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On general Kirchhoff type equations with steep potential well and critical growth in R2

  • In this paper, we study the following Kirchhoff-type equation:M(R2(|u|2+u2)dx)(Δu+u)+μV(x)u=K(x)f(u)  in  R2,where MC(R+,R+) is a general function, V0 and its zero set may have several disjoint connected components, μ>0 is a parameter, K is permitted to be unbounded above, and f has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhoff-type equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case M1.

    Citation: Zhenluo Lou, Jian Zhang. On general Kirchhoff type equations with steep potential well and critical growth in R2[J]. AIMS Mathematics, 2024, 9(8): 21433-21454. doi: 10.3934/math.20241041

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  • In this paper, we study the following Kirchhoff-type equation:M(R2(|u|2+u2)dx)(Δu+u)+μV(x)u=K(x)f(u)  in  R2,where MC(R+,R+) is a general function, V0 and its zero set may have several disjoint connected components, μ>0 is a parameter, K is permitted to be unbounded above, and f has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhoff-type equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case M1.


    The Kirchhoff-type problem appears as a model of several physical phenomena. For example, it is related to the stationary analog of the equation:

    ρ2ut2(P0h+E2LL0|ux|2dx)2ux2=0, (1.1)

    where u is the lateral displacement at x and t, E is the Young modulus, ρ is the mass density, h is the cross-section area, L is the length, and P0 is the initial axial tension. For more background, see [1,20] and the references therein. In this paper, we study the following Kirchhoff-type equation with steep potential well and exponential critical nonlinearity:

    M(R2(|u|2+u2)dx)(Δu+u)+μV(x)u=K(x)f(u)  in  R2, (1.2)

    where MC(R+,R+), VC(R2,R+) with Ω=int(V1(0)) having k connected components, μ>0 is a parameter. Because of the presence of the nonlocal term M(R2(|u|2+u2)dx), Eq (1.2) is no longer a pointwise identity, which causes additional mathematical difficulties. The motivation of the present paper arises from results for Schrödinger equations with steep potential well. In [6], Bartsch and Wang studied the following equation with steep potential well:

    Δu+(1+μV(x))u=up1  in  RN, (1.3)

    where N3 and 2<p<2=2NN2. Under appropriate conditions on V, the authors obtained the existence of positive ground state solutions for large μ and the concentration behavior of solutions as μ+. If p is close to 21, the authors also obtained multiple positive solutions. In [13], Ding and Tanaka constructed multi-bump positive solutions to Schrödinger equations with steep potential well. In [23], Sato and Tanaka obtained multiple positive and sign-changing solutions. For the critical case, Clapp and Ding [11] considered the following equation with steep potential well:

    Δu+μV(x)u=λu+u21  in  RN. (1.4)

    When N4, λ>0 is small and μ>0 is large, the authors obtained the existence and multiplicity of positive solutions. In [17,18], Guo and Tang constructed multi-bump solutions of (1.4) in the case that the potential is definite and indefinite. For other related results, see [4,5,12,24,25,26] and the references therein.

    There are relatively few results about Kirchhoff-type equations with steep potential well. In [19], Jia studied the ground-state solutions of the following equation with sign-changing potential well:

    (a+bR3|u|2dx)Δu+λV(x)u=|u|p2u  in  R3, (1.5)

    where 3<p<6. When V0 and 2<p<6, Zhang and Du [27] used the truncation technique to obtain the existence of solutions of (1.5). For the critical case, we [29] obtained the existence, multiplicity and concentration behavior of solutions to the following equation:

    (a+bR3|u|2dx)Δu+μV(x)u=λf(u)+(u+)5  in  R3. (1.6)

    To the best of our knowledge, there are no results about the existence and concentration behavior of Kirchhoff-type equations with steep potential wells and exponential critical growth nonlinearity in dimension two, especially when the zero set of the steep potential well admits more than one isolated connected component. This is the main motivation of the present paper. Here we say the nonlinearity f has exponential subcritical growth if for any α>0,

    limu+f(u)eαu2=0 (1.7)

    and the nonlinearity f has exponential critical growth if there exists α0>0 such that

    limu+f(u)eαu2={0,α>α0,+,α<α0. (1.8)

    In this paper, we study (1.2) and prove the existence of solutions trapped on one connected component of the potential well.

    To study the existence and concentration behavior of solutions, the main difficulty lies in the exponential critical growth of nonlinearity. The Trudinger–Moser inequality plays an important role in dealing with critical nonlinearity. When using this inequality, it is crucial to control the uniform H1-norm of the sequence. Compared with the classical Schrödinger equation, the nonlocal term of the Kirchhoff type equation prevents us from using the upper bound on energy and the Ambrosetti–Rabinowitz type condition to deduce the desired H1 norm estimate. If we use the Pohozaev identity, we must impose additional restrictions on V and K. In [3,22], the authors studied nonlinear scalar field equations in dimension two. We notice that the compactness lemma of Strauss in [7] plays an important role and cannot be used in a non-radial setting. In [10,16], the authors studied Kirchhoff-type equations with exponential critical growth in a bounded domain. To deal with the critical nonlinearity, a compactness lemma (Lemma 2.1 in [14]) was used. However, this lemma cannot be applied to study a non-radial problem in the whole space. In this paper, we give a compactness lemma restricted to a bounded domain (Lemma 2.5 in Section 2), which is motivated by Lemma 2.1 in [14]. Because this lemma cannot be applied to deal with the non-radial problem in the whole space and the coefficient of the nonlinearity may be unbounded above, we study the problem by penalizing the nonlinearity.

    When N=2, to deal with the exponential critical nonlinearity, we need to estimate an upper bound on the energy. In [3], the authors used the following condition:

    (f) There exist λ>0 and q>2 such that

    f(u)λuq1,   u0.

    When λ>0 is large, the upper bound on the energy can be controlled. In [14], the authors considered the following Dirichlet problem:

    Δu=f(x,u)  in  Ω,  u=0 on Ω,

    and introduced the following more natural condition:

    (f) There exists β>43α0d2 such that

    limu+f(x,u)ueα0u2β,

    where d is the radius of the largest open ball in Ω.

    By using the Moser sequence of functions, the authors deduced the desired upper bound. Related results can be found in [22,28] for nonlinear scalar field equations and in [10,16,28] for Kirchhoff type equations. Motivated by the above results, we use a direct argument to get the desired upper bound on the energy.

    Now we state our results. We assume the following conditions:

    (M1) MC(R+,R+), infR+M:=M0>0, and M(t) is strictly increasing for tR+.

    (M2) There exist θ, ε0>0 such that M(t)ε0tθ is decreasing for t>0.

    (M3) There exists ε0>0 such that ˆM(t)1θ+1M(t)tε0t is increasing for tR+, where ˆM(t)=t0M(s)ds.

    (V1) VC(R2,R+).

    (V2) Ω=int(V1(0)) is non-empty with smooth boundary and ˉΩ=V1(0).

    (V3) Ω consists of k connected components: Ω=ki=1Ωi and ¯Ωi¯Ωj= for all ij.

    (V4) There exists V0>0 such that |{xR2:V(x)V0}|<.

    (K1) KC(R2,R+) and k0:=infR2K>0.

    (K2) There exist k1, α>0 such that K(x)k1eα|x| for xR2.

    (f1) fC(R+,R+) and there exists l>1 such that limu0+f(u)ul<+.

    (f2) There exists α0>0 such that

    limu+f(u)eαu21={0,α>α0,+,α<α0.

    (f3) There exists β>0 such that

    βlimu+f(u)ueα0u2<+.

    (f4) There exists σ>2(θ+1) such that f(u)uσ1 is increasing for uR+{0}.

    (f5) There exist u0, L0>0 such that F(u)L0f(u) for uu0, where F(u)=u0f(s)ds.

    Theorem 1.1. Assume that (M1)(M3), (V1)(V4), (K1)(K2) and (f1)(f5) hold. Let i0{1,2,,k}. If β>2M(4πα0)k0r2α0er221, where r is the radius of an open ball contained in Ωi0, then there exists μ0>0 such that for μ>μ0, Eq (1.2) has a positive solution uμ. Moreover, there exist r0, c1, c2>0 independent of μ>0 large such that Ωdi0Br0(0) and

    uμ(x)c2ec1μ(|x|r0),   |x|r0. (1.9)

    Besides, for any sequence μn+, there exists u0H10(Ωi0) such that uμnu0 in H1(R2) as n, where u0H10(Ωi0) is a positive solution to the limiting problem:

    M(Ωi0(|u|2+u2)dx)(Δu+u)=K(x)f(u)  in  Ωi0. (1.10)

    Remark 1.1. If limu+f(u)ueα0u2=A(0,+), then there exists R>0 such that

    A2u1eα0u2f(u)3A2u1eα0u2,   uR.

    Moreover,

    limu+F(u)f(u)limu+R0f(s)ds+3A2uRs1eα0s2dsA2u1eα0u2=0,

    from which we get f satisfies (f5). If A=, one can prove it by the LHospital rule and the definition of ϵN.

    Remark 1.2. Let f1(u)=β(α0u21)eα0u2α0u3, where u>0. Then there exists u1>0 such that f1(u1)=uσ11. Define f(u)=uσ1 for u[0,u1] and f(u)=f1(u) for u>u1. Obviously, f satisfies (f1)(f3). We note that

    (f1(u)uσ1)=βeα0u2α0u3+σ[2α20u4(σ+2)α0u2+σ+2].

    If σ6, then f1(u)uσ1 is increasing for uu1. Moreover, f satisfies (f4). By Remark 1.1, we get f satisfies (f5).

    The outline of this paper is as follows: In Section 2, we study the truncated problem; in Section 3, we turn to the original problem and prove Theorem 1.1.

    We give some definitions. Denote C as universal positive constant (possibly different). Define us:=(R2|u(x)|sdx)1s, where s[1,). Define H1(R2) the Hilbert space with the norm uH1:=(u22+u22)12. It is well known that the embedding H1(R2)Lt(R2) is continuous for all t2. Let μ>0. Define

    Xμ:={uH1(R2):R2V(x)u2dx<}

    the Hilbert space equipped with the norm

    uμ:=(u22+R2(1+μV(x))u2dx)12.

    Obviously, the embedding XμH1(R2) is continuous. We give the following Trudinger–Moser inequality:

    Lemma 2.1. ([15,21,22]) If uH1(R2) and α>0, then

    R2(eαu21)dx<.

    Moreover, for any fixed τ>0, there exists a constant C>0 such that

    supuH1(R2):u22+τu221R2(e4πu21)dxC.

    Since we look for positive solutions, we assume that f(u)=0 for u0. For any d>0, define Ωd:={xR2:dist(x,Ω)<d}. By (V3), we can choose d>0 small such that Ω2diΩ2dj= for all ij. Let i0{1,2,,k}. Define

    χ(x)={1,  xΩdi0,0,  xR2Ωdi0.

    By (V4), we know that Ωdi0 is bounded. Let τ(0,1). For any xR2Ωdi0, define

    ˆf(x,u)=min{K(x)f(u),κu+},

    where u+=max{u,0} and κ(0,min{ε0,(θ+1)ε0θ,M0(1τ)}). Define

    g(x,u)=χ(x)K(x)f(u)+(1χ(x))ˆf(x,u). (2.1)

    Then

    G(x,u)=u0g(x,s)ds=χ(x)K(x)F(u)+(1χ(x))ˆF(x,u),

    where ˆF(x,u)=u0ˆf(x,s)ds. By (f4) and the structure of ˆf, we derive that for all (x,u)R2×R,

    K(x)f(u)uσK(x)F(u)0,  ˆf(x,u)u2ˆF(x,u)0. (2.2)

    Instead of studying (1.2), we consider the following truncated problem:

    M(u2H1)(Δu+u)+μV(x)u=g(x,u)  in  R2. (2.3)

    The functional associated with (2.3) is

    ˆIμ(u)=12ˆM(u2H1)+μ2R2V(x)u2dxR2G(x,u)dx,   uXμ. (2.4)

    Obviously, ˆIμC1(Xμ,R), and the critical points of ˆIμ are weak solutions of (2.3).

    Lemma 2.2. Let l(t)=ˆIμ(tu), where t0 and uXμ with |suppuΩdi0|>0. Then there exists a unique t0>0 such that l(t0)=0, l(t)>0 for t(0,t0), and l(t)<0 for t>t0.

    Proof. Obviously, l(0)=0. Let α>α0 and q>2. By (K1) and (f1)-(f2), for any ε>0, there exists Cε>0 such that

    |g(x,u)|(ε+κ)|u|+Cε|u|q1(eαu21),   (x,u)R2×R. (2.5)

    Then

    |G(x,u)|ε+κ2|u|2+Cεq|u|q(eαu21),   (x,u)R2×R. (2.6)

    By (2.6) and Lemma 2.1, we can choose ρ>0 small such that for uμρ,

    |R2G(x,u)dx|ε+κ2u22+Cεquq2q[R2(e2αu21)dx]12ε+κ2u22+Cuq2q. (2.7)

    By (M1), we get ˆM(s)M0s for sR+. Together with (2.7), the choice of κ and the Sobolev embedding theorem, we derive that l(t)>0 for t>0 small. Let s0>0. By (M1)-(M2), there exists C1>0 such that

    M(s)C1+M(s0)sθ0sθ,  sR+. (2.8)

    Let p>2θ+1. By (f1)-(f2), there exist c1, c2>0 such that

    f(u)c1upc2u,   uR. (2.9)

    By (2.8)-(2.9), we get l(t)<0 for t>0 large. Thus, maxt0l(t) is attained at t0>0 and l(t0)=0. Let

    y(t)=[ε0u22+μR2V(x)u2dxR2Ωdi0ˆf(x,tu)utdx]+[(M(t2u2H1)ε0)u22+M(t2u2H1)u22t2θΩdi0K(x)f(tu)ut2θ+1dx].

    Then y(t0)=0. Moreover, from the structure of g, we derive that for t>0,

    ε0u22+μR2V(x)u2dxR2Ωdi0ˆf(x,tu)utdx>0,(M(t2u2H1)ε0)u22+M(t2u2H1)u22t2θΩdi0K(x)f(tu)ut2θ+1dx<0.

    By (M2), we know (M(t2u2H1)ε0)u22+M(t2u2H1)u22t2θ is decreasing for t>0. By (f4), we know R2Ωdi0ˆf(x,tu)utdx is increasing for t>0 and Ωdi0K(x)f(tu)ut2θ+1dx is strictly increasing for t>0. Then y(t)>0 for t<t0 and y(t)<0 for t>t0. Moreover, l(t)>0 for t(0,t0) and l(t)<0 for t>t0.

    We consider the Moser sequence of functions

    ˉωn(x)=12π{(logn)12,  0|x|1n,log1|x|(logn)12,  1n|x|1,0,           |x|1.

    It is well known that ˉωn22=1 and ˉωn22=14logn+o(1logn). Choose x0Ωi0 and r>0 such that Br(x0)Ωi0, where r is the radius of an open ball contained in Ωi0. Define the functions ωn(x)=ˉωn(xx0r). Then, ωn22=1. Define the functional I0 as follows:

    I0(u)=12ˆM(Ωi0|u|2+u2dx)Ωi0K(x)F(u)dx,   uH10(Ωi0).

    Lemma 2.3. maxt0ˆIμ(tωn)=maxt0I0(tωn)<12ˆM(4πα0) for n large.

    Proof. Obviously, we have maxt0ˆIμ(tωn)=maxt0I0(tωn). By Lemma 2.2, we derive that maxt0ˆIμ(tωn) is attained at a tn>0. By (ˆIμ(tωn),tnωn)=0 and (K1),

    M(t2n+t2nωn22)(t2n+t2nωn22)=ΩK(x)f(tnωn)tnωndxk0r2B1(0)f(tnˉωn)tnˉωndx. (2.10)

    If limntn=0, then limnˆIμ(tnωn)=0. So we assume that limntn=l(0,+]. By a direct calculation, we have

    limt+F(t)t2eα0t2=limt+f(t)2α0t1eα0t2(1α10t2)=limt+f(t)2α0t1eα0t2.

    So by (f3), for any δ>0, there exists tδ>0 such that for ttδ,

    f(t)t(βδ)eα0t2,  F(t)t2βδ2α0eα0t2. (2.11)

    Since limntn2π(logn)12=+, by (2.10)-(2.11), we derive that

    M(t2n+r2t2n(14logn+o(1logn)))(t2n+r2t2n(14logn+o(1logn)))k0(βδ)r2πn2eα02πt2nlogn=k0(βδ)r2πe(α02πt2n2)logn.

    If limntn=+, by (M2), we get a contradiction. So limntn=l(0,+). Moreover, l(0,4πα0]. If l(0,4πα0), then

    limnˆIμ(tnωn)12limnˆM(t2nωn2H1)<12ˆM(4πα0). (2.12)

    Now we assume limntn=4πα0. Let

    An:={xBr(x0):tnωn(x)tδ}.

    By (K1) and (2.11), we have

    ΩK(x)F(tnωn)dx(βδ)k02α0Ant2nω2neα0t2nω2ndx.

    Let s(0,12). Then, for n large, we have

    tnωn(x)tδ,   |xx0|rns.

    Moreover,

    ΩK(x)F(tnωn)dx(βδ)k0r22α0B1ns(0)t2nˉω2neα0t2nˉω2ndx. (2.13)

    By direct calculation, we obtain

    B1ns(0)t2nˉω2neα0t2nˉω2ndx=|x|1n2πnα0t2n2πt2nlogndx+1n|x|1ns2πlogneα0t2n2πlognlog2|x|t2nlog2|x|dx=2π2t2nnα0t2n2π2logn+4π2lognt2n1ns1nxeα0t2n2πlognlog2xlog2xdx. (2.14)

    Let Cn=α0t2n2π. Then

    1ns1nxeα0t2n2πlognlog2xlog2xdx=CnlognCnsCnn2xCn+x2Cnx2dx1logn1sn2x+Cnx2dx. (2.15)

    Here

    1sn2x+Cnx2dx12πα0t2nn(α0t2nπ2)xα0t2n2πdx+2πα0t2nsn2xdx=nα0t2n2π(α0t2nπ2)logn(nα0t2nπ2n24πα0t2n)+12logn(n2sn4πα0t2n). (2.16)

    By (2.13)–(2.16), we derive that there exists C>0 such that

    ΩK(x)F(tnωn)dx(βδ)k0π2r2α0t2nnα0t2n2π2logn+(βδ)k0π2r2α0t2n1logn(n2sn4πα0t2n)+2(βδ)k0π2r2α0t2nnα0t2n2π(α0t2nπ2)logn(nα0t2nπ2n24πα0t2n)(βδ)k0π2r2α0t2n2πnα0t2n2π2logn+Cn2slogn. (2.17)

    Together with (M1), we have

    ˆIμ(tnωn)12ˆM(t2n+r2t2n4logn)+o(1logn)(βδ)k0π2r2α0t2n2πnα0t2n2π2lognCn2slogn. (2.18)

    By limntn=4πα0, we obtain that for any ε>0, there exists N1 such that α0t2n4π+ε for n>N1. Let

    ln(t)=12ˆM(t2+r2t24logn)(βδ)k0π2r22π+εnα0t22π2logn.

    Then

    ˆIμ(tnωn)supt0ln(t)+o(1logn). (2.19)

    Obviously, there exists tn>0 such that supt0ln(t)=ln(tn). Then (ln(tn),tn)=0, from which we get

    M((tn)2+r2(tn)24logn)(1+r24logn)=(βδ)k0πr2α02π+εnα0(tn)22π2. (2.20)

    By (2.19)-(2.20), we have

    ˆIμ(tnωn)12ˆM((tn)2+r2(tn)24logn)+o(1logn)πα0lognM((tn)2+r2(tn)24logn)(1+r24logn). (2.21)

    By (2.20) and (M1), we get limnα0(tn)2=4π. Moreover,

    (tn)2=4πα0+2πα0log(2π+ε)M((tn)2+r2(tn)24logn)(1+r24logn)(βδ)k0πr2α0logn:=4πα0+An, (2.22)

    where An=O(1logn). If An+r2(tn)24logn0, by (2.22) and (M2), we have

    ˆM((tn)2+r2(tn)24logn)=ˆM(4πα0)+(tn)2+r2(tn)24logn4πα0M(s)dsˆM(4πα0)+1θ+1M(4πα0)(4πα0)θ[(4πα0+An+r2(tn)24logn)θ+1(4πα0)θ+1]. (2.23)

    If An+r2(tn)24logn<0, by (2.22) and (M1), we have

    ˆM((tn)2+r2(tn)24logn)ˆM(4πα0). (2.24)

    By (2.21)–(2.24), we obtain that

    ˆIμ(tnωn)12ˆM(4πα0)+o(1logn)+12M(4πα0)(An+πr2α0logn)πα0lognM(4πα0+An+r2(tn)24logn). (2.25)

    Since β>2M(4πα0)k0r2α0er221, by choosing δ, ε small and n large, we can derive from (2.25) that ˆIμ(tnωn)<12ˆM(4πα0).

    Lemma 2.4. (Mountain pass geometry) There exist ρ, η>0 independent of μ such that ˆIμ(u)η for uμ=ρ. Also, there exists a non-negative function vXμ with vμ>ρ such that ˆIμ(v)<0.

    Proof. By (M1), we get ˆM(s)M0s for sR+. Thus, by choosing ε>0 small, we can derive from (2.7) and the Sobolev embedding theorem that ˆIμ(u)η for uμ=ρ. By (2.8)-(2.9), we get limt+ˆIμ(tv)=.

    Define

    cμ:=infγΓmaxt[0,1]ˆIμ(γ(t)),

    where Γ:={γC([0,1],Xμ):γ(0)=0,Iμ(γ(1))<0}. By Lemmas 2.3-2.4 and the mountain pass lemma in [2], there exist {un}Xμ and n0 such that

    limnˆIμ(un)=cμ[η,maxt0I0(tωn0)],  limnˆIμ(un)=0. (2.26)

    Moreover,

    maxt0I0(tωn0)<12ˆM(4πα0). (2.27)

    Now we give a compactness result.

    Lemma 2.5. Suppose Ω is a bounded domain in R2. Assume that h satisfies the following conditions:

    (h1) hC(¯Ω×R,R) and limu0h(x,u)u=0 uniformly in xΩ.

    (h2) There exists α0>0 such that for α>α0, limu+h(x,u)eαu21=0 uniformly in xΩ.

    If unH1(Ω), Ω|h(x,un)un|dx are bounded and un(x)u(x) a.e. xΩ, then limnΩ|h(x,un)h(x,u)|dx=0.

    Proof. Let α>α0 and q>2. By (h1)-(h2), for any ε>0, there exists Cε>0 such that

    |h(x,u)|ε|u|+Cε|u|q1(eαu21),   (x,u)R2×R.

    Then

    Ω|h(x,u)|2dxCΩ|u|2dx+CΩ|u|2(q1)(e2αu21)dxCΩ|u|2dx+C(Ω|u|4(q1)dx)12[Ω(e4αu21)dx]12.

    Together with Lemma 2.1, we get h(x,u)L2(Ω). Since unH1(Ω) is bounded, we get Ωu2ndx is bounded. Let M>0. Then

    {|un|M}Ω|h(x,un)h(x,u)|dx1M{|un|M}Ω|h(x,un)unh(x,u)un|dxCM. (2.28)

    Since unH1(Ω) is bounded and un(x)u(x) a.e. xΩ, we get unu in Lp(Ω) for any p>2. Thus, by the generalized Lebesgue- dominated convergence theorem, we derive that

    limn{|un|M}Ω|h(x,un)h(x,u)|dx=limnΩ|h(x,un)h(x,u)|χ{|un|M}(x)dx=0. (2.29)

    By (2.28)-(2.29), we obtain the result.

    Corollary 2.1. If, unH1(Ωdi0), Ωdi0|K(x)f(un)un|dx are bounded and un(x)u(x) a.e. xΩdi0, then limnΩdi0|K(x)f(un)K(x)f(u)|dx=0.

    Proof. Let h(x,u)=K(x)f(u), where (x,u)¯Ωdi0×R. By (K1) and (f1), we get hC(¯Ωdi0×R,R) and limu0h(x,u)u=0 uniformly in xΩdi0. By (K1) and (f2), we get limu+h(x,u)eαu21=0 uniformly in xΩdi0. Then, by Lemma 2.5, we get the result.

    Lemma 2.6. Let μ>0. If {un}Xμ is a sequence such that ˆIμ(un)cμ(0,12ˆM(4πα0)) and ˆIμ(un)0, then {un} converges strongly in Xμ up to a subsequence.

    Proof. By (2.2) and the structure of g, we have

    cμ+on(1)+on(1)unμ=ˆIμ(un)12(θ+1)(ˆIμ(un),un)12ˆM(un2H1)12(θ+1)M(un2H1)un2H1+θ2(θ+1)R2μV(x)u2ndxθκ2(θ+1)R2Ωdi0u2ndx+(12(θ+1)1σ)Ωdi0K(x)f(un)undx. (2.30)

    Since κ<(θ+1)ε0θ, by (M3), we get unμ is bounded. Assume that unuμ weakly in Xμ.

    We consider two cases.

    Case 1. un0 weakly in Xμ.

    By (2.30), we get Ωdi0K(x)f(un)undx is bounded. So by Corollary 2.1, we have limnΩdi0K(x)f(un)dx=0. Together with (K1), (f5), and the generalized Lebesgue-dominated convergence theorem, we obtain that

    limnΩdi0K(x)F(un)dx=0.

    By (M1), we get

    ˆM(t+s)ˆM(t)+M0s,    t,s0.

    Thus,

    cμ12ˆM(limnun22+τlimnun22)+M0(1τ)2limnun22κ2limnR2Ωdi0u2ndx12ˆM(limnun22+τlimnun22).

    By (M1), we have

    limn(un22+τun22)<4πα0. (2.31)

    Define ψC0([0,)) such that ψ(r)=1 on [1,), ψ(r)=0 on [0,12] and 0ψ(r)1 on [0,). Define ψR(x):=ψ(|x|R), where Ωdi0BR2(0). By (ˆIμ(un),ψ2Run)=on(1), we derive that

    R2[M(un2H1)(|un|2ψ2R+2unψRunψR+u2nψ2R)+μV(x)u2nψ2R]dx=R2g(x,un)unψ2Rdx+on(1)κR2|unψR|2dx+on(1).

    We note that

    R2|un|2|ψR|2dxψR2L(R2)R2|un|2dxCR2.

    Together with (M1), we obtain that

    limRlimn|x|R[|(unψR)|2+(1+μV(x))|unψR|2]dx=0. (2.32)

    Let A=limnM(un2H1). Define the functional

    Jμ(u)=A2u2H1+μ2R2V(x)u2dxR2G(x,u)dx,  uXμ.

    Then Jμ(un)=on(1). Let P(x,t)=g(x,t)t and Q(t)=t(eαt21), where α>α0. By (K1) and (f2), we have

    limtP(x,t)Q(t)=0  uniformly in xR2. (2.33)

    Also,

    limnP(x,un(x))=P(x,uμ(x)) a.e. xR2. (2.34)

    By (2.31), we can choose q>1(close to 1) and α>α0(close to α0) such that qα(un22+τun22)<4π for n large. Let q=qq1. By Lemma 2.1, we derive that for n large,

    R2Q(un)dxunq[R2(eqαu2n1)dx]1qC. (2.35)

    By (2.33)–(2.35) and Lemma 1.2 in [9], we have limnBR(0)g(x,un)undx=0. Together with (2.32), we derive that

    limnR2g(x,un)undx=0. (2.36)

    Since (Jμ(un),un)=on(1), by (2.36) and (M1), we get un0 in Xμ, a contradiction with cμ>0.

    Case 2. unuμ0 weakly in Xμ.

    By ˆIμ(un)=on(1), we get Jμ(un)=on(1). Then Jμ(uμ)=0. We claim that limnun2H1=uμ2H1. Otherwise, uμ2H1<limnun2H1. By (M1), we get (ˆIμ(uμ),uμ)<0. Since uμ0, we get |suppuμΩdi0|>0. By Lemma 2.2, there exists a unique tμ>0 such that (ˆIμ(tμuμ),tμuμ)=0. Moreover, tμ(0,1). By the structure of g, for xR2Ωdi0,

    ε02u2n+[12(θ+1)ˆf(x,un)unˆF(x,un)]0. (2.37)

    By (2.2), (2.37), (M3), and Fatou's lemma, we derive that

    cμ=ˆIμ(un)12(θ+1)(ˆIμ(un),un)+on(1)12ˆM(uμ2H1)12(θ+1)M(uμ2H1)uμ2H1+μθ2(θ+1)R2V(x)u2μdx+R2Ωdi0[12(θ+1)ˆf(x,uμ)uμˆF(x,uμ)]dx+Ωdi0[12(θ+1)K(x)f(uμ)uμK(x)F(uμ)]dx+on(1). (2.38)

    By (f4), we get f(u)u2θ+1 is strictly increasing for u0. Then for any xΩdi0 and u>v0,

    12(θ+1)K(x)f(u)uK(x)F(u)>12(θ+1)K(x)f(v)vK(x)F(v). (2.39)

    By (f4), we get f(u)u is strictly increasing for u0. Together with (K1) and (f1)-(f2), we derive that for any xR2Ωdi0, there exists a unique ux>0 such that K(x)f(u)=κu for u=ux, K(x)f(u)<κu for u<ux and K(x)f(u)>κu for u>ux. Then, for any xR2Ωdi0 and u>v0,

    ε02u2+12(θ+1)ˆf(x,u)uˆF(x,u)>ε02v2+12(θ+1)ˆf(x,v)vˆF(x,v). (2.40)

    By (2.38)–(2.40), (M3), Lemma 2.2, and the definition of cμ, we have

    cμ>12ˆM(t2μuμ2H1)12(θ+1)M(t2μuμ2H1)t2μuμ2H1+μθ2(θ+1)R2V(x)t2μu2μdx+R2Ωdi0[12(θ+1)ˆf(x,tμuμ)tμuμˆF(x,tμuμ)]dx+Ωdi0[12(θ+1)K(x)f(tμuμ)tμuμK(x)F(tμuμ)]dx=ˆIμ(tμuμ)=maxt0ˆIμ(tuμ)cμ, (2.41)

    a contradiction. So limnun2H1=uμ2H1. Moreover, ˆIμ(uμ)=0, from which we derive that

    cμ=limnˆIμ(un)12(θ+1)limn(ˆIμ(un),un)ˆIμ(uμ)12(θ+1)(Iμ(uμ),uμ)=ˆIμ(uμ)=maxt0ˆIμ(tuμ)cμ. (2.42)

    By (2.42), we get limnR2V(x)|unuμ|2dx=0. Then limnunuμμ=0.

    By (2.26)-(2.27) and Lemma 2.6, we get the following result:

    Lemma 2.7. There exists uμXμ such that ˆIμ(uμ)=cμ[η,maxt0I0(tωn0)] and ˆIμ(uμ)=0, where η>0 is independent of μ.

    Define the functional J on H10(Ωi0) by

    J(u)=12ˆM(Ωi0(|u|2+|u|2)dx)Ωi0K(x)F(u)dx.

    Lemma 3.1. For any sequence {μn} with μn as n, if ˆIμn(uμn)=cμn[η,maxt0I0(tωn0)] and ˆIμn(uμn)=0, then uμnu0 in H1(R2) as n, where u0H10(Ωi0) is a positive solution of the equation

    M(Ωi0(|u|2+u2)dx)(Δu+u)=K(x)f(u)  in Ωi0. (3.1)

    Proof. Similar to (2.30), we derive that uμnH1 is bounded. Assume that uμnu0 weakly in H1(R2). By Fatou's lemma, we get R2V(x)u20dx=0. Moreover, R2Ωu20dx=0. Then u0(x)=0 a.e. xR2Ω. By u0H1(R2), u0(x)=0 a.e. xR2Ω with Ω having a smooth boundary and Proposition 9.18 in [8], we get u0H10(Ω).

    Let E=limnM(uμn2H1). Define the functional ˜Iμ on Xμ by

    ˜Iμ(u)=E2u2H1+μ2R2V(x)u2dxR2G(x,u)dx.

    Then ˜Iμn(uμn)=on(1). For all φjH10(Ωj) with ji0, we get

    EΩj(u0φj+u0φj)dx=Ωjg(x,u0)φjdx.

    Since u0H10(Ω), we have u0|ΩjH10(Ωj). Then

    EΩj(|u0|2+|u0|2)dx=Ωjg(x,u0)u0dx. (3.2)

    By the structure of g, we get u0|Ωj=0. Then u0H10(Ωi0).

    We claim that limnuμn2H1>0. Otherwise, uμn0 in H1(R2). Choose q>1(close to 1) and α>α0(close to α0) such that qαuμn2H1<4π for n large. Let t>2. By (f1)-(f2), for any ε>0, there exists Cε>0 such that

    Ωdi0f(uμn)uμndxεuμn22+CεΩdi0|uμn|t(eαu2μn1)dx. (3.3)

    By Lemma 2.1, we have

    limnΩdi0|uμn|t(eαu2μn1)dxlimn(Ωdi0|uμn|tqq1dx)q1q[Ωdi0(eqαu2μn1)dx]1qClimn(Ωdi0|uμn|tqq1dx)q1q=0. (3.4)

    Since (ˆIμ(uμn),uμn)=0, by (3.3)-(3.4) and (M1), we get limnuμnμn=0. So limncμn0, a contradiction. Let D=limnˆM(uμn2H1)uμn2H1. Define the functional ˉIμ on Xμ by

    ˉIμ(u)=D2u2H1+μ2R2V(x)u2dxR2G(x,u)dx.

    Define the functionals ˉJ and ˜J on H10(Ωi0) by

    ˉJ(u)=D2Ωi0(|u|2+|u|2)dxΩi0K(x)F(u)dx,˜J(u)=E2Ωi0(|u|2+|u|2)dxΩi0K(x)F(u)dx.

    Then ˜J(u0)=0. By (M3), we have ˉJ(u0)0. Let wμn=uμnu0. Then wμn0 weakly in H1(R2) and

    cμn=ˉJ(u0)+ˉIμn(wμn)+on(1),  (˜Iμn(wμn),wμn)=on(1). (3.5)

    Similar to the argument in (2.30), we get Ωdi0K(x)f(wμn)wμndx is bounded. Together with Corollary 2.1 and the generalized Lebesgue-dominated convergence theorem, we derive that

    limnΩdi0K(x)F(wμn)dx=0. (3.6)

    By (3.5)-(3.6), the structure of g and ˆM(t+s)ˆM(t)+M0s for all t, s0, we have

    maxt0I0(tωn0)limncμn12limnˆM(wμn22+τwμn22).

    Together with (2.27), we get limn(wμn22+τwμn22)<4πα0. By (3.5) and (M1), we have

    M0wμn2μnΩdi0K(x)f(wμn)wμndx+κR2Ωdi0w2μndx+on(1). (3.7)

    Choose q>1(close to 1) and α>α0(close to α0) such that qα(wμn22+τwμn22)<4π for n large. By (K1), (f1)-(f2) and Lemma 2.1, we have

    limnΩdi0K(x)f(wμn)wμndx=0.

    Together with (3.7), we get limnwμnμn=0. So J(u0)=0. Since limncμnη, we have u00. The maximum principle shows that u0 is positive.

    Lemma 3.2. There exists μ>0 such that for μ>μ,

    uμL(R2Ωdi0)C0uμH1(R2Ωi0), (3.8)

    where C0>0 is a constant independent of μ.

    Proof. For i2, let ri=2+2i4r1, where r1(0,min{d,1}). For yR2Ωdi0, define ηiC0(Bri(y)) such that ηi(x)=1 for xBri+1(y), 0ηi(x)1 for xR2, and |ηi|2riri+1 for xR2. Let ulμ=min{uμ,l} and βi>1. By (Iμ(uμ),η2i|ulμ|2(βi1)uμ)=0 and (M1), we get

    M0R2[|uμ|2|ulμ|2(βi1)η2i+2(βi1)|ulμ|2|ulμ|2(βi1)η2i]dx+M0R2|uμ|2|ulμ|2(βi1)η2idxR2g(x,uμ)uμ|ulμ|2(βi1)η2idx+CR2|uμ||ηi||ηi||ulμ|2(βi1)|uμ|dx. (3.9)

    Let t2. By (2.5), (3.9), and Young's inequality, we have

    R2[|uμ|2|ulμ|2(βi1)η2i+2(βi1)|ulμ|2|ulμ|2|ulμ|2(βi1)η2i]dx+R2|uμ|2|ulμ|2(βi1)η2idxCR2|ηi|2|uμ|2|ulμ|2(βi1)dx+CR2|uμ|t(eαu2μ1)|ulμ|2(βi1)η2idx. (3.10)

    We note that

    R2|uμ|t(eαu2μ1)|ulμ|2(βi1)η2idx=R2|uμ|t(eαη21u2μ1)|ulμ|2(βi1)η2idx. (3.11)

    By a direct calculation,

    η1uμ2H12Br1(y)|uμ|2dx+(1+2η12L(R2))Br1(y)|uμ|2dx. (3.12)

    By (3.12) and Lemma 3.1, we can choose μ>0 large such that η1uμ2H1<4πα0 for μ>μ. Choose q>1(close to 1) and α>α0(close to α0) such that qαη1uμ2H1<4π. Then, by Lemma 2.1, there exists C>0 independent of μ such that

    R2(eαη21u2μ1)qdxR2(eqαη21u2μ1)dxC. (3.13)

    Let t=2 and p>2q with q=qq1. By (3.10)-(3.11), (3.13), and the Sobolev embedding theorem, we obtain that there exists Cp>0 such that

    ηiuμ(ulμ)βi12pCpR2[|[ηiuμ(ulμ)βi1]|2+|ηiuμ(ulμ)βi1|2]dx2CpR2[|uμ|2|ulμ|2(βi1)η2i+(βi1)2|ulμ|2|ulμ|2(βi1)η2i]dx+2CpR2|ηi|2|uμ|2|ulμ|2(βi1)dx+CpR2|uμ|2|ulμ|2(βi1)η2idxCβ2iR2|ηi|2|uμ|2|ulμ|2(βi1)dx+Cβ2iηiuμ(ulμ)βi122q. (3.14)

    By direct calculation, we obtain

    1riri+1=4r12i+1>1. (3.15)

    Let δ0=2qp and βi=δi0. Then, by (3.14)-(3.15), we have

    uμ(ulμ)βi1Lp(Bri+1(y))Cβiriri+1uμ(ulμ)βi1Lpδ0(Bri(y)). (3.16)

    Let l, we obtain

    uμLpβi(Bri+1(y))(Cβiriri+1)1βiuμLpβi1(Bri(y)). (3.17)

    By (3.17), we derive that

    uμLpβi(Bri+1(y))ij=2(Cβjrjrj+1)1βjuμLpβ1(Br2(y))=ij=2[8Cr1(2δ0)j]δj0uμLpβ1(Br2(y)).

    Let i, we have

    uμL(B12r1(y))CuμLpβ1(Br2(y))C0uμH1(R2Ωi0). (3.18)

    Since yR2Ωdi0 is arbitrary, we finish the proof.

    Lemma 3.3. There exist r0, c1, c2, μ>0 such that Ωdi0Br0(0) and for all μ>μ,

    uμ(x)c2ec1μ(|x|r0),   |x|r0, (3.19)

    where r0, c1, c2 are independent of μ.

    Proof. By (M1) and the structure of g, we obtain that for any xR2Ωdi0,

    M(uμ2H1)Δuμ+μV(x)uμ+(M0κ)uμ0.

    Similar to (2.30), we can derive from Lemma 2.7 to obtain that uμH1 is bounded. By (V4), there exist r0, c0>0 independent of μ such that Ωdi0Br0(0) and

    Δuμ+c0μuμ0,   |x|r0. (3.20)

    By Lemma 3.2, there exists c2>0 such that uμ(x)c2 for |x|=r0, where c2>0 is independent of μ>μ. Let vμ(x)=c2ec1μ(|x|r0). By choosing c1>0 as small, we obtain

    Δvμ+c0μvμ0,   |x|r0. (3.21)

    By (3.20)-(3.21) and the comparison principle, we obtain that uμ(x)vμ(x) for |x|r0.

    Proof of Theorem 1.1. By Lemma 2.7, there exists uμXμ such that ˆIμ(uμ)=cμ[η,maxt0I0(tωn0)] and ˆIμ(uμ)=0. Let q>2. By (K2) and (f1)-(f2), there exists C>0 such that

    K(x)f(uμ)uμCeα|x|[ul1μ+|uμ|q2(eαu2μ1)]. (3.22)

    By (3.22) and Lemma 3.3, we derive that there exists μ>0 such that for μμ,

    K(x)f(uμ)uμκ,   |x|2r0. (3.23)

    By (3.22) and Lemmas 3.1-3.2, we derive that there exists μ>0 such that for μμ,

    K(x)f(uμ)uμκ,   xB2r0(0)Ωdi0. (3.24)

    By (3.23)-(3.24), we know that uμ is the nonnegative solution of (1.2). The maximum principle shows that uμ is positive. Together with Lemma 3.1, we obtain the result.

    In this paper, we study the Kirchhoff type of elliptic equation, and we assume the nonlinear terms as K(x)f(u), where K is permitted to be unbounded above and f has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhoff-type equations with critical growth in the whole space, we get the existence and concentration behavior of solutions, where the solution satisfies the mountain pass geometry. The results are new even for the case M1.

    Prof. Zhang firstly have the idea of this paper and complete the part of introduction, he also provided the main references. Dr. Lou performed the calculation, and revised the final format of the paper.

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

    This work is supported by NSFC (No.12101192) and NSF of Shandong province (No.ZR2023MA037). The authors would like to thank the editors and referees for their useful suggestions and comments.

    The authors declare no conflicts of interest in this paper.



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