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# Solutions for a class of quasilinear Schrödinger equations with critical Sobolev exponents

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## Abstract

In this paper, we study a class of quasilinear Schrödinger equations involving a critical exponent which arises in plasma physics. By using the change of variable and variational approaches combining the concentration-compactness principle, the existence of a positive solution which has a local maximum point and decays exponentially is obtained.
Solutions for a class of quasilinear Schrödinger equations with critical
Sobolev exponents
Zhouxin Li and Yimin Zhang
Citation: J. Math. Phys. 58, 021501 (2017); doi: 10.1063/1.4975009
View online: http://dx.doi.org/10.1063/1.4975009
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JOURNAL OF MATHEMATICAL PHYSICS 58, 021501 (2017)
Solutions for a class of quasilinear Schrödinger equations
with critical Sobolev exponents
Zhouxin Li1,a) and Yimin Zhang2,a)
1School of Mathematics and Statistics, Central South University, Changsha 410083,
People’s Republic of China
2Department of Mathematics, School of Science, Wuhan University of Technology,
Wuhan 430070, People’s Republic of China
(Received 15 October 2015; accepted 11 January 2017; published online 2 February 2017)
In this paper, we study a class of quasilinear Schrödinger equations involving a critical
exponent which arises in plasma physics. By using the change of variable and varia-
tional approaches combining the concentration-compactness principle, the existence
of a positive solution which has a local maximum point and decays exponentially is
I. INTRODUCTION
In this paper, we are interested in the existence of a positive solution for the following quasilin-
ear Schrödinger equation:
ε2u+V(x)ukαε2((|u|2α))|u|2α2u=|u|q2u+|u|2(2α)2u,u>0,xRN,(1.1)
where V(x)is a given potential, ε > 0 is a real parameter, kis a real constant, q2, α > 1/2 is
a constant, 2(2α)=2×2α, and 2=2N
N2is the critical Sobolev exponent. Solutions of problem
(1.1) are related to the standing wave solutions of certain quasilinear Schrödinger equations which
arise in several physical phenomena such as the theory of superﬂuid ﬁlm in plasma physics, see
Refs. 7and 12 and the references therein for more backgrounds. For this reason, problem (1.1) is
still called a quasilinear Schrödinger equation.
For α=1, the existence of nontrivial solutions for the quasilinear schrödinger equation was
extensively studied in the past decades. We can see Refs. 4,5, and 13 for the subcritical case
and Refs. 14,16,19, and 20 for the critical case. In Ref. 13, by using a change of variable, they
transform the equation to a semilinear one, then the existence of solutions was obtained via varia-
tional methods under dierent types of potentials V(x). This method is signiﬁcant and was widely
used in the studies of this kind of problems. For general α, there are few results for this case, as far
as we know, just Refs. 1,2,12, and 15. In Ref. 12, for α > 1
2, 2 <q+1<2(2α), Liu and Wang
considered Equation (1.1) without a critical term, that is,
u+V(x)ukα((|u|2α))|u|2α2u=λ|u|q2u,u>0,xRN,(1.2)
where λ > 0 is a parameter. They obtained the existence of a solution for problem (1.2) by using
the method of Lagrange multiplier. In Ref. 15, the ﬁbering method was employed to obtain the
existence of at least one or sometimes two standing wave solutions for the following quasilinear
Schrödinger equation:
u+V(x)ukα((|u|2α))|u|2α2u=µf(x)|u|q2u,u>0,xRN,
where V(x)=λg(x)and g(x)ia a positive function and µand λare positive numbers. Employing
the change of variable method just as in Ref. 13, the authors of Ref. 1obtained the existence of at
least one positive solution for problem (1.2) by using variational approaches. Moreover, in Ref. 2,
0022-2488/2017/58(2)/021501/15/$30.00 58, 021501-1 Published by AIP Publishing. 021501-2 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) for V(x)=µ,α > 1 2, they obtained the unique existence of a positive radial solution of problem (1.2) under some suitable conditions and λ=1. All these results are obtained when the nonlinear term satisﬁes subcritical growth. If the right side of Equation (1.2) has a critical term, as far as we know, there are no results for general α.16 In this paper, our aim is to study the existence of positive solutions of (1.1) with general α > 1/2 and at 2(2α)growth. Problem of (1.1) at 2(2α)growth has two diculties. First, the embedding H1(RN)L2(2α)(RN)is not compact, so it is hard to prove the Palais-Smale (PS in short) condition. Second, even if we can obtain the compactness result of PS sequence, it only holds at some level of a positive upper bound and it is dicult for us to prove that the functional has such a minimax level. We assume that V(x)is locally Hölder continuous and (V)V>V0>0 such that minxRNV(x)=V0and lim|x|→∞ V(x)=V. Under assumption (V), we deﬁne a space XBuH1(RN):RN V(x)u2< with the norm u2 X=RN|u|2+RNV(x)u2. For the simplicity of notation, we let m=2α, ¯q=q/m, and kα=1 in this paper. Set g(t)=|t|q2t+|t|2m2tand G(t)=t 0 g(s)ds. We formulate problem (1.1) in the variational structure in the space Xas follows: I(u)=ε2 2RN (1+m|u|2(m1))|u|2+1 2RN V(x)u2RN G(u). Note that Iis lower semicontinuous on X; we deﬁne that uXis a weak solution for (1.1) if uXL(RN)and it is a critical point of I. First, for an arbitrary ε > 0, we have Theorem 1.1. Assume that q (2m,2m)and that condition (V) holds. Moreover, assume that one of the following conditions holds (i) 1<m<2and ¯q>4 N2+2 m; (ii) m 2and ¯q>21. Then for ε > 0small enough, problem (1.1) has a positive weak solution uεXL(RN)with lim ε0uεX=0and uε(x)Cexp(β ε|xxε|), where C >0, β > 0are constants, x εRNis a local maximum point of uε. Next, we consider the case ε=1. We have the following result: Theorem 1.2. Assume that all conditions in Theorem 1.1 hold and that ε=1, then problem (1.1) has a positive weak solution u1XL(RN). This paper is organized as follows. In Section II, we ﬁrst use a change of variable to reformulate the problem, then we modify the functional in order to regain the PS condition. In Section III, we prove that the functional satisﬁes the PS condition, which is a crucial job of this paper. Finally, in Section IV, we prove the main theorems, which involves the construction of a mountain pass level at a certain height. 021501-3 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) II. PRELIMINARIES Since Iis lower semicontinuous on X, we follow the idea in Refs. 5and 13 and make the change of variables v=f1(u), where fis deﬁned by f(0)=0, f(v)=(1+m|f(v)|2(m1))1/2,on [0,+), f(v)=f(v),on (−∞,0]. The above function f(t)and its derivative satisfy the following properties (see Refs. 1,2, and 13): Lemma 2.1. For m >1, we have (1) f is uniquely deﬁned, C2and invertible; (2) |f(t)| 1for all t R; (3) |f(t)| |t|for all t R; (4) f (t)/t1as t 0; (5) |f(t)| m1/2m|t|1/mfor all t R; (6) 1 mf(t)t f (t)f(t)for all t >0; (7) f (t)/m tm1/2mas t +. According to Ref. 6(see Corollary 2.1 and Proposition 2.2 in it, note that the embedding in Corollary 2.1 of Ref. 6is also compact.), we have Lemma 2.2. The map: v→ f(v)from X into Lr(RN)is continuous for 1r2m and is compact for 1r<2m. Using this change of variable, we rewrite the functional I(u)to J(v)=I(f(v)) =ε2 2RN |v|2+1 2RN V(x)f2(v)RN G(f(v)). The critical point of Jis the weak solution of the equation ε2v+V(x)f(v)f(v)=g(f(v)) f(v),xRN.(2.1) Now we deﬁne a suitable modiﬁcation of the functional Jin order to regain the Palais-Smale condition. In this time, we make use of the method in Ref. 17. Let lbe a positive constant such that l=sup{s>0 : g(t) tV0 kfor every 0 ts}(2.2) for some k> θ/(θ2)with θ(2m,q]. We deﬁne the functions, γ(s)= g(s),s>0, 0,s0¯γ(s)= γ(s),0sl, V0 ks,s>l and p(x,s)=χR(x)γ(s)+(1χR(x)) ¯γ(s), P(x,s)=s 0 p(x,t)dt, where χRdenotes the characteristic function of the set BR(the ball centered at 0 and with radius R in RN) and R>0 is suciently large such that min BR V(x)<min BR V(x). By deﬁnition, the function p(x,s)is measurable in x, of class Cin sand satisﬁes (p1) 0 < θP(x,s)p(x,s)sfor every xBRand sR+. 021501-4 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) (p2) 0 2P(x,s)p(x,s)s1 kV(x)s2for every xBc RBRN\BRand sR+. Now we study the existence of solutions for the deformed equation ε2v+V(x)f(v)f(v)=p(x,f(v)) f(v),xRN.(2.3) The corresponding functional of (2.3) is given by ¯ J(v)=ε2 2RN |v|2+1 2RN V(x)f2(v)RN P(x,f(v)). For vX, since |(| f(v)|m)|2=m2|f(v)|2(m1) 1+m|f(v)|2(m1)|v|2m|v|2,(2.4) we infer that |f(v)|mX. By Sobolev inequality, we have f(v)∥2m=| f(v)|m1/m 2C(| f(v)|m)∥1/m 2Cv1/m X.(2.5) It results that f(v)L2m(RN). Using interpolation inequality, we obtain that f(v)Lq(RN). Thus ¯ Jis well deﬁned on X. Let (vn)X, v Xwith vnvin X. Then from Lemma 2.2, we infer that V(x)f2(vn)V(x)f2(v)in L1(RN)and that f(vn)f(v)in Lq(RN). Thus ¯ Jis continuous on X.¯ J is Gateaux-dierentiable in Xand the G-derivative is ¯ J(v), ϕ=ε2RNvϕ+RN V(x)f(v)f(v)ϕRN p(x,f(v)) f(v)ϕ, ϕX. Then if vXL(RN)is a critical point of ¯ J, and v(x)aBf1(l),xBc R, we have u= f(v)XL(RN)(note that we have |u||v|and |u||v|by the properties of f) is a solution of (1.1). III. COMPACTNESS OF PS SEQUENCE In this section, we show that the functional ¯ Jsatisﬁes the PS condition which is a crucial job and its proof is composed of four steps. Let Sdenote the best Sobolev constant, we have Lemma 3.1. Assume that condition (V) holds and q (2m,2m). Then ¯ J satisﬁes the PS condi- tion at level cε<1 N m εNSN/2. Proof. Let (vn)Ebe a PS sequence of ¯ Jat level cε, that is, (vn)satisﬁes ¯ J(vn)=ε2 2RN |vn|2+1 2RN V(x)f2(vn)RN P(x,f(vn)) =cε+o(1)(3.1) and ¯ J(vn), ϕ=ε2RNvnϕ+RN V(x)f(vn)f(vn)ϕ RN p(x,f(vn)) f(vn)ϕ=o(1)∥ϕX,ϕX.(3.2) We divide the proof into four steps. Step 1: The sequence RN(|vn|2+V(x)f2(vn)) is bounded. Multiplying (3.1) by θ(θis given in Section II) and using (p1)-(p2), we get θε2 2RN |vn|2+θ 2RN V(x)f2(vn) BR p(x,f(vn)) f(vn)+θ 2kBc R V(x)f2(vn)+θcε+o(1). 021501-5 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) On the other hand, taking ϕ=f(vn)/f(vn)in (3.2), we get RN ε2(1+m(m1)| f(vn)|2(m1) 1+m|f(vn)|2(m1))|vn|2+RN V(x)f2(vn) =RN p(x,f(vn)) f(vn)+o(∥vnX)BR p(x,f(vn)) f(vn)+o(1)∥vnX. Combining the above two inequalities, we get (θ 2m)ε2RN |vn|2+(θ 2θ 2k1)RN V(x)f2(vn) θε2 2RN |vn|2RN ε2(1+m(m1)| f(vn)|2(m1) 1+m|f(vn)|2(m1))|vn|2 +θ 2RN V(x)f2(vn)θ 2kBc R V(x)f2(vn)RN V(x)f2(vn) θcε+o(1)+o(1)∥vnX.(3.3) Since θ > 2mand k>θ θ2, we get the conclusion from (3.3). Step 2: For every δ > 0, there exists R1R>0 such that lim sup nBc 2R1 (|vn|2+V(x)f2(vn)) < δ. (3.4) We consider a cutofunction ψR1=0 on BR1,ψR1=1 on Bc 2R1, and |ψR1|C/R1on RNfor some constant C>0. On the one hand, taking ϕ=f(vn)/f(vn), we compute ¯ J(vn), ϕψR1and get o(1)∥vnX=RN ε2(1+m(m1)| f(vn)|2(m1) 1+m|f(vn)|2(m1))|vn|2ψR1 +RN ε2ϕvnψR1+RN V(x)f2(vn)ψR1 RN p(x,f(vn)) f(vn)ψR1 RN ε2|vn|2ψR1+RN ε2ϕvnψR1 +(11 k)RN V(x)f2(vn)ψR1.(3.5) On the other hand, by Hölder inequality, RN ϕvnψR1C R1 vnL2(RN)ϕL2(RN).(3.6) Note that vnL2(RN)is bounded, and ϕ2 L2(RN)=RN f2(vn)(1+m|f(vn)|2(m1)) =RN f2(vn)+mRN |f(vn)|2m,(3.7) by (2.5), ϕL2(RN)is also bounded. Therefore, it follows from (3.5)-(3.7) that lim sup nBc 2R1 (|vn|2+V(x)f2(vn)) C R1 for R1suciently large, which yields (3.4). Step 3: There exists vXsuch that lim n→ ∞ RN p(x,f(vn)) f(vn)=RN p(x,f(v)) f(v).(3.8) 021501-6 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) First, by step 1, there exists vXsuch that up to a subsequence, vnvweakly in Xand vnv a.e. in RN. Since we may replace vnby |vn|, we assume vn0 and v0. By (3.4), for any δ > 0, there exists R1>0 suciently large such that lim sup nBc 2R1 (|vn|2+V(x)f2(vn)) kδ. Therefore, by (p2) we have lim sup nBc 2R1 p(x,f(vn)) f(vn)lim sup nBc 2R1 V(x) kf2(vn)δ, (3.9) and by Fatou lemma, Bc 2R1 p(x,f(v)) f(v)δ. (3.10) Second, we prove that B2R1 p(x,f(vn)) f(vn)B2R1 p(x,f(v)) f(v).(3.11) Then from this, (3.9)-(3.10), and the arbitrariness of δ, we get (3.8). In fact, since (vn)is bounded in X, we have (f(vn)) is also bounded. Thus there exists a wXsuch that f(vn)⇀ w in X, f(vn)win Lr(BR1)for 1 r<2, and f(vn)wa.e. in BR1. According to (2.4), (| f(vn)|m) is also bounded in X. By a normal argument, we have |f(vn)|m|w|min X,|f(vn)|m|w|min Lr(BR1)for 1 r<2, and |f(vn)|m|w|ma.e. in BR1. Applying Lions’ concentration compact- ness principle11 to (| f(vn)|m)on ¯ BR1, we obtain that there exist two nonnegative measures µ, ν, a countable index set K, positive constants {µk},{νk},kK, and a collection of points {xk},kK in ¯ BR1, such that (i) ν=|w|2m+ kK νkδxk; (ii) µ=|(|w|m)|2+ kK µkδxk; (iii) µkSν2/2 k, where δxkis the Dirac measure at xkand Sis the best Sobolev constant. We claim that νk=0 for all kK. In fact, let xkbe a singular point of measures µand ν, as in Ref. 9, we deﬁne a function φC 0(RN)by φ(x)= 1,Bρ(xk), 0,RN\B2ρ(xk), φ0,|φ|1 ρ,B2ρ(xk)\Bρ(xk), where Bρ(xk)is a ball centered at xkand with radius ρ > 0. We take ϕ=φf(vn)/f(vn)as test functions in ¯ J(vn), ϕand get RN ε2(1+m(m1)| f(vn)|2(m1) 1+m|f(vn)|2(m1))|vn|2·φ +RN ε2vnφ·f(vn)/f(vn)+RN V(x)f2(vn)φ RN p(x,f(vn)) f(vn)φ=o(1)∥vnφX.(3.12) Then Lions’ concentration compactness principle implies that BR1 ||f(vn)|m|2φBR1 φdµ, BR1 |f(vn)|2mφBR1 φdν. (3.13) 021501-7 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) Since xkis singular point of ν, by the continuity of f, we have f(vn(x))|(B2ρ\{xk}) → ∞ as ρ0. Thus 1+m(m1)| f(vn)|2(m1) 1+m|f(vn)|2(m1)=mo(ρ) on B2ρfor ρsuciently small. Then by (2.4) we get from (3.12) that BR1 ε2φdµBR1 φdν =lim nBR1 ε2||f(vn)|m|2φBR1 |f(vn)|2mφ lim nBR1 mε2|vn|2φBR1 |f(vn)|2mφ lim nBR1 ε2(1+m(m1)| f(vn)|2(m1) 1+m|f(vn)|2(m1))|vn|2φ +o(ρ)BR1 ε2|vn|2φBR1 |f(vn)|2mφ lim nBR1 ε2vnφ·f(vn)/f(vn)+λBR1 |f(vn)|qφ +o(ρ)BR1 ε2|vn|2φ+o(1)∥vnφX.(3.14) We prove that the last inequality in (3.14) tends to zero as ρ0. By Hölder inequality, we have lim n→ ∞ BR1vnφ·f(vn)/f(vn) lim sup n(BR1 |vn|2)1/2 ·(BR1 |[ f(vn)/f(vn)] · ∇φ|2)1/2 .(3.15) Since |f(vn)/f(vn)|2=f2(vn)+m|f(vn)|2m, using Hölder inequality we have lim n→ ∞ BR1 |[ f(vn)/f(vn)] · ∇φ|2 Cρw2 L2(B2ρ(xj)) +w2 L2m(B2ρ(xj))0 as ρ0. Thus we obtain that the right hand side of (3.15) tends to 0. On the other hand, since q(2m,2m), by Lemma 2.2, we can prove that g(x,h(vn))h(vn)φg(x, w )in L1(BR1)and BR1g(x, w)w φ 0 as ρ0. All these facts imply that the last inequality in (3.14) tends to zero as ρ0. Thus νkε2µk. This means that either νk=0 or νkεNSN/2by virtue of Lions’ concen- tration compactness principle. We claim that the latter is impossible. Indeed, if νkεNSN/2holds for some kK, then cε=lim n¯ J(vn)1 2m¯ J(vn),f(vn)/f(vn)⟩ lim n(1 2m1 2m)RN |f(vn)|2m(1 2m1 2m)RN dν (1 2m1 2m)RN |w|2m+(1 2m1 2m)SN/2εN1 N m εNSN/2, which is a contradiction. Thus νk=0 for all kK, and it implies that f(vn)L2m(BR1) wL2m(BR1). By the uniform convexity of L2m(BR1), we have f(vn)wstrongly in L2m(BR1). 021501-8 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) Finally, since p(x,f(vn)) f(vn)is sub-(2m)growth on B2R1\BR1, we conclude that (3.11) holds. This proves (3.8). Step 4: (vn)is compact in X. Since we have (3.8), the proof of the compactness is trivial. This completes the proof of the lemma. IV. PROOF OF MAIN RESULTS Before we prove Theorem 1.1, we will show ﬁrst some properties about the change of variable f. Lemma 4.1. Let f 1(v)=|f(v)|m/v , v ,0and f1(0)=0, then f 1is continuous, odd, nonde- creasing and lim v0f1(v)=0and lim |v|+|f1(v)| =m.(4.1) Proof. In fact, by (6) of Lemma 2.1, f 1(v)=v2(m|f(v)|m2f(v)f(v)v|f(v)|m)0, so f1is nondecreasing. By (4) of Lemma 2.1, f1(v)0 as v0. Finally, according to L’Hospital’s principle, lim v+f1(v)=lim v+ |f(v)|m v=lim v+m|f(v)|m2f(v)f(v)=m. This shows that (4.1) holds. Lemma 4.2. There exists d0>0such that lim v+(mvfm(v)) d0. Proof. Assume that v > 0. Since by (6) of Lemma 2.1, f(v)m f (v)v, we have mvfm(v)mvm f m1(v)f(v)v =1+m f 2(m1)(v)m f m1(v) 1+m f 2(m1)(v) mv =mv 1+m f 2(m1)(v)+m f m1(v)1+m f 2(m1)(v) mv 2(1+m f 2(m1)(v)) fm(v) 4m f 2(m1)(v) =1 4m f m2(v)Bd(m, v ).(4.2) In the last inequality, we have used the fact that mvfm(v)and that m f 2(m1)(v)>1 for v > 0 suciently large. If 1 <m<2, then d(m, v )+as v+. If m=2, then d(m, v )=1/8. If m>2, we claim that mvfm(v)0 as v+is impossible. In fact, assume on the contrary, then using L’Hospital’s principle, we get 0<lim v+ mvfm(v) f2m(v)=lim v+ mm f m1(v)f(v) (2m)f1m(v)f(v) =lim v+ m (2m)f1m(v)m1+m f 2(m1)(v)+m f m1(v) =1 2(2m)<0. 021501-9 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) This is a contradiction. Thus for all m>1, there exists d0>0 such that there holds lim v+(mvfm(v)) d0. This completes the proof. Lemma 4.3. We have (i) If 1<m<2, then lim v+ mvfm(v) f2m(v)=1 2(2m). (ii) If m 2, then lim v+ mvfm(v) log f(v) 1 2,m=2, 0,m>2. Proof. First, we prove part (i). According to (4.2) in Lemma 4.2, we have mvfm(v)+ as v+. Thus by L’Hospital’s principle, we get lim v+ mvfm(v) f2m(v)=lim v+ mm f m1(v)f(v) (2m)f1m(v)f(v)=1 2(2m). Next, we prove part (ii). If there exists a constant C>0 such that mvfm(v)C, then the conclusion holds. Otherwise, assume that mvfm(v)+as v+. Then again by L’Hospital’s principle, we have lim v+ mvfm(v) log f(v)=lim v+ mm f m1(v)f(v) f(v)/f(v)= 1 2,m=2, 0,m>2. This completes the proof. To prove Theorem 1.1, it is crucial to prove that ¯ Jhas the mountain pass level cε<1 N m εNSN/2. Let us consider the following family of functions in Ref. 3: v ω(x)=[N(N2)ω2](N2)/4 [ω2+|x|2](N2)/2, which solves the equation u=u21in RNand satisﬁes v ω2 L2=v ω2 L2=SN/2. Let ωbe such that 2ω < Rand let ηω(x)[0,1]be a positive smooth cutofunction with ηω(x)=1 in Bω, ηω(x)=0 in BR\B2ω. Let vω=ηωv ω. For all ω > 0, there exists tω>0 such that ¯ J(tωvω)<0 for all t>tω. Deﬁne the class of paths Γ={γC([0,1],X):γ(0)=0, γ(1)=tωvω}, and the minimax level cε=inf γΓmax t[0,1] ¯ J(γ(t)). Let tωbe such that ¯ J(tωvω)=max t0 ¯ J(tvω). Note that the sequence (vω)is uniformly bounded in X, then if ¯ J(tωvω)0 as tω0, we are done; on the other hand, if tω+, then ¯ J(tωvω)→ −∞, which is impossible, so it remains to consider the case where the sequence (tω)is upper and lower bounded by two positive constants. According to Ref. 3, we have, as ω0, vω2 L2=SN/2+O(ωN2),vω2 L2=SN/2+O(ωN). 021501-10 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) Let a(0,ε(N2)/2 2m),b(2ε(N2)/2 m,+)be such that tω[a,b],ω(00), where ω0>0 small enough. By computing d dt ¯ J(tvω)=0, we obtain tω=ε(N2)/2 m+o(1). Let H(v)=1 2V(x)f2(v)+λ q|f(v)|q1 2m|mv|2+1 2m|f(v)|2m, then by (4.1) and (4) of Lemma 2.1, for m>1, we have lim |v|+H(v)/|v|2=0 and lim v0H(v)/v2=1 2V(x). Thus H(v)is sub-(2)growth. The following proposition is important to the computation of a mountain pass level cε< 1 N m εnSN/2. Proposition 4.4. Under the assumptions of Theorem 1.1, there exists a function τ=τ(ω)such that limω0τ(ω)= +and for ωsmall enough, RN H(tωvω)τ(ω)·ωN2. Proof. We divide the proof into three steps. Step 1: We prove that 1 ωN2Bω H(tωvω)τ1(ω)(4.3) with limω0τ1(ω)= +. By the deﬁnition of vω, for xBω, there exist constants c2c1>0 such that for ωsmall enough, we have c1ω(N2)/2vω(x)c2ω(N2)/2 and c1ω(N2)/2fm(vω(x)) c2ω(N2)/2.(4.4) On the one hand, by (7) of Lemma 2.1, (4.4) and the continuity of V(x)in ¯ Bω, there exists C1>0 such that Bω V(x)f2(tωvω)C1ωN2 mN2 2=C1ω(2 21 m)(N2).(4.5) Similarly, there exists C2>0 such that Bω fq(tωvω)C2ωNq mN2 2=C2ω(2 2¯q 2)(N2),(4.6) where ¯q=q/m. On the other hand, since the function ζ(t)=t2is convex, using (5) of Lemma 2.1 and Hölder inequality, we have 1 2mBω(mtωvω)2(fm(tωvω))2 1 mBω (mtωvω)21[mtωvωfm(tωvω)] 1 m(Bω (mtωvω)2)(21)/2(Bω [mtωvωfm(tωvω)]2)1/2.(4.7) Case 1: 1 <m<2. From (4.7) and (i) of Lemma 4.3, we obtain that there exists C3>0 such that 1 2mBω(mtωvω)2(fm(tωvω))2C3ω[N(2 m1)N2 22]1 2=C3ω(11 m)(N2).(4.8) 021501-11 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) Combining (4.5), (4.6), and (4.8), we have 1 ωN2Bω H(tωvω)≥ −C1ω(2 21 m1)(N2)+C2ω(2 2¯q 21)(N2)C3ω1 m(N2)Bτ1(ω). It is obvious that 2 21 m1>1 m. Using the condition (i) in Theorem 1.1, we have 2 2¯q 21< 1 m. It results that τ1(ω)+as ω0. Case 2:m2. Note that for any δ(0,m), limv+log f(v)/fδ(v)=0,we have log f(v) fδ(v)for v > 0 large enough. Thus for ω > 0 small enough, from (4.7) and (ii) of Lemma 4.3, we get 1 2mBω(mtωvω)2(fm(tωvω))2C 3ω[Nδ mN2 22]1 2=C 3ω1 2(1δ m)(N2).(4.9) Combining (4.5), (4.6), and (4.9), we have 1 ωN2Bω H(tωvω)≥ −C1ω(2 21 m1)(N2)+C2ω(2 2¯q 21)(N2)C 3ω1 2(1+δ m)(N2)Bτ1(ω). Since m2, we have 2 21 m1>1 2(1+δ m). From condition (ii) in Theorem 1.1, there exists a δ=δ(N,¯q)>0 (depends on Nand ¯q) small enough such that 2 2¯q 21<1 2(1+δ m). It results that τ1(ω)+as ω0. Cases 1 and 2 show that (4.3) holds. Step 2: We prove that there exists C4>0 such that 1 ωN2B2ω\Bω H(tωvω)≥ −C4ω(2 21 m1)(N2)Bτ2(ω).(4.10) Note that for xB2ω\Bω, we have vω(x)v ω(x)c2ω(N2)/2.(4.11) Since ηωis a positive smooth cutofunction, without the loss of generality, we may assume that ηω is such that B2ω\Bω |vω|2B2ω\Bω V(x)f2(vω). By (4.1) and (4.11), we have fm(vω(x)) c2ω(N2)/2for xB2ω\Bω. Thus 1 ωN2B2ω\Bω H(tωvω) ≥ − 1 2ωN2B2ω\Bω V(x)f2(tωvω)1 2mωN2B2ω\Bω |mtωvω|2 ≥ − C5 ωN2B2ω\Bω V(x)f2(tωvω) ≥ −C4ωN2 mN2 2(N2)=C4ω(2 21 m1)(N2), where C4>0, C5>0 are constants. This shows that (4.10) holds. Step 3: To conclude, let τ(ω)=τ1(ω)+τ2(ω), we have τ(ω)+as ω0. This implies the conclusion of the proposition. Proof of Theorem 1.1. By (4) and (7) of Lemma 2.1, it is easy to verify that ¯ Jhas the moun- tain pass geometry. Lemma 3.1 shows that ¯ Jsatisﬁes the PS condition. We prove that ¯ Jhas the mountain pass level cε<1 N m εNSN/2. Let F(t)=ε2 2(tvω)∥2 L21 2mmtvω2 L2. Then we have F(t)F(t0)=1 N m εNSN/2+O(ωN2),t0,(4.12) 021501-12 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) where t0=ε(N2)/2 m. By (4.12) and Proposition 4.4, we have ¯ J(tωvω)=F(tωvω)RN H(tωvω) 1 N m εNSN/2+O(ωN2)τ(ω)ωN2 <1 N m εNSN/2. This shows that ¯ J(v)has a nontrivial critical point vεX, which is a weak solution of (2.3). We prove that vεis also a weak solution of (2.1). First, we can argue as the proof of Proposition 2.1 in Ref. 17 to obtain that lim ε0max xBR vε(x)=0. Thus there exists ε0>0 such that for all ε(0, ε0), we have vε(x)aBf1(l),|x|=R, where l is given in (2.2). Secondly, we prove that vε(x)a,ε(0, ε0)and xRN\BR.(4.13) Taking ϕ= (vεa)+,xRN\BR, 0,xBR as a test function in ¯ J(vε), ϕ=0, we get ε2RN\BR |(vεa)+|2+ε2RN\BR(V(x)p(x,f(vε)) f(vε))f(vε)f(vε)(vεa)+=0.(4.14) By (p2), we have V(x)p(x,f(vε)) f(vε)>0,xRN\BR. Therefore, all terms in (4.14) must be equal to zero. This implies vεain RN\BR. This proves (4.13). Thus vεis a solution of problem (2.1). To complete the proof, we deduce as the proof for Theorem 4.1 in Ref. 10 to obtain that vε|BRL(BR). Thus uε=f(vε)XL(RN)is a nontrivial weak solution of (1.1). Finally, by Proposition 4.5 in the following, we have limε0uεX=0 and uε(x)Ceβ ε|xxε|.This completes the proof. We prove the norm estimate and the exponential decay. Proposition 4.5. Let vεXL(RN)be a solution of (2.1) and let uε=f(vε), then we have lim ε0uεX=0and uε(x)Ceβ ε|xxε|, where C >0,β > 0are constants. Proof. First, let x0BRbe such that V(x0)=V0. Deﬁne J0:XRby J0(v)=1 2RN |v|2+1 2RN V0f2(v)RN G(f(v)). Let c0=inf γΓ0 sup t[0,1] J0(γ(t)), Γ0={vC([0,1],X):γ(0)=0,J0(γ(1)) <0}. 021501-13 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) Similar to the proof for estimate (2.4) in Ref. 17 (or Lemma 3.1 in Ref. 18), we can show that cεεNc0+o(εN)by using the change of coordinates y=(xx0). Arguing as for (3.3), and by virtue of this energy estimate, we obtain vεXθcε min{( θ 2m)ε2,(θ 2θ 2k1)} 2θc0 θ2mεN2+o(εN2) for ε > 0 suciently small. Let uε=f(vε), then uε,0. Note that |uε||vε|and |uε||vε|, we get limε0uεX=0. Second, similar to the proof for Theorem 4.1 in Ref. 10, we conclude that vεL(RN) and by Ref. 8, we have vεC1(BR). Now let xεdenote the maximum point of vεin BRand let σBsup{s>0 : g(t)<V0tfor every t[0,s]}. Then vε(xε)f1(σ)for ε > 0 small. In fact, assume that vε(xε)<f1(σ)for some ε > 0 su- ciently small. According to the deﬁnition of l(see (2.2)) and σ, we have vε(x)f1(l)<f1(σ) (note that k>1 in (2.2)), xRN\BR. Thus V(x)g(f(vε)) f(vε)>0,xRN. Since vε=f1(uε)is a critical point of Jε, we choose ϕ=f(vε)/f(vε)as a test function in J ε(vε), ϕ=0 and get 0=ε2RN(1+m(m1)| f(vn)|2(m1) 1+m|f(vn)|2(m1))|vε|2 +RN V(x)f2(vε)RN g(f(vε)) f(vε) =ε2RN(1+m(m1)| f(vn)|2(m1) 1+m|f(vn)|2(m1))|vε|2 +RN(V(x)g(f(vε)) f(vε))f2(vε). It turns out that all terms in the above equality must be equal to zero, which means that vε0, a contradiction. Now let wε(x)=vε(xε+εx), then wεsolves the equation wε+V(xε+εx)f(wε)f(wε)=g(f(wε)) f(wε),xRN. Note that limt0+f(t)f(t) t=1 by the properties of fand that wε(x)0 as |x|+; we have, there exists R0>0 such that for all |x|R0, f(wε(x)) f(wε(x)) 3 4wε(x)(4.15) and g(f(wε(x))) f(wε(x)) V0 2wε(x).(4.16) Let ϕ(x)=Meβ|x|with β2<V0 4and MeβR0wε(x)for all |x|=R0. It is easy to verify that for x,0, ϕβ2ϕ. (4.17) Now deﬁne ψε=ϕwε. Using (4.15)-(4.17), we have ψε+V0 4ψε0,in |x|R0, ψε0,in |x|=R0, lim|x|ψε=0. 021501-14 Z. Li and Y. Zhang J. Math. Phys. 58, 021501 (2017) By the maximum principle, we have ψε0 for all |x|R0. Thus, we obtain that for all |x| R0, wε(x)ϕ(x)Meβ|x|. Using the change of variable, we have that for all |x|R0, vε(x)=wε(ε1(xxε)) Meβ ε|xxε|. Then by the regularity of vεon BRand note that f(t)tfor all t0, we have uε(x)Ceβ ε|xxε| for some C>0. This completes the proof. Proof of Theorem 1.2. We consider the following equation: u+V(x)ukα((|u|2α))|u|2α2u=|u|q2u+|u|2(2α)2u,u>0,xRN.(4.18) Let y=εxwith ε(0, ε0),ε0is given by Theorem 1.1, then we can transform (4.18) into ε2u+¯ V(y)ukαε2((|u|2α))|u|2α2u=|u|q2u+|u|2(2α)2u,u>0, y RN.(4.19) Here ¯ V(y)=V(y ε)still has the properties given in assumption (V). Thus according to Theorem 1.1, (4.19) has a positive weak solution uε(y)in XL(RN), which implies that (4.18) has a positive weak solution u1(x)=uε(εx). ACKNOWLEDGMENTS The authors would like to express their sincere gratitude to the anonymous referee for his/her valuable suggestions and comments. The ﬁrst author was supported by the Natural Science Foundation of China (No. 11201488) and the Hunan Provincial Natural Science Foundation of China (No. 14JJ4002). 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M., and Zhou, H. S., “Positive solutions for a quasilinear Schrödinger equation involving Hardy potential and critical exponent,” Commun. Contemp. Math. 16, 1450034 (2014). 20 Zhang, Y. M., Wang, Y. J., and Shen, Y. T., “Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents,” Commun. Pure Appl. Anal. 10, 1037–1054 (2011). ... (1) was extensively considered in recent years [2,3,9,[14][15][16][19][20][21] since the change in [9,14] was introduced. Furthermore, using the change of variables, for general α > 1 2 , the existence of solutions of (1) have been studied; see [1,4,12] and the references therein. Comparing with the semilinear elliptic equations, it is much more challenging and interesting because of the existence of the term ( (|u| 2α ))|u| 2α-2 u. ... ... Hence, using Lebesgue dominated convergence theorem, it is easy to see that J γ (v γ ) = 0. Furthermore, we can replace v n by |v n |. Hence, we can assume that v n ≥ 0 in R N and v γ ≥ 0. If v γ = 0, then v γ is a positive solution of Eq. (12). By contradiction, we assume that v γ = 0. ... ... Then we get a contradiction as in a similar proof to [9,19,20] by using the compactness lemma [13]. Hence, v γ is a nontrivial solution of Eq. (12). By using the fact that G -1 γ (t) ∈ C 2 together with Lemma 2.1, a direct computation shows that u = G -1 ... Article Full-text available By a change of variables with cut-off functions, we study the existence and the asymptotic behavior of positive solutions for a general quasilinear Schrödinger equation which arises from plasma physics. We extend the results of (Adv. 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