Image reconstruction represents an important technique applied in various fields such as medicine, biology, materials science, nondestructive testing, and so forth. In this paper, we transform the problem of image reconstruction into the problem of solving linear systems with multiple right-hand sides. Based on the idea of K-means clustering, we propose the global randomized block Kaczmarz method, so as to solve the problem of the linear systems with multiple right-hand sides effectively and use this method to image reconstruction. Theoretical analysis proves the convergence of this method, and the simulation results demonstrate the performance of this method in image reconstruction.
Citation: Ranran Li, Hao Liu. On global randomized block Kaczmarz method for image reconstruction[J]. Electronic Research Archive, 2022, 30(4): 1442-1453. doi: 10.3934/era.2022075
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Image reconstruction represents an important technique applied in various fields such as medicine, biology, materials science, nondestructive testing, and so forth. In this paper, we transform the problem of image reconstruction into the problem of solving linear systems with multiple right-hand sides. Based on the idea of K-means clustering, we propose the global randomized block Kaczmarz method, so as to solve the problem of the linear systems with multiple right-hand sides effectively and use this method to image reconstruction. Theoretical analysis proves the convergence of this method, and the simulation results demonstrate the performance of this method in image reconstruction.
In this paper we will study the existence of nontrivial solutions of the following nonlocal elliptic problem
{−LKu=λℓu+g(x,u)x∈Ω,u=0x∈RN∖Ω | (1.1) |
where Ω is an open bounded subset of RN with a smooth boundary, g:Ω×R→R is a differential function whose properties will be given later, λℓ is an eigenvalue of LK and LK is a non-local elliptic operator formally defined as follows
LKu(x):=∫RN(u(x+y)+u(x−y)−2u(x))K(y)dy, x∈RN, | (1.2) |
where the kernel K:RN∖{0}→(0,∞) is a function with the properties that
{mK∈L1(RN) with m(x)=min{|x|2,1}, and there is θ>0 such thatK(x)⩾θ|x|−(N+2s) for any x∈RN∖{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(x−y)−2u(x)|y|N+2sdy, x∈RN. | (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=λℓu+g(x,u)x∈Ω,u=0,x∈RN∖Ω. | (1.5) |
The problem (1.5) can be regarded as the counterpart of the semilinear elliptic boundary value problem
{−Δu=λℓu+g(x,u)x∈Ω,u=0,x∈∂Ω, | (1.6) |
where λℓ is an eigenvalue of −Δ with a 0-Dirichlet boundary value.
A weak solution for (1.1) is a function u:RN→R such that
{∫R2N(u(x)−u(y))(φ(x)−φ(y))K(x−y)dxdy−λℓ∫Ωu(x)φ(x)dx =∫Ωg(x,u(x))φ(x)dx for all φ∈X0u∈X0. | (1.7) |
Here the linear space
X0={v∈X: v=0 a.e. in RN∖Ω}, |
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(x−y) is in L2(R2N∖(CΩ×CΩ),dxdy), |
where CΩ:=RN∖Ω. 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 g∈C1(ˉΩ×R,R) satisfies the following growth condition
(g) there is C>0 and p∈(2,2NN−2s) such that
|g′t(x,t)|⩽C(1+|t|p−2) for all (x,t)∈ˉΩ×R. | (1.8) |
We consider the situation that the problem (1.1) has the trivial solution u≡0 and is resonant at infinity in the sense that the function g satisfies the following assumptions
g(x,0)=0 uniformly in x∈ˉΩ, | (1.9) |
lim|t|→∞g(x,t)t=0 uniformaly in x∈ˉΩ. | (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=λux∈Ω,u=0x∈RN∖Ω, | (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 t∈R, x∈Ω. |
(g±2) There are c2>0 and r∈(0,1) given in (g1) such that
±g(x,t)t⩾0, ±g(x,t)t⩾c2(|t|1+r−1) for all t∈R, x∈Ω. |
We note here that (g1) implies (1.10) which characterizes the problem (1.1) as asymptotically linear resonant near infinity at the eigenvalue λℓ of the non-local operator −LK. As an example, we can take the function g(x,t)=±a(x)|t|r−1t with a∈L∞(Ω), infΩa>0 and r∈(0,1).
We will prove the following theorems. We first consider the case that g′t(x,0)+λℓ 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<g′t(x,0)+λℓ<λm+1, λm≠λℓ;(ii) (g−2), λm<g′t(x,0)+λℓ<λm+1, λm≠λℓ−1<λℓ. |
For the case that g′t(x,0)+λℓ=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), g′t(x,0)+λℓ=λm, λℓ<λm−1<λm or λm<λℓ;(ii) (g−2), g′t(x,0)+λℓ=λm, λm<λℓ−1<λℓ or λℓ−1<λm−1. |
Notice that in Theorem 1.2 there is a large difference between λℓ and λm. This can be reduced by imposing on g some local sign conditions near zero. We denote f(x,t):=λℓt+g(x,t) and F(x,t)=∫t0f(x,s)ds. We assume the following
(F±0) f′t(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), λℓ≠λm−1<λm; (ii) (g−2), (F+0), λℓ≠λm;(iii) (g+2), (F−0), λm<λℓ−1<λℓ; (iv) (g−2), (F−0), λℓ−1<λm−1. |
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:
{g∈C1(R), gl:=limt→−∞g(t)∈R, gr:=limt→+∞g(t)∈R with gl>gr;gr∫Ωϕ−dx−gl∫Ωϕ+dx<0<gl∫Ωϕ−dx−gr∫Ωϕ+dx, ∀ ϕ∈E(λℓ)∖{0}, | (1.13) |
where E(λℓ) is the linear space generated by the eigenfunctions corresponding to λℓ. In [7], there is a crucial assumption that all functions in E(λℓ) 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
‖v‖X0=:(∫R2N|v(x)−v(y)|2K(x−y)dxdy)12. | (2.1) |
Furthermore, (X0,‖⋅‖X0) is a Hilbert space with a scalar product (see [10, Lemmas 6 and 7]) defined by
⟨u,v⟩X0=∫R2N(u(x)−u(y))(v(x)−v(y))K(x−y)dxdy, u,v∈X0. | (2.2) |
The norm (2.1) on X0 is related to the so-called Gagliardo norm
‖v‖Hs(Ω)=:‖v‖L2(Ω)+(∫R2N|v(x)−v(y)|2|x−y|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∈[1,2NN−2s], the embedding X0↪Lq(RN) is continuous and there is Cq>0 such that
‖u‖Lq(RN)⩽Cq‖u‖X0, ∀ u∈X0. |
This embedding is compact whenever q∈[1,2NN−2s).
Next, we recall some basic facts about the eigenvalue problem associated with the integro-differential operator −LK
{−LKu=λux∈Ω,u=0,x∈RN∖Ω. | (2.3) |
The number λ∈R is an eigenvalue of (2.3) if there is a nontrivial function v:RN→R such that for all φ∈X0
{∫R2N(v(x)−v(y))(φ(x)−φ(y))K(x−y)dxdy=∫Ωv(x)φ(x)dxv∈X0. |
We denote by {λk}k∈N the sequence of the eigenvalue of the problem (2.3), with
0<λ1<λ2⩽⋯⩽λk⩽⋯ and λk→+∞ as k→+∞. | (2.4) |
We denote by ϕk the eigenfunction corresponding to λk. The sequence {ϕk}k∈N 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 ϕk∈L∞(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=minu∈X0,‖u‖L2(Ω)=1∫R2N|u(x)−u(y)|2K(x−y)dxdy. |
Each eigenvalue λk, k⩾2, has finite multiplicity. More precisely, we say that λk has the finite multiplicity νk∈N if
λk−1<λk=λk+1⋯=λk+νk−1<λk+νk. | (2.5) |
The set of all of the eigenfunctions corresponding to λk agrees with
E(λk):=span{ϕk,ϕk+1,⋯,ϕk+νk−1}, dimE(λk)=νk. |
The eigenvalue λ1 is achieved at a positive function ϕ1 with ‖ϕ1‖L2(Ω)=1. For each k⩾2, the eigenvalue λk can be characterized as follows:
λk=minu∈Pk,‖u‖L2(Ω)=1∫R2N|u(x)−u(y)|2K(x−y)dxdy, | (2.6) |
where
Pk:={u∈X0:⟨u,ϕj⟩X0=0 for all j=1,2,⋯,k−1}. |
Corresponding to the eigenvalue λk of −LK with multiplicity νk, the space X0 can be split as follows:
X0=W−k⊕Vk⊕W+k=Vk⊕Wk, Wk=W−k⊕W+k, |
where
W−k=⨁λj<λkE(λj), Vk=E(λk), W+k=(W−k⊕Vk)⊥=¯⨁λj>λkE(λj). |
For each eigenvalue λk, we can define a linear operator Ak:X0→X∗0 by
⟨Aku,v⟩:=∫R2N(u(x)−u(y))(v(x)−v(y))K(x−y)dxdy−λk∫Ωuvdx. | (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 ϕ∈Vk=: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⟩⩽(1−λkλk−1)‖u‖2X0, ∀ u∈W−k, | (2.8) |
⟨Akv,v⟩⩾(1−λkλk+νk)‖v‖2X0, ∀ v∈W+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:X0→R defined as
J(u)=12∫R2N|u(x)−u(y)|2K(x−y)dxdy−12λℓ∫Ω|u|2dx−∫ΩG(x,u)dx, u∈X0, | (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(x−y)dxdy −λℓ∫Ωuvdx−∫Ωg(x,u)vdx, ∀ u,v∈X0, | (3.2) |
⟨J″(u)v,w⟩=∫R2N(v(x)−v(y))(w(x)−w(y))K(x−y)dxdy −λℓ∫Ωvwdx−∫Ωg′t(x,u)vwdx, ∀ u,v,w∈X0. | (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:X0→R by
F(u)=−∫ΩG(x,u(x))dx, u∈X0. | (3.4) |
According to (2.7) with λℓ, the functional J can be written as
J(u)=12⟨Aℓu,u⟩+F(u), u∈X0. | (3.5) |
Using the assumption (g1) we can deduce that
‖F′(u)‖=o(‖u‖X0) as ‖u‖X0→∞. | (3.6) |
Therefore J fits the basic assumptions in the abstract framework required by [26, Proposition 2.5] with respect to X0=Vℓ⊕Wℓ.
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)vdx⩾β‖v‖1+rX0. | (3.7) |
for any u=v+w∈X0=Vℓ⊕Wℓ with ‖u‖X0⩾M and ‖w‖X0⩽ϵ‖u‖X0.
Proof. We give the proof for the case that (g1) and (g+2) hold.
For u=v+w∈X0=Vℓ⊕Wℓ, we set
C(M,ϵ)={u=v+w: ‖u‖X0⩾M, ‖w‖X0⩽ϵ‖u‖X0}, |
where M>0 and ϵ∈(0,1) will be chosen below.
For u∈C(M,ϵ), we have
|v‖X0⩾√1−ϵ2 ‖u‖X0, ‖w‖X0⩽ϵ√1−ϵ2‖v‖X0. | (3.8) |
It follows from (g1) and (g+2) that
∫Ωg(x,u)vdx=∫Ωg(x,u)udx−∫Ωg(x,u)wdx⩾∫Ωg(x,u)udx−∫Ω|g(x,u)||w|dx⩾c2∫Ω(|u|1+r−1)dx−c1∫Ω(|u|r+1)|w|dx.⩾c2∫Ω|u|1+rdx−c1∫Ω|u|r|w|dx−c1C1‖w‖X0−c2|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+r⩽C1+r1+r‖u‖rX0‖w‖X0⩽C1+r1+rϵ‖u‖1+rX0. | (3.10) |
Since Vℓ is finite dimensional, by the elementary inequality |a+b|q⩽2q−1(|a|q+|b|q) for all a,b∈R, we have that
∫Ω|u|1+rdx=∫Ω|v+w|1+rdx⩾12r∫Ω|v|1+rdx−∫Ω|w|1+rdx⩾12r‖v‖1+rL1+r(Ω)−C1+r1+r‖w‖1+rX0⩾12rˆc1+r(1−ϵ2)1+r2‖u‖1+rX0−C1+r1+rϵ1+r‖u‖1+rX0. | (3.11) |
here ˆc is the embedding constant of L1+r(Ω)↪Vℓ. Therefore for u∈C(M,ϵ), it follows from (3.9)–(3.11) that
∫Ωg(x,u)vdx⩾(12rc2ˆc1+r(1−ϵ2)1+r2−c2C1+r1+rϵ1+r−c1C1+r1+rϵ)‖u‖1+rX0 −c1C1ϵ‖u‖X0−c2|Ω|⩾(c22rˆc1+r(1−ϵ2)1+r2−c2C1+r1+rϵ1+r−c1C1+r1+rϵ−c1C1ϵMr−c2|Ω|M1+r)‖u‖1+rX0=:β‖u‖1+rX0. | (3.12) |
Now we can take M>0 large enough and 0<ϵ<1 small enough so that
β=c22rˆc1+r(1−ϵ2)1+r2−c2C1+r1+rϵ1+r−c1C1+r1+rϵ−c1C1ϵM−c2|Ω|M1+r>0; |
hence,
∫Ωg(x,u)vdx⩾β‖u‖1+rX0⩾β‖v‖1+rX0 for u∈C(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
‖un‖X0→∞ as n→∞. | (3.15) |
Write un=vn+wn, where vn∈Vℓ and wn∈Wℓ. By the variational inequalities (2.8) and (2.9), we have
|⟨Aℓwn,wn⟩|⩾σ‖wn‖2X0, ∀ n∈N, | (3.16) |
where
σ=min{1−λℓλℓ+νℓ, λℓλℓ−1−1}. |
By (3.11), there is N1∈N such that
|⟨J′(un),wn⟩|⩽‖wn‖X0, ∀ n⩾N1. | (3.17) |
By (3.2) and (3.5), we have
⟨J′(un),wn⟩=⟨Aℓwn,wn⟩X0+⟨F′(un),wn⟩. | (3.18) |
By (3.6) and (3.15) we have
‖F′(un)‖=o(‖un‖X0), n→∞. | (3.19) |
It follows that, for any given δ>0 sufficiently small and all n sufficiently large,
σ‖wn‖2X0⩽|⟨Aℓwn,wn⟩X0|⩽|⟨J′(un),wn⟩|+|⟨F′(un),wn⟩|⩽‖wn‖X0+δ‖un‖X0‖wn‖X0. | (3.20) |
Since δ>0 was chosen arbitrarily, from (3.15) and (3.20) we deduce that
‖wn‖X0‖un‖X0→0 as n→∞. | (3.21) |
It follows that there is N2∈N with N2⩾N1 such that
‖un‖X0⩾M and ‖wn‖X0⩽ϵ‖un‖X0 for n⩾N2, | (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)vn‖vn‖X0dx⩾β‖vn‖rX0⩾β(1−ϵ2)r2Mr>0 for n⩾N2. | (3.23) |
On the other hand, by (3.14), we get
limn→∞|∫Ωg(x,un)vn‖vn‖X0dx|=limn→∞|⟨J′(un),vn‖vn‖X0⟩|=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 u∗∈X0, such that
un⇀u∗ weakly in X0 as n→∞. | (3.25) |
By Proposition 2.1, up to a subsequence, it holds that
un→u∗in Lq(RN) ∀ q∈[1,2NN−2s),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∗))(un−u∗)dx|⩽2c1‖un−u∗‖L1(Ω)+c1Cr1+r(‖u∗‖rX0+‖un‖rX0)‖un−u∗‖L1+r(Ω)→0 as n→∞. | (3.27) |
From (3.14) we deduce that
⟨J′(un),un⟩=∫R2N|un(x)−un(y)|2K(|x−y|)dxdy−λℓ∫Ω|un|2dx−∫Ωg(x,un(x))un(x)dx→0 | (3.28) |
as n→∞, and
⟨J′(un),u∗⟩=∫R2N(un(x)−un(y))(u∗(x)−u∗(y))K(|x−y|)dxdy−λℓ∫Ωunu∗dx−∫Ωg(x,un(x))u∗(x)dx→0 | (3.29) |
as n→∞. Now by (3.14) and (3.26)–(3.29), we deduce from
⟨J′(un)−J′(u∗),un−u∗⟩→0 n→∞ |
that
‖un−u∗‖2X0→0, 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, ∀ v∈Vℓ, |
it follows from Lemma 3.1 that J satisfies the angle condition (AC−∞) in [26, Proposition 2.5] at infinity with respect to X0=Vℓ⊕Wℓ. Thus by [26, Proposition 2.5(ⅱ)] we have
Cq(J,∞)≅δq,μℓ+νℓZ, q∈Z, | (3.30) |
where
μℓ=dim⨁λk<λℓkerE(λk), νℓ=dimE(λℓ). |
Therefore J has a critical point u∗ satisfying
Cμℓ+νℓ(J,u∗)≇0. | (3.31) |
The second derivative of J at the trivial solution u=0 can be written as
⟨J″(0)ϕ,ϕ⟩=‖ϕ‖2X0−∫Ω(λℓ+g′t(x,0))ϕ2dx, ϕ∈X0. | (3.32) |
By the condition we see that u=0 is a nondegenerate critical point of J with the Morse index
ˉμ0=dim⨁λk⩽λmkerE(λk). | (3.33) |
Hence
Cq(J,0)≅δq,ˉμ0Z. | (3.34) |
Since λm≠λℓ, we get that μℓ+νℓ≠ˉμ0, and we see from (3.33) and (3.34) that u∗≠0. 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=Vℓ⊕Wℓ. Thus by [26, Proposition 2.5(ⅱ)] we have
Cq(J,∞)≅δq,μℓZ, q∈Z, | (3.35) |
and J has a critical point u∗ satisfying
Cμℓ(J,u∗)≇0. | (3.36) |
Now J″(0) takes the form
⟨J″(0)ϕ,ϕ⟩=‖ϕ‖2X0−λm∫Ωϕ2dx, ϕ∈X0. | (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=dim⨁λk⩽λm−1kerE(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<λℓ−1<λℓ or λℓ−1<λm−1 that μ0+ν0<μℓ or μ0>μℓ, and we see from (3.36) and (3.39) that u∗≠0. 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 (F−0) 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 (F−0) holds. We will prove that J has the local linking structure at 0 as with respect to X0=E−⊕E+, where E−=W−m and E+=Vm⊕W+m. We refer the readers to [28,29] for the concept of the local linking.
1) Take u∈E−=W−m. Since W−m is finite dimensional, there is ρ>0 such that
‖u‖X0⩽ρ ⇒ |u(x)|⩽δ, a.e. x∈Ω. |
Consequently, thanks to (2.8) with λm and (F−0), for any u∈E− with ‖u‖⩽ρ, we get
J(u)⩽12(1−λmλm−1)‖u‖2X0−∫ΩF0(x,u)dx⩽14(1−λmλm−1)‖u‖2X0⩽0. | (3.40) |
2) For u∈E+=Vm⊕W+m, we write u=w+z, where w∈W+m and z∈Vm. Then
J(u)⩾12(1−λmλm+νm)‖w‖2X0−∫ΩF0(x,u)dx. | (3.41) |
Since Vm is finite dimensional, there is ρ>0 such that
‖z‖⩽‖u‖⩽ρ ⇒ |z(x)|<13δ, a.e. x∈Ω. |
Consequently,
|u(x)|>δ ⇒ |w(x)|=|u(x)−z(x)|⩾|u(x)|−|z(x)|>23|u(x)|. |
By (F−0), we have
∫{|u(x)|⩽δ}F0(x,u)dx⩽0. | (3.42) |
By (g1), we get that, for each given σ∈(2,2NN−2s], there is κ=κ(σ,δ)>0 such that
|F0(x,t)|⩽κ|u|σ, x∈Ω, |t|>δ. | (3.43) |
Hence
∫{|u(x)|>δ}F0(x,u)dx⩽κ∫{|u(x)|>δ}|u|σdx⩽κ(3/2)σ∫Ω|w|σdx⩽C(σ,δ)‖w‖σX0. | (3.44) |
Now, by (3.41), (3.42) and (3.44) we get
J(u)⩾12(1−λmλm+νm)‖w‖2X0−∫{|u(x)|⩽δ}F0(x,u)dx−C(κ,σ)‖w‖σX0. | (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, ∀ ‖u‖⩽ρ with w≠0. | (3.46) |
For z∈Vm with ‖z‖⩽ρ, we have by (F−0) that
2F0(x,z(x))=2F(x,z(x))−λmz(x)2⩽0, ∀ a.e. x∈Ω. |
Thus for all z∈Bρ∩Vm,
J(z)=−12∫Ω(2F(x,z(x))−λmz(x)2)dx⩾0. | (3.47) |
To apply [24, Proposition 2.3], we need to show that the above inequality holds strictly for z≠0. Assume, for contradiction, that for any 0<ϵ⩽ρ, there is zϵ∈Vm such that 0<‖zϵ‖<ϵ and J(zϵ)=0. Then, the following holds:
2F(x,zϵ(x))=λmzϵ(x)2, a.e. x∈Ω, |
and then
f(x,zϵ(x))=λmzϵ(x), a.e. x∈Ω. |
Given that zϵ∈Vm, 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<‖u‖⩽ρ, u∈E+. |
Therefore, J has a local linking structure with respect to E=E−⊕E+ 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=Vℓ⊕Wℓ, and then that J has a critical point u∗ satisfying
Cμℓ(J,u∗)≇0. | (3.48) |
By (F−0) and Lemma 3.3, J has a local linking at 0 with respect to X0=E−⊕E+. Thus it follows from [24, Proposition 2.3] that
Cq(J,0)≅δq,μ0Z. | (3.49) |
By λℓ−1<λm−1, we have that λℓ<μ0. It follows from (3.48) and (3.49) that u∗≠0. 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 limt→0f(x,t)t=λm. In this case, we have only J∈C2−0(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 (F−0) holds. |
2) In the case that λℓ=λ1 and (g−2) holds, we have that μℓ=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 q⩾1, 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.
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