Existence of a ground state solution for a class of singular elliptic problem in \(\mathbb{R}^{N}\)

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R E S E A R C H

Open Access

Existence of a ground state solution for a

class of singular elliptic problem in

R

N

Yanjun Liu

*

*_{Correspondence: xbsdlyj@126.com}

Department of Mathematics and Information Engineering, Chongqing University of Education, Chongqing, 400067, China

Abstract

In this paper, we prove the existence of a ground state solution for a quasi-linear singular elliptic equation inRN_{with exponential growth by using the mountain-pass}

theorem and the Vitali convergence theorem.

Keywords: singular term; exponential subcritical growth; lack of compactness; Trudinger-Moser inequality

1 Introduction and main results

Let⊂RNbe a bounded smooth domain; there are many results on the following prob-lem:

⎧ ⎨ ⎩

–div(|u|p–_u_{) =}_f₍_x_,_u_), _x_∈_,

u∈W_,p(), (.)

whenp= ,|f(x,u)| ≤c(|u|+|u|q–_{),  <}_q_≤_∗₌ N

N–,N≥, for the corresponding results

one may refer to Brézis [], Brézis and Nirenberg [], Bartsch and Willem [] and Capozzi, Fortunato and Palmieri []. Garcia and Alonso [] generalized Brézis, Nirenberg’s exis-tence and nonexisexis-tence results top-Laplace equation. Moreover, let us consider the fol-lowing semilinear Schrödinger equation:

⎧ ⎨ ⎩

–u+V(x)u=f(x,u), x∈RN,

u∈H (RN),

(.)

where|f(x,u)| ≤c(|u|+|u|q–_{),  <}_q_≤_∗ ₌ N

N–. Problem (.) is considered in many

papers such as by Kryszewski and Szulkin [], Alama and Li [], Ding and Ni [] and Jeanjean []. The Sobolev embedding theorems and critical point theory, in particular the mountain-pass theorem would play an important role in studying problems (.) and (.) since both of them have a variational structure. Whenp=Nandf(x,u) behaves like

eα|u|NN–

as|u| → ∞, problem (.) was studied by Adimurthi [], Adimurthi and Yadava [], Rufet al.[, ], do Ó [], Panda [], and the references therein, all these results are based on the Trudinger-Moser inequality [–] and critical point theory.

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Let us consider the problem

–div|u|N–u+V(x)|u|N–u=f(x,u)

|x|η , x∈R

N_, _(.)

whereN≥, ≤η<N,V :RN →Ris a continuous function, f(x,s) is continuous in

RN _×_R_{and behaves like}_eα|s|NN–

as|s| → ∞. The problems of this type are important in many fields of sciences, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, grav-itational, and fluid potentials. Many works focus on the subcritical and critical growth of the nonlinearity which allows us to treat the problem using general critical point theory. Problem (.) can be compared with (.) in this way: the Sobolev embedding theorem can be applied to (.), while the Trudinger-Moser type embedding theorem can be applied to (.). Whenη= , problem (.) was studied by Cao [] in the caseN= , by Panda [], do Ó [] and Alves and Figueiredo [] in the general dimensional case. When  <η<N, problem (.) is closely related to a singular Trudinger-Moser type inequality [], Theo-rem . or [], TheoTheo-rem .

For the problem

–div|u|N–u+V(x)|u|N–u=f(x,u)

|x|η +εh(x), x∈R

N_, _(.)

when η= , the multiplicity of the solutions of the problem (.) is proved by do Ó, Medeiros and Severo [] using Ekeland variational principle and the mountain-pass the-orem. When η> , the existence of a nontrival weak solution of the problem (.) is proved by Adimurthi and Yang [], furthermore, they get a weak solution of negative energy when εis small enough, however, the diﬀerence of these two solutions has not been proved in Adimurthi and Yang []. In Yang [], the author derives similar results for the bi-Laplacian operator in dimension four and Yang [] constructs the existence and multiplicity of a weak solution for the N-Laplacian elliptic equation. Lam and Lu [] considered the existence and multiplicity of a nontrivial weak solution for non-uniformly N-Laplacian elliptic equation.

For the problem

⎧ ⎨ ⎩

–u=Q(x)f(u), x∈RN_,

u∈D,₍_RN_), (.)

whereN≥,f is continuous with subcritical growth andQ∈L∞(RN₎_∩_Lr₍_RN_{), for some} r≥, is positive almost everywhere inRN_{, the existence of a ground state solution of}

problem (.) is proved by Alves, Montenegro and Souto []. Motivated by [], Abreu [] considered the problem

⎧ ⎨ ⎩

–u=Q(x)f_|_x(u_|η), x∈R,

u∈H_(R),

(.)

where ≤η<_,f(s) is continuous and behaves likeeα|s| _when_|_s_{| → ∞}_,_Q_∈_L∞₍_RN₎_∩ Lr₍_RN_{), for some}_r_≥_{, is positive almost everywhere in}_RN_{, Abreu [] showed the}

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In this paper, we consider the problem

⎧ ⎨ ⎩

–div(|u|N–_u_{) +}_V₍_x₎_|_u_|N–_u₌_Q₍_x₎f(u)

|x|η, x∈RN,

u∈W,N₍_RN_), (Pη)

whereN≥, ≤η<N,V:RN _→_R_{is a continuous function satisfying the following}

hypothesis:

(V) V(x)≥V> .

SupposeQ:RN_→_R_{∪ {}_–_∞_{, +}_∞}_satisfying:

(Q) Q> is positive almost everywhere inRN.

(Q) Q∈L∞(RN).

Furthermore, we assume thatf :R→Ris a continuous function satisfying:

(f) f(t) =o(|t|N–)ast→,i.e.limt→_|_tf_|N(t–) = .

(f) f(t)has exponential subcritical growth at+∞,i.e.for allα> , we have

lim

|t|→+∞

f(t)

eα|t|NN–

= .

(f) There existsθ>Nsuch that

 <θF(t)≤tf(t), ∀t∈R\{},

whereF(t) =_tf(s)ds, this is the well-known Ambrosetti-Rabinowitz condition.

Deﬁne a function space

E=

u∈W,NRN

RNV(x)|u|

N_dx_<_∞

,

which is equipped with the norm

u=

RN|u|

N₊_V₍_x₎_|_u_|N_dx 

N

,

then the assumptionV(x)≥V>  impliesEis a reﬂexive Banach space. For ≤η<N,

we deﬁne a singular eigenvalue by

λη= inf

u∈E\{}

uN

RN|Q(x)||u| N

|x|ηdx

.

Ifη= , obviously we haveλ> . If  <η<N, the continuous embedding ofE→Lq(RN)

(q≥N) and the Hölder inequality imply

RN

Q(x) |u|

N

|x|η dx

≤ QL∞(RN₎

{|x|>}

|u|Ndx+QL∞(RN₎

{|x|≤} |u|Ntdx



t

{|x|≤}



|x|ηt dx 

t

≤C

RN

|u|N+|u|Ndx,

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For allq≥N, the embedding

E→W,NRN→LqRN

is continuous. Furthermore, we see that the embeddingE→Lq₍_RN_{) (}_q_≥_N_{) is compact}

(see []) ifVsatisﬁes the following hypothesis: (V) _V ∈L(RN).

Deﬁnition  We say thatu∈Eis a weak solution of problem (Pη) if

RN

|u|N–uφ+V(x)|u|N–uφdx–

RN

Q(x)f(u)

|x|ηφdx= , ∀φ∈E.

DeﬁneI:E→Rby

I(u) = 

Nu–

RNQ(x) F(u)

|x|η dx, u∈E. (.)

From condition (f), it follows that there exist positive constantsαandCsuch that

RN

Q(x)F(u)

|x|η

dx≤CQL∞(RN₎

RN

(eα|u|NN–

–SN–(α,u))

|x|η dx, ∀u∈E,

whereSN–(α,u) =

N–

k=

αk|u|kN/(N–)

k! , thus,Iis well deﬁned thanks to the Trudinger-Moser

inequality andI∈C₍_E_,_R_{). A straightforward calculation shows that}

I(u),φ=

RN|u|

N–_u_φ_dx₊

RNV(x)|u|

N–_uφ_dx_–

RNQ(x) f(u)

|x|ηφdx,

for allu,φ∈E, hence, a critical point of (.) is a weak solution of (Pη).

Deﬁnition ([]) A solutionuof problem (Pη) is said to be ground state if

I(u) =infI(ω) :ω∈W,NRN\{},I(ω)ω= , whereI:E→Ris the functional associated to (Pη).

In this paper, we prove the existence of a ground state solution about problem (Pη) under

weak conditions.

Theorem . Suppose V(x)≥V> inRN(N≥), (V), (f)-(f),ηandσ are two

num-bers which satisfy

≤η<N– 

N , σ>

N–η

N–  –Nη. (.) Let the function Q satisfy(Q)-(Q)and

(Q) Q∈Lr(RN),for somer> N _σ

σ(N––η)+η–N,

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(f) s→_sfN(s–) is increasing fors> 

then this solution is a ground state.

This paper is organized as follows: in Section , we introduce some preliminary results. In Section , we demonstrate Theorem ..

2 Preliminaries

Lemma . Letθ be the number given by the condition(f)and{un}be a sequence

satis-fying

lim sup n→∞

unNN– ≤

Nθ θ–N



N–

cN–_,

for some c> .Then there exist constantsα> ,t> ,C> ,independent of n,such that

RN

eα|un|NN–

–SN–(α,un) |x|η

t dx≤C,

for n large enough.

Proof Let

α=αN

σ

 – η

N

θ–N Nθ



c, (.)

whereαN=NωN/(–N–)andωN–is the measure of the unit sphere inRN.

Since ≤η<N, we haveα> . Moreover, let

m:= 

N

N–  +σ· N N–η

Nθ θ–N



N–

cN–_,

then by (.)

Nθ θ–N



N–

cN–<_m,

hence, passing to a subsequence, there existsn∈Nsuch that un

N

N– _<_m_, ∀_n≥_n .

From (.), we also have

N

N– <

N_σ

N–η+Nσ+Nσ η.

Lettsatisfy

N

N– <t<

Nσ

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For sucht, we have

N

N– <t<

 –ηt

N

N_σ

N–η+Nσ.

Letβ∈Rsuch that

t<β<

 –ηt

N

N_σ

N–η+Nσ =

 –ηt

N

αN αm.

Notice m

unNN–

> , using Lemma . in Yang [], we have

RN

(eα|un|NN–

–SN–(α,un))t |x|ηt dx≤C

RN

eβαm(|unun|)

N N–

–SN–(βαm,|un_un|)

|x|ηt dx,

for eachn≥n. By the choice ofβ, we have

βαm<

 –ηt

N

αN.

Then we conclude the proof by using the Trudinger-Moser inequality [, ].

Lemma . Suppose{un}is bounded in E and the assumptions of(Q)-(Q)and(f)-(f)

are satisﬁed.If

Q(x)|f(un(x))un(x)|

|x|η →Q(x)

|f(u(x))u(x)|

|x|η a.e. inR

N_,

then

RNQ(x) f(un)un

|x|η dx→

RNQ(x) f(u)u

|x|η dx. (.)

Proof From (f) and (f), for allε>  andα, there exist positive constantsδ,K,C, such

that

f(s)s ≤ε|s|N+εC|s|N+eαsNN–

–SN–(α,s)

+ max

δ≤|s|≤K

f(s)s , ∀s∈R.

ForR> , then

BR()

Q(x)|f(un)un|

|x|η dx

≤εQ_L∞₍_RN₎

RN

|un|N

|x|η dx+εCQL∞(RN)

RN|un|

(eαu N N–

n _–_SN

–(α,un))

|x|η dx

+ max

δ≤|s|≤K

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Here,|BR()|denotes the volume of the ballBR(). We can considerα>  given by (.) and Lemma . implies that there existst>  such that, up to a subsequence,

eαu N N–

n _–_SN

–(α,un)

|x|η ∈L

t_RN_, _∀_n_∈_N_,

and there existsC>  such that

RN

eα|un|NN–

–SN–(α,un) |x|η

t

dx≤C, ∀n∈N.

By applying Hölder’s inequality with exponentstand its conjugatetand using a contin-uous embedding, we see that there exist positive constantsCandCsuch that

BR()

Q(x)|f(un)un|

|x|η dx≤εC+εC+_δ_≤|max_s_|≤_K f(s)s BR() . (.)

On the other hand, for allε>  and allα> , there existsC=C(ε,α) >  such that

f(s)s ≤ε|s|N+C|s|N+eαsNN–

–SN–(α,s)

, ∀s∈R.

Then we have

RN_\_B R()

Q(x)|f(un)un|

|x|η dx

≤εQ_L∞₍_RN₎

RN_\_BR_()

|un|N

|x|η dx

+C

RN_\_BR_()Q(x)|un| N+(eαu

N N–

n _–_SN

–(α,un)) |x|η dx.

By consideringα>  given by (.) and applying the Holder inequality with exponentsr

given by (Q),tgiven by (.), andNsuch that



r+



t+



N = ,

from the Hölder inequality and there being a continuous embedding, there exist positive constantsCandCsuch that

RN_\_BR_()Q(x)

|f(un)un|

|x|η dx≤εC+C

RN_\_BR_()Q(x) r_dx



r

.

Using (Q), we have

RN_\_BR_()Q(x)

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Thus, forR>  large enough,

RN_\_BR_()Q(x)

|f(un)un|

|x|η dx≤εC+εC. (.)

From (.) and (.), the sequence

Q(x)|f(un)un|

|x|η

is equi-integrable. If

Q(x)|f(un(x))un(x)|

|x|η →Q(x)

|f(u(x))u(x)|

|x|η a.e. inR

N_,

Vitali’s theorem implies (.). Firstly one proves that functionalI has the geometry of the mountain-pass theorem, more exactly, we have the following lemma.

Lemma .([]) If V(x)≥V> inRN, (V), (f)-(f)are satisﬁed,then the functional I

veriﬁes the following properties:

(i) There existr,ρ> ,such that whenu=r,

I(u)≥ρ.

(ii) There existse∈Bc

r()withI(e) < .

By Lemma ., using a version of the mountain-pass theorem without the Palais-Smale condition (see []), we obtain the existence of a sequence{un}inEsatisfying

I(un)→c and I(un)→ inX, (.) wherec=infγ∈maxt∈[,]I(γ(t)) > ,={γ∈C([, ],X) :γ() = ,I(γ()) < }.

Lemma .([]) Suppose V(x)≥V> inRN, (V), (f)-(f)are satisﬁed,let{un}be an arbitrary Palais-Smale sequence,then{un}is bounded and there exists a subsequence of

{un}(still denoted by{un}),u∈E such that ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Q(x)f_|(_xun_|η) →Q(x)

f(u)

|x|η strongly in L_loc(RN), un→u almost everywhere inRN,

|un|N–_un_|_u_|N–_u _{weakly in}₍_LN/(N–)₍_RN₎₎N_.

Furthermore,uis a weak solution of (Pη).

3 Proof of the theorem

Proof of Theorem. By Lemma . and Lemma ., we see that the Palais-Smale sequence

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We will show thatuis nonzero. Since{un}is a Palais-Smale sequence, Lemma . implies that

lim n→∞un

N₌ _lim n→∞

RNQ(x) f(un)un

|x|η dx=

RNQ(x) f(u)u

|x|η dx.

Recalling thatuis a critical point ofI, we conclude that

lim n→∞un

N₌

RN

Q(x)f(u)u

|x|η dx=u

N_.

Ifu≡, then

lim n→∞un

N_{= .}

SinceI∈C₍_E_,_R_{), we have}

I(un)→,

and it is a contradiction becauseI(un)→candc> . This way, we conclude thatuis nonzero.

Now, we will show thatuis a ground state. Setting

m=inf

u∈I(u), :=

u∈E\{}:I(u)u= . Since{un}is a Palais-Smale sequence, we see that

c=lim inf

n→∞ I(un) =lim infn→∞

I(un) –I(un)un

=lim inf n→∞

RNQ(x)

(f(un)un– F(un))

|x|η dx.

By Fatou’s lemma,

c≥

RN

Q(x)(f(u)u– F(u))

|x|η dx.

Ifuis a critical point ofI, we have

I(u) =I(u) –I(u)u=

RN

Q(x)(f(u)u– F(u))

|x|η dx.

Hence, we can conclude thatI(u)≤c, thusm≤c. On the other hand, as in the proof of Theorem . in [], letu∈and deﬁneh: (, +∞)→Rbyh(t) =I(tu). We see thathis diﬀerentiable and

h(t) =I(tu)u=tN–uN–

RNQ(x) f(tu)u

|x|η dx, ∀t> .

SinceI(u)u= , we get

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so

h(t) =tN–

RN Q(x)

|x|η

f(u)

uN––

f(tu) (tu)N–

uNdx, ∀t> .

Using the condition (f), we conclude thath(t) >  for  <t<  andh(t) <  fort> , since

h() = , thus,

I(u) =max t≥ I(tu).

Now, deﬁneγ : [, ]→E,γ(t) =ttu, wheretis a real number which satisﬁesI(tu) < ,

we haveγ ∈, and therefore

c≤max t∈[,]I

γ(t)≤max

t≥ I(tu) =I(u),

u∈is arbitrary,c≤m, thusI(u) =c=m. This ends the proof of Theorem ..

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author would like to thank the referee for his/her valuable comments, which have led to an improvement of the presentation of this paper. This work was supported by the National Natural Science Foundation of China (11471147) and the college project of Chongqing University of Education in China (KY201548C).

Received: 19 March 2016 Accepted: 6 December 2016

References

1. Brézis, H: Elliptic equations with limiting Sobolev exponents. Commun. Pure Appl. Math.39, 517-539 (1986) 2. Brézis, H, Nirenberg, L: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents.

Commun. Pure Appl. Math.36, 437-477 (1983)

3. Bartsch, T, Willem, M: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc.123, 3555-3561 (1995)

4. Capozzi, A, Fortunato, D, Palmieri, G: An existence result for nonlinear elliptic problems involving critical Sobolev exponent. Ann. Inst. Henri Poincaré, Anal. Non Linéaire2, 463-470 (1985)

5. Garcia, A, Alonso, P: Existence and non-uniqueness for the p-Laplacian. Commun. Partial Diﬀer. Equ.12, 1389-1430 (1987)

6. Kryszewski, W, Szulkin, A: Generalized linking theorem with an application to semilinear Schrödinger equation. Adv. Diﬀer. Equ.3, 441-472 (1998)

7. Alama, S, Li, YY: Existence of solutions for semilinear elliptic equations with indeﬁnite linear part. J. Diﬀer. Equ.96, 89-115 (1992)

8. Ding, WY, Ni, WM: On the existence of positive entire solutions of a semilinear elliptic equation. Arch. Ration. Math. Anal.31, 283-308 (1986)

9. Jeanjean, L: Solutions in spectral gaps for a nonlinear equation of Schrödinger type. J. Diﬀer. Equ.112, 53-80 (1994) 10. Adimurthi, A: Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the

N-Laplacian. Ann. Sc. Norm. Super. PisaXVII, 393-413 (1990)

11. Adimurthi, A, Yadava, SL: Multiplicity results for semilinear elliptic equations in a bounded domain ofR2_involving

critical exponent. Ann. Sc. Norm. Super. PisaXVII, 481-504 (1990)

12. De Figueiredo, DG, Do Ó, JM, Ruf, B: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Commun. Pure Appl. Math.IV, 135-152 (2002)

13. De Figueiredo, DG, Miyagaki, OH, Ruf, B: Elliptic equations inR2_{with nonlinearities in the critical growth range. Calc.}

Var.3, 139-153 (1995)

14. Do Ó, JM: Semilinear Dirichlet problems for the N-Laplacian inRN_{with nonlinearities in the critical growth range.} Diﬀer. Integral Equ.9, 967-979 (1996)

15. Panda, R: On semilinear Neumann problems with critical growth for the N-Laplacian. Nonlinear Anal.26, 1347-1366 (1996)

16. Moser, J: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J.20, 1077-1091 (1971) 17. Pohozaev, S: The Sobolev embedding in the special case pl = n. In: Proceedings of the Technical Scientiﬁc

Conference on Advances of Scientiﬁc Research 1964–1965. Mathematics Sections, pp. 158-170. Moscov. Energet. Inst., Moscow (1965)

18. Trudinger, NS: On embeddings into Orlicz spaces and some applications. J. Math. Mech.17, 473-484 (1967) 19. Cao, DM: Nontrivial solution of semilinear elliptic equations with critical exponent inR2_{. Commun. Partial Diﬀer. Equ.}

(11)

20. Panda, R: Nontrivial solution of a quasilinear elliptic equation with critical growth in_RN_{. Proc. Indian Acad. Sci. Math.} Sci.105, 425-444 (1995)

21. Do Ó, JM: N-Laplacian equations inRN_{with critical growth. Abstr. Appl. Anal.}_{2, 301-315 (1997)}

22. Alves, CO, Figueiredo, GM: On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth inRN_{. J. Diﬀer. Equ.}_{246, 1288-1311 (2009)}

23. Adimurthi, A, Yang, Y: An interpolation of Hardy inequality and Trudinger-Moser inequality in_RN_{and its applications.} Int. Math. Res. Not.13, 2394-2426 (2010)

24. Souza, M, Do Ó, JM: On singular Trudinger-Moser type inequalities for unbounded domains and their best exponents. Potential Anal.38, 1091-1101 (2013)

25. Do Ó, JM, Medeiros, E, Severo, U: On a quasilinear nonhomogeneous elliptic equation with critical growth in_RN_. J. Diﬀer. Equ.246, 1363-1386 (2009)

26. Yang, Y: Adams type inequalities and related elliptic partial diﬀerential equations in dimension four. J. Diﬀer. Equ.252, 2266-2295 (2012)

27. Yang, Y: Existence of positive solutions to quasilinear elliptic equations with exponential growth in the whole Euclidean space. J. Funct. Anal.262, 1679-1704 (2012)

28. Lam, N, Lu, G: Existence and multiplicity of solutions to equations of n-Laplacian type with critical exponential growth in_RN_{. J. Funct. Anal.}_{262, 1132-1165 (2012)}

29. Alves, CO, Souto, M, Montenegro, M: Existence of solution for two classes of elliptic problems inRN_{with zero mass.} J. Diﬀer. Equ.252(10), 5735-5750 (2012)

30. Abreu, R: Existence of a ground state solution for a singular elliptic problem in unbounded domain and dimension 2. Nonlinear Anal.98, 104-109 (2014)

31. Willem, M: Minimax Theorems. Birkhauser, Basel (1996)