Existence of weak solutions for quasilinear Schrödinger equations with a parameter

Existence of weak solutions for quasilinear Schrödinger equations with a parameter

Yunfeng Wei

, Caisheng Chen

, Hongwei Yang

and Hongwang Yu

1School of Statistics and Mathematics, Nanjing Audit University, Nanjing 211815, China

2College of Science, Hohai University, Nanjing 210098, China

3College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Received 7 January 2020, appeared 27 June 2020 Communicated by Dimitri Mugnai

Abstract. In this paper, we study the following quasilinear Schrödinger equation of the form

−∆pu+V(x)|u|^p−2u−^h∆p(₁+u²)^{α/2i αu}

2(1+u²)^(2−α)/2 =k(u)_, x∈R^N,

where p-Laplace operator∆pu=div(|∇u|^p−2∇u) (1< p ≤ N)andα≥1 is a param- eter. Under some appropriate assumptions on the potentialV and the nonlinear term k, using some special techniques, we establish the existence of a nontrivial solution in C_loc^1,β(_R^N) (₀<β<₁), we also show that the solution is in L^∞(_R^N)and decays to zero at infinity when 1<p<N.

Keywords: quasilinear Schrödinger equation, variational method, mountain-pass theo- rem, p-Laplace operator.

2020 Mathematics Subject Classification: 35J62, 35J20, 35Q55.

1 Introduction

In this work, we are interested in the existence of nontrivial solution to the following quasi- linear Schrödinger equation

−_∆pu+V(x)|u|^p−2u−^h_∆p(1+u²)^{α/2i αu}

2(1+u²)^(2−α)/2 =k(u), x∈_R^N, (1.1) where p-Laplace operator ∆pu =div(|∇u|^p−2∇u) (1 < p ≤ N)andα≥ 1 is a parameter. V is a positive continuous potential andk(u)is a nonlinear term of subcritical type.

BCorresponding author. Email: weiyunfeng@nau.eu.cn

Such equations arise in various branches of mathematical physics. For instance, solutions of equation (1.1), in the case p = 2 andα = 1 are closed related to the existence of solitary wave solutions for quasilinear Schrödinger equations

izt =−_∆z+W(x)z−k^˜(|z|²)z−_∆l(|z|²)l⁰(|z|²)z, x∈_R^N, (1.2) where z : R×_R^N → _C,W : R^N → _R is a given potential, ˜k, l : R⁺ → _R are real functions.

The form of (1.2) has been derived as models of several physical phenomena corresponding to various types ofl. For instance, the casel(s) =s models the time evolution of the condensate wave function in super-fluid film [15,16], and is called the superfluid film equation in fluid mechanics by Kurihara [15]. In the case l(s) = (1+s)^1/2, problem (1.2) 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 equation (see [2,4,8,28]). For more physical motivations and more references dealing with applications, we refer the reader to [1,13,17, 25–27] and references therein.

It is well known that, via the ansatzz(t,x) =exp(−iEt)u(x), where E∈ _Randuis a real function, (1.2) can be reduced to the following elliptic equation

−_∆u+V(x)u−_∆(l(u²))l⁰(u²)u=k(u), x∈ _R^N, (1.3) whereV(x) =W(x)−Eandk(u) =k^˜(u²)u.

If we takel(s) =sin (1.3), then we obtain the superfluid film equation in plasma physics

−_∆u+V(x)u−_∆(u²)u=k(u), x ∈_R^N. (1.4) Clearly, when p = 2 and α = 2, equation (1.1) turns into equation (1.4). Equation (1.4) has been paid much attention in the past two decades. Many existence and multiplicity re- sults of nontrivial solutions have been established by differential methods such as constrained minimization argument, changes of variables, Nehari method, a dual approach, perturbation method, see [7,12,14,20–24,26,29,31] and references therein.

If we takel(s) = (1+s)^1/2 in (1.3), then we get the equation

−_∆u+V(x)u−^h_∆(₁+u²)^{1/2i u}

2(1+u²)^1/2 =k(u)_, x∈_R^N, (1.5) which models the self-channeling of a high-power ultrashort laser in matter. Obviously, equa- tion (1.1) turns into (1.5) for the case p=2 andα=1.

The existence of positive solutions for (1.5) has been studied recently. In [32], by a change of variables and the Ambrosetti–Rabinowitz mountain-pass theorem, the authors proved that (1.5) has a positive solution. They assume that the potentialV ∈C(_R^N,R)and the nonlinear- ityk:R→_Ris Hölder continuous and satisfy the following conditions:

(V₁) V(x)≥V₀ >0, for allx∈_R^N;

(_V2) _lim|x|→∞V(x) =V(_∞)<_∞andV(x)≤V(_∞)_{, for all} x∈_R^N; (H₁) k(s) =0 if s≤0;

(_H2) k(s) =o(s)_ass →₀⁺_;

(H₃) There exists 2<θ <2^∗ such that|k(s)| ≤C(1+|s|^θ−1); (_H4) There existsµ>√

6 such that 0<µK(s)≤sk(s)_{for all}s>_{0, where} K(s) =Rs 0 k(t)dt.

In [5], by a dual approach, the authors studied the existence of positive solution for the fol-

lowing equation

−_∆u+Ku−^h_∆(1+u²)^{α/2i αu}

2(1+u²)^(2−α)/2 = |u|^q−1u+|u|^p−1u, x ∈_R^N, (1.6) where K > 0, N ≥ 3,α ≥ 1 and 2 < q+1 < p+1 < α2^∗. Similar works can be found in [3,6,18,22] and reference therein.

However, to the best of our knowledge, in all works mentioned above, there are no exis- tence results in the literature on the casep 6=2,α≥1 and the nonlinear term becomes general function. Motivated by the works mentioned above and [5,7,20,22,31,32], our purpose in this paper is to study the existence of nontrivial weak solutions of (1.1) under some assumptions on the potentialV(x)and nonlinear termk(s).

Definition 1.1. We say thatu:R^N →_Ris a weak solution of (1.1) ifu∈W^1,p(_R^N)∩L^∞_loc(_R^N) and

Z

R^N

1+ ^α

p|u|^p 2(1+u²)^(2−α)p/2

|∇u|^p−2∇u∇ψdx +^α

p

2 Z

R^N

1+ (α−1)u²

(₁+u²)^{1+(2−α)p/2}|∇u|^p|u|^p−2uψdx

=

Z

R^Nη(x,u)ψdx, ∀ψ∈C₀^∞(_R^N),

(1.7)

whereη(x,u) =k(u)−V(x)|u|^p−2u.

In such a case, we can deduce formally that the Euler–Lagrange functional associated with the equation (1.1) is

J(u) = ¹ p

Z

R^N

1+ ^α

p|u|^p 2(1+u²)^(2−α)p/2

|∇u|^pdx+ ¹ p

Z

R^NV(x)|u|^pdx−

Z

R^NK(u)dx, where K(s) =Rs

0 k(t)dt.

For (1.1), due to the appearance of the nonlocal term R

R^N α^p|u|^p

2(1+u²)^(2−α)p/2|∇u|^pdx, J may be not well defined. To overcome this difficulty, enlightened by [7,20,32], we make a change of variables as

v =H(u) =

Z _u

0 h(t)dt, (1.8)

where h(t) = ^h1+ ^αp|t|p

2(1+t²)^(2−α)p/2

i1/p

, t∈ _{R. Since} H(t)is strictly increasing onR, the inverse function H⁻¹(t)_of H(t)exists. Then after the change of variables, J(u)can be written by

F(v) =J(H⁻¹(v)) = ¹ p

Z

R^N|∇v|^pdx+ ¹ p

Z

R^NV(x)|H⁻¹(v)|^pdx−

Z

R^NK(H⁻¹(v))dx. (1.9) According to Lemma2.1and our hypotheses onV(x)andk(s)below, it is clear thatF is well defined inW^1,p(_R^N)andF ∈ C¹. The Euler–Lagrange equation associated to the functional F is

−_∆pv = ^η(x,H⁻¹(v))

h(H⁻¹(v)) ^{, x}∈_R^N. (1.10) In Proposition 2.2, we will show the relationship between the solutions of (1.10) and the solutions of (1.1).

Throughout this paper, let 1 < p ≤ N, α ≥ 1. Besides, we assume that the potential V(x) ∈ C(_R^N,R)and satisfies (V₁)−(V₂), the nonlinearity k(s)∈ C(_R,R) and satisfies the following conditions:

(K₁) kis odd andk(s) =o(|s|^p−2s)ass→0;

(K2) There exists a constantC>0 such that

|k(s)| ≤C(1+|s|^θ−1), ∀s∈ _R, whereαp<θ <αp^∗ if 1< p< Nandθ >αpif p= N;

(K₃) There existsµ≥Te(p,α)psuch that 0<µK(s)≤sk(s)for alls >0, whereK(s) =Rs

0 k(t)dt,Te(p,α) =1+T(p,α)and T(p,α) =sup

t≥0

th⁰(t)

h(t) =sup

t≥0

α^pt^p

1+ (α−1)t²

(₁+t²)₂(₁+t²)^(2−α)p/2+α^pt^p >0. (1.11) Our main result is the following.

Theorem 1.2. Let1< p ≤N, α≥1.Suppose(V₁)–(V₂)and(K₁)–(K₂)hold. Then(1.1) admits a nontrivial weak solution u ∈ C_loc^1,β(_R^N) (0 < β< 1)provided that one of the following conditions is satisfied:

(a) (_K3)holds withµ> Te(p,α)p;

(b) (K₃)holds withµ= Te(p,α)p=2p and p <θ < p^∗ if1< p<N orθ > p if p= N in(K₂). Furthermore, if1< p< N,then u∈L^∞(_R^N)and u(x)→0as|x| →_∞.

Remark 1.3. It is not difficult to verify thatT(p,α) =α−_{1 if}α≥_{2 and}α−₁≤T(p,α)<_{1 if} 1≤α<2. If p=2, thenT(p,α) =T(2,α), which equals to theT(α)in [5]. If p=2 andα=1, we obtain T(2, 1) = 5−2√

6. Thus, µ ≥ Te(2, 1)2 = (1+T(2, 1))2 ≈ 2.202 in (K₃) is better thanµ>₂√

6≈_{2.449 in}(_H4)_{. If} p=_{2 and}α=_{2, we have}Te(_{2, 2})₂=4, which coincides with that in [7]. Therefore, our conclusion in Theorem 1.2 can be viewed as an extension result in [5,7,20,32].

The organization of this paper is as follows. In Section 2, we give some properties of H(t) and some preliminary results. In Section 3, we present an auxiliary problem and some related results. In Section 4, we complete the proof of Theorem1.2.

Throughout this paper, C and C_i stand for positive constants which may take different values at different places. BR denotes the open ball centered at the origin and radius R > 0, C₀^∞(_R^N)denotes functions infinitely differentiable with impact support inR^N. For 1≤ p≤ _∞, L^p(_R^N)denotes the usual Lebesgue space with the norms

kuk_p=

Z

R^N|u|^pdx1/p

, 1≤ p <_∞;

kuk_∞ =_infM >_{0 :}|u(x)| ≤ Malmost everywhere in R^N . W^1,p(_R^N)denotes the Sobolev spaces modelled inL^p(_R^N)with its usual norm

kuk=

Z

R^N(|∇u|^p+|u|^p)dx1/p

.

h·,·idenotes the duality pairing between X and its dualX^∗. The weak (strong) convergence inXis denoted by *(→), respectively.

2 Preliminaries

We first give some properties of the change of variables H : R → _R defined by (1.8), which will be used frequently in the sequel of the paper.

Lemma 2.1. For functions h,H and H⁻¹, the following properties hold:

(1) H is odd, strictly increasing, invertible and C²inR;

(2) 0< (H⁻¹)⁰(t)≤1,∀t ∈_R;

(3) |H⁻¹(t)| ≤ |t|,∀t ∈_R;

(4) lim_t→0H⁻¹(t) t =1;

(5) limt→+_∞

H⁻¹(t)α

t =







p

q2

3, α=1,

√p

2, α>1;

(6) h(H⁻¹(t))H⁻¹(t)≤Te(p,α)t ≤Te(p,α)h(H⁻¹(t))H⁻¹(t),∀t ≥0;

(7) h(H⁻¹(t)) H⁻¹(t)²≤ Te(p,α)tH⁻¹(t)≤Te(p,α)h(H⁻¹(t)) H⁻¹(t)²,∀t∈ _R; (8) |H⁻¹(t)| ≤C|t|^1/αfor some C >0and∀t ∈_R;

(9) There exists C>0such that

|H⁻¹(t)| ≥

(C|t|, |t| ≤1, C|t|^1/α, |t| ≥1.

Proof. By the definition ofH, it is easy to verify that (1)–(4) hold.

(5) Ifα>1, since h(t) =

1+ ^α

pt^p 2(1+t²)^(2−α)p/2

1/p

=

1+ ^α

pt^p

2(1+t²)^p/2(1+t²)^(α−1)p/2) 1/p

, t>0, one has h(t) ∼ (^α₂^pt^p(α−1))^1/p = ^√_p^α

2t^α−1 as t → +∞. Moreover, H(t) = Rt

0h(s)ds ∼ ^√_p¹

2t^α as t → +_∞. Remember the fact H⁻¹(t)is the inverse of H(t), so we get H⁻¹(t) ∼ (√^p

2t)^1/α as t→+∞, which implies limt→+_∞(H⁻¹(t))^α

t =√^p

2. Ifα=1, the result is obvious sinceh(t)is an increasing bounded function whent >_0.

(6) Denote g1(_t) =_h(_H⁻¹(_t))_H⁻¹(_t)−t, t ≥0. Obviouslyg1(₀) =_{0. Since}α≥1, one has g⁰₁(t) = ^H

−1(t)h⁰(H⁻¹(t)) h(H⁻¹(t))

= ^α

p(H⁻¹(t))^p1+ (α−1)(H⁻¹(t))²

1+ (H⁻¹(t))^2h2 1+ (H⁻¹(t))^2(2−α)p/2+α^p(H⁻¹(t))^pi

≥0, ∀t ≥0,

which implies

h(H⁻¹(t))H⁻¹(t)≥t, ∀t ≥_0.

Consequently,

Te(p,α)t ≤Te(p,α)h(H⁻¹(t))H⁻¹(t), ∀t ≥0.

Set g₂(t) = Te(p,α)t−h(H⁻¹(t))H⁻¹(t), t ≥ 0. Clearlyg₂(0) = 0. By virtue of H⁻¹(t)≥ 0, t≥0 and (1.11), we can deduce that

g⁰₂(t) =T(p,α)− ^H

−1(t)h⁰(H⁻¹(t)) h(H⁻¹(t))

= T(p,α)− ^sh

0(s)

h(s) |_s=H−1(t)

≥0, ∀t ≥0, which implies

h(H⁻¹(t))H⁻¹(t)≤Te(p,α)t, ∀t≥0.

(7) SincetH⁻¹(t)≥0, ∀t ∈R, utilizing (6), we have

h(H⁻¹(t)) H⁻¹(t)²≤ Te(p,α)tH⁻¹(t)≤ Te(p,α)h(H⁻¹(t)) H⁻¹(t)², ∀t∈ _R.

It is not difficult to verify that (8) and (9) are right from (1), (4) and (5).

Under the hypotheses(_V1)_–(_V2)_and(_K1)_–(_K3), we readily derive thatF ∈ C¹(_W^1,p(_R^N)) and

hF⁰(v),ωi=

Z

R^N|∇v|^p−2∇v∇ωdx−

Z

R^N

η(x,H⁻¹(v)) h(H⁻¹(v)) ^ωdx

forv,ω ∈W^1,p(_R^N). Thus, the critical points ofF correspond exactly to the weak solutions of (1.10). The following results characterize the relationship between the solutions of (1.10) and (1.1).

Proposition 2.2.

(i) If v∈ W^1,p(_R^N)∩L_loc^∞ (_R^N)is a critical point of the functionalF, then u= H⁻¹(v)is a weak solution of(1.1);

(ii) if v is a classical solution of(1.10), then u= H⁻¹(v)is a classical solution of(1.1).

Proof. (i) It is easy to see that |u|^p = |H⁻¹(v)|^p ≤ |v|^p and |∇u|^p = |(H⁻¹)⁰(v)|^p|∇v|^p ≤

|∇v|^p. Hence,u∈W^1,p(_R^N)∩L_loc^∞ (_R^N). Sincevis a critical point ofF, we get Z

R^N|∇v|^p−2∇v∇ωdx=

Z

R^N

η(x,H⁻¹(v))

h(H⁻¹(v)) ^ωdx, ∀ω ∈W^1,p(_R^N). (2.1) Note that

∇v= H⁰(u)∇u=h(u)∇u=

1+ ^α

p|u|^p 2(1+u²)^(2−α)p/2

1/p

∇u. (2.2)

For allψ∈C₀^∞(_R^N), one can achieve h(H⁻¹(v))ψ= h(u)ψ=

1+ ^α

p|u|^p 2(₁+u²)^(2−α)p/2

1/p

ψ∈W^1,p(_R^N),

and

∇h(H⁻¹(v))ψ

=h⁰(u)ψ∇u+h(u)∇ψ

= ^α

p

2

1+ ^α

p|u|^p 2(₁+u²)^(2−α)p/2

(1−p)/p 1+ (α−1)u²

(₁+u²)^{1+(2−α)p/2}|u|^p−2uψ∇u +

1+ ^α

p|u|^p 2(1+u²)^(2−α)p/2

1/p

∇ψ.

(2.3)

Letting ω = h(H⁻¹(v))ψ in (2.1) and combining (2.2)–(2.3) enable us to deduce (1.7), which means thatu= H⁻¹(v)is a weak solution of (1.1).

(ii)From

∆pv=

∑

∂

∂x_i

|∇v|^p−2∂v

∂x_i

=

∑

∂

∂x_i

h^p−1(u)|∇u|^p−2∂u

∂x_i

, we deduce that

∆pv= h^p−1(u)

∑

∂

∂x_i

|∇u|^p−2∂u

∂x_i

+|∇u|^p−2

∑

∂u

∂x_i

∂

∂x_i

h^p−1(u)

= h^p−1(u)_∆pu+ (p−1)h^p−2(u)h⁰(u)|∇u|^p

=

1+ ^α

p|u|^p 2(1+u²)^(2−α)p/2

(p−1)/p

∆pu

+

1+ ^α

p|u|^p 2(1+u²)^(2−α)p/2

−1/p (p−₁)α^p 1+ (α−₁)u²

2(1+u²)^{1+(2−α)p/2} |u|^p−2u|∇u|^p. Consequently,

1+ ^α

p|u|^p 2(1+u²)^(2−α)p/2

(p−1)/p

∆pu

+

1+ ^α

p|u|^p 2(1+u²)^(2−α)p/2

−1/p (p−1)α^p 1+ (α−1)u²

2(1+u²)^{1+(2−α)p/2} |u|^p−2u|∇u|^p

= −

1+ ^α

p|u|^p 2(1+u²)^(2−α)p/2

−1/p

η(x,u), that is,

∆pu+ ^α

p|u|^p

2(1+u²)^{(2−α)p/2∆pu}+(p−1)α^p 1+ (α−1)u²

2(1+u²)^{1+(2−α)p/2} |u|^p−2u|∇u|^p= −η(x,u). (2.4) Noticing that

α^p|u|^p

2(1+u²)^{(2−α)p/2∆pu}+ (p−1)α^p 1+ (α−1)u²

2(1+u²)^{1+(2−α)p/2} |u|^p−2u|∇u|^p

=^h_∆p(1+u²)^{α/2i αu}

2(1+u²)^(2−α)/2. This together with the (2.4) derive

−_∆pu−^h_∆p(1+u²)^{α/2i αu}

2(1+u²)^(2−α)/2 =η(x,u). The proof is finished.

3 Auxiliary problem

To prove the main result, we employ the results [9] for the equation

−_∆pv= g(v), x∈_R^N. (3.1)

The energy functional associated to (3.1) is I(v) = ¹

p Z

R^N|∇v|^pdx−

Z

R^NG(v)dx, whereG(s) =Rs

0 g(t)dt. Obviously, I ∈C¹(W^1,p(_R^N))under the assumptions ong(s)below:

(G₀) gis odd andg ∈C(_R,R); (G₁) −_∞<lim inf

s→0 g(_s)

|s|^p−2s ≤lim sup

s→0 g(_s)

|s|^p−2s =−σ<0 if 1< p< N,

−_∞<lim

s→0 g(s)

|s|^N−2s =−σ<0 if p =N;

(G₂) When 1 < p < N, lim

s→_∞

|g(s)|

|s|^{p∗ −1} = 0, where p^∗ = _N^{N p}_−p; when p = N, for some positive constantsCand β₀, that

|g(s)| ≤Ch

exp β₀|s|^N/(N−1)−S_N−2(β₀,s)ⁱ for all|s| ≥R>0, where

S_N−2(β₀,s) =

N−2 k

∑

β^k₀

k!|s|^kN/(N−1); (G₃) There existsξ >0 such thatG(ξ)>0.

We recall that a solutionv(x)of (3.1) is said to bea least energy solution (or ground state solution) if and only if

I(v) =a, where a=_inf{I(w)_:w∈W^1,p(_R^N)\ {0}is a solution of(_3.1)}. (3.2) Theorem 3.1([9, Theorem 1.4]). Let1< p≤ N and suppose(G0)–(G2)hold. Then setting

Λ=ⁿγ∈C [0, 1],W^1,p(_R^N) : γ(0) =0, I(γ(1))<0o

, b= inf

γ∈_Λ max

0≤t≤1I(γ(t)), we haveΛ6= _∅and b = a.Furthermore, for each least energy solution w of(3.1), there exists a path γ∈_Λsuch that w∈ γ([0, 1])and

tmax∈[0,1]I(γ(t)) =I(w).

Theorem 3.2 ([9, Theorem 1.6]). Let 1 < p ≤ N and assume that (G₀)–(G₃) are satisfied, then equation(3.1)has a least energy solution v which is positive.

Theorem 3.3 ([9, Theorem 1.8]). Assume that all conditions of Theorem3.1 hold, then there exist λ>₀andδ>₀such that I(v)≥ λkvk^pifkvk ≤_δ.

Lemma 3.4. Assume that (V₁)–(V₂) and (K₁)–(K₂) are satisfied, then the functional F has a mountain-pass geometry.

Proof. Let the energy functionals corresponding to the equations−_∆pv= m₀(v)and−_∆pv= m_∞(v)be

F₀(v) = ¹ p

Z

R^N|∇v|^pdx+ ¹ p

Z

R^NV₀|H⁻¹(v)|^pdx−

Z

R^NK(H⁻¹(v))dx, F_∞(v) = ¹

p Z

R^N|∇v|^pdx+ ¹ p

Z

R^NV(_∞)|H⁻¹(v)|^pdx−

Z

R^NK(H⁻¹(v))dx, respectively, where

m₀(v) = ¹ h(H⁻¹(v))

h

k(H⁻¹(v))−V₀|H⁻¹(v)|^p−2H⁻¹(v)ⁱ, m_∞(v) = ¹

h(H⁻¹(v)) h

k(H⁻¹(v))−V(_∞)|H⁻¹(v)|^p−2H⁻¹(v)ⁱ. Notice thatF₀(v)≤ F(v)≤ F_∞(v)for allv∈W^1,p(_R^N).

Now, we claim thatm₀andm_∞ satisfy(G₀)–(G₂). Obviously,m0andm_∞ satisfy(G0).

By use ofk(s) =o(|s|^p−2s)ass→0 and Lemma2.1(4), we derive that lims→0

m₀(s)

|s|^p−2s =−V₀ <0, lim

s→0

m_∞(s)

|s|^p−2s =−V(_∞)<0, if 1< p< N, and

lims→0

m₀(s)

|s|^N−2s =−V₀<0, lim

s→0

m_∞(s)

|s|^N−2s =−V(_∞)<0, if p= N.

Hence,m₀ andm_∞satisfy(_G1)_.

Similarly to the argument in the proof of Lemma2.1(5), we can show that

slim→_∞

|H⁻¹(s)|^α−1 h(H⁻¹(s)) =







p

q2

3, α=1,

√p

2

α , α>1.

(3.3)

When 1< p< N, it follows from(_K2)_{and Lemma 2.1}(2), (3), (8) that

|m₀(s)| ≤ ¹

h(H⁻¹(s))(C+C|H⁻¹(s)|^θ−1+V₀|H⁻¹(s)|^p−1)

≤C+C|H⁻¹(s)|^θ−1

h(H⁻¹(s)) +V₀|s|^p−1

=C+C|H⁻¹(s)|^θ−α|H⁻¹(s)|^α−1

h(H⁻¹(s)) +V₀|s|^p−1

≤C+C|s|^(θ−α)/α|H⁻¹(s)|^α−1

h(H⁻¹(s)) +V₀|s|^p−1,

(3.4)

whereαp <θ< αp^∗. Combining (3.3) and (3.4), we can deduce

slim→_∞

|m₀(s)|

|s|^p∗−1 =0.

On the other hand, when p=N, applying(K₂)and Lemma2.1(2), (3), we conclude that

|m₀(s)| ≤C₁+C₂|s|^θ−1. Then there exist positive constantsCandβ₀ such that

|m₀(s)| ≤Ch

exp β₀|s|^N/(N−1)−S_N−2(β₀,s)ⁱ

for all |s| ≥ R > 0, where S_N−2(β₀,s) = _∑k^N₌⁻₀^2β_k!^k0|s|^kN/(N−1). Therefore, m₀ satisfies (G₂). Analogously,m_∞ also satisfies(_G2)_.

Based upon Theorem3.3, there existλ₁>0 andδ₁>0 such that F(v)≥ F₀(v)≥λ₁kvk^p _ifkvk ≤δ₁.

Moreover, for the functional F_∞, by virtue of Theorem 3.1, we obtain that there exists e ∈ W^1,p(_R^N)_withkek>δ₁ such thatF_∞(e)<0, which impliesF(e)<_{0. Thus Γ}6=_{∅, where}

Γ=ⁿγ∈C [0, 1],W^1,p(_R^N) : γ(0) =0, F(γ(1))<0o . The proof is complete.

Remark 3.5. By (K₃), for any given s₀ > 0, there exists C > 0 depending on s₀ such that K(s) ≥ Cs^µ for all s ≥ s₀. Particularly, we have lim_s→+_∞K(s)/s^p = +∞. Thus, there exists ξ >0 such thatM0(ξ)>0 andM_∞(ξ)>0, where

M₀(s) =

Z _s

0 m₀(t)dt= K(H⁻¹(s))− ^V0

p |H⁻¹(s)|^p, M_∞(s) =

Z _s

0 m_∞(t)dt=K(H⁻¹(s))−^V(_∞)

p |H⁻¹(s)|^p.

Hence,m₀ andm_∞ also satisfy(G₃). Taking advantage of Theorem3.2, the equations

−_∆pv=m0(v) and −_∆pv=m_∞(v), x∈ _R^N have least energy solutions inW^1,p(_R^N)which are positive.

4 Proof of Theorem 1.2

SinceF has the mountain-pass geometry, we know (see [10]) that for the constant c= inf

γ∈_Γmax

t∈[0,1]F(γ(t))>0, where

Γ=ⁿγ∈C [0, 1],W^1,p(_R^N) : γ(0) =0, F(γ(1))<0o , there exists a Cerami sequence{vn}forF at the levelc, that is,

F(v_n)→c and kF⁰(v_n)k(₁+kv_nk)→_{0 as}n→_{∞. (4.1)} Lemma 4.1. Assume that(V₁)–(V₂)and(K₁)–(K₃)are satisfied. Let{v_n} ⊂W^1,p(_R^N)be a Cerami sequence forF at the level c>0, then{v_n}is bounded in W^1,p(_R^N)_.

Proof. First, we will prove that if{v_n}satisfies Z

R^N|∇v_n|^pdx+

Z

R^NV(x)|H⁻¹(v_n)|^pdx≤C (4.2) for some constant C>0, then it is bounded inW^1,p(_R^N). In fact, we only need to verify that R

R^N|v_n|^pdxis bounded. We start splitting Z

R^N|vn|^pdx=

Z

{x:|vn(x)|≤1}|vn|^pdx+

Z

{x:|vn(x)|>1}|vn|^pdx.

Note that µ≥ Te(p,α)p ≥ αp, then it follows from Lemma 2.1 (9) and Remark 3.5 that there existsC>0 such thatK(H⁻¹(s))≥C|s|^p for all|s|>1. Consequently,

Z

{x:|vn(x)|>1}|v_n|^pdx ≤C⁻¹ Z

{x:|vn(x)|>1}K(H⁻¹(v_n))dx≤C⁻¹ Z

R^NK(H⁻¹(v_n))dx. (4.3) Using Lemma2.1(9) again, we derive that

Z

{x:|vn(x)|≤1}

|v_n|^pdx≤ C^−p Z

{x:|vn(x)|≤1}

|H⁻¹(v_n)|^pdx

≤ C^−pV₀⁻¹ Z

R^NV(x)|H⁻¹(vn)|^pdx.

(4.4)

Combining (4.1)–(4.4), we can achieve that{v_n}is bounded inW^1,p(_R^N). Next, we will show that (4.2) holds. By (4.1), we obtain

1 p

Z

R^N|∇v_n|^pdx+ ¹ p

Z

R^NV(x)|H⁻¹(v_n)|^pdx−

Z

R^NK(H⁻¹(v_n))dx=c+o_n(1), (4.5) and for allψ∈W^1,p(_R^N)_,

hF⁰(vn),ψi=

Z

R^N|∇vn|^p−2∇vn∇ψdx+

Z

R^NV(x)|H⁻¹(v_n)|^p−2H⁻¹(v_n) h(H⁻¹(vn)) ^ψdx

−

Z

R^N

k(H⁻¹(v_n)) h(H⁻¹(vn))^ψdx.

(4.6)

Denote ψ_n = h(H⁻¹(v_n))H⁻¹(v_n), taking advantage of Lemma 2.1 (6), one can find |ψ_n| ≤ Te(p,α)|vn|and

|∇ψ_n|=

"

1+ ^α

pt^p[1+ (α−1)t²]

(1+t²) 2(1+t²)^(2−α)p/2+α^pt^p|_t=|H−1(vn)|

#

|∇v_n| ≤Te(p,α)|∇v_n|.

Thus, kψ_nk ≤Te(p,α)kv_nk. By choosingψ=ψ_nin (4.6), we deduce that Z

R^N

"

1+ ^α

pt^p[₁+ (α−1)t²]

(1+t²) 2(1+t²)^(2−α)p/2+α^pt^p|_t=|H−1(vn)|

#

|∇vn|^pdx +

Z

R^NV(x)|H⁻¹(v_n)|^pdx−

Z

R^Nk(H⁻¹(v_n))H⁻¹(v_n)dx

=hF⁰(v_n),ψ_ni=o_n(1).

(4.7)

Combining (4.5), (4.7) and(K₃), one has Z

R^N

(1 p − ¹

µ

"

1+ ^α

pt^p[1+ (α−1)t²]

(1+t²) 2(1+t²)^(2−α)p/2+α^pt^p|_t=|H−1(vn)|

#)

|∇v_n|^pdx +

1 p − ¹

µ _Z

R^NV(x)|H⁻¹(v_n)|^pdx

≤c+o_n(1).

(4.8)

Ifµ>Te(p,α)pin(_K3), by virtue of (1.11), it follows that h

µ−Te(_p,α)_pⁱ pµ

Z

R^N|∇vn|^pdx+ ^T(p,α) µ

Z

R^NV(_x)|H⁻¹(_vn)|^pdx≤c+_on(₁)_,

which implies that (4.2) holds and hence {v_n} is bounded. If µ = Te(p,α)p = 2p, applying Remark1.3, we deriveα=2. In this case, we can apply the estimate (4.8) to derive

1 2p

Z

R^N

|∇vn|^p

1+₂^p−1|H⁻¹(v_n)|^pdx+ ¹ 2p

Z

R^NV(x)|H⁻¹(v_n)|^pdx ≤c+o_n(1). (4.9) Setun= H⁻¹(vn), we get that

|∇vn|^p= 1+2^p−1|H⁻¹(vn)|^p|∇un|^p. (4.10) According to(V₁)and (4.9)–(4.10), it holds that

1

2pmin{1,V0}kunk^p ≤ ¹ 2p

Z

R^N|∇un|^pdx+ ¹ 2p

Z

R^NV(x)|un|^pdx≤c+on(1). This implies{u_n}is bounded inW^1,p(_R^N). The conditions(K₁)–(K₂)yield that

K(s)≤ |s|^p+C|s|^θ. (4.11) Combining the condition (b) in Theorem 1.2 with (4.11), we can apply Sobolev embedding theorem to achieve thatR

R^NK(H⁻¹(vn))dx = R

R^NK(un)dx is bounded. Thus, utilizing (4.5), we derive (4.2), which implies{v_n}is bounded inW^1,p(_R^N). The proof is finished.

4.1 Existence of nontrivial critical points forF

According to Lemma4.1, {vn}is a bounded Cerami sequence in W^1,p(_R^N). SinceW^1,p(_R^N) is a reflexive Banach space, up to a subsequence, still denoted by{v_n}, such thatv_n *v. We assert thatF⁰(v) =0. In fact, sinceC₀^∞(_R^N)is dense inW^1,p(_R^N), we only need to verify that hF⁰(v),ψi=0 for allψ∈C₀^∞(_R^N). Note that

hF⁰(v_n),ψi − hF⁰(v),ψi

=

Z

R^N(|∇v_n|^p−2∇v_n− |∇v|^p−2∇v)∇ψdx +

Z

R^N

h|H⁻¹(v_n)|^p−2H⁻¹(v_n)

h(H⁻¹(vn)) − |H⁻¹(v)|^p−2H⁻¹(v) h(H⁻¹(v))

i

V(x)ψdx

−

Z

R^N

hk(H⁻¹(v_n))

h(H⁻¹(v_n))− ^k(H⁻¹(v)) h(H⁻¹(v)) i

ψdx.