A class of fourth-order elliptic equations with concave and convex nonlinearities in R N

A class of fourth-order elliptic equations with concave and convex nonlinearities in R ^N

Zijian Wu and Haibo Chen

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P. R. China Received 15 October 2020, appeared 15 September 2021

Communicated by Roberto Livrea

Abstract.In this article, we study the multiplicity of solutions for a class of fourth-order elliptic equations with concave and convex nonlinearities inR^N. Under the appropriate assumption, we prove that there are at least two solutions for the equation by Nehari manifold and Ekeland variational principle, one of which is the ground state solution.

Keywords:fourth-order elliptic equation, multiple solutions, Nehari manifold, Ekeland variational principle.

2020 Mathematics Subject Classification: 35J35, 35J60.

1 Introduction and main results

In this article, we consider the multiplicity results of solutions of the following fourth-order elliptic equation:

(∆²u−_∆u+u= f(x)|u|^q−2u+|u|^p−2u, inR^N,

u∈ H²(_R^N), (1.1)

where N > 4, 1 < q < 2 < p < 2∗(2∗ = 2N/(N−4)), the weight function f satisfies the following condition:

(F) f ≥0, f ∈ L^rq(_R^N)∩L^∞(_R^N)wherer_q= _r−^r_q for somer∈ (2, 2∗). Associated with (1.1), we consider theC¹-functional I_f, for each u∈ H²(_R^N),

I_f(u) = ¹

2kuk²−¹ q

Z

R^N f(x)|u|^qdx− ¹ p

Z

R^N|u|^pdx, where kuk= R

R^N |_∆u|²+|∇u|²+u² dx1/2

is the norm in H²(_R^N). It is well known that the solutions of (1.1) are the critical points of the energy functionalI_f [14].

BCorresponding author. Email: math_chb@163.com

In reality, elliptic equations with concave ang convex nonlinearities in bounded domains have been the focus of a great deal of research in recent years. Ambrosetti et al. [1], for example, considered the following equation:











−_∆u=λu^q−1+u^p−1, in Ω,

u>0, in Ω,

u∈ H¹₀(_Ω)_,

(1.2)

where Ω is a bounded domain in R^N with 1 < q < 2 < p < 2^∗ (2^∗ = _N^2N₋₂ if N ≥ 3; 2^∗ =

∞ifN = 1, 2)and λ> 0. They found that there is λ₀ > 0 such that (1.2) admits at least two positive solutions for λ∈ (_0,λ₀), has a positive solution for λ= λ₀ and no positive solution exists forλ> λ₀. Actually, many scholars have also obtained the same results in the unit ball B^N(0; 1), see [2,6,10,13].

Furthermore, it is also an important subject to deal with elliptic equation with concave- convex nonlinearities when a bounded domain Ω is replaced by R^N. Wu [18] studied the concave-convex elliptic problem:











−_∆u+u= f_λ(x)u^q−1+g_µ(x)u^p−1, inR^N,

u>0, inR^N,

u∈ H¹(_R^N),

(1.3)

where 1<q<2< p<2^∗ (2^∗ =2N/(N−2)if N≥3, 2^∗ =_∞if N=1, 2), f_λ= λf++ f−(f±=±max{0,±f} 6=0)

is sign-changing, gµ = a+µband the parameters λ,µ > 0. When the functions f+, f−, a, b satisfy appropriate hypotheses, author obtained the multiplicity of positive solutions for the problem (1.3). Hsu and Lin [9] dealt with the existence and multiplicity of positive solutions for the following semilinear elliptic equation:











−_∆u+u=λa(x)|u|^q−2u+b(x)|u|^p−2u, inR^N,

u>0, inR^N,

u∈ H¹(_R^N),

(1.4)

wherea, bare measurable functions and meet the right conditions. They obtained the result of multiple solutions of the equation (1.4).

Inspired by the existing literature [5,8,9,11,15,18–20], the main aim of this article is to study (1.1) involving concave-convex nonlinearities on the whole space R^N. As far as we know, there are few articles dealing with this type of fourth-order elliptic equation (1.1) involving concave-convex nonlinearities. Using arguments similar to those used in [16], we will prove the existence of two nontrivial solutions by using Ekeland variational principle [7].

Let

σ=

p−2 p−q

2−q p−q

²_p⁻₋^q₂ S

p(2−q) 2(p−2)

p S

q

r2 >_0,

whereSpandSr are the best Sobolev constant. Now, we state the main result.

Theorem 1.1. Assume that(F)holds. If|f|_rq ∈ (0,σ), then(1.1)has at least two nontrivial solutions, one of which is the ground state solution.

This paper is organized as follows. In Section 2, we give some notations and preliminaries.

In Section 3, we are concerned with the proof of Theorem1.1.

2 Notations and preliminaries

We shall throughout use the Sobolev space H²(_R^N) with standard norm. The dual space of H²(_R^N) will be denoted by H⁻²(_R^N)_. h·,·i denotes the usual scalar product in H²(_R^N)_. L^r(_R^N) is the usual Lebesgue space whose norms we denote by |u|_r = R

R^N|u|^rdx1/r

for 1 ≤ p < ∞. Moreover, we denote by S_r the best Sobolev constant for the embedding of H²(_R^N)in L^r(_R^N).

Now, we consider the Nehari minimization problem:

α_f =inf{I_f(u)|u∈ N_f}, whereN_f ={u∈ H²(_R^N)\{0}|hI⁰_f(u),ui=0}. Define

ψ_f(u) =hI⁰_f(u),ui=kuk²−

Z

R^N f(x)|u|^qdx−

Z

R^N|u|^pdx.

Then foru∈ N_f,

hψ⁰_f(u),ui=hψ⁰_f(u),ui − hI⁰_f(u),ui

=kuk²−(q−1)

Z

R^N f(x)|u|^qdx−(p−1)

Z

R^N|u|^pdx.

Similarly to the skill used in Tarantello [16], we splitN_f into three parts:

N_f⁺={u∈ N_f | hψ⁰_f(u),ui>0}, N⁰_f ={u∈ N_f | hψ⁰_f(u),ui=0}, N_f⁻={u∈ N_f | hψ⁰_f(u),ui<0} and note that ifu ∈ N_f, that is,hI⁰_f(u),ui=0, then

hψ⁰_f(u),ui= (2−p)kuk²−(q−p)

Z

R^N f(x)|u|^qdx

= (2−q)kuk²−(p−q)

Z

R^N|u|^pdx.

(2.1)

Then, we have the following results.

Lemma 2.1. If|f|_rq ∈(0,σ), then the submanifoldN⁰ =_∅.

Proof. Suppose the contrary. ThenN_f⁰ 6= ∅, i.e., there existu ∈ N_f such that hψ⁰_f(u),ui= 0.

Then foru∈ N⁰by (2.1) and Sobolev inequality, we have (2−q)kuk² = (p−q)

Z

R^N|u|^pdx≤ (p−q)S⁻

p

p 2kuk^p, and so

kuk ≥





(2−q)S

p

p2

p−q





1 p−2

. (2.2)

Similarly, using (2.1), Sobolev and Hölder inequalities, we have (p−2)kuk² = (p−q)

Z

R^N f(x)|u|^qdx≤ (p−q)|f|_rqS⁻

q

r 2kuk^q,

which implies that

kuk ≤ (p−q)|f|_rq (p−₂)S

q

r2

!₂₋¹_q

. (2.3)

Combining (2.2) and (2.3) we deduce that

|f|_rq ≥

p−2 p−q

2−q p−q

²_p⁻₋^q₂ S

p(2−q) 2(p−2)

p S

q

r2 =σ, which is a contradiction. This completes the proof.

Lemma 2.2. If|f|_rq ∈ (0,σ), then the setN_f⁻is closed in H²(_R^N).

Proof. Let {un} ⊂ N_f⁻ such thatun → u in H²(_R^N). In the following we show u ∈ N_f⁻. In fact, byhI⁰_f(u_n),u_ni=0 and

hI⁰_f(u_n),u_ni − hI⁰_f(u),ui=hI⁰_f(u_n)−I⁰_f(u),ui+hI⁰_f(u_n),u_n−ui →0 asn→_∞, we havehI⁰_f(u)_,ui=_{0. So}u∈ N_{f. For any}u ∈ N_f⁻_{, that is,}hψ⁰_f(u)_,ui<0, from (2.1) we have

(₂−q)kuk²<(p−q)

Z

R^N|u|^pdx ≤(p−q)S⁻

p

p 2kuk^p_, and so

kuk>





(2−q)S

p

p2

p−q





1 p−2

>0.

Hence N_f⁻ is bounded away from 0. Obviously, by (2.1), it follows that hψ⁰_f(u_n),u_ni → hψ⁰_f(u),ui as n → +_{∞. From} hψ⁰_f(un),uni < 0, we have hψ⁰_f(u),ui ≤ 0. By Lemma 2.1, for|f|_rq ∈ (0,σ), N_f⁰ =_{∅, then} hψ⁰_f(u),ui<0. Thus we deduce u∈ N_f⁻. This completes the proof.

Lemma 2.3. The energy functional I_f is coercive and bounded below onN_f. Proof. Foru ∈ N_f, then, by Sobolev and Hölder inequalities,

I_f(u) = I_f(u)− ¹

phI⁰_f(u),ui

= ^p−2

2p kuk²− ^p−q pq

Z

R^N f(x)|u|^qdx

≥ ^p−2

2p kuk²− ^p−q

pq |f|_rqS⁻

q

r 2kuk^q. This completes the proof.

The following lemma shows that the minimizers onN_f are “usually” critical points forI_f. The details of the proof can be referred to Brown and Zhang [4].

Lemma 2.4. Suppose thatu is a local minimizer for Ib _f on N_f. Then, if ub∈ N/ _f⁰,u is a critical pointb of I_f.

For eachu∈ H²(_R^N)\{0}, we write

t_max:= (2−q)kuk² (p−q)R

R^N|u|^pdx

!_p−¹₂

>0.

Then, we have the following lemma.

Lemma 2.5. For each u∈ H²(_R^N)\{0}and|f|_rq ∈ (0,σ), we have

(i) there exist unique 0 < t⁺ := t⁺(u) < tmax < t⁻ := t⁻(u)such that t⁺u ∈ N_f⁺, t⁻u ∈ N_f⁻ and

I_f(t⁺u) = inf

tmax≥t≥0I_f(tu), I_f(t⁻u) = sup

t≥tmax

I_f(tu). (ii) t⁻is a continuous function for nonzero u.

(iii) N_f⁻=ⁿu∈ H²(_R^N)\{0}|_k¹

ukt⁻

u kuk

=1o . Proof. (i)Fixu∈ H²(_R^N)\{0}. Let

s(t) =t^2−qkuk²−t^p−q Z

R^N|u|^pdx fort≥0.

We have s(0) = 0, s(t) → −_∞ as t → _∞, s(t)is concave and achieves its maximum at t_max. Moreover, for |f|_rq ∈(0,σ),

s(t_max) = (2−q)kuk² (p−q)R

R^N|u|^pdx

!²_p⁻₋^q₂

kuk²− (2−q)kuk² (p−q)R

R^N|u|^pdx

!^p_p⁻₋^q₂ Z

R^N|u|^pdx

=kuk^q kuk^p R

R^N|u|^pdx

!²_p⁻₋^q₂

2−q p−q

²_p⁻₋^q₂ p−2 p−q

≥ kuk^q





kuk^p S⁻

p

p2kuk^p





2−q p−2

2−q p−q

²_p⁻₋^q₂ p−2 p−q

=kuk^q





(2−q)S

p

p2

p−q





2−q p−2

p−2 p−q

>|f|_rqS⁻

q

r 2kuk^q

≥

Z

R^N f(x)|u|^qdx>0.

(2.4)

Hence, there are uniquet⁺andt⁻such that 0< t⁺ <t_max<t⁻, s(t⁺) =

Z

R^N f(x)|u|^qdx =s(t⁻) and

s⁰(t⁺)>0>s⁰(t⁻). Note that

hI⁰_f(tu),tui=t^q−1

s(t)−

Z

R^N f(x)|u|^qdx

and

hψ⁰_f(tu)_,tui=t^q+1s⁰(t) _fortu∈ N_f.

We have t⁺u ∈ N_f⁺, t⁻u ∈ N_f⁻, and I_f(t⁻u) ≥ I_f(tu) ≥ I_f(t⁺u) for each t ∈ [t⁺,t⁻] and I_f(t⁺u)≥ I_f(tu)for each t∈[0,t⁺]. Thus,

I_f(t⁺u) = inf

tmax≥t≥0I_f(tu), I_f(t⁻u) = sup

t≥tmax

I_f(tu).

(ii)By the uniqueness oft⁻and the external property oft⁻, we have thatt⁻is a continuous function ofu6=0.

(iii)Foru∈ N_f⁻, letv= _k^u_uk. By part (i), there is a uniquet⁻(v)>0 such thatt⁻(v)v∈ N_f⁻, that ist⁻(_k^u_uk)_k^u_uk ∈ N_f⁻. Sinceu∈ N_f⁻, we havet⁻(_k^u_uk)_k^u_uk =1, which implies

N_f⁻⊂

u∈ H²(_R^N)\{₀}| ¹ kuk^t

−

u kuk

=₁

. Conversely, letu∈ H²(_R^N)\{0}such that _ku¹_kt⁻

u kuk

=1. Then t⁻

u kuk

u

kuk ∈ N_f⁻. Thus, N_f⁻=

u∈ H²(_R^N)\{0}

1 kuk^t

− u kuk

=1

. This completes the proof.

By Lemma2.1, for|f|_rq ∈ (0,σ)we writeN_f =N_f⁺∪ N_f⁻and define α⁺_f = inf

u∈N_f⁺I_f(u), α⁻_f = inf

u∈N_f⁻I_f(u). Lemma 2.6. For|f|_rq ∈(0,σ), we haveα_f ≤α⁺_f <0.

Proof. Letu∈ N_f⁺. By (2.1) we have Z

R^N|u|^pdx< ²−q p−qkuk², and so

I_f(u) = 1

2− ¹ q

kuk²+ 1

q− ¹ p

_Z

R^N|u|^pdx

<

1 2 −¹

q

+ 1

q− ¹ p

2−q p−q

kuk²

=−(p−2)(2−q)

2pq kuk² <0.

Therefore,α_f ≤α⁺_f <0.

3 Proof of Theorem 1.1

First, we will use the idea of Ni and Takagi [12] to get the following lemmas.

Lemma 3.1. If|f|_rq ∈ (0,σ), then for every u ∈ N_f, there existe > 0and a differentiable function g:Be(0)⊂ H²(_R^N)→_R⁺:= (0,+_∞)such that

g(0) =1, g(ω)(u−ω)∈ N_f, ∀ω∈ Be(0) and

hg⁰(0),vi= ²(u,v)−qR

R^N f(x)|u|^q−2uvdx−pR

R^N|u|^p−2uvdx

hψ⁰_f(u),ui ^(3.1)

for all v∈ H²(_R^N). Moreover, if0<C₁≤ kuk ≤C₂, then there exists C>0such that

|hg⁰(0),vi| ≤Ckvk. (3.2) Proof. We defineF:R×H²(_R^N)→_Rby

F(t,ω) =tku−ωk²−t^q−1 Z

R^N f(x)|u−ω|^qdx−t^p−1 Z

R^N|u−ω|^pdx,

it is easy to see F is differentiable. Since F(1, 0) =0 and Ft(1, 0) = hψ⁰_f(u),ui 6= 0, we apply the implicit function theorem at point (1, 0) to get the existence of e > 0 and differentiable functiong: Be(0)→_R⁺ such thatg(0) =1 andF(g(ω),ω) =0 for∀ω∈ Be(0). Thus,

g(ω)(u−ω)∈ N_f, ∀ω ∈B_e(0).

Also by the differentiability of the implicit function theorem, for allv∈ H²(_R^N), we know that

hg⁰(0),vi=−hF_ω(1, 0),vi F_t(_{1, 0}) ^. Note that

−hFω(1, 0),vi=2(u,v)−q Z

R^N f(x)|u|^q−2uvdx−p Z

R^N|u|^p−2uvdx andFt(1, 0) =hψ⁰_f(u),ui. So (3.1) holds.

Moreover, by (3.1), 0<C₁≤ kuk ≤C₂ and Hölder’s inequality, we have

|hg⁰(0),vi| ≤ ^Cekvk hψ⁰_f(u),ui

for some Ce > 0. To prove (3.2), therefore, we only need to show that |hψ⁰_f(u),ui| > d for some d > 0. We argue by contradiction. Assume that there exists a sequence {u_n} ∈ N_f, C₁ ≤ kunk ≤ C2, we havehψ⁰_f(un),uni = on(1). Then by (2.1) and Sobolev’s inequality, we have

(2−q)kunk²= (p−q)

Z

R^N|un|^pdx+on(1)

≤ (p−q)S⁻

p

p 2ku_nk^p+o_n(1), and so

ku_nk ≥





(2−q)S

p

p2

p−q





1 p−2

+o_n(1). (3.3)

Similarly, using (2.1) and Hölder and Sobolev inequalities, we have (p−2)ku_nk² = (p−q)

Z

R^N f(x)|u_n|^qdx+o_n(₁)

≤(p−q)|f|_rqS⁻

q

r 2kunk^q+on(1), which implies that

kunk ≤ (p−q)|f|_rq (p−2)S

q

r2

!₂₋¹_q

+on(1). (3.4)

Combining (3.3) and (3.4) asn→+_∞, we deduce that

|f|_rq ≥

p−2 p−q

2−q p−q

²_p⁻₋^q₂ S

p(2−q) 2(p−2)

p S

q

r2 =σ,

which is a contradiction. Thus if 0<C₁≤ kuk ≤C₂, there existsC>0 such that

|hg⁰(₀)_,vi| ≤Ckvk_. This completes the proof.

Lemma 3.2. If |f|_rq ∈ (0,σ)∈ (0,σ), then for every u ∈ N_f⁻, there exist e> 0and a differentiable function g⁻: Be(0)⊂ H²(_R^N)→_R⁺ such that

g⁻(0) =1, g⁻(ω)(u−ω)∈ N_f⁻, ∀ω∈ Be(0) and

h(g⁻)⁰(0),vi= ²(_u,v)−qR

R^N f(_x)|u|^q−2uvdx−pR

R^N|u|^p−2uvdx

hψ⁰_f(u),ui ^(3.5)

for all v∈ H²(_R^N). Moreover, if0<C₁≤ kuk ≤C₂, then there exists C>0such that

|h(g⁻)⁰(0),vi| ≤Ckvk. (3.6) Proof. Similar to the argument in Lemma 3.2, there exist e > 0 and a differentiable function g⁻: Be(0)→_R⁺such thatg⁻(0) =1 andg⁻(ω)(u−ω)∈ N_f for allω ∈ Be(0). By u∈ N_f⁻, we have

hψ⁰_f(u),ui=kuk²−(q−1)

Z

R^N f(x)|u|^qdx−(p−1)

Z

R^N|u|^pdx<0.

Since g⁻(ω)(u−ω)is continuous with respect to ω, when e is small enough, we know for ω∈ Be(0)

kg⁻(ω)(u−ω)k²−(q−1)

Z

R^N f(x)|g⁻(ω)(u−ω)|^qdx−(p−1)

Z

R^N|g⁻(ω)(u−ω)|^pdx<0.

Thus, g⁻(ω)(u−ω) ∈ N_f⁻, ∀ω ∈ Be(0). Moreover, the proof details of (3.5) and (3.6) are similar to Lemma3.1.

Lemma 3.3. If|f|_rq ∈ (0,σ), then

(i) there exists a minimizing sequence{u_n} ∈ N_f such that I_f(u_n) =α_f +o_n(₁)_,

I⁰_f(un) =on(1) in H⁻²(_R^N);

(ii) there exists a minimizing sequence{u_n} ∈ N_f⁻such that I_f(un) =α⁻_f +on(1),

I⁰_f(un) =on(1) in H⁻²(_R^N).

Proof. (i)By Lemma2.3and the Ekeland variational principle onN_f, there exists a minimizing sequence {un} ⊂ N_f such that

α_f ≤ I_f(un)<α_f + ¹

n (3.7)

and

I_f(un)≤ I_f(v) + ¹

nkv−unk for each v∈ N_f. (3.8) And we can show that there exists C₁,C₂ > 0 such that 0 < C₁ ≤ ku_nk ≤ C₂. Indeed, if not, that is, u_n → 0 in H²(_R^N), then I_f(u_n) would converge to zero, which contradict with I_f(un) → α_f < 0. Moreover, by Lemma 2.3 we know that I_f(u) is coercive on N_f, {un} is bounded in N_f.

Now, we show that

kI⁰_f(un)k_H−2(_R^N) →0 asn→_∞.

Applying Lemma 3.1 with u_n to obtain the functions g_n(ω) _: B_en(₀)→ _R⁺ _{for some} e_n > _0, such that

gn(0) =1, gn(ω)(un−ω)∈ N_f, ∀ω ∈Ben(0).

We choose 0 < ρ< en. Let u∈ H²(_R^N)\{0}andωρ = _k^ρu_uk. Sincegn(ωρ)(un−ωρ)∈ N_f, we deduce from (3.8) that

1

n[|gn(ωρ)−1|kunk+ρgn(ωρ)]

≥ ¹

nkgn(ωρ)(un−ωρ)−unk

≥ I_f(u_n)−I_f(g_n(ωρ)(u_n−ωρ))

= ¹

2ku_nk²− ¹ q

Z

R^N f(x)|u_n|^qdx− ¹ p

Z

R^N|u_n|^pdx−¹

2 g_n(ωρ)²ku_n−ωρk² +¹

q g_n(ω_ρ)^q

Z

R^N f(x)|u_n−ω_ρ|^qdx+ ¹

p g_n(ω_ρ)^p

Z

R^N|u_n−ω_ρ|^pdx

= − ^gn(ω_ρ)²−1

2 ku_n−ω_ρk²− ¹

2(ku_n−ω_ρk²− ku_nk²) + ^gn(ωρ)^q−1

q

Z

R^N f(x)|u_n−ωρ|^qdx +¹

q _Z

R^N f(x)|u_n−ω_ρ|^qdx−

Z

R^N f(x)|u_n|^qdx

+ ^gn(ω_ρ)^p−1 p

Z

R^N|u_n−ω_ρ|^pdx+ ¹ p

_Z

R^N|u_n−ω_ρ|^pdx−

Z

R^N|u_n|^pdx

.

(3.9)

Note that

lim

ρ→0⁺

g_n(ωρ)−1

ρ = lim

ρ→0⁺

g_n(0+ρ_k^u_uk)−g_n(0)

ρ =

(g_n)⁰(0), u kuk

.

If we divide the ends of (3.9) byρand letρ→0⁺, we have 1

n

(g_n)⁰(0), u kuk

ku_nk+1

≥ −

(g_n)⁰(0), u kuk

ku_nk²−

Z

R^N∆u_n∆

− ^u kuk

+∇u_n∇

− ^u kuk

+u_n

− ^u kuk

dx +

(g_n)⁰(₀)_, ^u kuk

Z

R^N f(x)|u_n|^qdx+

Z

R^N f(x)|u_n|^q−2u_n

− ^u kuk

dx +

(gn)⁰(0), u kuk

_Z

R^N|un|^pdx+

Z

R^N|un|^p−2un

− ^u kuk

dx

= −

(g_n)⁰(0), u

kuk ku_nk²−

Z

R^N f(x)|u_n|^qdx−

Z

R^N|u_n|^pdx

− ¹ kuk

Z

R^N|u_n|^p−2u_nudx + ¹

kuk

Z

R^N(_∆un∆u+∇u_n∇u+u_nu)dx− ¹ kuk

Z

R^N f(x)|u_n|^q−2u_nudx

= −

(g_n)⁰(0), u kuk

hI⁰_f(u_n),u_ni+ ¹

kukhI⁰_f(u_n),ui

= ¹ kuk

D

I⁰_f(u_n),uE , that is,

1 n

|h(g_n)⁰(0),ui|ku_nk+kuk≥ hI⁰_f(u_n),ui. By the boundedness ofkunkand Lemma3.2, there exists ˆC>0 such that

Cˆ n ≥

I⁰_f(un), u kuk

. Therefore, we have

kI⁰_f(un)k_H−2(_R^N)= sup

u∈H²(_R^N)\{0}

hI⁰_f(u_n)_,ui kuk ≤ ^Cˆ

n, that is,I⁰_f(un) =on(1)as n→+∞. This completes the proof of (i).

(ii)Similarly, by using Lemma3.2, we can prove (ii). We will omit the details here.

Now, we establish the existence of minimum for I_f on N_f⁺_.

Theorem 3.4. Assume that(F)holds. If|f|_rq ∈ (0,σ), then the functional I_f has a minimizer u⁺in N_f⁺and it satisfies

(i) I_f(u⁺) =α_f =α⁺_f ;

(ii) u⁺is a solution of equation(1.1).

Proof. From Lemma3.3, let{un}be a(PS)_αf sequence for I_f on N_f, i.e.,

I_f(u_n) =α_f +o_n(₁)_, I⁰_f(u_n) =o_n(₁) _in H⁻²(_R^N)_{. (3.10)}

Then it follows from Lemma2.3that{u_n}is bounded inH²(_R^N). Hence, up to a subsequence, there existsu⁺∈ H²(_R^N)such that











un*u⁺ inH²(_R^N);

u_n→u⁺ inL^s_loc(_R^N) (2≤s<2∗); un(x)→u⁺(x) a.e. inR^N.

(3.11)

By(F), Hölder inequality and (3.11), we can infer that Z

R^N f(x)|u_n|^qdx=

Z

R^N f(x)|u⁺|^qdx+o_n(1) asn→_∞. (3.12) In fact, for any e>0, there exists Msufficiently large such that

_Z

|x|>M

|f(x)|^rqdx ¹

rq < e.

And from{u_n} ⊂ N_f inH²(_R^N)is bounded, we obtain that R

R^N|u_n−u⁺|^rdx^q_r

is bounded.

Therefore, we have Z

R^N|f(x)(|u_n|^q− |u⁺|^q)|dx≤

Z

R^N f(x)|u_n−u⁺|^qdx

=

Z

|x|≤M f(x)|un−u⁺|^qdx+

Z

|x|>M f(x)|un−u⁺|^qdx

≤ _Z

|x|≤M

|f(x)|^rqdx _rq¹ _Z

|x|≤M

|un−u⁺|^rdx ^q_r

+ Z

|x|>M

|f(x)|^rqdx _rq¹ Z

|x|>M

|un−u⁺|^rdx ^q_r

→ 0 asn→_∞.

First, we can claim that u⁺ is a nontrivial solution of (1.1). Indeed, by (3.10) and (3.11), it is easy to see that u⁺ is a solution of (1.1). Next we show that u⁺ is nontrivial. Fromu_n ∈ N_f, we have that

I_f(u_n) = 1

2− ¹ p

ku_nk²− 1

q− ¹ p

_Z

R^N f(x)|u_n|^qdx. (3.13) Letn→_∞in (3.13), we can get

α_f ≥ −^p−q pq

Z

R^N f(x)|u⁺|^qdx.

In view of Lemma 2.6, we have 0 > α⁺_f ≥ α_f, which implies R

R^N f(x)|u⁺|^qdx > 0. Thus, u⁺ is a nontrivial solution of (1.1). Now we prove that un → u⁺ strongly in H²(_R^N) and I_f(u⁺) =α. In fact, byu_n,u∈ N_f, (3.12) and weak lower semicontinuity of norm, we have

α_f ≤ I_f(u⁺) = 1

2 − ¹ p

ku⁺k²− 1

q− ¹ p

_Z

R^N f(x)|u⁺|^qdx

≤ lim

n→_∞

1 2− ¹

p

ku_nk²− 1

q− ¹ p

_Z

R^N f(x)|u_n|^qdx

= lim

n→_∞I_f(u_n) =α_f,

which implies thatI_f(u⁺) =α_f and lim_n→_∞ku_nk²= ku⁺k². Noting thatu_n *u⁺in H²(_R^N), so un → u⁺ strongly in H²(_R^N). Furthermore, we have u⁺ ∈ N_f⁺. On the contrary, if u⁺ ∈ N_f⁻, then by Lemma 2.5 (i), there are unique t⁺ and t⁻ such that t⁺u⁺ ∈ N_f⁺ and t⁻u⁺ ∈ N_f⁻. In particular, we have t⁺ < t⁻ = 1 and so I_f(t⁺u⁺)< I_f(t⁻u⁺) = I_f(u⁺) = α_f, which is a contradiction. By Lemma2.4 we may assume that u⁺ is a solution of (1.1). This completes the proof.

In order to obtain the existence of the second local minimum, we consider the following minimization problem:

S₀=inf{I₀(u)|u∈ H²(_R^N)\{0},I₀⁰(u) =0}, where

I0(u) = ¹

2kuk²− ¹ p

Z

R^N|u|^pdx.

From [17,21], we know thatS₀ is achieved atu₀∈ H²(_R^N). Moreover, S0 = I0(u0) =sup

t≥0

I0(tu0). Then, we have the following lemma.

Lemma 3.5. If|f|_rq ∈ (0,σ), thenα⁻_f <α_f +S₀.

Proof. From Lemma2.5(iii), N_f⁻disconnects H²(_R^N)\{0}in exactly two components:

Λ1=

u

1 kuk^t

−

u kuk

>1

, Λ2=

u

1 kuk^t

− u kuk

<1

,

and N_f⁺ ⊂ _Λ1. Moreover, there exists t₁ such that u⁺+t₁u₀ ∈ _Λ2. Indeed, denote t₀ = t⁻((u⁺+tu₀)/ku⁺+tu₀k). Since

t⁻

u⁺+tu₀ ku⁺+tu₀k

u⁺+tu₀ ku⁺+tu₀k

∈ N_f⁻, we have

0≤ ^t

q 0

R

R^N f(x)|u⁺+tu₀|^qdx

ku⁺+tu0k^q = t²₀− ^t

p 0

R

R^N|u⁺+tu₀|^pdx ku⁺+tu0k^{p .} Thus

t₀ ≤

"

ku⁺/t+u₀k R

R^N|u⁺/t+u₀|^p1/p

#p/(p−2)

→ ku₀k ast →_∞.

Therefore, there existst₂ >0 such thatt₀ < lku₀k, for somel > 1 andt ≥ t₂. Set t₁ > t₂+l, then

t⁻

u⁺+t₁u₀ ku⁺+t₁u₀k

2

<l²ku₀k²

≤ ku⁺k²+t²₁ku₀k²+2t₁ Z

R^N(_∆u⁺∆u₀+∇u⁺∇u₀+u⁺u₀)dx

=ku⁺+t₁u₀k²,

that is,u⁺+t₁u₀∈ _Λ2. So there existsk ∈(0, 1)such thatu⁺+kt₁u₀ ∈ N_f⁻. Furthermore, we have

α⁻_f ≤ I_f(u⁺+kt₁u₀)

= ¹

2ku⁺+kt₁u0k²− ¹ q

Z

R^N f(x)|u⁺+kt₁u0|^qdx− ¹ p

Z

R^N|u⁺+kt₁u0|^pdx

< I_f(u⁺) + ¹

2kkt₁u₀k²− ¹ p

Z

R^N|kt₁u₀|^pdx

= I_f(u⁺) +I0(kt₁u0)

≤α_f +I₀(u₀)

=α_f +S₀. This completes the proof.

Next, we establish the existence of minimum forI_f onN_f⁻.

Theorem 3.6. Assume that(F)holds. If|f|_rq ∈(0,σ), then the functional I_f has a minimizer u⁻in N_f⁻and it satisfies

(i) I_f(u⁻) =α⁻_f ;

(ii) u⁻is a solution of equation(1.1).

Proof. From Lemma3.3, let{un}be a(PS)_α⁻

f sequence for I_f on N⁻_f , i.e.,

I_f(u_n) =α⁻_f +o_n(₁)_, I⁰_f(u_n) =o_n(₁) _in H⁻²(_R^N)_{. (3.14)} From Lemma 2.3 we have {un} is bounded in H²(_R^N). Hence, up to a subsequence, there exists u⁻∈ H²(_R^N)such that











u_n*u⁻ inH²(_R^N);

un→u⁻ inL^s_loc(_R^N) (2≤s<2∗); u_n(x)→u⁻(x) _{a.e. inR}^N_.

(3.15)

From (3.14) and (3.15), we havehI⁰_f(u⁻),vi = 0, ∀v ∈ H²(_R^N), that is, u⁻is a weak solution of (1.1) andu⁻∈ N_f. Letvn=un−u⁻. Then











v_n*0 in H²(_R^N);

vn→0 in L^s_loc(_R^N) (2≤ s<2∗); v_n(x)→0 a.e. inR^N.

(3.16)

Now we prove thatu_n→u⁻strongly in H²(_R^N), that is,v_n →0 strongly inH²(_R^N). Arguing by contradiction, we assume that there is c > 0 such that kv_nk ≥ c > 0. By the Brézis–Lieb theorem [3],

I_f(u_n) = ¹

2ku_nk²− ¹ q Z

R^N f(x)|u_n|^qdx− ¹ p

Z

R^N|u_n|^pdx

= I_f(u⁻) + ¹

2kvnk²− ¹ q

Z

R^N f(x)|vn|^qdx− ¹ p

Z

R^N|vn|^pdx+on(1)

= I_f(u⁻) + ¹

2kv_nk²− ¹ p

Z

R^N|v_n|^pdx+o_n(1),

(3.17)