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Research article Special Issues

Resonant problems for non-local elliptic operators with unbounded nonlinearites

  • In this paper we study the existence of nontrivial solutions of a class of asymptotically resonant problems driven by a non-local integro-differential operator with homogeneous Dirichlet boundary conditions by applying Morse theory and critical groups for a C2 functional at both isolated critical points and infinity.

    Citation: Yutong Chen, Jiabao Su. Resonant problems for non-local elliptic operators with unbounded nonlinearites[J]. Electronic Research Archive, 2023, 31(9): 5716-5731. doi: 10.3934/era.2023290

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  • In this paper we study the existence of nontrivial solutions of a class of asymptotically resonant problems driven by a non-local integro-differential operator with homogeneous Dirichlet boundary conditions by applying Morse theory and critical groups for a C2 functional at both isolated critical points and infinity.



    In this paper we will study the existence of nontrivial solutions of the following nonlocal elliptic problem

    {LKu=λu+g(x,u)xΩ,u=0xRNΩ (1.1)

    where Ω is an open bounded subset of RN with a smooth boundary, g:Ω×RR is a differential function whose properties will be given later, λ is an eigenvalue of LK and LK is a non-local elliptic operator formally defined as follows

    LKu(x):=RN(u(x+y)+u(xy)2u(x))K(y)dy,   xRN, (1.2)

    where the kernel K:RN{0}(0,) is a function with the properties that

    {mKL1(RN) with m(x)=min{|x|2,1}, and there is θ>0 such thatK(x)θ|x|(N+2s) for any xRN{0}, and s(0,1) is fixed. (1.3)

    The integro-differential operator LK is a generalization of the fractional Laplacian (Δ)s which is defined as

    (Δ)su(x):=RNu(x+y)+u(xy)2u(x)|y|N+2sdy,   xRN. (1.4)

    When one takes the kernel K(x)=|x|(N+2s) then LK=(Δ)s. In this case the problem (1.1) becomes

    {(Δ)su=λu+g(x,u)xΩ,u=0,xRNΩ. (1.5)

    The problem (1.5) can be regarded as the counterpart of the semilinear elliptic boundary value problem

    {Δu=λu+g(x,u)xΩ,u=0,xΩ, (1.6)

    where λ is an eigenvalue of Δ with a 0-Dirichlet boundary value.

    A weak solution for (1.1) is a function u:RNR such that

    {R2N(u(x)u(y))(φ(x)φ(y))K(xy)dxdyλΩu(x)φ(x)dx                                      =Ωg(x,u(x))φ(x)dx      for all  φX0uX0. (1.7)

    Here the linear space

    X0={vX: v=0  a.e. in RNΩ},

    and the functional space X denotes the linear space of Lebesgue measurable functions from RN to R such that the restriction to Ω of any function v in X belongs to L2(Ω) and

    the map (x,y)(v(x)v(y))K(xy)  is in  L2(R2N(CΩ×CΩ),dxdy),

    where CΩ:=RNΩ. The properties of the functional space X0 will be introduced in the next section.

    The non-local equations have been experiencing impressive applications in different subjects, such as the thin obstacle problem, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes and flame propagation, conservation laws, ultrarelativistic limits of quantum mechanics, quasigeostrophic flows, multiple scattering, minimal surfaces, materials science, water waves, elliptic problems with measured data, optimization, finance, etc. See [1] and the references therein. The non-local problems and operators have been widely studied in the literature and have attracted the attention of lot of mathematicians coming from different research areas due to the interesting analytical structure and broad applicability. Many mathematicians have applied variational methods [2] such as the mountain pass theorem[3], the saddle-point theorem[2] or other linking type of critical point theorem in the study of non-local equations with various nonlinearities that exhibit subcritical or critical growth; see [1,4,5,6,7,8,9,10,11,12,13] and references therein.

    In the present paper we will apply the Morse theory to find weak solutions to (1.1). We assume, throughout the whole paper, that the nonlinear function gC1(ˉΩ×R,R) satisfies the following growth condition

    (g) there is C>0 and p(2,2NN2s) such that

    |gt(x,t)|C(1+|t|p2)    for all (x,t)ˉΩ×R. (1.8)

    We consider the situation that the problem (1.1) has the trivial solution u0 and is resonant at infinity in the sense that the function g satisfies the following assumptions

    g(x,0)=0   uniformly in xˉΩ, (1.9)
    lim|t|g(x,t)t=0   uniformaly  in xˉΩ. (1.10)

    We refer the reader to [12, Proposition 9 and Appendix A], [14, Propositions 2.3 and 2.4] and [11, Proposition 4] for the existence and basic properties of the eigenvalue of the linear non-local eigenvalue problem given by

    {LKu=λuxΩ,u=0xRNΩ, (1.11)

    that will be collected in the next section.

    We make some further conditions on g.

    (g1) There are c1>0 and r(0,1) such that

    |g(x,t)|c1(|t|r+1)   for all tR, xΩ.

    (g±2) There are c2>0 and r(0,1) given in (g1) such that

    ±g(x,t)t0,  ±g(x,t)tc2(|t|1+r1)    for all tR, xΩ.

    We note here that (g1) implies (1.10) which characterizes the problem (1.1) as asymptotically linear resonant near infinity at the eigenvalue λ of the non-local operator LK. As an example, we can take the function g(x,t)=±a(x)|t|r1t with aL(Ω), infΩa>0 and r(0,1).

    We will prove the following theorems. We first consider the case that gt(x,0)+λ is not an eigenvalue of (1.11). We have the following conclusions.

    Theorem 1.1. Assume (1.3), (1.9) and (g1). Then the problem (1.1) admits at least one nontrivial weak solution in each of the following cases:

    (i)  (g+2),  λm<gt(x,0)+λ<λm+1,  λmλ;(ii)  (g2),  λm<gt(x,0)+λ<λm+1,  λmλ1<λ.

    For the case that gt(x,0)+λ=λm, an eigenvalue of (1.11), i.e., the trivial solution u=0 of (1.1), is degenerate. In this case the problem (1.1) is double resonant at both infinity and zero. We have the following conclusions.

    Theorem 1.2. Assume (1.3), (1.9) and (g1). Then the problem (1.1) admits at least one nontrivial weak solution in each of the following cases:

    (i)  (g+2),  gt(x,0)+λ=λm,   λ<λm1<λm or  λm<λ;(ii)  (g2),  gt(x,0)+λ=λm,   λm<λ1<λ or  λ1<λm1.

    Notice that in Theorem 1.2 there is a large difference between λ and λm. This can be reduced by imposing on g some local sign conditions near zero. We denote f(x,t):=λt+g(x,t) and F(x,t)=t0f(x,s)ds. We assume the following

    (F±0) ft(x,0)λm and there is δ>0 such that

    ±2F0(x,t):=±(2F(x,t)λmt2)0,  x¯Ω,  |t|δ.

    Theorem 1.3. Assume (1.3), (1.9) and (g1). Then the problem (1.1) admits at least one nontrivial weak solution in each of the following cases:

    (i)  (g+2),  (F+0),   λλm1<λm;   (ii)  (g2),  (F+0),   λλm;(iii)  (g+2),  (F0),   λm<λ1<λ;   (iv)  (g2),  (F0),  λ1<λm1.

    We give some remarks and comparisons. The non-local equations with resonance at infinity have been studied in some recent works. In [6,7], the famous saddle-point theorem [2] has been applied in the existence of solutions of the non-local problem related to (1.1) for the Landesman-Lazer resonance condition [15]. In [6], the authors treated a case in which one version of the Landesman-Lazer resonance condition [15] was formulated as follows:

    {g(x,t) is bounded for all (x,t)ˉΩ×R;G(x,t)=t0f(x,ς)dς as |t|. (1.12)

    In [7], the authors treated an autonomous case in which another version of the Landesman-Lazer resonance condition [15] was formulated as follows:

    {gC1(R), gl:=limtg(t)R, gr:=limt+g(t)R with gl>gr;grΩϕdxglΩϕ+dx<0<glΩϕdxgrΩϕ+dx,   ϕE(λ){0}, (1.13)

    where E(λ) is the linear space generated by the eigenfunctions corresponding to λ. In [7], there is a crucial assumption that all functions in E(λ) having a nodal set with the zero Lebesgue measure, which is valid for the fractional Laplacian (Δ)s (see [16]) and is still open for the general non-local elliptic operator LK (see [7, Equation (1.12)] and remarks therein).

    We note here that the common feature in (1.12) and (1.13) is that the nonlinear term g is bounded. Motivated by previous works [6,7], we treat, in the present paper, the completely resonant case via the application of Morse theory and critical groups. The results in this paper are new in two aspects. On one hand, the nonlinear term g is indeed unbounded and by imposing on g the global conditions (g1) and (g±2), we do not make the same assumption on the eigenfunctions of (1.11) as that in [7]. On the other hand, we explore a new application of the abstract results about critical groups at infinity that were built in [17] and modified in [18]. The conditions on g used here were first constructed in [19] for semilinear elliptic problems at resonance. Some of the above theorems may be regarded as the natural extension of local setting (1.6) to the non-local fractional setting.

    We prove the main results via Morse theory [20,21] and critical group computations. Precisely, we will work under the abstract framework built in [17] and modified in [18]. In Section 2, we collect some preliminaries about the variational formulas related to (1.1). In Section 3, we give the proofs of the main theorems including some technical lemmas.

    In this section we will give the preliminaries for the variational structure of (1.1) and preliminary results in Morse theory.

    We first recall some basic results on the functional X0 mentioned in Section 1. The functional space X0 is non-empty because C20(Ω)X0 (see [22, Lemma 11]), and it is endowed with the norm defined as

    vX0=:(R2N|v(x)v(y)|2K(xy)dxdy)12. (2.1)

    Furthermore, (X0,X0) is a Hilbert space with a scalar product (see [10, Lemmas 6 and 7]) defined by

    u,vX0=R2N(u(x)u(y))(v(x)v(y))K(xy)dxdy,  u,vX0. (2.2)

    The norm (2.1) on X0 is related to the so-called Gagliardo norm

    vHs(Ω)=:vL2(Ω)+(R2N|v(x)v(y)|2|xy|N+2sdxdy)12

    of the usual fractional Sobolev space Hs(Ω). For further details related to the fractional Sobolev spaces one can see [1,10,13] and the references therein.

    By [10, Lemma 8] and [13, Lemma 9], we have following embedding results.

    Proposition 2.1. For each q[1,2NN2s], the embedding X0Lq(RN) is continuous and there is Cq>0 such that

    uLq(RN)CquX0,    uX0.

    This embedding is compact whenever q[1,2NN2s).

    Next, we recall some basic facts about the eigenvalue problem associated with the integro-differential operator LK

    {LKu=λuxΩ,u=0,xRNΩ. (2.3)

    The number λR is an eigenvalue of (2.3) if there is a nontrivial function v:RNR such that for all φX0

    {R2N(v(x)v(y))(φ(x)φ(y))K(xy)dxdy=Ωv(x)φ(x)dxvX0.

    We denote by {λk}kN the sequence of the eigenvalue of the problem (2.3), with

    0<λ1<λ2λk   and  λk+  as  k+. (2.4)

    We denote by ϕk the eigenfunction corresponding to λk. The sequence {ϕk}kN can be normalized in such a way that the sequence provides an orthonormal basis of L2(Ω) and an orthogonal basis of X0. By [14, Proposition 2.4] one has that all ϕkL(Ω). One can refer to [12, Proposition 9 and Appendix A], [14, Proposition 2.3] and [11, Proposition 4] for a complete study of the spectrum of the integro-differential operator LK.

    The first eigenvalue λ1 is simple and can be characterized as

    λ1=minuX0,uL2(Ω)=1R2N|u(x)u(y)|2K(xy)dxdy.

    Each eigenvalue λk, k2, has finite multiplicity. More precisely, we say that λk has the finite multiplicity νkN if

    λk1<λk=λk+1=λk+νk1<λk+νk. (2.5)

    The set of all of the eigenfunctions corresponding to λk agrees with

    E(λk):=span{ϕk,ϕk+1,,ϕk+νk1},  dimE(λk)=νk.

    The eigenvalue λ1 is achieved at a positive function ϕ1 with ϕ1L2(Ω)=1. For each k2, the eigenvalue λk can be characterized as follows:

    λk=minuPk,uL2(Ω)=1R2N|u(x)u(y)|2K(xy)dxdy, (2.6)

    where

    Pk:={uX0:u,ϕjX0=0  for all  j=1,2,,k1}.

    Corresponding to the eigenvalue λk of LK with multiplicity νk, the space X0 can be split as follows:

    X0=WkVkW+k=VkWk,    Wk=WkW+k,

    where

    Wk=λj<λkE(λj),  Vk=E(λk),  W+k=(WkVk)=¯λj>λkE(λj).

    For each eigenvalue λk, we can define a linear operator Ak:X0X0 by

    Aku,v:=R2N(u(x)u(y))(v(x)v(y))K(xy)dxdyλkΩuvdx. (2.7)

    By the continuous embedding from X0 into L2(Ω) in Proposition 2.1, one can deduce that Ak is a bounded self-adjoint linear operator so that Akϕ,ϕ=0 for all ϕVk=:ker(Ak).

    Finally, we conclude this subsection with the following variational inequalities which can be deduced by the variational characterization of the eigenvalues and the standard Fourier decomposition:

    Aku,u(1λkλk1)u2X0,    uWk, (2.8)
    Akv,v(1λkλk+νk)v2X0,    vW+k. (2.9)

    In this section we give the proofs of the main results in this paper via some abstract results on Morse theory [20,21] for a C2 functional J defined on a Hilbert space. These results come from [17,18,20,21,23,24,25], etc. We refer the readers to [26] for a brief summary of the concepts, definitions and the abstract results about critical groups and Morse theory.

    First of all, we observe that the problem (1.1) has a variational structure; indeed, it is the Euler-Lagrange equation of the functional J:X0R defined as

    J(u)=12R2N|u(x)u(y)|2K(xy)dxdy12λΩ|u|2dxΩG(x,u)dx, uX0, (3.1)

    where G(x,t)=t0g(x,ς)dς. Since the nonlinear function g satisfies the assumption (g), by Proposition 2.1, the functional J is well defined on X0 and is of class C2 (see a detailed proof in [26]) with the derivatives given by

    J(u),v=R2N(u(x)u(y))(v(x)v(y))K(xy)dxdy  λΩuvdxΩg(x,u)vdx,      u,vX0, (3.2)
    J(u)v,w=R2N(v(x)v(y))(w(x)w(y))K(xy)dxdy    λΩvwdxΩgt(x,u)vwdx,     u,v,wX0. (3.3)

    From (3.2) and (1.7), one sees that critical points of J are exactly weak solutions to (1.1).

    Define the functional F:X0R by

    F(u)=ΩG(x,u(x))dx,  uX0. (3.4)

    According to (2.7) with λ, the functional J can be written as

    J(u)=12Au,u+F(u),  uX0. (3.5)

    Using the assumption (g1) we can deduce that

    F(u)=o(uX0)   as  uX0. (3.6)

    Therefore J fits the basic assumptions in the abstract framework required by [26, Proposition 2.5] with respect to X0=VW.

    Next we prove one technical lemma that will be used to verify the angle conditions required by [26, Proposition 2.5] for computation of the critical groups at infinity.

    Lemma 3.1. Assume (g1) and (g±2). Then there exist M>0, ϵ(0,1) and β>0 such that

    ±Ωg(x,u)vdxβv1+rX0. (3.7)

    for any u=v+wX0=VW with uX0M and wX0ϵuX0.

    Proof. We give the proof for the case that (g1) and (g+2) hold.

    For u=v+wX0=VW, we set

    C(M,ϵ)={u=v+w: uX0M, wX0ϵuX0},

    where M>0 and ϵ(0,1) will be chosen below.

    For uC(M,ϵ), we have

    |vX01ϵ2 uX0,   wX0ϵ1ϵ2vX0. (3.8)

    It follows from (g1) and (g+2) that

    Ωg(x,u)vdx=Ωg(x,u)udxΩg(x,u)wdxΩg(x,u)udxΩ|g(x,u)||w|dxc2Ω(|u|1+r1)dxc1Ω(|u|r+1)|w|dx.c2Ω|u|1+rdxc1Ω|u|r|w|dxc1C1wX0c2|Ω|. (3.9)

    By Proposition 2.1 and the Hölder inequality we have

    Ω|u|r|w|dx(Ω|u|1+r)r1+r(Ω|w|1+r)11+rC1+r1+rurX0wX0C1+r1+rϵu1+rX0. (3.10)

    Since V is finite dimensional, by the elementary inequality |a+b|q2q1(|a|q+|b|q) for all a,bR, we have that

    Ω|u|1+rdx=Ω|v+w|1+rdx12rΩ|v|1+rdxΩ|w|1+rdx12rv1+rL1+r(Ω)C1+r1+rw1+rX012rˆc1+r(1ϵ2)1+r2u1+rX0C1+r1+rϵ1+ru1+rX0. (3.11)

    here ˆc is the embedding constant of L1+r(Ω)V. Therefore for uC(M,ϵ), it follows from (3.9)–(3.11) that

    Ωg(x,u)vdx(12rc2ˆc1+r(1ϵ2)1+r2c2C1+r1+rϵ1+rc1C1+r1+rϵ)u1+rX0   c1C1ϵuX0c2|Ω|(c22rˆc1+r(1ϵ2)1+r2c2C1+r1+rϵ1+rc1C1+r1+rϵc1C1ϵMrc2|Ω|M1+r)u1+rX0=:βu1+rX0. (3.12)

    Now we can take M>0 large enough and 0<ϵ<1 small enough so that

    β=c22rˆc1+r(1ϵ2)1+r2c2C1+r1+rϵ1+rc1C1+r1+rϵc1C1ϵMc2|Ω|M1+r>0;

    hence,

    Ωg(x,u)vdxβu1+rX0βv1+rX0  for  uC(M,ϵ). (3.13)

    The proof is complete.

    In order to apply [26, Proposition 2.5] and Morse theory to prove our results, we have to verify that J satisfies the Palais-Smale condition.

    Lemma 3.2. Assume (g1) and (g±2). Then the functional J defined by (3.1) satisfies the Palais-Smale condition.

    Proof. Let the sequence {un}X0 be such that

    J(un)0,  n. (3.14)

    We show that {un} is bounded in X0. Suppose, by the way of contradiction, that

    unX0  as  n. (3.15)

    Write un=vn+wn, where vnV and wnW. By the variational inequalities (2.8) and (2.9), we have

    |Awn,wn|σwn2X0,   nN, (3.16)

    where

    σ=min{1λλ+ν, λλ11}.

    By (3.11), there is N1N such that

    |J(un),wn|wnX0,   nN1. (3.17)

    By (3.2) and (3.5), we have

    J(un),wn=Awn,wnX0+F(un),wn. (3.18)

    By (3.6) and (3.15) we have

    F(un)=o(unX0),  n. (3.19)

    It follows that, for any given δ>0 sufficiently small and all n sufficiently large,

    σwn2X0|Awn,wnX0||J(un),wn|+|F(un),wn|wnX0+δunX0wnX0. (3.20)

    Since δ>0 was chosen arbitrarily, from (3.15) and (3.20) we deduce that

    wnX0unX00  as  n. (3.21)

    It follows that there is N2N with N2N1 such that

    unX0M  and  wnX0ϵunX0  for  nN2, (3.22)

    where M>0 and ϵ(0,1) was given in Lemma 3.1. Therefore by Lemma 3.1 we have that

    ±Ωg(x,un)vnvnX0dxβvnrX0β(1ϵ2)r2Mr>0  for   nN2. (3.23)

    On the other hand, by (3.14), we get

    limn|Ωg(x,un)vnvnX0dx|=limn|J(un),vnvnX0|=0. (3.24)

    This contradicts (3.23). Hence {un} is bounded in X0.

    Since X0 is a Hilbert space, there is a subsequence of {un}, still denoted by {un}, and there exists uX0, such that

    unu   weakly in  X0   as  n. (3.25)

    By Proposition 2.1, up to a subsequence, it holds that

    unuin Lq(RN)   q[1,2NN2s),un(x)u(x)a.e. in RN. (3.26)

    as n. By (g1), Proposition 2.1 and (3.26) we get

    |Ω(g(x,un)g(x,u))(unu)dx|2c1unuL1(Ω)+c1Cr1+r(urX0+unrX0)unuL1+r(Ω)0   as  n. (3.27)

    From (3.14) we deduce that

    J(un),un=R2N|un(x)un(y)|2K(|xy|)dxdyλΩ|un|2dxΩg(x,un(x))un(x)dx0 (3.28)

    as n, and

    J(un),u=R2N(un(x)un(y))(u(x)u(y))K(|xy|)dxdyλΩunudxΩg(x,un(x))u(x)dx0 (3.29)

    as n. Now by (3.14) and (3.26)–(3.29), we deduce from

    J(un)J(u),unu0   n

    that

    unu2X00,   n.

    This completes the proof for verifying the Palais-Smale condition.

    Notice here that only (3.14) is used for verifying the Palais-Smale condition, it follows that the critical point set of J is compact and is then bounded.

    Now we are ready to give the proofs of the main results in this paper.

    Proof of Theorem 1.1. We give the proof of the case (ⅰ). Since

    J(u),v=Ωg(x,u)vdx,   vV,

    it follows from Lemma 3.1 that J satisfies the angle condition (AC) in [26, Proposition 2.5] at infinity with respect to X0=VW. Thus by [26, Proposition 2.5(ⅱ)] we have

    Cq(J,)δq,μ+νZ, qZ, (3.30)

    where

    μ=dimλk<λkerE(λk),    ν=dimE(λ).

    Therefore J has a critical point u satisfying

    Cμ+ν(J,u)0. (3.31)

    The second derivative of J at the trivial solution u=0 can be written as

    J(0)ϕ,ϕ=ϕ2X0Ω(λ+gt(x,0))ϕ2dx,    ϕX0. (3.32)

    By the condition we see that u=0 is a nondegenerate critical point of J with the Morse index

    ˉμ0=dimλkλmkerE(λk). (3.33)

    Hence

    Cq(J,0)δq,ˉμ0Z. (3.34)

    Since λmλ, we get that μ+νˉμ0, and we see from (3.33) and (3.34) that u0. The case (ⅱ) can be proved in the same way. The proof is complete.

    Proof of Theorem 1.2 We give the proof of the case (ⅱ). It follows from Lemma 3.1 that J satisfies the angle condition (AC+) in [26, Proposition 2.5] at infinity with respect to X0=VW. Thus by [26, Proposition 2.5(ⅱ)] we have

    Cq(J,)δq,μZ, qZ, (3.35)

    and J has a critical point u satisfying

    Cμ(J,u)0. (3.36)

    Now J(0) takes the form

    J(0)ϕ,ϕ=ϕ2X0λmΩϕ2dx,    ϕX0. (3.37)

    It follows that 0 is a degenerate critical point of J with the Morse index μ0 and the nullity ν0 given by

    μ0=dimλkλm1kerE(λk),   ν0=dimE(λm). (3.38)

    By the Gromoll-Meyer result[27], we have that

    Cq(J,0)0,  for q[μ0,μ0+ν0]. (3.39)

    It follows from λm<λ1<λ or λ1<λm1 that μ0+ν0<μ or μ0>μ, and we see from (3.36) and (3.39) that u0. The case (ⅰ) can be proved in the same way. The proof is complete.

    Lemma 3.3. Assume (1.3), (1.9), (g1) and (F±0). Then

    (ⅰ) Cq(J,0)δq,μ0+ν0Z for (F+0) holds,

    (ⅱ) Cq(J,0)δq,μ0Z for (F0) holds,

    where μ0 and ν0 are given by (3.38).

    Proof. We will apply [24, Proposition 2.3] to prove the results. We first note that by (g1) and the last part in the proof of Lemma 3.2, the functional J verifies the bounded Palais-Smale condition which ensures the deformation property for computing Cq(J,0) (see [20,21]).

    We treat the case (ⅱ) for which (F0) holds. We will prove that J has the local linking structure at 0 as with respect to X0=EE+, where E=Wm and E+=VmW+m. We refer the readers to [28,29] for the concept of the local linking.

    1) Take uE=Wm. Since Wm is finite dimensional, there is ρ>0 such that

    uX0ρ    |u(x)|δ,  a.e. xΩ.

    Consequently, thanks to (2.8) with λm and (F0), for any uE with uρ, we get

    J(u)12(1λmλm1)u2X0ΩF0(x,u)dx14(1λmλm1)u2X00. (3.40)

    2) For uE+=VmW+m, we write u=w+z, where wW+m and zVm. Then

    J(u)12(1λmλm+νm)w2X0ΩF0(x,u)dx. (3.41)

    Since Vm is finite dimensional, there is ρ>0 such that

    zuρ    |z(x)|<13δ,  a.e. xΩ.

    Consequently,

    |u(x)|>δ  |w(x)|=|u(x)z(x)||u(x)||z(x)|>23|u(x)|.

    By (F0), we have

    {|u(x)|δ}F0(x,u)dx0. (3.42)

    By (g1), we get that, for each given σ(2,2NN2s], there is κ=κ(σ,δ)>0 such that

    |F0(x,t)|κ|u|σ,   xΩ,  |t|>δ. (3.43)

    Hence

    {|u(x)|>δ}F0(x,u)dxκ{|u(x)|>δ}|u|σdxκ(3/2)σΩ|w|σdxC(σ,δ)wσX0. (3.44)

    Now, by (3.41), (3.42) and (3.44) we get

    J(u)12(1λmλm+νm)w2X0{|u(x)|δ}F0(x,u)dxC(κ,σ)wσX0. (3.45)

    Since σ>2, one sees from (3.42) and (3.45) that for ρ>0 small enough once again, it holds that

    Φ(u)>0,  uρ  with w0. (3.46)

    For zVm with zρ, we have by (F0) that

    2F0(x,z(x))=2F(x,z(x))λmz(x)20,    a.e. xΩ.

    Thus for all zBρVm,

    J(z)=12Ω(2F(x,z(x))λmz(x)2)dx0. (3.47)

    To apply [24, Proposition 2.3], we need to show that the above inequality holds strictly for z0. Assume, for contradiction, that for any 0<ϵρ, there is zϵVm such that 0<zϵ<ϵ and J(zϵ)=0. Then, the following holds:

    2F(x,zϵ(x))=λmzϵ(x)2,    a.e. xΩ,

    and then

    f(x,zϵ(x))=λmzϵ(x),    a.e. xΩ.

    Given that zϵVm, going back to (1.1), we see that zϵ is a nontrivial solution of (1.1). This contradicts the conventional assumption that 0 is an isolated solution of (1.1). In summary, we obtain by (3.46) and (3.47) that

    J(u)>0, 0<uρ, uE+.

    Therefore, J has a local linking structure with respect to E=EE+ with μ0=dimE. It follows from [24, Proposition 2.3] that Cq(J,0)δq,μ0Z.

    The case (ⅰ) is proved in a similar and simpler way. The proof is complete.

    Proof of Theorem 1.3. We give the proof of the case (ⅳ). As in the proof of Theorem 1.1(ⅱ), we have gotten the following conclusion that J satisfies the angle condition (AC+) in [26, Proposition 2.5] at infinity with respect to X0=VW, and then that J has a critical point u satisfying

    Cμ(J,u)0. (3.48)

    By (F0) and Lemma 3.3, J has a local linking at 0 with respect to X0=EE+. Thus it follows from [24, Proposition 2.3] that

    Cq(J,0)δq,μ0Z. (3.49)

    By λ1<λm1, we have that λ<μ0. It follows from (3.48) and (3.49) that u0. The other cases can be proved in the same way. The proof is complete.

    Remark 3.4. We conclude the paper with some remarks.

    1) In Theorem 1.3, the result for one nontrivial solution is valid for f that is locally Lipschitz continuous with f(x,0)λm being replaced by satisfying limt0f(x,t)t=λm. In this case, we have only JC20(X0,R) and no Morse index is involved. We have the critical groups at zero by applying the local linking theorem in [23] as follows:

    Cμ0+ν0(J,0)0  for (F+0) holds;   Cμ0(J,0)0  for (F0) holds.

    2) In the case that λ=λ1 and (g2) holds, we have that μ=0 and

    Cq(J,)δq,0Z. (3.50)

    Thus J has a critical point u with

    C0(J,u)0. (3.51)

    It follows that

    Cq(J,u)δq,0Z. (3.52)

    Indeed, (3.50) is equivalent to J being bounded from below and (3.51) is equivalent to u being a local minimizer of J. Furthermore, in the case that Cq(J,0)0 for some q1, we can apply [30, Theorem 2.1], i.e., the most general version of the three critical point theorem, to get two nontrivial solutions of (1.1).

    The authors declare that they have not used Artificial Intelligence tools in the creation of this article.

    The authors appreciate the reviewers for carefully reading the manuscript and giving valuable comments to improve the exposition of the paper. This work was supported by the NSFC (12001382, 12271373, 12171326).

    The authors declare there is no conflicts of interest.



    [1] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
    [2] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, Providence, RI 1986.
    [3] A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349–381. https://doi.org/10.1016/0022-1236(73)90051-7 doi: 10.1016/0022-1236(73)90051-7
    [4] G. M. Bisci, R. Servadei, A Brezis-Nirenberg splitting approach for nonlocal fractional equations, Nonlinear Anal. Theory Methods Appl., 119 (2015), 341–353. https://doi.org/10.1016/j.na.2014.10.025 doi: 10.1016/j.na.2014.10.025
    [5] G. M. Bisci, D. Mugnai, R. Servadei, On multiple solutions for nonlocal fractional problems via -theorems, Differ. Integr. Equations, 30 (2017), 641–666.
    [6] A. Fiscella, R. Servadei, E. Valdinoci, A resonance problem for non-Local elliptic operators, Zeitschrift für Analysis und ihre Anwendungen, 32 (2013), 411–431. https://doi.org/10.4171/ZAA/1492 doi: 10.4171/ZAA/1492
    [7] A. Fiscella, R. Servadei, E. Valdinoci, Asymptotically linear problems driven by fractionl operators, Math. Methods Appl. Sci., 38 (2015), 3551–3563. https://doi.org/10.1002/mma.3438 doi: 10.1002/mma.3438
    [8] D. Mugnai, D. Pagliardini, Existence and multiplicity results for the fractional Laplacian in bounded domains, Adv. Calc. Var., 10 (2017), 111–124. https://doi.org/10.1515/acv-2015-0032 doi: 10.1515/acv-2015-0032
    [9] R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal., 43 (2014), 251–267.
    [10] R. Servadei, E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887–898. https://doi.org/10.1016/j.jmaa.2011.12.032 doi: 10.1016/j.jmaa.2011.12.032
    [11] R. Servadei, E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445–2464. https://doi.org/10.3934/cpaa.2013.12.2445 doi: 10.3934/cpaa.2013.12.2445
    [12] R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105–2137. https://doi.org/10.3934/10.3934/dcds.2013.33.2105 doi: 10.3934/10.3934/dcds.2013.33.2105
    [13] R. Servadei, E. Valdinoci, The Brézis-Nirenberg result for the fractional Laplacian, Trans. Am. Math. Soc., 367 (2015), 67–102. https://doi.org/10.1090/S0002-9947-2014-05884-4 doi: 10.1090/S0002-9947-2014-05884-4
    [14] R. Servadei, The Yamabe equation in a non-local setting, Adv. Nonlinear Anal., 2 (2013), 235–270. https://doi.org/10.1515/anona-2013-0008 doi: 10.1515/anona-2013-0008
    [15] E. M. Landesman, A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609–623.
    [16] M. M. Fall, V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Commun. Partial Differ. Equations, 39 (2014), 354–397. https://doi.org/10.1080/03605302.2013.825918 doi: 10.1080/03605302.2013.825918
    [17] T. Bartsch, S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. Theory Methods Appl., 28 (1997), 419–441. https://doi.org/10.1016/0362-546X(95)00167-T doi: 10.1016/0362-546X(95)00167-T
    [18] J. Su, L. Zhao, An elliptic resonance problem with multiple solutions, J. Math. Appl. Appl., 319 (2006), 604–616. https://doi.org/10.1016/j.jmaa.2005.10.059 doi: 10.1016/j.jmaa.2005.10.059
    [19] J. Su, Semilinear elliptic resonant problems at higher eigenvalue with unbounded nonlinear terms, Acta Math. Sin., 14 (1998), 411–419. https://doi.org/10.1007/BF02580445 doi: 10.1007/BF02580445
    [20] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solutions Problems, Birkhauser, Boston, 1993.
    [21] J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Springer, Berlin, 1989.
    [22] R. Servadei, E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091–1126. https://doi.org/10.4171/RMI/750 doi: 10.4171/RMI/750
    [23] J. Liu, A Morse index for a saddle point, Syst. Sci. Math. Sci., 2 (1989), 32–39.
    [24] J. Su, Semilinear elliptic boundary value problems with double resonance between two consecutive eigenvalues, Nonlinear Anal. Theory Methods Appl., 48 (2002), 881–895. https://doi.org/10.1016/S0362-546X(00)00221-2 doi: 10.1016/S0362-546X(00)00221-2
    [25] Z. Q. Wang, Multiple solutions for indefinite functionals and applications to asymptotically linear problems, Acta Math. Sin., 5 (1989), 101–113. https://doi.org/10.1007/BF02107664 doi: 10.1007/BF02107664
    [26] Y. Chen, J. Su, Resonant problems for fractional Laplacian, Commun. Pure Appl. Anal., 16 (2017), 163–187. https://doi.org/10.3934/cpaa.2017008 doi: 10.3934/cpaa.2017008
    [27] D. Gromoll, M. Meyer, On differential functions with isolated point, Topology, 8 (1969), 361–369. https://doi.org/10.1016/0040-9383(69)90022-6 doi: 10.1016/0040-9383(69)90022-6
    [28] S. Li, J. Liu, Some existence theorems on multiple critical points and their applications, Kexue Tongbao, 17 (1984), 1025–1027.
    [29] S. Li, M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl., 189 (1995), 6–32.
    [30] J. Liu, J. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258 (2001), 209–222. https://doi.org/10.1006/jmaa.2000.7374 doi: 10.1006/jmaa.2000.7374
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