In this paper, the pattern formation of a volume-filling chemotaxis model with bistable source terms was studied. First, it was shown that self-diffusion does not induce Turing patterns, but chemotaxis-driven instability occurs. Then, the asymptotic behavior of the chemotaxis model was analyzed by weakly nonlinear analysis with the method of multiple scales. When the chemotaxis coefficient exceeded a threshold value and there was a single unstable mode, the supercritical and subcritical bifurcation of the model was discussed. The amplitude equations and the asymptotic expressions of the patterns were obtained. When the chemotaxis coefficient was large enough, the two-mode competition behavior of the model with two unstable modes was analyzed, and the corresponding amplitude equations and the asymptotic expressions of the patterns were obtained. Finally, numerical simulations were provided to further illuminate the above analytical results.
Citation: Zuojun Ma. Pattern formation of a volume-filling chemotaxis model with a bistable source[J]. AIMS Mathematics, 2024, 9(11): 30816-30837. doi: 10.3934/math.20241488
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In this paper, the pattern formation of a volume-filling chemotaxis model with bistable source terms was studied. First, it was shown that self-diffusion does not induce Turing patterns, but chemotaxis-driven instability occurs. Then, the asymptotic behavior of the chemotaxis model was analyzed by weakly nonlinear analysis with the method of multiple scales. When the chemotaxis coefficient exceeded a threshold value and there was a single unstable mode, the supercritical and subcritical bifurcation of the model was discussed. The amplitude equations and the asymptotic expressions of the patterns were obtained. When the chemotaxis coefficient was large enough, the two-mode competition behavior of the model with two unstable modes was analyzed, and the corresponding amplitude equations and the asymptotic expressions of the patterns were obtained. Finally, numerical simulations were provided to further illuminate the above analytical results.
The Kirchhoff-type problem appears as a model of several physical phenomena. For example, it is related to the stationary analog of the equation:
ρ∂2u∂t2−(P0h+E2L∫L0|∂u∂x|2dx)∂2u∂x2=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 M∈C(R+,R+), V∈C(R2,R+) with Ω=int(V−1(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=up−1 in RN, | (1.3) |
where N≥3 and 2<p<2∗=2NN−2. 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 2∗−1, 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+u2∗−1 in RN. | (1.4) |
When N≥4, 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+b∫R3|∇u|2dx)Δu+λV(x)u=|u|p−2u in R3, | (1.5) |
where 3<p<6. When V≥0 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+b∫R3|∇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)e−αu2=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)≥λuq−1, ∀ u≥0. |
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)ueα0u2≥β, |
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) M∈C(R+,R+), infR+M:=M0>0, and M(t) is strictly increasing for t∈R+.
(M2) There exist θ, ε0>0 such that M(t)−ε0tθ is decreasing for t>0.
(M3) There exists ε′0>0 such that ˆM(t)−1θ+1M(t)t−ε′0t is increasing for t∈R+, where ˆM(t)=∫t0M(s)ds.
(V1) V∈C(R2,R+).
(V2) Ω=int(V−1(0)) is non-empty with smooth boundary and ˉΩ=V−1(0).
(V3) Ω consists of k connected components: Ω=∪ki=1Ωi and ¯Ωi∩¯Ωj=∅ for all i≠j.
(V4) There exists V0>0 such that |{x∈R2:V(x)≤V0}|<∞.
(K1) K∈C(R2,R+) and k0:=infR2K>0.
(K2) There exist k1, α>0 such that K(x)≤k1eα|x| for x∈R2.
(f1) f∈C(R+,R+) and there exists l>1 such that limu→0+f(u)ul<+∞.
(f2) There exists α0>0 such that
limu→+∞f(u)eαu2−1={0,∀α>α0,+∞,∀α<α0. |
(f3) There exists β>0 such that
β≤limu→+∞f(u)ueα0u2<+∞. |
(f4) There exists σ>2(θ+1) such that f(u)uσ−1 is increasing for u∈R+∖{0}.
(f5) There exist u0, L0>0 such that F(u)≤L0f(u) for u≥u0, 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)k0r2α0er22−1, 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 Ωdi0⊂Br0(0) and
uμ(x)≤c2e−c1√μ(|x|−r0), ∀ |x|≥r0. | (1.9) |
Besides, for any sequence μn→+∞, there exists u0∈H10(Ωi0) such that uμn→u0 in H1(R2) as n→∞, where u0∈H10(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
A2u−1eα0u2≤f(u)≤3A2u−1eα0u2, ∀ u≥R. |
Moreover,
limu→+∞F(u)f(u)≤limu→+∞∫R0f(s)ds+3A2∫uRs−1eα0s2dsA2u−1eα0u2=0, |
from which we get f satisfies (f5). If A=∞, one can prove it by the L′Hospital rule and the definition of ϵ−N.
Remark 1.2. Let f1(u)=β(α0u2−1)eα0u2α0u3, where u>0. Then there exists u1>0 such that f1(u1)=uσ−11. Define f(u)=uσ−1 for u∈[0,u1] and f(u)=f1(u) for u>u1. Obviously, f satisfies (f1)–(f3). We note that
(f1(u)uσ−1)′=βeα0u2α0u3+σ[2α20u4−(σ+2)α0u2+σ+2]. |
If σ≤6, then f1(u)uσ−1 is increasing for u≥u1. 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 ‖u‖s:=(∫R2|u(x)|sdx)1s, where s∈[1,∞). Define H1(R2) the Hilbert space with the norm ‖u‖H1:=(‖∇u‖22+‖u‖22)12. It is well known that the embedding H1(R2)↪Lt(R2) is continuous for all t≥2. Let μ>0. Define
Xμ:={u∈H1(R2):∫R2V(x)u2dx<∞} |
the Hilbert space equipped with the norm
‖u‖μ:=(‖∇u‖22+∫R2(1+μV(x))u2dx)12. |
Obviously, the embedding Xμ↪H1(R2) is continuous. We give the following Trudinger–Moser inequality:
Lemma 2.1. ([15,21,22]) If u∈H1(R2) and α>0, then
∫R2(eαu2−1)dx<∞. |
Moreover, for any fixed τ>0, there exists a constant C>0 such that
supu∈H1(R2):‖∇u‖22+τ‖u‖22≤1∫R2(e4πu2−1)dx≤C. |
Since we look for positive solutions, we assume that f(u)=0 for u≤0. For any d>0, define Ωd:={x∈R2:dist(x,Ω)<d}. By (V3), we can choose d>0 small such that Ω2di∩Ω2dj=∅ for all i≠j. Let i0∈{1,2,…,k}. Define
χ(x)={1, x∈Ωdi0,0, x∈R2∖Ωdi0. |
By (V4), we know that Ωdi0 is bounded. Let τ∈(0,1). For any x∈R2∖Ωdi0, define
ˆf(x,u)=min{K(x)f(u),κu+}, |
where u+=max{u,0} and κ∈(0,min{ε0,(θ+1)ε′0θ,M0(1−τ)}). Define
g(x,u)=χ(x)K(x)f(u)+(1−χ(x))ˆf(x,u). | (2.1) |
Then
G(x,u)=∫u0g(x,s)ds=χ(x)K(x)F(u)+(1−χ(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)u−σK(x)F(u)≥0, ˆf(x,u)u−2ˆF(x,u)≥0. | (2.2) |
Instead of studying (1.2), we consider the following truncated problem:
M(‖u‖2H1)(−Δu+u)+μV(x)u=g(x,u) in R2. | (2.3) |
The functional associated with (2.3) is
ˆIμ(u)=12ˆM(‖u‖2H1)+μ2∫R2V(x)u2dx−∫R2G(x,u)dx, u∈Xμ. | (2.4) |
Obviously, ˆIμ∈C1(Xμ,R), and the critical points of ˆIμ are weak solutions of (2.3).
Lemma 2.2. Let l(t)=ˆIμ(tu), where t≥0 and u∈Xμ with |suppu∩Ωdi0|>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|q−1(eαu2−1), ∀ (x,u)∈R2×R. | (2.5) |
Then
|G(x,u)|≤ε+κ2|u|2+Cεq|u|q(eαu2−1), ∀ (x,u)∈R2×R. | (2.6) |
By (2.6) and Lemma 2.1, we can choose ρ>0 small such that for ‖u‖μ≤ρ,
|∫R2G(x,u)dx|≤ε+κ2‖u‖22+Cεq‖u‖q2q[∫R2(e2αu2−1)dx]12≤ε+κ2‖u‖22+C‖u‖q2q. | (2.7) |
By (M1), we get ˆM(s)≥M0s for s∈R+. 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, s∈R+. | (2.8) |
Let p>2θ+1. By (f1)-(f2), there exist c1, c2>0 such that
f(u)≥c1up−c2u, ∀ u∈R. | (2.9) |
By (2.8)-(2.9), we get l(t)<0 for t>0 large. Thus, maxt≥0l(t) is attained at t0>0 and l′(t0)=0. Let
y(t)=[ε0‖u‖22+μ∫R2V(x)u2dx−∫R2∖Ωdi0ˆf(x,tu)utdx]+[(M(t2‖u‖2H1)−ε0)‖u‖22+M(t2‖u‖2H1)‖∇u‖22−t2θ∫Ωdi0K(x)f(tu)ut2θ+1dx]. |
Then y(t0)=0. Moreover, from the structure of g, we derive that for t>0,
ε0‖u‖22+μ∫R2V(x)u2dx−∫R2∖Ωdi0ˆf(x,tu)utdx>0,(M(t2‖u‖2H1)−ε0)‖u‖22+M(t2‖u‖2H1)‖∇u‖22−t2θ∫Ωdi0K(x)f(tu)ut2θ+1dx<0. |
By (M2), we know (M(t2‖u‖2H1)−ε0)‖u‖22+M(t2‖u‖2H1)‖∇u‖22t2θ is decreasing for t>0. By (f4), we know ∫R2∖Ωdi0ˆf(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)=1√2π{(logn)12, 0≤|x|≤1n,log1|x|(logn)12, 1n≤|x|≤1,0, |x|≥1. |
It is well known that ‖∇ˉωn‖22=1 and ‖ˉωn‖22=14logn+o(1logn). Choose x0∈Ωi0 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(x−x0r). Then, ‖∇ωn‖22=1. Define the functional I0 as follows:
I0(u)=12ˆM(∫Ωi0|∇u|2+u2dx)−∫Ωi0K(x)F(u)dx, u∈H10(Ωi0). |
Lemma 2.3. maxt≥0ˆIμ(tωn)=maxt≥0I0(tωn)<12ˆM(4πα0) for n large.
Proof. Obviously, we have maxt≥0ˆIμ(tωn)=maxt≥0I0(tωn). By Lemma 2.2, we derive that maxt≥0ˆIμ(tωn) is attained at a tn>0. By (ˆI′μ(tωn),tnωn)=0 and (K1),
M(t2n+t2n‖ωn‖22)(t2n+t2n‖ωn‖22)=∫ΩK(x)f(tnωn)tnωndx≥k0r2∫B1(0)f(tnˉωn)tnˉωndx. | (2.10) |
If limn→∞tn=0, then limn→∞ˆIμ(tnωn)=0. So we assume that limn→∞tn=l∈(0,+∞]. By a direct calculation, we have
limt→+∞F(t)t−2eα0t2=limt→+∞f(t)2α0t−1eα0t2(1−α−10t−2)=limt→+∞f(t)2α0t−1eα0t2. |
So by (f3), for any δ>0, there exists tδ>0 such that for t≥tδ,
f(t)t≥(β−δ)eα0t2, F(t)t2≥β−δ2α0eα0t2. | (2.11) |
Since limn→∞tn√2π(logn)12=+∞, by (2.10)-(2.11), we derive that
M(t2n+r2t2n(14logn+o(1logn)))(t2n+r2t2n(14logn+o(1logn)))≥k0(β−δ)r2πn−2eα02πt2nlogn=k0(β−δ)r2πe(α02πt2n−2)logn. |
If limn→∞tn=+∞, by (M2), we get a contradiction. So limn→∞tn=l∈(0,+∞). Moreover, l∈(0,√4πα0]. If l∈(0,√4πα0), then
limn→∞ˆIμ(tnωn)≤12limn→∞ˆM(t2n‖ωn‖2H1)<12ˆM(4πα0). | (2.12) |
Now we assume limn→∞tn=√4πα0. Let
An:={x∈Br(x0):tnωn(x)≥tδ}. |
By (K1) and (2.11), we have
∫ΩK(x)F(tnωn)dx≥(β−δ)k02α0∫Ant−2nω−2neα0t2nω2ndx. |
Let s∈(0,12). Then, for n large, we have
tnωn(x)≥tδ, ∀ |x−x0|≤rns. |
Moreover,
∫ΩK(x)F(tnωn)dx≥(β−δ)k0r22α0∫B1ns(0)t−2nˉω−2neα0t2nˉω2ndx. | (2.13) |
By direct calculation, we obtain
∫B1ns(0)t−2nˉω−2neα0t2nˉω2ndx=∫|x|≤1n2πnα0t2n2πt2nlogndx+∫1n≤|x|≤1ns2πlogneα0t2n2πlognlog2|x|t2nlog2|x|dx=2π2t2nnα0t2n2π−2logn+4π2lognt2n∫1ns1nxeα0t2n2πlognlog2xlog2xdx. | (2.14) |
Let Cn=α0t2n2π. Then
∫1ns1nxeα0t2n2πlognlog2xlog2xdx=Cnlogn∫CnsCnn−2xCn+x2Cnx−2dx≥1logn∫1sn−2x+Cnx2dx. | (2.15) |
Here
∫1sn−2x+Cnx2dx≥∫12πα0t2nn(α0t2nπ−2)x−α0t2n2πdx+∫2πα0t2nsn−2xdx=n−α0t2n2π(α0t2nπ−2)logn(nα0t2nπ−2−n2−4πα0t2n)+12logn(n−2s−n−4πα0t2n). | (2.16) |
By (2.13)–(2.16), we derive that there exists C′>0 such that
∫ΩK(x)F(tnωn)dx≥(β−δ)k0π2r2α0t2nnα0t2n2π−2logn+(β−δ)k0π2r2α0t2n1logn(n−2s−n−4πα0t2n)+2(β−δ)k0π2r2α0t2nn−α0t2n2π(α0t2nπ−2)logn(nα0t2nπ−2−n2−4πα0t2n)≥(β−δ)k0π2r2α0t2n−2πnα0t2n2π−2logn+C′n−2slogn. | (2.17) |
Together with (M1), we have
ˆIμ(tnωn)≤12ˆM(t2n+r2t2n4logn)+o(1logn)−(β−δ)k0π2r2α0t2n−2πnα0t2n2π−2logn−C′n−2slogn. | (2.18) |
By limn→∞tn=√4πα0, we obtain that for any ε>0, there exists N1 such that α0t2n≤4π+ε for n>N1. Let
ln(t)=12ˆM(t2+r2t24logn)−(β−δ)k0π2r22π+εnα0t22π−2logn. |
Then
ˆIμ(tnωn)≤supt≥0ln(t)+o(1logn). | (2.19) |
Obviously, there exists t′n>0 such that supt≥0ln(t)=ln(t′n). Then (l′n(t′n),t′n)=0, from which we get
M((t′n)2+r2(t′n)24logn)(1+r24logn)=(β−δ)k0πr2α02π+εnα0(t′n)22π−2. | (2.20) |
By (2.19)-(2.20), we have
ˆIμ(tnωn)≤12ˆM((t′n)2+r2(t′n)24logn)+o(1logn)−πα0lognM((t′n)2+r2(t′n)24logn)(1+r24logn). | (2.21) |
By (2.20) and (M1), we get limn→∞α0(t′n)2=4π. Moreover,
(t′n)2=4πα0+2πα0log(2π+ε)M((t′n)2+r2(t′n)24logn)(1+r24logn)(β−δ)k0πr2α0logn:=4πα0+An, | (2.22) |
where An=O(1logn). If An+r2(t′n)24logn≥0, by (2.22) and (M2), we have
ˆM((t′n)2+r2(t′n)24logn)=ˆM(4πα0)+∫(t′n)2+r2(t′n)24logn4πα0M(s)ds≤ˆM(4πα0)+1θ+1M(4πα0)(4πα0)θ[(4πα0+An+r2(t′n)24logn)θ+1−(4πα0)θ+1]. | (2.23) |
If An+r2(t′n)24logn<0, by (2.22) and (M1), we have
ˆM((t′n)2+r2(t′n)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(t′n)24logn). | (2.25) |
Since β>2M(4πα0)k0r2α0er22−1, 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 ‖u‖μ=ρ. Also, there exists a non-negative function v∈Xμ with ‖v‖μ>ρ such that ˆIμ(v)<0.
Proof. By (M1), we get ˆM(s)≥M0s for s∈R+. Thus, by choosing ε>0 small, we can derive from (2.7) and the Sobolev embedding theorem that ˆIμ(u)≥η for ‖u‖μ=ρ. By (2.8)-(2.9), we get limt→+∞ˆIμ(tv)=−∞.
Define
cμ:=infγ∈Γmaxt∈[0,1]ˆIμ(γ(t)), |
where Γ:={γ∈C([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
limn→∞ˆIμ(un)=cμ∈[η,maxt≥0I0(tωn0)], limn→∞ˆI′μ(un)=0. | (2.26) |
Moreover,
maxt≥0I0(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) h∈C(¯Ω×R,R) and limu→0h(x,u)u=0 uniformly in x∈Ω.
(h2) There exists α0>0 such that for α>α0, limu→+∞h(x,u)eαu2−1=0 uniformly in x∈Ω.
If ‖un‖H1(Ω), ∫Ω|h(x,un)un|dx are bounded and un(x)→u(x) a.e. x∈Ω, then limn→∞∫Ω|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|q−1(eαu2−1), ∀ (x,u)∈R2×R. |
Then
∫Ω|h(x,u)|2dx≤C∫Ω|u|2dx+C∫Ω|u|2(q−1)(e2αu2−1)dx≤C∫Ω|u|2dx+C(∫Ω|u|4(q−1)dx)12[∫Ω(e4αu2−1)dx]12. |
Together with Lemma 2.1, we get h(x,u)∈L2(Ω). Since ‖un‖H1(Ω) is bounded, we get ∫Ωu2ndx is bounded. Let M>0. Then
∫{|un|≥M}∩Ω|h(x,un)−h(x,u)|dx≤1M∫{|un|≥M}∩Ω|h(x,un)un−h(x,u)un|dx≤CM. | (2.28) |
Since ‖un‖H1(Ω) is bounded and un(x)→u(x) a.e. x∈Ω, we get un→u 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=limn→∞∫Ω|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, ‖un‖H1(Ωdi0), ∫Ωdi0|K(x)f(un)un|dx are bounded and un(x)→u(x) a.e. x∈Ωdi0, then limn→∞∫Ωdi0|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 h∈C(¯Ωdi0×R,R) and limu→0h(x,u)u=0 uniformly in x∈Ωdi0. By (K1) and (f2), we get limu→+∞h(x,u)eαu2−1=0 uniformly in x∈Ωdi0. 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 ˆI′μ(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)‖un‖μ=ˆIμ(un)−12(θ+1)(ˆI′μ(un),un)≥12ˆM(‖un‖2H1)−12(θ+1)M(‖un‖2H1)‖un‖2H1+θ2(θ+1)∫R2μV(x)u2ndx−θκ2(θ+1)∫R2∖Ωdi0u2ndx+(12(θ+1)−1σ)∫Ωdi0K(x)f(un)undx. | (2.30) |
Since κ<(θ+1)ε′0θ, by (M3), we get ‖un‖μ is bounded. Assume that un⇀uμ weakly in Xμ.
We consider two cases.
Case 1. un⇀0 weakly in Xμ.
By (2.30), we get ∫Ωdi0K(x)f(un)undx is bounded. So by Corollary 2.1, we have limn→∞∫Ωdi0K(x)f(un)dx=0. Together with (K1), (f5), and the generalized Lebesgue-dominated convergence theorem, we obtain that
limn→∞∫Ωdi0K(x)F(un)dx=0. |
By (M1), we get
ˆM(t+s)≥ˆM(t)+M0s, ∀ t,s≥0. |
Thus,
cμ≥12ˆM(limn→∞‖∇un‖22+τlimn→∞‖un‖22)+M0(1−τ)2limn→∞‖un‖22−κ2limn→∞∫R2∖Ωdi0u2ndx≥12ˆM(limn→∞‖∇un‖22+τlimn→∞‖un‖22). |
By (M1), we have
limn→∞(‖∇un‖22+τ‖un‖22)<4πα0. | (2.31) |
Define \psi \in C_{0}^{\infty} ([0, \infty)) such that \psi(r) = 1 on [1, \infty) , \psi(r) = 0 on [0, \frac{1}{2}] and 0 \le \psi(r) \le 1 on [0, \infty) . Define \psi_R(x): = \psi\left(\frac{|x|}{R}\right) , where \Omega_{i_0}^d \subset B_{\frac{R}{2}}(0) . By (\hat{I}_{\mu}'(u_n), \psi_R^2 u_n) = o_n(1) , we derive that
\begin{align*} &\int_{\mathbb{R}^2} \left[M(\|u_n\|_{H^1}^2)\left(|\nabla u_n|^2 \psi_R^2 +2\nabla u_n \nabla \psi_R u_n \psi_R+u_n^2\psi_R^2 \right)+\mu V(x) u_n^2\psi_R^2\right]\mathrm{d} x\notag\\ & = \int_{\mathbb{R}^2} g(x,u_n)u_n\psi_R^2\mathrm{d} x +o_n(1) \le \kappa \int_{\mathbb{R}^2}|u_n \psi_R|^2 \mathrm{d}x + o_n(1). \end{align*} |
We note that
\begin{align*} \int_{ \mathbb R^2}|u_n|^2 |\nabla \psi_R|^2 \mathrm{d}x \le \|\nabla \psi_R\|_{L^\infty( \mathbb R^2)}^2 \int_{ \mathbb R^2}|u_n|^2 \mathrm{d}x \le \frac{C}{R^2}. \end{align*} |
Together with (M_1) , we obtain that
\begin{align} \lim\limits_{R \rightarrow \infty}\lim\limits_{n \rightarrow \infty}\int_{|x| \ge R}\left[|\nabla (u_n \psi_R)|^2 + (1+\mu V(x)) |u_n \psi_R|^2\right]\mathrm{d}x = 0. \end{align} | (2.32) |
Let A = \lim_{n \rightarrow \infty}M(\|u_n\|_{H^1}^2) . Define the functional
\begin{align*} J_\mu(u) = \frac{A}{2}\|u\|_{H^1}^2+\frac{\mu}{2}\int_{ \mathbb R^2} V(x)u^2\mathrm{d}x-\int_{ \mathbb R^2}G(x,u)\mathrm{d}x, \ \ u \in X_\mu. \end{align*} |
Then J_\mu'(u_n) = o_n(1) . Let P(x, t) = g(x, t)t and Q(t) = t\left(e^{\alpha t^2}-1\right) , where \alpha > \alpha_0 . By (K_1) and (f_2) , we have
\begin{align} \lim\limits_{t \rightarrow \infty}\frac{P(x,t)}{Q(t)} = 0 \ \ \mathrm{uniformly} \ \mathrm{in} \ x \in \mathbb R^2. \end{align} | (2.33) |
Also,
\begin{align} \lim\limits_{n \rightarrow \infty}P(x,u_n(x)) = P(x,u_\mu(x)) \ a.e. \ x \in \mathbb R^2. \end{align} | (2.34) |
By (2.31), we can choose q > 1 (close to 1) and \alpha > \alpha_0 (close to \alpha_0 ) such that q \alpha (\|\nabla u_n\|_2^2+\tau\|u_n\|_2^2) < 4\pi for n large. Let q' = \frac{q}{q-1} . By Lemma 2.1 , we derive that for n large,
\begin{align} \int_{ \mathbb R^2}Q(u_n)\mathrm{d}x \le \|u_n\|_{q'}\left[\int_{ \mathbb R^2}(e^{q\alpha u_n^2}-1)\mathrm{d}x\right]^{\frac{1}{q}} \le C. \end{align} | (2.35) |
By (2.33)–(2.35) and Lemma 1.2 in [9], we have \lim_{n\rightarrow \infty}\int_{B_R(0)}g(x, u_n)u_n\mathrm{d}x = 0 . Together with (2.32), we derive that
\begin{align} \lim\limits_{n\rightarrow \infty}\int_{ \mathbb R^2}g(x,u_n)u_n\mathrm{d}x = 0. \end{align} | (2.36) |
Since (J_\mu'(u_n), u_n) = o_n(1) , by (2.36) and (M_1) , we get u_n\rightarrow 0 in X_\mu , a contradiction with c_\mu > 0 .
Case 2 . u_n \rightharpoonup u_\mu \ne 0 weakly in X_\mu .
By \hat{I}_\mu'(u_n) = o_n(1) , we get J_\mu'(u_n) = o_n(1) . Then J_\mu'(u_\mu) = 0 . We claim that \lim_{n \rightarrow \infty}\|u_n\|_{H^1}^2 = \|u_\mu\|_{H^1}^2 . Otherwise, \|u_\mu\|_{H^1}^2 < \lim_{n \rightarrow \infty}\|u_n\|_{H^1}^2 . By (M_1) , we get (\hat{I}_\mu'(u_\mu), u_\mu) < 0 . Since u_\mu \ne 0 , we get \left|\mathrm{supp} u_\mu \cap \Omega_{i_0}^d\right| > 0 . By Lemma 2.2 , there exists a unique t_\mu > 0 such that (\hat{I}_\mu'(t_\mu u_\mu), t_\mu u_\mu) = 0 . Moreover, t_\mu \in (0, 1) . By the structure of g , for x \in \mathbb R^2 \setminus \Omega_{i_0}^d ,
\begin{align} \frac{\varepsilon_0'}{2}u_n^2+\left[\frac{1}{2(\theta+1)}\hat{f}(x,u_n)u_n-\hat{F}(x,u_n)\right] \ge 0. \end{align} | (2.37) |
By (2.2), (2.37), (M_3) , and Fatou's lemma, we derive that
\begin{align} c_\mu& = \hat{I}_{\mu}(u_n) -\frac{1}{2(\theta+1)}(\hat{I}_{\mu}'(u_n),u_n)+ o_n(1) \\ &\ge \frac{1}{2}\hat{M}(\|u_\mu\|_{H^1}^2)-\frac{1}{2(\theta+1)}M(\|u_\mu\|_{H^1}^2)\|u_\mu\|_{H^1}^2\\ &\quad +\frac{\mu \theta}{2(\theta+1)}\int_{ \mathbb R^2}V(x)u_\mu^2 \mathrm{d}x+ \int_{ \mathbb R^2 \setminus \Omega_{i_0}^d}\left[\frac{1}{2(\theta+1)}\hat{f}(x,u_\mu)u_\mu-\hat{F}(x,u_\mu)\right]\mathrm{d} x\\ &\quad+\int_{\Omega_{i_0}^d}\left[\frac{1}{2(\theta+1)}K(x)f(u_\mu)u_\mu-K(x)F(u_\mu)\right] \mathrm{d}x+o_n(1). \end{align} | (2.38) |
By (f_4) , we get \frac{f(u)}{u^{2 \theta+1}} is strictly increasing for u \ge 0 . Then for any x \in \Omega_{i_0}^d and u > v \ge 0 ,
\begin{align} &\frac{1}{2(\theta+1)}K(x)f(u)u-K(x)F(u) \\ & > \frac{1}{2(\theta+1)}K(x)f(v)v-K(x)F(v). \end{align} | (2.39) |
By (f_4) , we get \frac{f(u)}{u} is strictly increasing for u \ge 0 . Together with (K_1) and (f_1) - (f_2) , we derive that for any x \in \mathbb R^2 \setminus \Omega_{i_0}^d , there exists a unique u_x > 0 such that K(x)f(u) = \kappa u for u = u_x , K(x)f(u) < \kappa u for u < u_x and K(x)f(u) > \kappa u for u > u_x . Then, for any x \in \mathbb R^2 \setminus \Omega_{i_0}^d and u > v \ge 0 ,
\begin{align} &\frac{\varepsilon_0'}{2}u^2 + \frac{1}{2(\theta+1)}\hat{f}(x,u)u-\hat{F}(x,u) \\ & > \frac{\varepsilon_0'}{2}v^2 + \frac{1}{2(\theta+1)}\hat{f}(x,v)v-\hat{F}(x,v). \end{align} | (2.40) |
By (2.38)–(2.40), (M_3) , Lemma 2.2 , and the definition of c_\mu , we have
\begin{align} c_\mu > & \frac{1}{2}\hat{M}(t_\mu^2\|u_\mu\|_{H^1}^2)-\frac{1}{2(\theta+1)}M(t_\mu^2\|u_\mu\|_{H^1}^2)t_\mu^2\|u_\mu\|_{H^1}^2+\frac{\mu \theta}{2(\theta+1)}\int_{ \mathbb R^2}V(x)t_\mu^2 u_\mu^2 \mathrm{d}x\\ &+ \int_{ \mathbb R^2 \setminus \Omega_{i_0}^d}\left[\frac{1}{2(\theta+1)}\hat{f}(x,t_\mu u_\mu)t_\mu u_\mu-\hat{F}(x,t_\mu u_\mu)\right]\mathrm{d} x\\ &+\int_{\Omega_{i_0}^d}\left[\frac{1}{2(\theta+1)}K(x)f(t_\mu u_\mu)t_\mu u_\mu-K(x)F(t_\mu u_\mu)\right] \mathrm{d}x\\ = &\hat{I}_\mu(t_\mu u_\mu) = \max\limits_{t \ge 0}\hat{I}_\mu(t u_\mu) \ge c_\mu, \end{align} | (2.41) |
a contradiction. So \lim_{n \rightarrow \infty}\|u_n\|_{H^1}^2 = \|u_\mu\|_{H^1}^2 . Moreover, \hat{I}_\mu'(u_\mu) = 0 , from which we derive that
\begin{align} c_\mu & = \lim\limits_{n \rightarrow \infty}\hat{I}_{\mu}(u_n) -\frac{1}{2(\theta+1)} \lim\limits_{n \rightarrow \infty}(\hat{I}_{\mu}'(u_n),u_n) \\ &\ge \hat{I}_\mu(u_\mu)-\frac{1}{2(\theta+1)}(I_\mu'(u_\mu),u_\mu) = \hat{I}_\mu(u_\mu) = \max\limits_{t \ge 0}\hat{I}_\mu(t u_\mu) \ge c_\mu. \end{align} | (2.42) |
By (2.42), we get \lim_{n \rightarrow \infty}\int_{ \mathbb R^2}V(x)|u_n-u_\mu|^2 \mathrm{d}x = 0 . Then \lim_{n \rightarrow \infty}\|u_n-u_\mu\|_\mu = 0 .
By (2.26)-(2.27) and Lemma 2.6 , we get the following result:
Lemma 2.7. There exists u_\mu \in X_\mu such that \hat{I}_\mu(u_\mu) = c_\mu \in [\eta, \max_{t \ge 0}I_0(t\omega_{n_0})] and \hat{I}'_\mu(u_\mu) = 0 , where \eta > 0 is independent of \mu .
Define the functional J on H_0^1(\Omega_{i_0}) by
\begin{align*} J(u) = \frac{1}{2}\hat{M}\left(\int_{\Omega_{i_0}}(|\nabla u|^2+|u|^2) \mathrm{d}x\right)-\int_{\Omega_{i_0}}K(x)F(u) \mathrm{d}x. \end{align*} |
Lemma 3.1. For any sequence \{\mu_n\} with \mu_n \rightarrow \infty as n \rightarrow \infty , if \hat{I}_{\mu_n}(u_{\mu_n}) = c_{\mu_n} \in [\eta, \max_{t \ge 0}I_0(t\omega_{n_0})] and \hat{I}_{\mu_n}'(u_{\mu_n}) = 0 , then u_{\mu_n} \rightarrow u_0 in H^1(\mathbb R^2) as n \rightarrow \infty , where u_0 \in H_0^1(\Omega_{i_0}) is a positive solution of the equation
\begin{align} M\left(\int_{\Omega_{i_0}}(|\nabla u|^2+u^2) \mathrm{d}x\right)(-\Delta u+ u) = K(x)f(u) \ \ \mathrm{in} \ \Omega_{i_0}. \end{align} | (3.1) |
Proof. Similar to (2.30), we derive that \|u_{\mu_n}\|_{H^1} is bounded. Assume that u_{\mu_n} \rightharpoonup u_0 weakly in H^1(\mathbb R^2) . By Fatou's lemma, we get \int_{\mathbb{R}^2}V(x)u_0^2\mathrm{d}x = 0 . Moreover, \int_{\mathbb{R}^2 \setminus \Omega}u_0^2 \mathrm{d}x = 0 . Then u_0(x) = 0 a.e. x \in \mathbb{R}^2 \setminus \Omega . By u_0 \in H^1(\mathbb R^2) , u_0(x) = 0 a.e. x \in \mathbb{R}^2 \setminus \Omega with \Omega having a smooth boundary and Proposition 9.18 in [8], we get u_0 \in H_0^1(\Omega) .
Let E = \lim_{n \rightarrow \infty}M(\|u_{\mu_n}\|_{H^1}^2) . Define the functional \tilde{I}_{\mu} on X_\mu by
\begin{align*} \tilde{I}_{\mu}(u) = \frac{E}{2}\|u\|_{H^1}^2+\frac{\mu}{2}\int_{ \mathbb R^2}V(x)u^2 \mathrm{d}x-\int_{ \mathbb R^2}G(x,u) \mathrm{d}x. \end{align*} |
Then \tilde{I}_{\mu_n}'(u_{\mu_n}) = o_n(1) . For all \varphi_j \in H_0^1(\Omega_j) with j \ne i_0 , we get
\begin{align*} E\int_{\Omega_j}(\nabla u_0 \nabla \varphi_j+ u_0 \varphi_j)\mathrm{d}x = \int_{\Omega_j}g(x,u_0)\varphi_j \mathrm{d}x. \end{align*} |
Since u_0 \in H_0^1(\Omega) , we have u_0|_{\Omega_j} \in H_0^1(\Omega_j) . Then
\begin{align} E\int_{\Omega_j}(|\nabla u_0|^2+ |u_0|^2)\mathrm{d}x = \int_{\Omega_j}g(x,u_0)u_0\mathrm{d}x. \end{align} | (3.2) |
By the structure of g , we get u_0|_{\Omega_j} = 0 . Then u_0 \in H_0^1(\Omega_{i_0}) .
We claim that \lim_{n \rightarrow \infty}\|u_{\mu_n}\|_{H^1}^2 > 0 . Otherwise, u_{\mu_n} \rightarrow 0 in H^1(\mathbb R^2) . Choose q > 1 (close to 1) and \alpha > \alpha_0 (close to \alpha_0 ) such that q \alpha \|u_{\mu_n}\|_{H^1}^2 < 4\pi for n large. Let t > 2 . By (f_1) - (f_2) , for any \varepsilon > 0 , there exists C_\varepsilon > 0 such that
\begin{align} \int_{\Omega_{i_0}^d}f(u_{\mu_n})u_{\mu_n} \mathrm{d}x \le \varepsilon \|u_{\mu_n}\|_2^2+C_\varepsilon\int_{\Omega_{i_0}^d} |u_{\mu_n}|^t(e^{\alpha u_{\mu_n}^2}-1)\mathrm{d}x. \end{align} | (3.3) |
By Lemma 2.1, we have
\begin{align} &\lim\limits_{n \rightarrow \infty}\int_{\Omega_{i_0}^d} |u_{\mu_n}|^t(e^{\alpha u_{\mu_n}^2}-1)\mathrm{d}x\\ & \le \lim\limits_{n \rightarrow \infty}\left(\int_{\Omega_{i_0}^d}|u_{\mu_n}|^{\frac{tq}{q-1}}\mathrm{d}x\right)^{\frac{q-1}{q}}\left[\int_{\Omega_{i_0}^d}(e^{q\alpha u_{\mu_n}^2}-1)\mathrm{d}x\right]^{\frac{1}{q}}\\ & \le C \lim\limits_{n \rightarrow \infty}\left(\int_{\Omega_{i_0}^d}|u_{\mu_n}|^{\frac{tq}{q-1}}\mathrm{d}x\right)^{\frac{q-1}{q}} = 0. \end{align} | (3.4) |
Since (\hat{I}_\mu'(u_{\mu_n}), u_{\mu_n}) = 0 , by (3.3)-(3.4) and (M_1) , we get \lim_{n \rightarrow \infty}\|u_{\mu_n}\|_{\mu_n} = 0 . So \lim_{n \rightarrow \infty}c_{\mu_n} \le 0 , a contradiction. Let D = \lim_{n \rightarrow \infty}\frac{\hat{M}(\|u_{\mu_n}\|_{H^1}^2)}{\|u_{\mu_n}\|_{H^1}^2} . Define the functional \bar{I}_{\mu} on X_\mu by
\begin{align*} \bar{I}_{\mu}(u) = \frac{D}{2}\|u\|_{H^1}^2+\frac{\mu}{2}\int_{ \mathbb R^2}V(x)u^2 \mathrm{d}x-\int_{ \mathbb R^2}G(x,u) \mathrm{d}x. \end{align*} |
Define the functionals \bar{J} and \tilde{J} on H_0^1(\Omega_{i_0}) by
\begin{align*} &\bar{J}(u) = \frac{D}{2} \int_{\Omega_{i_0}}(|\nabla u|^2+|u|^2) \mathrm{d}x-\int_{\Omega_{i_0}}K(x)F(u) \mathrm{d}x,\\ &\tilde{J}(u) = \frac{E}{2} \int_{\Omega_{i_0}}(|\nabla u|^2 +|u|^2) \mathrm{d}x-\int_{\Omega_{i_0}}K(x)F(u) \mathrm{d}x. \end{align*} |
Then \tilde{J}'(u_0) = 0 . By (M_3) , we have \bar{J}(u_0) \ge 0 . Let w_{\mu_n} = u_{\mu_n}-u_0 . Then w_{\mu_n} \rightharpoonup 0 weakly in H^1(\mathbb R^2) and
\begin{align} c_{\mu_n} = \bar{J}(u_0)+\bar{I}_{\mu_n}(w_{\mu_n})+o_n(1), \ \ \left(\tilde{I}_{\mu_n}'(w_{\mu_n}),w_{\mu_n}\right) = o_n(1). \end{align} | (3.5) |
Similar to the argument in (2.30), we get \int_{\Omega_{i_0}^d}K(x)f(w_{\mu_n})w_{\mu_n} \mathrm{d}x is bounded. Together with Corollary 2.1 and the generalized Lebesgue-dominated convergence theorem, we derive that
\begin{align} \lim\limits_{n \rightarrow \infty}\int_{\Omega_{i_0}^d}K(x)F(w_{\mu_n})\mathrm{d}x = 0. \end{align} | (3.6) |
By (3.5)-(3.6), the structure of g and \hat{M}(t+s) \ge \hat{M}(t)+ M_0 s for all t , s \ge 0 , we have
\begin{align*} \max\limits_{t \ge 0}I_0(t\omega_{n_0}) \ge \lim\limits_{n \rightarrow \infty}c_{\mu_n}\ge \frac{1}{2}\lim\limits_{n \rightarrow \infty}\hat{M}(\|\nabla w_{\mu_n}\|_2^2+ \tau \|w_{\mu_n}\|_2^2). \end{align*} |
Together with (2.27), we get \lim_{n \rightarrow \infty}(\|\nabla w_{\mu_n}\|_2^2+ \tau \|w_{\mu_n}\|_2^2) < \frac{4 \pi}{\alpha_0} . By (3.5) and (M_1) , we have
\begin{align} M_0 \|w_{\mu_n}\|_{\mu_n}^2 \le \int_{\Omega_{i_0}^d}K(x)f(w_{\mu_n})w_{\mu_n} \mathrm{d}x + \kappa\int_{ \mathbb R^2 \setminus \Omega_{i_0}^d}w_{\mu_n}^2 \mathrm{d}x + o_n(1). \end{align} | (3.7) |
Choose q > 1 (close to 1) and \alpha > \alpha_0 (close to \alpha_0 ) such that q \alpha (\|\nabla w_{\mu_n}\|_2^2+ \tau \|w_{\mu_n}\|_2^2) < 4\pi for n large. By (K_1) , (f_1) - (f_2) and Lemma 2.1 , we have
\begin{align*} \lim\limits_{n \rightarrow \infty}\int_{\Omega_{i_0}^d}K(x)f(w_{\mu_n})w_{\mu_n} \mathrm{d}x = 0. \end{align*} |
Together with (3.7), we get \lim_{n \rightarrow \infty}\|w_{\mu_n}\|_{\mu_n} = 0 . So J'(u_0) = 0 . Since \lim_{n \rightarrow \infty}c_{\mu_n} \ge \eta , we have u_0 \ne 0 . The maximum principle shows that u_0 is positive.
Lemma 3.2. There exists \mu' > 0 such that for \mu > \mu' ,
\begin{align} \|u_\mu\|_{L^\infty( \mathbb R^2 \setminus \Omega_{i_0}^d)}\leq C_0\|u_\mu\|_{H^1( \mathbb R^2 \setminus \Omega_{i_0})}, \end{align} | (3.8) |
where C_0 > 0 is a constant independent of \mu .
Proof. For i\geq2 , let r_i = \frac{2+2^{-i}}{4}r_1 , where r_1 \in (0, \min\{d, 1\}) . For y \in \mathbb R^2 \setminus \Omega_{i_0}^d , define \eta_i\in C_0^\infty(B_{r_i}(y)) such that \eta_i(x) = 1 for x\in B_{r_{i+1}}(y) , 0\leq\eta_i(x) \leq1 for x\in \mathbb R^2 , and |\nabla\eta_i|\leq\frac{2}{r_i-r_{i+1}} for x\in \mathbb R^2 . Let u_\mu^l = \min\{u_\mu, l\} and \beta_i > 1 . By (I'_\mu(u_\mu), \eta_i^2 |u_\mu^l|^{2(\beta_i-1)}u_\mu) = 0 and (M_1) , we get
\begin{align} &M_0\int_{ \mathbb R^2}[|\nabla u_\mu|^2|u_\mu^l|^{2(\beta_i-1)}\eta_i^2+2(\beta_i-1)|\nabla u_\mu^l|^2| u_\mu^l|^{2(\beta_i-1)}\eta_i^2]\mathrm{d}x\\ &+M_0\int_{ \mathbb R^2}|u_\mu|^2|u_\mu^l|^{2(\beta_i-1)}\eta_i^2\mathrm{d}x\\ &\leq\int_{ \mathbb R^2}g(x,u_\mu)u_\mu|u_\mu^l|^{2(\beta_i-1)}\eta_i^2\mathrm{d}x+C\int_{ \mathbb R^2}|\nabla u_\mu||\nabla \eta_i||\eta_i||u_\mu^l|^{2(\beta_i-1)}|u_\mu|\mathrm{d}x. \end{align} | (3.9) |
Let t \ge 2 . By (2.5), (3.9), and Young's inequality, we have
\begin{align} &\int_{ \mathbb R^2}[|\nabla u_\mu|^2|u_\mu^l|^{2(\beta_i-1)}\eta_i^2+2(\beta_i-1)|\nabla u_\mu^l|^2|u_\mu^l|^2| u_\mu^l|^{2(\beta_i-1)}\eta_i^2]\mathrm{d}x\\ &+\int_{ \mathbb R^2}|u_\mu|^2|u_\mu^l|^{2(\beta_i-1)}\eta_i^2\mathrm{d}x\\ &\le C\int_{ \mathbb R^2}|\nabla\eta_i|^2|u_\mu|^2|u_\mu^l|^{2(\beta_i-1)}\mathrm{d}x+C\int_{ \mathbb R^2}|u_\mu|^t(e^{\alpha u_\mu^2}-1)|u_\mu^l|^{2(\beta_i-1)}\eta_i^2\mathrm{d}x. \end{align} | (3.10) |
We note that
\begin{align} \int_{ \mathbb R^2}|u_\mu|^t(e^{\alpha u_\mu^2}-1)|u_\mu^l|^{2(\beta_i-1)}\eta_i^2\mathrm{d}x = \int_{ \mathbb R^2}|u_\mu|^t(e^{\alpha \eta_1^2 u_\mu^2}-1)|u_\mu^l|^{2(\beta_i-1)}\eta_i^2\mathrm{d}x. \end{align} | (3.11) |
By a direct calculation,
\begin{align} \|\eta_1 u_\mu\|_{H^1}^2 \le 2\int_{B_{r_1}(y)}|\nabla u_\mu|^2 \mathrm{d}x+\left(1+2\|\nabla \eta_1\|_{L^\infty( \mathbb R^2)}^2\right)\int_{B_{r_1}(y)}|u_\mu|^2 \mathrm{d}x. \end{align} | (3.12) |
By (3.12) and Lemma 3.1, we can choose \mu' > 0 large such that \|\eta_1 u_\mu\|_{H^1}^2 < \frac{4 \pi}{\alpha_0} for \mu > \mu' . Choose q > 1 (close to 1 ) and \alpha > \alpha_0 (close to \alpha_0 ) such that q \alpha \|\eta_1 u_\mu\|_{H^1}^2 < 4\pi . Then, by Lemma 2.1, there exists C > 0 independent of \mu such that
\begin{align} \int_{ \mathbb R^2}(e^{\alpha \eta_1^2 u_\mu^2}-1)^q\mathrm{d}x \le \int_{ \mathbb R^2}(e^{q\alpha \eta_1^2 u_\mu^2}-1)\mathrm{d}x \le C. \end{align} | (3.13) |
Let t = 2 and p > 2q' with q' = \frac{q}{q-1} . By (3.10)-(3.11), (3.13), and the Sobolev embedding theorem, we obtain that there exists C_p > 0 such that
\begin{align} &\|\eta_i u_\mu(u_\mu^l)^{\beta_i-1}\|_p^2\\ &\leq C_p\int_{ \mathbb R^2}\left[|\nabla[\eta_i u_\mu(u_\mu^l)^{\beta_i-1}]|^2+|\eta_i u_\mu(u_\mu^l)^{\beta_i-1}|^2\right]\mathrm{d}x\\ &\leq 2 C_p \int_{ \mathbb R^2}[|\nabla u_\mu|^2|u_\mu^l|^{2(\beta_i-1)}\eta_i^2+(\beta_i-1)^2|\nabla u_\mu^l|^2|u_\mu^l|^{2(\beta_i-1)}\eta_i^2]\mathrm{d}x\\ &\quad +2 C_p\int_{ \mathbb R^2}|\nabla\eta_i|^2|u_\mu|^2|u_\mu^l|^{2(\beta_i-1)}\mathrm{d}x+C_p\int_{ \mathbb R^2}|u_\mu|^2|u_\mu^l|^{2(\beta_i-1)}\eta_i^2\mathrm{d}x \\ &\le C \beta_i^2 \int_{ \mathbb R^2}|\nabla\eta_i|^2|u_\mu|^2|u_\mu^l|^{2(\beta_i-1)}\mathrm{d}x+C \beta_i^2\|\eta_i u_\mu(u_\mu^l)^{\beta_i-1}\|_{2 q'}^2. \end{align} | (3.14) |
By direct calculation, we obtain
\begin{align} \frac{1}{r_i-r_{i+1}} = \frac{4}{r_1}2^{i+1} > 1. \end{align} | (3.15) |
Let \delta_0 = \frac{2 q'}{p} and \beta_i = \delta_0^{-i} . Then, by (3.14)-(3.15), we have
\begin{align} \|u_\mu(u_\mu^l)^{\beta_i-1}\|_{L^p(B_{r_{i+1}}(y))} \le \frac{C\beta_i}{r_i-r_{i+1}} \|u_\mu(u_\mu^l)^{\beta_i-1}\|_{L^{p \delta_0}(B_{r_i}(y))}. \end{align} | (3.16) |
Let l\rightarrow \infty , we obtain
\begin{align} \|u_\mu\|_{L^{p\beta_i}(B_{r_{i+1}}(y))}&\leq \left(\frac{C \beta_i}{r_i-r_{i+1}}\right)^{\frac{1}{\beta_i}}\|u_\mu\|_{L^{p\beta_{i-1}}(B_{r_i}(y))}. \end{align} | (3.17) |
By (3.17), we derive that
\begin{align*} \|u_\mu\|_{L^{p\beta_i}(B_{r_{i+1}}(y))} &\leq\prod^{i}_{j = 2}(\frac{C\beta_j}{r_j-r_{j+1}})^{\frac{1}{\beta_j}}\|u_\mu\|_{L^{p\beta_{1}}(B_{r_2}(y))}\\ & = \prod^{i}_{j = 2}[\frac{8C}{r_1}(\frac{2}{\delta_0})^j]^{\delta_0^j}\|u_\mu\|_{L^{p\beta_{1}}(B_{r_2}(y))}. \end{align*} |
Let i\rightarrow \infty , we have
\begin{align} \|u_\mu\|_{L^\infty(B_{\frac{1}{2}r_1}(y))} \le C \|u_\mu\|_{L^{p\beta_{1}}(B_{r_2}(y))} \le C_0 \|u_\mu\|_{H^1( \mathbb R^2 \setminus \Omega_{i_0})}. \end{align} | (3.18) |
Since y \in \mathbb R^2 \setminus \Omega_{i_0}^d is arbitrary, we finish the proof.
Lemma 3.3. There exist r_0 , c_1 , c_2 , \mu'' > 0 such that \Omega_{i_0}^d \subset B_{r_0}(0) and for all \mu > \mu'' ,
\begin{align} u_\mu(x) \leq c_2 e^{-c_1 \sqrt{\mu} (|x|-r_0)}, \ \ \forall \ |x| \ge r_0, \end{align} | (3.19) |
where r_0 , c_1 , c_2 are independent of \mu .
Proof. By (M_1) and the structure of g , we obtain that for any x \in \mathbb R^2 \setminus \Omega_{i_0}^d ,
\begin{align*} -M(\|u_\mu\|_{H^1}^2)\Delta u_\mu + \mu V(x) u_\mu +(M_0 - \kappa) u_\mu \le 0. \end{align*} |
Similar to (2.30), we can derive from Lemma 2.7 to obtain that \|u_{\mu}\|_{H^1} is bounded. By (V_4) , there exist r_0 , c_0 > 0 independent of \mu such that \Omega_{i_0}^d \subset B_{r_0}(0) and
\begin{align} -\Delta u_\mu +c_0 \mu u_\mu \le 0, \ \ \forall \ |x| \ge r_0. \end{align} | (3.20) |
By Lemma 3.2, there exists c_2 > 0 such that u_\mu(x) \le c_2 for |x| = r_0 , where c_2 > 0 is independent of \mu > \mu' . Let v_\mu(x) = c_2 e^{-c_1 \sqrt{\mu} (|x|-r_0)} . By choosing c_1 > 0 as small, we obtain
\begin{align} -\Delta v_\mu +c_0 \mu v_\mu \ge 0, \ \ \forall \ |x| \ge r_0. \end{align} | (3.21) |
By (3.20)-(3.21) and the comparison principle, we obtain that u_\mu(x) \le v_\mu(x) for |x| \ge r_0 .
Proof of Theorem 1.1. By Lemma 2.7, there exists u_\mu \in X_\mu such that \hat{I}_\mu(u_\mu) = c_\mu \in [\eta, \max_{t \ge 0}I_0(t\omega_{n_0})] and \hat{I}'_\mu(u_\mu) = 0 . Let q > 2 . By (K_2) and (f_1) - (f_2) , there exists C > 0 such that
\begin{align} \frac{K(x)f(u_\mu)}{u_\mu} \leq C e^{\alpha |x|}\left[u_\mu^{l-1}+|u_\mu|^{q-2}(e^{\alpha u_\mu^2}-1)\right]. \end{align} | (3.22) |
By (3.22) and Lemma 3.3, we derive that there exists \mu'' > 0 such that for \mu \ge \mu'' ,
\begin{align} \frac{K(x)f(u_\mu)}{u_\mu} \le \kappa, \ \ \forall \ |x| \ge 2 r_0. \end{align} | (3.23) |
By (3.22) and Lemmas 3.1-3.2, we derive that there exists \mu''' > 0 such that for \mu \ge \mu''' ,
\begin{align} \frac{K(x)f(u_\mu)}{u_\mu} \le \kappa, \ \ \forall\ x \in B_{2 r_0}(0) \setminus \Omega_{i_0}^d. \end{align} | (3.24) |
By (3.23)-(3.24), we know that u_{\mu} is the nonnegative solution of (1.2). The maximum principle shows that u_\mu 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 M \equiv 1 .
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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