Research article

Probabilistic machine learning for breast cancer classification


  • A probabilistic neural network has been implemented to predict the malignancy of breast cancer cells, based on a data set, the features of which are used for the formulation and training of a model for a binary classification problem. The focus is placed on considerations when building the model, in order to achieve not only accuracy but also a safe quantification of the expected uncertainty of the calculated network parameters and the medical prognosis. The source code is included to make the results reproducible, also in accordance with the latest trending in machine learning research, named Papers with Code. The various steps taken for the code development are introduced in detail but also the results are visually displayed and critically analyzed also in the sense of explainable artificial intelligence. In statistical-classification problems, the decision boundary is the region of the problem space in which the classification label of the classifier is ambiguous. Problem aspects and model parameters which influence the decision boundary are a special aspect of practical investigation considered in this work. Classification results issued by technically transparent machine learning software can inspire more confidence, as regards their trustworthiness which is very important, especially in the case of medical prognosis. Furthermore, transparency allows the user to adapt models and learning processes to the specific needs of a problem and has a boosting influence on the development of new methods in relevant machine learning fields (transfer learning).

    Citation: Anastasia-Maria Leventi-Peetz, Kai Weber. Probabilistic machine learning for breast cancer classification[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 624-655. doi: 10.3934/mbe.2023029

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  • A probabilistic neural network has been implemented to predict the malignancy of breast cancer cells, based on a data set, the features of which are used for the formulation and training of a model for a binary classification problem. The focus is placed on considerations when building the model, in order to achieve not only accuracy but also a safe quantification of the expected uncertainty of the calculated network parameters and the medical prognosis. The source code is included to make the results reproducible, also in accordance with the latest trending in machine learning research, named Papers with Code. The various steps taken for the code development are introduced in detail but also the results are visually displayed and critically analyzed also in the sense of explainable artificial intelligence. In statistical-classification problems, the decision boundary is the region of the problem space in which the classification label of the classifier is ambiguous. Problem aspects and model parameters which influence the decision boundary are a special aspect of practical investigation considered in this work. Classification results issued by technically transparent machine learning software can inspire more confidence, as regards their trustworthiness which is very important, especially in the case of medical prognosis. Furthermore, transparency allows the user to adapt models and learning processes to the specific needs of a problem and has a boosting influence on the development of new methods in relevant machine learning fields (transfer learning).



    Let ΩRN be a bounded domain with a C2-boundary Ω. In this paper, we study the following quasilinear Robin problem

    {ΔuΔ(u2)u+a(x)u=λf(x,u)inΩ,un+β(x)u=0onΩ, (1.1)

    where un=un with n(x) being the outward unit normed vector to Ω at its point x, a(x)L(Ω) satisfying ess infxΩa(x)>0, β(x)C0,τ(Ω,R+0) for some τ(0,1) and β(x)0. λ>0 is a real parameter.

    The Robin boundary problems with Laplacian operator have been widely investigated in recent years ([1,2,3,4,5,6]). For example, Papageorgiou et al. in [4] considered the Robin problems driven by negative Laplacian with a superlinear reaction and proved the existence and multiplicity theorems by variational methods. However, the Robin boundary problems with quasilinear Schrödinger operator have not been dealt with yet. Based on the above issues, this paper aims to study the existence and multiplicity of solutions to problem (1.1). In our work, one of the main difficulties is that there is no suitable space on which the energy functional is well defined. To the best of our knowledge, the first existence result for the equation of the following form

    ΔuΔ(u2)u+a(x)u=λf(x,u),xRN (1.2)

    is due to Poppenberg, Schmitt and Wang([7]), and their approach to the problem is the constrained minimization argument. Since then, some ideas and approaches were developed to overcome these difficulties. Liu and Wang, in [8], reduced the quasilinear equation to a semilinear one by the change of variable. In [9], using the same methods in [8], Colin and Jeanjean considered the problem (1.2) on the usual Sobolev space. In this paper, we will use the change of variable to overcome the main difficulties.

    Let f:Ω×RR be a Carathéodory function such that f(x,0)=0 for a.e. xΩ and let

    F(x,t):=t0f(x,ξ)dξ,(x,t)Ω×R.

    We first posit the following assumptions on f.

    (f1)|f(x,t)|c0(1+|t|r1) for a.e. xΩ, where c0>0, 4<r<22;

    (f2)limt+F(x,t)t4=0 and limt+F(x,t)= uniformly for xΩ;

    (f3) There exists ˜uLr(Ω) such that ΩF(x,˜u)dx>0;

    (f4) There is a constant δ>0 such that f(x,t)0 for a.e. xΩ and t(0,δ);

    (f5) For every ρ>0, there exists μρ>0 such that tf(x,t)1+2t2+μρt is nondecreasing on [0,ρ];

    Remark 1.1. The following function satisfies hypotheses (f1)-(f5), for simplicity we drop the x-dependence,

    f(t)={0,if t0,tk11+2tk21,if t(0,1],2tk31tk41,if t>1,

    where 3<k1<k2<2k1<+ and 1<k3<k4<4.

    The theorem below is the first result of our paper.

    Theorem 1.1. If hypotheses (f1)-(f5) hold, then there exists a critical parameter value λ>0 such that for all λ>λ, problem (1.1) has at least two positive solutions v0,v1int(C+) with v0v1 in Ω.

    To study further the multiplicity of solutions for problem (1.1) under the assumption (f1), we give some other conditions on f as follows:

    (f6)lim|t|+F(x,t)t4=+ uniformly with respect to xΩ, and there exist α(max{1,(r4)N4},2), ˆc0>0 such that

    0<ˆc0lim inf|t|+f(x,t)t4F(x,t)|t|2α

    uniformly for xΩ.

    (f7)f(x,t)=f(x,t).

    Remark 1.2. The following function satisfies hypotheses (f1), (f6) and (f7):

    f(t)=t3ln(1+|t|) for all tR.

    Theorem 1.2. If hypotheses (f1), (f6), (f7) hold and λ>1, then problem (1.1) has infinitely many high energy solutions in H1(Ω).

    The paper is organized as follows. In Section 2, we give some preliminary results, which will be used in this paper. We prove Theorem 1.1 and Theorem 1.2 in Section 3 and Section 4 respectively.

    In this paper, the main working spaces are H1(Ω), C1(ˉΩ) and Lq(Ω), 1q+. We denote the norm of Lq(Ω) and H1(Ω) by

    uq=(Ω|u|qdx)1/q,u=(Ω|u|2dx+Ω|u|2dx)1/2.

    The Banach space C1(ˉΩ) is an ordered Banach space with positive (order) cone

    C+:={uC1(ˉΩ):u(x)0,  xˉΩ}.

    This cone has a nonempty interior given by

    int(C+):={uC+:u(x)>0,  xˉΩ}.

    On Ω we will employ the (N1)-dimensional Hausdorff measure σ. By applying this measure we can define the Lebesgue spaces Lq(Ω) (1q+). The Trace Theorem says that there exists a unique continuous linear map γ0:H1(Ω)L2(Ω), known as "trace map", such that

    γ0(u)=u|Ω, uH1(Ω)C(ˉΩ).

    Consequently, the trace map extends the notion of boundary values to any Sobolev function, and

    imγ0=H12,2(Ω),   kerγ0=H10(Ω).

    Moreover, the trace map γ0 is compact from H1(Ω) into Lq(Ω) with q[1,2(N1)N2) if N3 and q[1,+) if N=1, 2. In the sequel, for the sake of notational simplicity, the use of the trace map γ0 will be dropped. The restrictions of all Sobolev functions on the boundary Ω are understood in the sense of traces.

    We know that (1.1) is the Euler-Lagrange equation associated to the energy functional

    I(u)=12(Ω|u|2(1+2u2)dx+Ωa(x)u2dx+Ωβ(x)u2(1+u2)dσ)λΩF(x,u)dx.

    Unfortunately, the functional I could not be well defined in H1(Ω) for N3. To overcome this difficulty, we use the argument developed in [9]. More precisely, we make the change of variable u=g(v), which is defined by

    g(t)=11+2g2(t)on[0,+),g(t)=g(t)on(,0].

    Now we present some important results about the change of variable u=g(v).

    Lemma 2.1. ([10]) The function g and its derivative satisfy the following properties:

    (ⅰ) g is uniquely defined, C2 and invertible;

    (ⅱ) |g(t)|1 for all tR;

    (ⅲ) |g(t)|t for all tR;

    (ⅳ) g(t)t1, as t0;

    (ⅴ) |g(t)|21/4|t|1/2 for all tR;

    (ⅵ) g(t)/2<tg(t)<g(t) for all t>0, and the reverse inequalities hold for t<0;

    (ⅶ) g(t)ta1>0, as t+;

    (ⅷ) |g(t)g(t)|1/2 for all tR;

    (ⅸ) There exists a positive constant C1 such that

    |g(t)|{C1|t|,|t|1,C1|t|1/2,|t|1.

    Therefore, after the change of variable, the functional I(u) can be rewritten in the following way

    J(v)=12Ω|v|2dx+12Ωa(x)g2(v)dx+12Ωβ(x)g2(v)(1+g2(v))dσλΩF(x,g(v))dx. (2.1)

    From Lemma 2.1, by a standard argument, we see that J is well defined in H1(Ω) and JC1. In addition,

    J(v)φ=Ωvφdx+Ωa(x)g(v)g(v)φdx+Ωβ(x)g(v)g(v)(1+2g2(v))φdσλΩf(x,g(v))g(v)φdx (2.2)

    for all v,φH1(Ω).

    It is easy to see that the critical points of J correspond exactly to the weak solutions of the semilinear problem

    {Δv+a(x)g(v)g(v)=λf(x,g(v))g(v)inΩ,vn+β(x)g(v)g(v)(1+2g2(v))=0onΩ. (2.3)

    Hence, to prove our main results, we shall look for solutions of problem (2.3), i.e., the critical points of the functional J.

    Let X be a Banach space and X be its topological dual. By , we denote the duality brackets for the pair (X,X). We now give the definitions of the (PS)c condition and Cerami condition in X as follows.

    Definition 2.2. Let ψC1(X,R) and cR. We say that ψ satisfies the (PS)c condition, if every sequence {un}X such that

    ψ(un)c, ψ(un)0 in X as n

    admits a strongly convergent subsequence.

    Definition 2.3. Let ψC1(X,R) and cR. We say that ψ satisfies the Cerami condition, if every sequence {un}X such that

    ψ(un)c, (1+unX)ψ(un)0 in X as n

    admits a strongly convergent subsequence.

    The following two lemmas are known as Mountain Pass Theorem and Fountain Theorem.

    Lemma 2.4. (Mountain Pass Theorem, [11]) If ψC1(X,R) satisfies the (PS)c-condition, there are u0, u1X with u1u0X>ϱ>0,

    max{ψ(u0), ψ(u1)}<inf{ψ(u):uu0X=ϱ}=mϱ

    and c=infγΓmax0τ1ψ(Γ(τ)) where Γ={γC([0,1], X):γ(0)=u0, γ(1)=u1}, then cmϱ and c is a critical value of ψ.

    If X is a reflexive and separable Banach space, then there are ejX and ejX such that

    X=¯span{ej|j=1, 2, }, X=¯span{ej|j=1, 2, },
    ei,ej={1,i=j,0,ij.

    For convenience, we write

    Xj:=span{ej}, Yk:=kj=0Xj, Zk:=¯j=kXj. (2.4)

    Lemma 2.5. (Fountain Theorem, [12]) Let ψC1(X,R) be an even functional. If, for every kN, there exists ρk>γk>0 such that

    (A1)ak:=supuYk,uX=ρkψ(u)0,

    (A2)bk:=infuZk,uX=γkψ(u) as k,

    (A3)ψ satisfies the Cerami condition for every c>0.

    Then ψ has an unbounded sequence of critical values.

    In this section we prove the existence of positive solutions of problem (1.1). For simplicity, we may take f(x,t)=0 for a.e. xΩ, all t0.

    Proposition 1. If (f1)–(f3) and (f5) hold, then problem (2.3) admits a solution v0int(C+).

    Proof. We consider the C1-functional J+:H1(Ω)R defined by

    J+(v)=12(v22+v22+Ωa(x)g2(v)dx+Ωβ(x)g2(v)(1+g2(v))dσ)λΩF(x,g(v+))dx. (3.1)

    We claim that J+ is coercive. Arguing by contradiction, assume that we can find a sequence {vn}n1H1(Ω) and a constant M such that

    vn+ as n and J+(vn)M for all n1. (3.2)

    Then from (3.2) we have

    M12(vn22+vn22+Ωa(x)g2(vn)dx+Ωβ(x)g2(vn)(1+g2(vn))dσ)λΩF(x,g(v+n))dx12(vn22+vn22)λΩF(x,g(v+n))dx (3.3)

    for all n1. Hypotheses (f1) and (ⅴ) of Lemma 2.1 imply that there exist c2, c3>0 such that

    |F(x,g(v+n))|c2(1+|g(v+n)|r)c3(1+|v+n|r2).

    Combining this with (3.2), (3.3), we get

    v+n+ as n.

    Let wn=v+nv+n, n1. Then wn=1 for all n1, and so we may assume that

    wnw in H1(Ω), wnw in Lθ(Ω) and in Lθ(Ω) for each θ(1,2). (3.4)

    From (3.3) we have

    12wn22λΩF(x,g(v+n))v+n2dxMv+n2for all n1. (3.5)

    Invoking (f1), (f2) and (ⅴ) of Lemma 2.1, we can find cϵ>0 such that

    |F(x,g(t))|ϵ2t2+cϵfor a.e. xΩ, all t0,

    which yields, for n1 large enough,

    Ω|F(x,g(v+n))|v+n2dxϵ2+cϵ|Ω|Nv+n2ϵ. (3.6)

    Passing to the limsup as n in (3.5), and using (3.6), we have

    lim supn+wn220. (3.7)

    In addition, from (3.4) we have w22lim infn+wn22. Combining this with (3.7), we obtain that wnw=0 in L2(Ω). Hence,

    wnw in H1(Ω),

    and so w=1, w0. Since w=0, we have w=1/|Ω|N, and v+n(x)+ for a.e. xΩ.

    On the other hand, combining with (f1), (f2), (3.3), (ⅶ) of Lemma 2.1 and Fatou's lemma, we get

    MJ+(vn)λΩF(x,g(v+n))dx+ as n,

    which is impossible. Thus, J+ is coercive.

    Furthermore, from Lemma 2.1, Sobolev embedding theorem and trace theorem, we see that J+ is sequentially weak lower semicontinuous. By Weierstrass-Tonelli theorem we can find v0H1(Ω) such that

    J+(v0)=inf{J+(v):vH1(Ω)}. (3.8)

    Consider the integral functional JF:Lr(Ω)R defined by

    JF(v)=ΩF(x,g(v))dx.

    From hypothesis (f1), (ⅰ) of Lemma 2.1 and the dominated convergence theorem, we see that JF is continuous. By hypothesis (f3) and (ⅸ) of Lemma 2.1, there exists ˜vLr(Ω) such that

    JF(˜v)>0.

    Exploiting the density of H1(Ω) in Lr(Ω), we can find ˉvH1(Ω) such that

    JF(ˉv)>0. (3.9)

    Therefore, from (3.1) and (3.9), there exists λ>0 such that

    J+(v0)<J+(ˉv)<0=J+(0)

    for all λ>λ. This ensure that v00. From (3.8), we have J+(v0)=0, that is,

    Ωv0φdxΩv0φdx+Ωa(x)g(v0)g(v0)φdx
    +Ωβ(x)g(v0)g(v0)(1+2g2(v0))φdσ=λΩf(x,g(v+0))g(v+0)φdx (3.10)

    for all φH1(Ω). In (3.10) choosing φ=v0, we obtain

    v022+v0220,

    which implies v00 a.e. in Ω. Hence,

    Ωv0φdx+Ωa(x)g(v0)g(v0)φdx+Ωβ(x)g(v0)g(v0)(1+2g2(v0))φdσ=λΩf(x,g(v0))g(v0)φdx (3.11)

    for all φH1(Ω). That is to say v0 is a weak solution of problem (2.3). By Theorem 4.1 in [13], we know that v0L(Ω). Furthermore, applying Theorem 2 in [14], we have v0C+{0}. Taking ρ=g(v0), from (f5), there exists μρ>0 such that

    f(x,g(v0))g(v0)+μρg(v0)0for a. e. xΩ. (3.12)

    From (2.3), (3.12) and (ⅲ) of Lemma 2.1 we obtain

    Δv0+(λμρ+a(x))v00for a. e. xΩ.

    Therefore, v0int(C+) by the strong maximum principle(see [15]).

    To look for another positive solution to problem (2.3), we consider the functional ˆJ+C1(H1(Ω),R) defined by

    ˆJ+(v)=12(v22+v22+(vv0)+22+ΩˆK(x,v)dσ)ΩˆF(x,v)dx, (3.13)

    where v0 is given in Proposition 1, ˆF(x,t)=t0ˆf(x,s)ds, ˆK(x,t)=t0ˆk(x,s)ds, and

    ˆf(x,s)={0if s0,λf(x,g(s))g(s)a(x)g(s)g(s)if 0<s<v0(x),λf(x,g(v0))g(v0)a(x)g(v0)g(v0)if sv0(x), (3.14)

    for all (x,s)Ω×R.

    ˆk(x,s)={0if s0,β(x)g(s)g(s)(1+2g2(s))if 0<s<v0(x),β(x)g(v0)g(v0)(1+2g2(v0))if sv0(x), (3.15)

    for all (x,s)Ω×R. It is easy to see that ˆf(x,s) and ˆk(x,s) are Carathéodory functions.

    Proposition 2. If (f1), (f5) hold and v is a critical point of ˆJ+(v), then v is a solution of problem (2.3), and v[0,v0]int(C+).

    Proof. The assumption yields the following equality:

    ΩvφdxΩvφdx+Ω(vv0)+φdx+Ωˆk(x,v)φdσ=Ωˆf(x,v)φdx (3.16)

    for all φH1(Ω). Taking φ=v in (3.16), we obtain (arguing as in Proposition 1) v0 a.e. in Ω.

    On the one hand, letting φ=(vv0)+ in (3.16) again, we have

    Ωv(vv0)+dx+(vv0)+22+Ωa(x)g(v0)g(v0)(vv0)+dx+Ωβ(x)g(v0)g(v0)(1+2g2(v0))(vv0)+dσ=λΩf(x,g(v0))g(v0)(vv0)+dx. (3.17)

    On the other hand, the fact that v0>0 is a critical point of J+ yields

    Ωv0(vv0)+dx+Ωa(x)g(v0)g(v0)(vv0)+dx+Ωβ(x)g(v0)g(v0)(1+2g2(v0))(vv0)+dσ=λΩf(x,g(v0))g(v0)(vv0)+dx. (3.18)

    Then from (3.17) and (3.18), we obtain

    (vv0)+22+(vv0)+22=0,

    and (vv0)+=0. Therefore v[0,v0] a.e. in Ω. Now, (3.16) becomes

    Ωvφdx+Ωa(x)g(v)g(v)φdx+Ωβ(x)g(v)g(v)(1+2g2(v))φdσ=λΩf(x,g(v))g(v)φdx

    for all φH1(Ω). So v is a weak solution of problem (2.3). Reasoning as in the proof of Proposition 1, we have vint(C+).

    Lemma 3.1. If hypotheses (f1)–(f5) hold, then ˆJ+ satisfies the (PS)c condition.

    Proof. We first check that ˆJ+ is coercive. Since v0C1¯(Ω), using (f1), (ⅲ) of Lemma 2.1 and (3.14), we can find a constant c4>0 such that

    |ΩˆF(x,v)dx|λc4v2for all vH1(Ω). (3.19)

    Because of (3.15) we have

    ΩˆK(x,v)dσ0for all vL2(Ω). (3.20)

    Moreover, for every vH1(Ω) we see that

    (vv0)+22=(v+v0)+22=vv022{v+v0}(v0v)2dx12v+2212v022{v+v0}v20dx12v+222v022. (3.21)

    From (3.13), (3.19), (3.20) and (3.21) we derive

    ˆJ+(v)14v2v022λc4v2,  vH1(Ω).

    It is easy to see that ˆJ+ is coercive.

    Now let {vn}n1 be a sequence such that

    ˆJ+(vn)c, ˆJ+(vn)0 in H1(Ω) as n. (3.22)

    The coercivity of ˆJ+ and (3.22) imply that {vn}n1 is bounded in H1(Ω). Hence, there is vH1(Ω) such that, along a relabeled subsequence {vn}n1,

    vnv in H1(Ω), vnv in Lθ(Ω) and in Lθ(Ω) for each θ(1,2). (3.23)

    The second part of (3.22) yields

    ΩvnφdxΩvnφdx+Ω(vnv0)+φdx+Ωˆk(x,vn)φdσΩˆf(x,vn)φdx0 (3.24)

    as n, for all φH1(Ω). In (3.24) choosing φ=vnv, using (f1), (3.23) and Lemma 2.1, we find

    limnΩvn(vnv)dx=0.

    Therefore vn2v2 as n. Recalling that the Hilbert space H1(Ω) is locally uniformly convex, we obtain vnv in H1(Ω). This shows that ˆJ+ satisfies the (PS)c condition.

    In the next lemma, we state a consequence which depends on the regularity results in [13] and Theorem 2 in [14]. The proof is similar to Proposition 3 in [2].

    Lemma 3.2. If (f1) holds and vH1(Ω) is a local C1(ˉΩ)-minimizer of ˆJ+, then vC1,τ(ˉΩ) and it is a local H1(Ω)-minimizer of ˆJ+.

    Proposition 3. If (f1)-(f5) hold, then ˆJ+ admits a critical point v1H1(Ω) different from 0 and v0.

    Proof. Let vC1(ˉΩ) with 0<vC1(ˉΩ)δ. From (f4) and (ⅳ) of Lemma 2.1, we get ˆF(x,g(v))0 for a.e. xΩ which implies

    ˆJ+(v)λΩˆF(x,g(v))dx>0.

    This shows that 0 is a local C1(ˉΩ)-minimizer of ˆJ+. Applying Lemma 3.2, we derive that 0 is a local H1(Ω)-minimizer of ˆJ+.

    If 0 is not a strict local minimizer of ˆJ+, the result is obvious because any neighborhood of 0 in H1(Ω) contains another critical point of ˆJ+.

    We next only need to discuss that 0 is a strict local minimizer of ˆJ+. In such a condition, we can find a sufficiently small ϱ(0,1) such that

    ˆJ+(0)<inf{ˆJ+(v):v=ϱ}=ˆmϱ. (3.25)

    Note that

    ˆJ+(v0)=J+(v0)<J+(0)=ˆJ+(0)=0.

    This fact, together with (3.25) and Lemma 3.1 permit the use of the Mountain Pass Theorem, which yields a critical point v1 of ˆJ+ different from 0 and v0.

    Theorem 1.1 follows immediately from Proposition 1, Proposition 2 and Proposition 3.

    Lemma 4.1. If (f1) and (f6) hold, then, for any c>0, the functional J satisfies the Cerami condition.

    Proof. Let {vn}n1H1(Ω) be a Cerami sequence of J, that is,

    J(vn)c, (1+vn)J(vn)0 in H1(Ω) as n. (4.1)

    First we prove the boundedness of {vn}n1. By (4.1) we have

    c+112(Ω|vn|2dx+Ωa(x)g2(vn)dx+Ωβ(x)g2(vn)(1+g2(vn))dσ)λΩF(x,g(vn))dx (4.2)

    and

    ϵnφ1+vnJ(vn)φ=Ωvnφdx+Ωa(x)g(vn)g(vn)φdx+Ωβ(x)g(vn)g(vn)(1+2g2(vn))φdσλΩf(x,g(vn))g(vn)φdx (4.3)

    for all φH1(Ω) with ϵn1. Choosing φ=φn=g(vn)g(vn), we obtain

    |φn|=|vn|(1+2(g(vn)g(vn))2).

    Because of Lemma 2.1(ⅵ, ⅷ), φnH1(Ω). Therefore,

    2ϵnJ(vn)φn=Ω|vn|2((1+2(g(vn)g(vn))2)dxΩa(x)g2(vn)dxΩβ(x)g2(vn)(1+2g2(vn))dσ+λΩf(x,g(vn))g(vn)dx. (4.4)

    Now using (f6) and (ⅷ) of Lemma 2.1, it follows from (4.2) and (4.4) that

    c+1+ϵn2Ω(1214(1+2(g(vn)g(vn))2)|vn|2dx+14Ωa(x)g2(vn)dx+14Ωβ(x)g2(vn)dσ+λ4Ω(f(x,g(vn))g(vn)4F(x,g(vn)))dxλ4Ω(f(x,g(vn))g(vn)4F(x,g(vn)))dx. (4.5)

    On the other hand, from (f1) and (f6) we can find ˜c0(0,ˆc0) and c5>0 such that

    f(x,t)4F(x,t)˜c0|t|2αc5a.e. xΩ. (4.6)

    By (4.5) and (4.6), we conclude that {g(vn)}n1 is bounded in L2α(Ω). Furthermore, due to (ix) of Lemma 2.1,

    {vn}n1 is bounded in Lα(Ω). (4.7)

    Suppose that N2. By (f6), without loss of generality, we may assume that α<r2<2. So, we can find z(0,1) such that

    2r=1zα+z2. (4.8)

    Invoking the interpolation inequality, Sobolev embedding theorem and (4.7) we have

    vnr2vn1zαvnz2c6vnz (4.9)

    for some c6>0. Taking φ=φn=g(vn)g(vn) in (4.3), we have

    Ω|vn|2((1+2(g(vn)g(vn))2)dx+Ωa(x)g2(vn)dx+Ωβ(x)g2(vn)(1+2g2(vn))dσ2ϵn+λΩf(x,g(vn))g(vn)dx. (4.10)

    By use of (f1) and (ⅴ) of Lemma 2.1, we can obtain that

    f(x,g(vn))g(vn)c4(1+|g(vn)|r)c5(1+|vn|r2). (4.11)

    From (f1), we assume that r is close to 22, hence α2. Then {vn}n1 is bounded in L2(Ω). Combining this with (4.9), (4.10) and (4.11), we have

    vn2c6(1+vnzr2). (4.12)

    Hypothesis (f6) and (4.8) lead to zr<4, and consequently, {vn}n1 is bounded in H1(Ω).

    If N=2, then 2=+ and the Sobolev embedding theorem says that H1(Ω)Lη(Ω) for all η[1,+). Let η>r2>α and z(0,1) such that

    2r=1zα+zη, (4.13)

    or zr=η(r2α)ηα. Note that

    limη+η(r2α)ηα=r2α and r2α<4.

    Repeating the previous proof method, for η>1 large enough, we again obtain that

    {vn}n1 is bounded in H1(Ω).

    Next, we show that {vn}n1 is strongly convergent in H1(Ω). Since {vn}n1 is bounded, up to a subsequence(which we still denote by {vn}n1), we assume that there exists vH1(Ω) such that

    vnv in H1(Ω), vnv in Lθ(Ω) and Lθ(Ω) for each θ(1,2). (4.14)

    Choosing φ=vnvH1(Ω) in (4.3) and combining (f1) with Lemma 2.1(ⅱ, ⅲ, ⅸ), we obtain limnΩvn(vnv)dx=0. Hence vn2v2 as n. Recalling that the Hilbert space H1(Ω) is locally uniformly convex, we get vnv in H1(Ω).

    Lemma 4.2. ([16]) Let X=H1(Ω) and define Yk, Zk as in (2.4). If 1q<2, then

    dk:=supuZk,u=1uq0 as k.

    Proof of Theorem 1.2. Since g(t) and f(x,t) are odd respect to t, functional JC1(H1(Ω),R) is even obviously. Further, from Lemma 4.1, we know that J satisfies the Cerami condition. So we need only to verify J satisfying the conditions (A1) and (A2) in Lemma 2.5.

    Since Yk is finite-dimensional and all norms are equivalent on Yk, for vYk, we have

    v2C2v22.

    From the assumption (f6) and (ⅸ) of Lemma 2.1, there exists R>0 such that

    F(x,g(t))2C2|t|2C3for all xΩ, |t|R.

    For vYk, λ>1, we have

    J(v)v2ΩF(x,g(v))dxC2v22+C3|Ω|Nv2+C3|Ω|N.

    So (A1) is satisfied for sufficiently large v.

    From (f1) and (ⅴ) of Lemma 2.1, there exists constant C4>0 such that

    F(x,g(t))C4(1+|t|r2), tR. (4.15)

    Let us define

    dk:=supvZk,v=1vr2.

    For vZk, using (4.15), we get

    J(v)12v22+12Ωa(x)g2(v)dx+12Ωβ(x)g2(v)(1+g2(v))dσC4λvr2r2C4λ|Ω|N12v2C4λvr2r212v22C4λ|Ω|N12v2C5dr2kvr2C6|Ω|N.

    Choosing γk=(12C5rdr2k)24r, if v=γk, we obtain

    J(v)(122r)(12C5rdr2k)44rC6|Ω|N.

    By Lemma 4.2, dk0 and γk as k, the condition (A2) is verified. In summary, all conditions of Fountain Theorem are satisfied. Thus J has an unbounded sequence of critical values. Theorem 1.2 is proved.

    In this paper, we have studied a class of quasilinear Schrödinger equation in a bounded domain with Robin boundary. By giving different conditions on the reaction, we obtained two existence results of solutions to the equation. The Mountain Pass Theorem and Fountain Theorem were also employed in this study.

    The authors wish to thank the referees and the editor for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11171220).

    All authors declare no conflicts of interest in this paper.



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