u-Ku=P, then

Lu+(λ1-η)u=LKu+(λ1-η)Ku+p=g0(,u)+(λ1-η)u+p,

and

uν|Ω=(Ku)ν|Ω+Pν|Ω=γ(Ku+P)=γu.

In other words, u is a solution of (1.1)–(1.2), which completes the proof. ∎

Remark 5.3.

Using the Sard–Smale theorem (see [8]), we deduce that the problem has generically at least three solutions, namely, there exists a residual set ΣL2(Ω,N) such that if pΣBr(0), then the problem has at least three solutions.

6 Further discussion and open problems

The existence of solutions for problem (1.1)–(1.2) was obtained under condition (1.5). Remarkably, the bounds of Section 4.1 imply by themselves that the degree of the operator I-K defined in the proof of Theorem 5.2 over a large ball is equal to 1, thus proving the existence of at least one solution. Since the H1 bounds require only that μ-λ1, one might be tempted to believe that the latter condition is sufficient for all our purposes; however, the operator K is not well defined for arbitrary g and this is why we needed also the L bounds. As an immediate consequence, we conclude that the condition μ=-λ1 is optimal when n=1. This is still true for n=2, provided that g has polynomial growth and also when n3, provided that g behaves asymptotically as |u|q with qn+2n-2. So it is an open problem to determine, for the general case, the optimal value of μ when n2. The value given by (4.1) is clearly larger than -λ1 and also satisfies the following lower bound: let

κ=φ1φ12,

then

|φ1|2κ|φ1|2and|φ1|2κ.

Since λ1=|φ1|2-Ωγφ12, it is deduced that

φ1φ12-λ1Ωγφ12.

On the other hand, observe that the use of a variable R in the Hartman condition can be avoided if μ=0 and γ0. The fact that γ0 is compensated by enlarging the value of μ. It is worth mentioning that, as shown in [5], the problem can always be reformulated into an equivalent one with γ0; however, under this transformation the relation between g0 and the eigenvalues becomes less clear.

Also, it would be interesting to analyse the case in which Dg0(x,0) is not necessarily symmetric. It is worth noticing that symmetry was used only in order to apply the lemma on bilinear symmetric forms used in [7], but, at first sight, more general results are possible. This is left as a second open problem.

Another matter concerns further multiplicity. In the same line of Theorem 5.2, one may try to see, when ν1++νN is even, if it is possible to obtain five solutions, as it happens for the ODE case presented in [2]. This is true for the scalar case N=1, for which the function R(x) defined in Section 3 can be used as an upper solution and, if g0u(x,0)<0 in Ω¯, then α:=rφ1, with r>0 small enough serves as a lower solution. Taking

Uα,β:={uC(Ω¯,):α<u<β},

we have that

deg(I-K,Uα,β,0)=1.

In the same way, an ordered couple α~<β~<0 of a lower and an upper solution is readily obtained. Furthermore, we may take ρ>0 small enough such that

Uα,βBρ(0)=Uα~,β~Bρ(0)=;

thus, if condition (1.4) is satisfied with k>0 even, then

deg(Uα,β(Bρ(0)Uα,βUα~,β~)=-2,

so generically the problem has at least five solutions. Moreover, observe that the condition gu(x,0)<0 is fulfilled automatically if k is sufficiently large: for instance, for the ODE studied in [2] with a0,a1>0, it suffices to take k2 because it is verified that λ2>0.

Finally, we remark that condition (1.6) makes sense only when λνkλνk+1. Unlike the case n=1, for which an elementary study of the Wronskian determinant shows that all the eigenvalues are simple, nothing is known about the multiplicity of higher order eigenvalues when n>1. This might reduce the number of possible different situations in Theorem 5.2: for example, for the periodic ODE one has

0=λ1<λ2=λ3<λ4=λ5<.

In particular, all the eigenvalues, except the first one, have even multiplicity. This means that, if (1.6) holds, then νk=0 or νk is odd. Thus, the assumption that ν1++νN is odd simply says, in this case, that #{k:νk0} is odd.

Acknowledgements

The author thanks Professor Mónica Clapp for her thoughtful comments and to the anonymous referee for the very careful reading of the manuscript and fruitful corrections and remarks.

References

  • [1]

    H. Amann, Dual semigroups and second order linear elliptic boundary value problems, Israel J. Math. 45 (1983), no. 2–3, 225–254.  CrossrefGoogle Scholar

  • [2]

    P. Amster and M. P. Kuna, Multiple solutions for a second order equation with radiation boundary conditions, Electron. J. Qual. Theory Differ. Equ. (2017), 10.14232/ejqtde.2017.1.37.  Web of ScienceGoogle Scholar

  • [3]

    P. Amster and M. P. Kuna, On exact multiplicity for a second order equation with radiation boundary conditions, submitted.  

  • [4]

    P. Amster, M. K. Kwong and C. Rogers, A Painlevé II model in two-ion electrodiffusion with radiation boundary conditions, Nonlinear Anal. Real World Appl. 16 (2014), 120–131.  CrossrefGoogle Scholar

  • [5]

    D. Daners, Inverse positivity for general Robin problems on Lipschitz domains, Arch. Math. (Basel) 92 (2009), no. 1, 57–69.  CrossrefGoogle Scholar

  • [6]

    P. Hartman, On boundary value problems for systems of ordinary, nonlinear, second order differential equations, Trans. Amer. Math. Soc. 96 (1960), 493–509.  CrossrefGoogle Scholar

  • [7]

    A. C. Lazer, Application of a lemma on bilinear forms to a problem in nonlinear oscillations, Proc. Amer. Math. Soc. 33 (1972), 89–94.  CrossrefGoogle Scholar

  • [8]

    S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866.  CrossrefGoogle Scholar

  • [9]

    K. Umezu, On eigenvalue problems with Robin type boundary conditions having indefinite coefficients, Appl. Anal. 85 (2006), no. 11, 1313–1325.  CrossrefGoogle Scholar

About the article

Received: 2017-02-18

Revised: 2017-05-11

Accepted: 2017-05-17

Published Online: 2017-07-08


Funding Source: Secretaria de Ciencia y Tecnica, Universidad de Buenos Aires

Award identifier / Grant number: 20020120100029BA

Funding Source: Consejo Nacional de Investigaciones Científicas y Técnicas

Award identifier / Grant number: PIP11220130100006CO

This work was partially supported by projects UBACyT 20020120100029BA and CONICET PIP 11220130100006CO.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 603–614, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0034.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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