# Existence of Nontrivial Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth

Authors: Quanqing Li; Kaimin Teng; Xian Wu
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We study the following generalized quasilinear Schrödinger equations with critical growth-divg2u∇u+gug′u∇u|2+Vxu=λfx,u+guGu|2⁎-2Gu,x∈RN, where λ>0, N≥3, g(s):R→R+ is a C1 even function, g(0)=1, and g′(s)≥0 for all s≥0, where G(u)≔∫0ug(t)dt. Under some suitable conditions, we prove that the equation has a nontrivial solution by variational method.


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## Body

## 1. Introduction and Preliminaries

Consider the following generalized quasilinear Schrödinger equations with critical growth:(1)-divg2u∇u+gug′u∇u2+Vxu=λfx,u+guGu2∗-2Gu,x∈RN,where λ>0, N≥3, g(s):R→R+ is a C1 even function, g(0)=1, and g′(s)≥0 for all s≥0.The equations are related to the existence of solitary wave solutions for quasilinear Schrödinger equations(2)izt=-Δz+Wxz-kx,zz-Δlz2l′z2z,where z:R×RN→C, W:RN→R is a given potential, l:R→R, and k:RN×R→R are suitable functions. The form of (2) has been derived as models of several physical phenomena corresponding to various types of l(s). For instance, the case l(s)=s models the time evolution of the condensate wave function in superfluid film [1, 2] and is called the superfluid film equation in fluid mechanics by Kurihara [1]. In the case l(s)=(1+s)1/2, problem (2) models the self-channeling of a high-power ultra short laser in matter, the propagation of a high-irradiance laser in a plasma creates an optical index depending nonlinearly on the light intensity, and this leads to interesting new nonlinear wave equations; see [3–6]. For more physical motivations and more references dealing with applications, we can refer to [7–14] and references therein.Setz(t,x)=exp(-iEt)u(x), where E∈R and u is a real function. Then (2) can be reduced to the corresponding equation of elliptic type (see [15]): (3)-Δu+Vxu-Δlu2l′u2u=fx,u,x∈RN,where f(x,u)=k(x,|u|)u. If we take (4)g2u=1+lu2′22,then (1) turns into (3) (see [16]).Moreover, problem (3) also arises in biological models and propagation of laser beams when g(u) is a positive constant (see [17, 18]). In (3), if we set l(u)=u, that is, g2(u)=1+2u2, then we get the superfluid film equation in plasma physics: (5)-Δu+Vxu-Δu2u=fx,u,x∈RN;if we set l(u)=(1+u)1/2, that is, g2(u)=1+u2/2(1+u2), then we get the equation (6)-Δu+Vxu-Δ1+u21/2u21+u21/2=fx,u,x∈RN,which models the self-channeling of a high-power ultrashort laser in matter.In the past, the research on the existence of solitary wave solutions of Schrödinger equations (2) is for some given special function l(s). In this paper, we will use a unified new variable replacement to study (2), constructed by Shen and Wang in [16]. Define the energy functional associated with (1) by (7)Iλu=12∫RNg2u∇u2dx+12∫RNVxu2dx-λ∫RNFx,udx-12∗∫RNGu2∗dx,where F(x,u)≔∫0uf(x,t)dt. However, Iλ is not well defined in H1(RN) because of the term ∫RNg2(u)|∇u|2dx. To overcome this difficulty, we make a change of variable constructed by Shen and Wang in [16]: v≔G(u)≔∫0ug(t)dt. Then we obtain (8)Jλv=12∫RN∇v2dx+12∫RNVxG-1v2dx-λ∫RNFx,G-1vdx-12∗∫RNv2∗dx.If u is a nontrivial solution of (1), then (9)Iλ′u,φ=∫RNg2u∇u∇φ+gug′u∇u2φ+Vxuφ-λfx,uφ-guGu2∗-2Guφdx=0for all φ∈C0∞(RN). Let φ=(1/g(u))ψ. By [16] we know that (9) is equivalent to (10)Jλ′v,ψ=∫RN∇v∇ψ+VxG-1vgG-1vψ-λfx,G-1vgG-1vψ-v2∗-2vψdx=0for all ψ∈C0∞(RN). Therefore, in order to find the nontrivial solution of (1), it suffices to study the existence of the nontrivial solutions of the following equations: (11)-Δv+VxG-1vgG-1v-λfx,G-1vgG-1v-v2∗-2v=0.Recently, the authors studied generalized quasilinear Schrödinger equations with subcritical growth [19, 20], critical growth [21], and supercritical growth [22].In order to reduce the statements for main results, we list the assumptions as follows:(V1)
V(x)≥V0≔infx∈RN⁡V(x)⁡>0 for all x∈RN.(V2)
lim|x|→∞⁡V(x)=V∞<+∞ and V(x)≤V∞ for all x∈RN.(f1)
f∈C(RN×R,R) and there exists 2<p<2∗ such that (12)fx,t≤C1+gtGtp-1for all(x,t)∈RN×R.(f2)
f(x,t)=o(|t|) uniformly in x∈RN as |t|→0.(f3)
(fx,G-1t/gG-1t)t-2F(x,G-1(t))≥(fx,G-1st/gG-1st)st-2F(x,G-1(st)) for all t∈R and s∈[0,1].(f4)
f(x,t)t>0 for all (x,t)∈RN×R\{0}.(f5)
lim|t|→+∞⁡F(x,G-1(t))/t2=+∞ uniformly in x∈RN.SetE=H1(RN) with the norm (13)uE=∫RN∇u2+u2dx1/2.It is easy to prove that Jλ is well defined on E and Jλ∈C1(E,R) under our assumptions and its Gateaux derivative is given by (14)Jλ′v,φ=∫RN∇v∇φ+VxG-1vgG-1vφ-λfx,G-1vgG-1vφ-v2∗-2vφdxfor all v,φ∈E.Our main result of this paper is as follows.Theorem 1.
Suppose that(V1), (V2), and (f1)–(f5) are satisfied. Then if N≥5, (1) admits a nontrivial solution for all λ>0; if N=3,4, (1) admits a nontrivial solution for large λ.Remark 2.
Condition(f3) is weaker than the following condition (f6). (f6)
f(x,G-1(t))/g(G-1(t))t is nonincreasing on (-∞,0) and nondecreasing on (0,+∞).
Indeed, setl(s)=s2tf(x,G-1(t))/g(G-1(t))-2F(x,G-1(st)), ∀s∈[0,1]. Then (15)l′s=2stfx,G-1tgG-1t-2fx,G-1stgG-1stt=2stfx,G-1tgG-1t-2tfx,G-1stgG-1ststst.If (f6) holds, then (16)l′s≥2stfx,G-1tgG-1t-2tfx,G-1tgG-1ttst=0whenever t>0 or t<0. Hence l(s) is nondecreasing on [0,1], and hence l(1)≥l(s) for all s∈[0,1]. Consequently, (f6) implies that (17)tfx,G-1tgG-1t-2Fx,G-1t≥s2tfx,G-1tgG-1t-2Fx,G-1st=s2ttfx,G-1tgG-1tt-2Fx,G-1st≥s2ttfx,G-1stgG-1stst-2Fx,G-1st=stfx,G-1stgG-1st-2Fx,G-1stfor all s∈[0,1]; that is, the condition (f3) holds.From Remark2 we obtain Corollary 3.Corollary 3.
Suppose that(V1), (V2), (f1)-(f2), (f4)-(f5), and (f6) are satisfied. Then if N≥5, (1) admits a nontrivial solution for all λ>0; if N=3,4, (1) admits a nontrivial solution for large λ.Remark 4.
In [16], Shen and Wang studied the existence of nontrivial solutions for generalized quasilinear Schrödinger equations (18)-divg2u∇u+gug′u∇u2+Vxu=hu,x∈RN,where h is a subcritical nonlinearity satisfying the following conditions: (h0)
h(t)=0 if t≤0.(h1)
h(t)=o(t) as t→0+.(h2)
There exists2<p<2∗ such that (19)ht≤C1+gtGtp-1for allt>0.(h3)
There existsμ>2 such that, for any t>0, there holds (20)0<μgG-1tHG-1t≤hG-1tt.As mentioned above, if we setg2(u)=1+2u2, then we get the superfluid film equation in plasma physics (21)-Δu+Vxu-Δu2u=hu,x∈RN,whose nontrivial solutions were studied in [23]. But our problem (1) is elliptic problem involving the critical exponent, so our result extends the results of the work [16, 23] to a critical setting. Moreover, the assumptions about the nonlinearity in this paper are different from the assumptions about the nonlinearity in [16, 23].Remark 5.
In [24], Deng et al. studied problem (1) and their result based on more harsh conditions: (f1)∗
f(x,t)≥0 is differentiable with respect to t∈[0,+∞) for all x∈RN and continuous with respect to x∈RN for all t∈[0,+∞). Moreover, f(x,t)≡0 for all (x,t)∈RN×R-.(f3)∗
There existsδ∈(0,2∗-2) such that, for any t>0, there holds (1+δ)f(x,t)≤G(t)[f(x,t)/g(t)]′, which implies that there exists μ∈(2,2∗) such that f(x,t)G(t)≥μg(t)F(x,t) for all (x,t)∈RN×R.In this paper, we just assume thatf is a continuous function. Moreover, there are functionals f(x,t) satisfying (f3) but not satisfying the above Ambrosetti-Rabinowitz type condition (see Remark  1.2 in [25]). Hence, our result is different from the result there.


## 2. Proof of Theorem1

To begin with, we give some lemmas.Lemma 6.
For the functionsg, G, and G-1, the following properties hold: (1)
the functionsG(·) and G-1(·) are strictly increasing and odd;(2)
G(s)≤g(s)s for all s≥0; G(s)≥g(s)s for all s≤0;(3)
g(G-1(s))≥g(0)=1 for all s∈R;(4)
G-1(s)/s is decreasing on (0,+∞) and increasing on (-∞,0);(5)
|G-1(s)|≤1/g(0)|s|=|s| for all s∈R;(6)
|G-1(s)|/g(G-1(s))≤1/g2(0)|s|=|s| for all s∈R;(7)
G-1(s)s/g(G-1(s))≤|G-1(s)|2 for all s∈R;(8)
lims→0⁡(G-1s/s)=1/g(0)=1 and (22)lims→∞G-1ss=1g∞,ifgis  bounded,0,ifgis  unbounded.Proof.
Properties (1)–(3) are obvious. By (2), we have(23)G-1ss′=s-G-1sgG-1sgG-1ss2≤0for all s>0 and (24)G-1ss′=s-G-1sgG-1sgG-1ss2≥0for all s<0. Consequently, we obtain (4). By mean value theorem and (3), one has (25)G-1s=G-1s-G-10=1gG-1θss≤1g0sfor all s∈R, where θ∈(0,1); that is, (5) is proved. Obviously, (6) is a consequence of (3) and (5). Moreover, (7) is a consequence of (2). Finally, using L’ Hospital’s rule, we know that (8) is satisfied. This completes the proof.Denote(26)hλx,s=Vxs-VxG-1sgG-1s+λfx,G-1sgG-1s.Then (27)Hλx,s≔∫0shλx,tdt=12Vxs2-G-1s2+λFx,G-1s.Consequently, (28)Jλv=12∫RN∇v2+Vxv2dx-∫RNHλx,vdx-12∗∫RNv2∗dx.Lemma 7.
The functionshλ(x,s) and Hλ(x,s) enjoy the following properties under (f1)–(f5): (1)
lim|s|→0⁡(hλx,s/s)=0 and lim|s|→0⁡(Hλx,s/s2)=0 uniformly in x∈RN;(2)
lim|s|→∞⁡(hλx,s/|s|2∗-1)=0 and lim|s|→∞⁡(Hλx,s/|s|2∗)=0 uniformly in x∈RN;(3)
thλ(x,t)-2Hλ(x,t)≥sthλ(x,st)-2Hλ(x,st) for all t∈R and s∈[0,1];(4)
Hλ(x,s)≥0 for all (x,s)∈RN×R;(5)
lim|s|→+∞⁡(Hλx,s/s2)=+∞ uniformly in x∈RN.Proof.
By(f1)-(f2), for any ε>0, there exists Cε>0 such that (29)fx,G-1sgG-1s≤εs+Cεsp-1for all (x,s)∈RN×R. Set G-1(s)=t. Then Lemma 6(8) implies that (30)lims→0hλx,ss=Vx1-1g20+λlimt→0fx,tgtGt=0uniformly in x∈RN. Moreover, by Lemma 6(6) one has (31)lims→∞hλx,ss2∗-1=-Vxlims→∞G-1ssgG-1sss2∗-1+λlimt→∞fx,tgtGt2∗-1=0uniformly in x∈RN. Similarly, we have (32)lims→0Hλx,ss2=0uniformly in x∈RN and (33)lims→∞Hλx,ss2∗=0uniformly in x∈RN. Hence, (1) and (2) hold.
In the following, we setl(t)=G-1(t)2-G-1(t)t/g(G-1(t)),∀t∈R. If t≥0, by Lemma 6(2) and g′(t)≥0 for t≥0, we have (34)Gt1g2tgt-g′tt≤tfor t≥0, which implies that (35)Gttgt′1gt≤tgtfor all t≥0. Let r=G(t). Then (36)Gtddrtgt≤tgtand hence (37)rG-1rgG-1r′≤G-1rgG-1rfor r≥0. Consequently, (38)l′t=2G-1tgG-1t-G-1tgG-1t′t-G-1tgG-1t=G-1tgG-1t-G-1tgG-1t′t≥0for all t≥0, that is, l(t) is increasing with respect to t≥0. Hence l(st)≤l(t) for all s∈[0,1] and t≥0; that is, (39)G-1st2-G-1ststgG-1st≤G-1t2-G-1ttgG-1tfor all s∈[0,1] and t≥0. Note that Lemma 6(1) implies that l(t) is an even function. Therefore, if t<0, we easily obtain that l(st)≤l(t) for all s∈[0,1] and t<0. Consequently, (40)G-1st2-G-1ststgG-1st≤G-1t2-G-1ttgG-1tfor all s∈[0,1] and t∈R. Combining with (f3), we can conclude (3). Moreover, (f4) and Lemma 6(5) imply that H(x,s)≥0 for all (x,s)∈RN×R. Clearly, (f5) and Lemma 6(5) imply that (5) is satisfied. This completes the proof.Lemma 8.
Suppose that(V1), (V2), and (f1)-(f2) are satisfied. Then the energy functional Jλ satisfies the following conditions: (i)
There existβ,ρ>0 such that Jλ(v)≥β for vE=ρ.(ii)
There existse∈E with eE>ρ such that Jλ(e)<0.Proof.
(i) SetSρ≔{u∈E:uE=ρ}. By (f1)-(f2), Lemmas 6(6) and 7(1), and (2), for any ε>0, there exists Cε>0 such that (41)Hλx,s≤εs2+s2∗+Cεspfor all (x,s)∈RN×R. Consequently, for v∈Sρ, we have (42)Jλv≥12C1vE2-C2εvE2-C3εvE2∗-C4CεvEp≥12C1ρ2-C2ερ2-C3ερ2∗-C4Cερp≔β>0for small ε>0 and ρ>0.
(ii) Takev∗∈E\{0}. Then (43)Jλtv∗≤12C5t2v∗E2-12∗t2∗∫RNv∗2∗dx+εt2∫RNv∗2dx+εt2∗∫RNv∗2∗dx+Cεtp∫RNv∗pdx<0for large t>0 and small ε>0. Consequently, we can take e≔t∗v∗ for some large t∗>0 such that (ii) holds. This completes the proof.Lemma 9.
Suppose that(V1), (V2), and (f1)–(f4) are satisfied. Then there exists a bounded Cerami sequence {vn}⊂E for Jλ with Jλ(vn)→cλ≥β>0, where (44)cλ≔infγ∈Γsupt∈0,1Jλγt,Γ≔γ∈C0,1,E:γ0=0,Jλγ1<0,β is the constant appearing in Lemma 8.Proof.
By Lemma8 and the mountain pass theorem without (PS) condition (see Theorem  4.1 in [26]), there exists a Cerami sequence {vn}⊂E satisfying (45)Jλvn⟶cλ≥β>0,1+vnEJλ′vnE∗⟶0,where (46)cλ≔infγ∈Γsupt∈0,1Jλγt,Γ≔γ∈C0,1,E:γ0=0,Jλγ1<0,β is the constant appearing in Lemma 8.
Lettn∈[0,1] be such that Jλ(tnvn)=maxt∈[0,1]⁡Jλ(tvn). Then {Jλ(tnvn)} is bounded from above. Indeed, without loss of the generality, we may assume that tn∈(0,1) for all n∈N. Hence, by Lemma 7(3) we have (47)Jλtnvn=Jλtnvn-12Jλ′tnvn,tnvn=12-12∗tn2∗∫RNvn2∗dx+∫RN12tnvnhλx,tnvn-Hλx,tnvndx≤12-12∗∫RNvn2∗dx+∫RN12vnhλx,vn-Hλx,vndx=Jλvn-12Jλ′vn,vn=cλ+o1.This shows that {Jλ(tnvn)} is bounded from above.
Now, we prove that{vn} is bounded in E. Otherwise, if vnE is unbounded, then, up to a subsequence, we may assume that vnE→+∞. Set wn=vn/vnE. Then there exists w∈E such that wn⇀w in E. By Jλ(vn)→cλ, we have (48)o1+12max⁡1,V∞≥12∗vn2∗2∗vnE2+∫RNHλx,vnvnE2dx.Set Ω={x∈RN:w(x)≠0}. If meas(Ω)>0, then by Lemma 7(4) and Fatou Lemma, one has (49)o1+12max⁡1,V∞≥12∗vn2∗2∗vnE2+∫RNHλx,vnvnE2dx≥12∗∫Ωwn2vn2∗-2dx⟶+∞as n→∞. This is a contradiction. Hence |Ω|=0, that is, w=0 a.e. on RN. For any B>0, by vnE→+∞ we have (50)Jλtnvn≥JλBvnEvn=JλBwn≥B22min⁡1,V0-∫RNHλx,Bwndx-B2∗2∗∫RNwn2∗dxfor n sufficiently large. By (29), Lemmas 6(6) and 7(1), and (2), for any ε>0, there exists Cε>0 such that (51)hλx,ss≤εs2+s2∗+Cεspfor all (x,s)∈RN×R. Consequently, (52)∫RNwn2∗dx≤max⁡1,V∞vnE2∗-2-1vnE2∗∫RNhλx,vnvndx+o1⟶0as n→∞ and so ∫RN|wn|pdx→0 as n→∞ by using interpolation inequality. Moreover, (41) implies that (53)∫RNHλx,Bwndx≤εB2∫RNwn2dx+εB2∗∫RNwn2∗dx+CεBp∫RNwnpdx.By the arbitrariness of ε, we obtain ∫RNHλ(x,Bwn)dx→0 as n→∞. Hence (54)liminfn→∞Jλtnvn≥B22min⁡1,V0,∀B>0.This contradicts the fact that {Jλ(tnvn)} is bounded from above. Consequently, {vn} is bounded in E. This completes the proof of Lemma 9.Lemma 10.
Suppose that(V1), (V2), and (f1)–(f5) are satisfied. Then if N≥5, the minimax level cλ satisfies cλ<1/NSN/2 for all λ>0; if N=3,4, the minimax level cλ satisfies cλ<1/NSN/2 for large λ, where S is the best constant of the embedding D1,2(RN)↪L2∗(RN).Proof.
From the minimax characterization ofcλ we see that it is sufficient to show that there exists v0∈E\{0} such that supt≥0⁡Jλ(tv0)<1/NSN/2.
We follow the strategy used in [24] but need to modify some process. Given ε>0, we consider the function (55)wεx=NN-2εN-2/4ε+x2N-2/2,which satisfies the following equations:(56)-Δu=u2∗-1,inRN,u∈D1,2RN,ux>0,inRN.Moreover, wε(x) satisfies (57)∇wε22=wε2∗2∗=SN/2.Let φ∈C0∞(RN,[0,1]) be such that φ(x)≡1 for |x|≤ρε and φ(x)≡0 for |x|≥2ρε, where ρε≔ετ with τ∈(1/4,1/2). Set ψε(x)=φ(x)wε(x). Then (58)∫RN∇ψε2dx=SN/2+OεN-2/2,∫RNψε2∗dx=SN/2+OεN/2,∫RNψεdx≤CεN-2/4,∫RNψε2∗-1dx≤CεN-2/4,∫RN∇ψεdx≤CεN-2/4,∫RNψε2dx=Cε+OεN-2/2,ifN≥5,Cεlnε+Oε,ifN=4,Oε1/2,ifN=3.Since Jλ(0)=0 and limt→∞⁡Jλ(tψε)=-∞, there exists tε>0 such that Jλ(tεψε)=maxt≥0⁡Jλ(tψε). We claim that there exist two positive constants t1, t2 independent of ε such that (59)t1≤tε≤t2for small ε>0. Indeed, by 〈Jλ′(tεψε),ψε〉=0 we have (60)∫RN∇ψε2+Vxψε2dxψε2∗2∗-tε2∗-2-∫RNhλx,tεψεtεψεdxtε2ψε2∗2∗=0.By (29), Lemmas 6(6) and 7(1), and (2), for any δ>0, there exists Cδ>0 such that (61)hλx,ss≤δs2∗+Cδs2for all (x,s)∈RN×R. Consequently, (62)∫RNhλx,tεψεtεψεdxtε2ψε2∗2∗≤∫RNδtε2∗ψε2∗+Cδtε2ψε2dxtε2ψε2∗2∗=δtε2∗-2+Cδψε22ψε2∗2∗=δtε2∗-2+CδSN/2+OεN/2-1ψε22≤δtε2∗-2+CS-N/2ψε22=δtε2∗-2+CS-N/2Cε+OεN-2/2,ifN≥5,Cεlnε+Oε,ifN=4,Oε1/2,ifN=3=δtε2∗-2+o1as ε→0. Note that (63)ψεE2ψε2∗2∗=∇ψε22+ψε22ψε2∗2∗=1SN/2+OεN/2SN/2+OεN-2/2+Cε+OεN-2/2,ifN≥5,SN/2+OεN-2/2+Cεlnε+Oε,ifN=4,SN/2+OεN-2/2+Oε1/2,ifN=3⟶1as ε→0. Hence by (60) one has (64)0≥min⁡1,V01+o1-tε2∗-2-δtε2∗-2+o1as ε→0, which implies that (65)tε≥min⁡1,V021+δ1/2∗-2≔t1>0for ε>0 small enough. On the other hand, (60) leads to (66)tε2∗-2≤max⁡1,V∞ψεE2ψε2∗2∗+∫RNhλx,tεψεtεψεdxtε2ψε2∗2∗≤max⁡1,V∞1+o1+δtε2∗-2+o1as ε→0, which implies that (67)tε≤2max⁡1,V∞1-δ1/2∗-2≔t2<+∞for δ>0 and ε>0 small enough.
SinceQ(t)≔t2/2-t2∗/2∗ has only maximum at t=1, one has (68)Jλtεψε=12tε2∫RN∇ψε2dx+12tε2∫RNVxψε2dx-∫RNHλx,tεψεdx-12∗tε2∗∫RNψε2∗dx=tε22-tε2∗2∗SN/2+OεN-2/2+12tε2∫RNVxψε2dx-∫RNHλx,tεψεdx≤1NSN/2+OεN-2/2+12t22V∞∫RNψε2dx-∫RNHλx,tεψεdx.Notice that, for x∈Bρε, we have (69)tεψε=tεwε=tεNN-2εN-2/4ε+x2N-2/2≥CtεNN-2N-2/4εN-2/4ετN-2≥Ct1NN-2N-2/4εN-21/4-τ⟶+∞as ε→0, which combining with Lemma 7(4) and (5) implies that for any M>0(70)∫RNHλx,tεψεdx≥Mtε2∫Bρεψε2dxfor ε>0 small enough. Note that (71)∫Bρεψε2dx=NN-2N-2/2NωNε∫0ρε/εsN-11+s2N-2ds,∫0+∞sN-11+s2N-2ds≥∫01sN-11+s2N-2ds≥1N·12N-2≔C>0.Consequently, (72)∫RNHλx,tεψεdx≥MCεfor ε>0 small enough. Hence by (68) (73)Jλtεψε≤1NSN/2+OεN-2/2+12t22V∞∫RNψε2dx-∫RNHλx,tεψεdx≤1NSN/2+OεN-2/2-MCε+Cε+OεN-2/2,ifN≥5,εlnε+Oε,ifN=4,Oε1/2,ifN=3.From this, we see that Jλ(tεψε)<(1/N)SN/2 for ε>0 small enough and M big enough if N≥5. Consequently, cλ<(1/N)SN/2 for all λ>0 if N≥5.
In the following, we consider the caseN=3,4. Indeed, if the conclusion is false, then there exists a sequence {λn} with λn→+∞ such that cλn≥(1/N)SN/2. Take v∈E\{0}. Then by the proof of Lemma 8, there exists a unique tλn>0 such that maxt>0⁡Jλn(tv)=Jλn(tλnv). Hence (74)tλn2∫RN∇v2dx+∫RNVxG-1tλnvgG-1tλnvtλnvdx=tλn2∗∫RNv2∗dx+λn∫RNfx,G-1tλnvgG-1tλnvtλnvdx.By Lemma 6(6) and (f4) we get (75)max⁡1,V∞vE2≥tλn2∗-2∫RNv2∗dx,which implies that {tλn} is bounded. Hence, up to a subsequence, there exists t0≥0 such that tλn→t0 as n→∞. If t0>0, then by (f4) and Fatou lemma we have (76)limn→∞tλn2∗∫RNv2∗dx+λn∫RNfx,G-1tλnvgG-1tλnvtλnvdx=+∞.But, on the other hand, by Lemma 6(6) one has (77)tλn2∗∫RNv2∗dx+λn∫RNfx,G-1tλnvgG-1tλnvtλnvdx≤max⁡1,V∞tλn2vE2⟶max⁡1,V∞t02vE2,a contradiction. Hence t0=0 and by Lemma 7(4) we know that (78)maxt>0⁡Jλntv=Jλntλnv≤12max⁡1,V∞tλn2vE2-12∗tλn2∗∫RNv2∗dx⟶0as n→∞. Consequently, (79)0<1NSN/2≤cλn≤infu∈E\0maxt>0⁡Jλntu≤maxt>0⁡Jλntv⟶0,a contradiction. This completes the proof.Proof of Theorem1.
Since{vn}⊂E is a bounded Cerami sequence for Jλ at the level cλ>0, there exists v∈E such that (80)vn⇀vinE,vn⟶vinLlocqRNfor1≤q<2∗,vnx⟶vxa.e.onRN.Using a standard argument, we know that Jλ′(v)=0, that is, v is a weak solution of (11). Indeed, for any ψ∈C0∞(RN), we have (81)o1=Jλ′vn,ψ=∫RN∇vn∇ψdx+∫RNVxvnψdx-∫RNvn2∗-2vnψdx-∫RNhλx,vnψdx.Since vn⇀v in E, one has (82)∫RN∇vn∇ψdx⟶∫RN∇v∇ψdx,∫RNVxvnψdx⟶∫RNVxvψdx,∫RNvn2∗-2vnψdx⟶∫RNv2∗-2vψdx,∫RNhλx,vnψdx⟶∫RNhλx,vψdx.Consequently, (83)0=∫RN∇v∇ψdx+∫RNVxvψdx-∫RNv2∗-2vψdx-∫RNhλx,vψdxfor all ψ∈C0∞(RN). For any φ∈E, there exists a sequence {ψn}⊂C0∞(RN) such that ψn→φ in E. Hence (84)0=∫RN∇v∇ψndx+∫RNVxvψndx-∫RNv2∗-2vψndx-∫RNhλx,vψndx.Let n→∞, we get (85)0=∫RN∇v∇φdx+∫RNVxvφdx-∫RNv2∗-2vφdx-∫RNhλx,vφdx;that is, 〈Jλ′(v),φ〉=0 for all φ∈E. Hence Jλ′(v)=0; that is, v is a weak solution of (11).
In the following, we prove thatv is nontrivial. With the aid of Lemma 10, the proof follows essentially the proof of Theorem  1.1 in [16]. For completeness, we present the proof as follows. If the conclusion is false, we may assume v=0. We divide the proof into four steps.
Step  1. We prove that {vn}⊂E is also a Cerami sequence for the functional Jλ∞:E→R, where (86)Jλ∞vn=12∫RN∇vn2+V∞vn2dx-∫RNHλx,vndx-12∗∫RNvn2∗dx.By (V2) and vn⇀0 in E, one has (87)Jλvn-Jλ∞vn=12∫RNVx-V∞vn2dx⟶0as n→∞. Similarly, we have (88)Jλ′vn-Jλ∞′vnE∗=sup∥φ∥E≤1Jλ′vn-Jλ∞′vn,φ=supφE≤1∫RNVx-V∞vnφdx⟶0as n→∞. Consequently, {vn} is also a Cerami sequence of Jλ∞.
Step  2. There exist α,R>0 and {yn}⊂RN such that (89)limn→∞∫BRynvn2dx≥α>0.Indeed, by contradiction, then by Lemma  1.21 in [27], one has vn→0 in Lq(RN) for 2<q<2∗. Notice that (90)o1=Jλ′vn,vn=∫RN∇vn2+Vxvn2dx-∫RNhλx,vnvndx-∫RNvn2∗dx,which combining with (51) leads to (91)∫RN∇vn2+Vxvn2dx-∫RNvn2∗dx⟶0as n→∞. Consequently, there exists a constant l≥0 such that (92)∫RN∇vn2+Vxvn2dx⟶l,∫RNvn2∗dx⟶l.Obviously, l>0. Otherwise, Jλ(vn)→0 as n→∞, which contradicts with cλ>0. Hence by the definition of S, we have (93)S≤∫RN∇vn2dx∫RNvn2∗dx2/2∗≤∫RN∇vn2+Vxvn2dx∫RNvn2∗dx2/2∗⟶ll2/2∗=l2/N;that is, l≥SN/2. Therefore, (41) implies that (94)cλ+o1=Jλvn=12∫RN∇vn2+Vxvn2dx-∫RNHλx,vndx-12∗∫RNvn2∗dx⟶12-12∗l=1Nl≥1NSN/2,as n→∞, which implies that cλ≥(1/N)SN/2, a contradiction.
Step  3. After a translation of {vn} called {v~n}, then v~n converges weakly to a nonzero critical point of Jλ∞.
Setv~n(x)=vn(x+yn). Since {vn}⊂E is a Cerami sequence of Jλ∞ and v~nE=vnE, arguing as in the case of {vn}, we may assume v~n⇀v~ in E and (Jλ∞)′(v~)=0. So by Step 2 we know v~≠0. By Lemma 7(3) and Fatou Lemma, one has (95)2cλ=liminfn→∞2Jλ∞v~n-Jλ∞′v~n,v~n≥liminfn→∞∫RNhλx,v~nv~n-2Hλx,v~ndx+1-22∗liminfn→∞∫RNv~n2∗dx≥∫RNhλx,v~v~-2Hλx,v~dx+1-22∗∫RNv~2∗dx=2Jλ∞v~-Jλ∞′v~,v~=2Jλ∞v~,which implies that Jλ∞(v~)≤cλ.
Step  4. We use v~ to construct a path which allows us to obtain a contradiction with the definition of mountain pass level cλ.
Define the mountain pass levelcλ∞≔infγ∈Γ∞⁡supt∈[0,1]⁡Jλ∞(γ(t))>0, where Γ∞≔{γ∈C([0,1],E):γ(0)=0,Jλ∞(γ(1))<0}. It follows the arguments used in [28, 29], we can construct a path γ:[0,1]→E such that (96)γ0=0,Jλ∞γ1<0,v~∈γ0,1,γtx>0,∀x∈RN,t∈0,1,maxt∈0,1⁡Jλ∞γt=Jλ∞v~.Then cλ∞≤maxt∈[0,1]⁡Jλ∞(γ(t))=Jλ∞(v~). If V(x)≡V∞, we have already proved Theorem 1. If V(x)≤V∞ but V(x)≢V∞, we take the path γ given by above, and by γ∈Γ∞⊂Γ, we have (97)cλ≤maxt∈0,1⁡Jλγt=Jλγt-<Jλ∞γt-≤maxt∈0,1⁡Jλ∞γt=Jλ∞v~≤cλ,a contradiction. Consequently, v≢0. This completes the proof of Theorem 1.



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*Source: 3615085-2018-01-03.xml*