Existence of multiple positive solutions for nonhomogeneous elliptic problems in R

Contents lists available atSciVerse ScienceDirect

Nonlinear Analysis

journal homepage:www.elsevier.com/locate/na

Existence of multiple positive solutions for nonhomogeneous elliptic problems in R

^N

Tiexiang Li

^a

, Tsung-fang Wu

^b,∗

aDepartment of Mathematics, Southeast University, Nanjing 211189, PR China

bDepartment of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

a r t i c l e i n f o

Article history:

Received 31 October 2011 Accepted 17 May 2012 Communicated by S. Ahmad Keywords:

Nonhomogeneous elliptic problems Palais–Smale

Multiple positive solutions

a b s t r a c t

In this paper, we study the multiplicity of positive solutions for the nonhomogeneous elliptic problem:−∆^u+λ^u = f(^x)^up−1+µ^h(^x)^inRN. We will show how the shape of the graph off(^x)affects the number of positive solutions.

©2012 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we study the multiplicity of positive solutions for the following nonhomogeneous elliptic problem:







−

_∆u

+ λ

^u

=

f

(

^x

)

^up−1

+ µ

^h

(

^x

)

ⁱⁿR^N

,

u

>

^{0 in}R^N

,

u

∈

H¹



R^N



, (

^Eλ,µ

)

where 2

<

^p

<

_N^2N−2

(

^N

≥

3

) ,

the parameters

λ, µ >

⁰

,

^f

∈

C



R^N



andh

∈

L²



R^N



\ {

0

}

withh

≥

0

.

Under the assumption

µ ̸=

0, our equation(^Eλ,µ)can be regarded as a perturbation problem of the following semilinear elliptic equation:







−

_∆u

+ λ

^u

=

f

(

^x

)

^{up−1 in}R^N

,

u

>

^{0 in}R^N

,

u

∈

H¹



R^N



.

(1)

It is known that the existence of positive solutions of Eq.(1)is affected by the shape of the graph off

(

^x

)

. This has been the focus of a great deal of research by several authors (see [1–7] etc.). Furthermore, iff is a positive constant, then Eq.(1) has a unique radially symmetric positive solution (see [8,9]).

Whenf

≡

1

,

^his the exponential decay and

µ

small, Hirano [10] and Zhu [11] showed that Eq.(^Eλ,µ)admits at least two positive solutions. Generalizations of the results of [10,11] were made by Cao and Zhou [12], Jeanjean [13] and Adachi and Tanaka [14,15]. In [15] Adachi and Tanaka showed the existence of at least four positive solutions of Eq.(^Eλ,µ)^{under the} following assumptions:

∗Corresponding author. Tel.: +886 7 591 9519; fax: +886 7 591 9344.

E-mail addresses:txli@seu.edu.cn(T. Li),tfwu@nuk.edu.tw(T.-f. Wu).

0362-546X/$ – see front matter©2012 Elsevier Ltd. All rights reserved.

doi:10.1016/j.na.2012.05.012

(f1) f

(

^x

) →

1 as

|

_x

| → ∞ ,

(f2) f

(

^x

) ∈ (

⁰

,

¹

]

for allx

∈

_R^Nandf

(

^x

) ̸≡

1

,

(f3) there exist

δ >

^{0 andC}

>

0 such that

f

(

^x

) −

1

≥ −

Cexp

( − (

²

+ δ) |

x

| )

^{for allx}

∈

_R^N

,

and

µ

sufficiently small. In [12–14], the general equations







−

_∆u

+ λ

^u

=

g

(

^x

,

^u

) + µ

^h

(

^x

)

ⁱⁿR^N

,

u

>

^{0 in}R^N

,

u

∈

H¹



R^N



were studied, wheregsatisfies some suitable conditions andh

∈

H⁻¹



R^N



\ {

0

}

is non-negative, and the existence of at least two positive solutions when

µ

sufficiently small was proved.

The main purpose of this paper is to use the shape of the graph off

(

^x

)

to prove the multiplicity of positive solutions for Eq.(^Eλ,µ). First, we consider the following assumptions:

(

^f3

)

there exist

δ >

^{0 andC}

>

0 such that

f

(

^x

) −

1

≥ −

Cexp

( − (

¹

+ δ) |

x

| )

^{for allx}

∈

_R^N

; (

^f4

)

there exist some pointsx¹

,

^x2

, . . . ,

^xkinR^Nsuch thatf



xⁱ



are strict maximums onR^Nwith 1

=

f



xⁱ



≡

max



f

(

^x

) |

x

∈

_R^N



for alli

=

1

,

²

, . . . ,

^k

.

Then we have the following results.

Theorem 1.1. Suppose that the function f satisfies the conditions

(

^f1

)

^,

(

^f2

)

^and

(

^f3

)

. Then for each h

∈

L²



R^N



\ {

0

}

with h

≥

0 there exists a positive numberΛ^∗such that for any

λ, µ >

^0with

µλ

^{N4−pp−−12}

<

Λ^{∗, Eq.}(^Eλ,µ)has at least four positive solutions.

Theorem 1.2. Suppose that the function f satisfies the conditions

(

^f1

)

^,

(

^f2

)

^,

(

^f3

)

^and

(

^f4

)

. Then for each h

∈

L²



R^N



\ {

0

}

_with h

≥

0there exist positive numbersΛ∗and

λ

0such that for any

λ > λ

0and

µ >

^0with

µλ

^{N4−pp−−12}

<

Λ∗, Eq.(^Eλ,µ)has at least k

+

3positive solutions.

This paper is organized as follows. In Section2, we prove the existence of four positive solutions. In Section3, we prove the existence of otherkpositive solutions. Based on this result, we can proveTheorem 1.2.

2. Existence of four solutions By the change of variables

η =

^√1

λ

, v (

^x

) = η

^2/(p−2)u

(η

^x

)

^{, Eq.}(^Eλ,µ)is transformed to



 

 

−

_∆

v + v =

f_η

v

^p−1

+ µη

^{2(pp−−21)h}η inR^N

,

v >

^{0 in}R^N

,

v ∈

H¹



R^N

 ,

(2)

where f_η

=

f

(η

^x

)

^{and h}η

=

h

(η

^x

)

. Associated with Eq. (2), we consider the following minimization problem: for

η >

⁰

, µ ≥

0 andu

∈

H¹



R^N



define I_η,µ

(

^u

) =

¹

2

∥

u

∥

²

H¹

−

¹ p



R^N

f_ηu^p₊dx

− µη

^{2(pp−−21)}



R^N

h_ηudx

;

M_η,µ

= 

u

∈

H¹



R^N



\ {

0

} | 

I_η,µ^′

(

^u

) ,

^u



=

0



; α

η,µ

=

inf



I_η,µ

(

^u

) |

u

∈

M_η,µ



where

∥

u

∥

_H1

= 

R^N

|∇

u

|

²

+

u²



1/2

is a standard norm inH¹



R^N



andu+

=

max

{

0

,

^u

}

. It is well known that the solutions of Eq.(2)are the critical points of the energy functionalI_η,µ(see [16]).

Now, we study the break down of the (PS)-condition forI_η,µ

.

First, we introduce the following elliptic equation:







−

_∆u

+

u

=

u^p−1 inR^N

,

u

>

^{0 in}R^N

,

u

∈

H¹



R^N



.

(3)

We define the energy functionalI^∞

:

H¹



R^N



→

_Ras follows I^∞

(

^u

) =

¹

2

∥

u

∥

²

H¹

−

¹ p



R^N

u^p₊dx

.

Using the results of Berestycki and Lions [8] and Kwong [9], Eq.(3)has a unique radially symmetric positive solution

w (

^x

)

such that

α

^∞

=

I^∞

(w) =

inf



I^∞

(

^u

) |

for any positive solutionuof Eq.(3)



and for any

δ >

^{0 andx}

∈

_R^N

,

w (

^x

) ≤

Cexp

( − (

¹

− δ) |

_x

| )

^and

|∇ w (

^x

) | ≤

_Cexp

( − (

¹

− δ) |

_x

| )

for someC

>

0. Moreover, the unique solution

w (

^x

)

^{of Eq.(3)}plays an important role in describing the asymptotic behavior of a (PS)-sequence forI_η,µ

.

Proposition 2.1. Let

{

u_n

}

be a (PS)-sequence in H¹

(

R^N

)

^{for I}η,µ. Then there exist a subsequence

{

u_n

}

, an integer m

∈

_N

∪ {

0

} ,

^m sequences



x¹_n

 , 

x²_n

 , . . . , 

x^m_n



⊂

_R^Nand a critical point u₀

∈

H¹

(

R^N

)

^{of I}η,µsuch that





xⁱ_n



 → ∞

for 1

≤

i

≤

m

,





xⁱ_n

−

x^j_n



 → ∞

for 1

≤

i

,

^j

≤

m and i

̸=

j

,

u_n

⇀

^u0 weakly in H¹

(

R^N

),

u_n

=

_u0

+

m



i=1

w( · −

_xⁱ_n

) +

_o

(

¹

)

strongly in H¹

(

R^N

),

I_η,µ

(

^un

) =

I_η,µ

(

^u0

) +

mI^∞

(w) +

o

(

¹

).

Proof. This is a standard result. See Lions [6,7] and Benci and Cerami [17] for analogous arguments.

2.1. Existence of a local minimum Define

ψ

η,µ

(

^u

) = 

I_η,µ^′

(

^u

) ,

^u



= ∥

u

∥

²

H¹

−



R^N

f_ηu^p₊dx

− µη

^{2(pp−−21)}



R^N

h_ηudx

.

Clearly, foru

∈

M_fη,hη, we have

 ψ

_η,µ^′

(

^u

) ,

^u



= ∥

u

∥

²

H¹

− (

^p

−

1

) 

R^N

f_ηu^p₊dx (4)

= (

²

−

p

) ∥

_u

∥

²

H¹

− (

¹

−

p

) µη

^{2(pp−−21)}



R^N

h_ηudx

.

⁽⁵⁾

Let

Λ0

=

^p

−

2 p

−

1



S_p^p/2 p

−

1



_p−¹₂

∥

h

∥

⁻¹

L²

.

Then we have the following result.

Lemma 2.2. For each

η, µ >

^0with

µη

^{2(pp−−21)−N2}

<

Λ0, we have

 ψ

_η,µ^′

(

^u

) ,

^u



̸=

_{0 for all u}

∈

M_η,µ

.

Proof. Our proof is almost the same as that in [18, Lemma 2.3]. Suppose the contrary. Then there exist

η, µ >

^{0 with}

µη

^{2(pp−−21)−N2}

<

Λ0such that

 ψ

_η,µ^′

(

^u

) ,

^u



=

0

.

Then, foru

∈

M_η,µwith



ψ

_η,µ^′

(

^u

) ,

^u



=

0, by(5)and the Hölder and Sobolev inequalities we have

∥

_u

∥

²

H¹

= µη

^{2(pp−−21)p}

−

1 p

−

2



R^N

h_ηudx

≤ µη

^{2(pp−−21)−N2p}

−

1

p

−

2

∥

_h

∥

_L2

∥

_u

∥

_H1

and so

∥

_u

∥

_H1

≤ µη

^{2(pp−−21)−N2p}

−

1 p

−

2

∥

_h

∥

_L2

.

Similarly, using(4),

(

^f2

)

and the Sobolev inequality we have 1

p

−

1

∥

u

∥

²

H¹

=



R^N

f_ηu^p₊dx

≤

S

−p p2

∥

u

∥

^p

H¹

,

which implies

∥

u

∥

_H1

≥



S_p^p/2 p

−

1



_p−¹₂

for all

µ ≥

0

.

Hence, we must have

µη

^{2(pp−−21)−N2}

≥

^p

−

2 p

−

1



S_p^p/2 p

−

1



_p−¹₂

∥

_h

∥

⁻¹

L²

=

_Λ0 which is a contradiction. This completes the proof.

ByLemma 2.2, we may writeM_η,µ

=

M⁺_η,µ

∪

M⁻_η,µ, where M⁺_η,µ

=



u

∈

M_η,µ



 ∥

_u

∥

²

H¹

− (

^p

−

1

)



R^N

f_η

|

_u+

|

^p_dx

>

⁰



;

M⁻_η,µ

=



u

∈

M_η,µ



 ∥

u

∥

²

H¹

− (

^p

−

1

)



R^N

f_η

|

u+

|

^pdx

<

⁰



and define

α

_η,µ⁺

=

inf

u∈M+ η,µ

I_η,µ

(

^u

)

^and

α

_η,µ⁻

=

inf

u∈M− η,µ

I_η,µ

(

^u

) .

Then we have the following result.

Theorem 2.3. We have the following.

(i)

α

_η,µ⁺

<

^{0for all}

η, µ >

^0with

µη

^{2(pp−−21)−N2}

<

Λ0

.

(ii) If

η, µ >

^0with

µη

^{2(pp−−21)−N2}

<

^Λ₂^{0, then}

α

⁻_η,µ

>

^c0for some c₀

>

⁰

.

In particular, for each

η, µ >

^0with

µη

^{2(pp−−21)−N2}

<

^Λ₂^{0, Eq.(2)}has a unique positive solution u¹_η,µ

∈

M⁺_η,µsuch that I_η,µ



u¹_η,µ



= α

_η,µ⁺

= α

η,µ

.

Proof. Our proof is almost the same as that in [14, Lemma 1.4] and [19, Theorem 3.1].

2.2. Existence of two solutions

First, we establish the decay estimate for solutions of Eq.(2).

Lemma 2.4. Let u₀

∈

H¹



R^N



be a positive solution of Eq.(^Eλ,µ)^{. Then}

v

0

(

^x

) = η

^2/(p−2)u0

(η

^x

)

is a positive solution of Eq.(2)and

v

0

(

^x

) ≥

C

η

^{2(6p−−p2)exp}

( − (

¹

+ ε) η |

x

| ) ,

^{for all}

|

x

| ≥

R₀for some C

>

⁰

.

⁽⁶⁾ Proof. Our proof is almost the same as that in [20,21].

Forc

>

0, we define I_η,^c₀

(

^u

) =

¹

2

∥

u

∥

²

H¹

−

¹ p



R^N

cf_η

|

u+

|

^pdx

;

M^c_η,0

=



u

∈

H¹



R^N



\ {

0

} | 



I_η,^c₀



′

(

^u

) ,

^u



=

0



;

M_η,0

= 

u

∈

H¹



R^N



\ {

0

} | 

I_η,^′₀

(

^u

) ,

^u



=

0



.

Note thatI_η,0

=

I_η,^c₀forc

=

1, and for eachu

∈

M⁻_η,µthere is a uniquet¹

=

t¹

(

^u

) >

0 such thatt¹u

∈

M_η,0. Then we have the following results.

Lemma 2.5. Suppose that

η, µ >

^0with

µη

^{2(pp−−21)−N2}

<

^Λ₂⁰. Then for each u

∈

M⁻_η,µ, we have the following.

(i) There is a unique t^c

(

^u

) >

⁰such that t^c

(

^u

)

^u

∈

M^c_η,0and

max

t≥0 I_η,^c₀

(

^tu

) =

_Iη,^c₀



t^c

(

^u

)

^u



=



1 2

−

¹

p



c

−2 p−2

 ∥

_u

∥

^p

H¹



R^Nf_η

|

u+

|

^pdx



_p−²₂

.

(ii)For

σ ∈



0

, µ

⁻¹

η

^{N2−2(pp−−21)}

 ,

I_η,µ

(

^u

) ≥



1

− σ µη

^{2(pp−−21)−N2}



p p−2

I_η,0



t¹u



− η

^{2(pp−−21)−N2} 2

σ ∥

h

∥

²

L²

and

I_η,µ

(

^u

) ≤



1

+ σ µη

^{2(pp−−21)−N2}



p p−2

I_η,0



t¹u



+ η

^{2(pp−−21)−N2} 2

σ ∥

h

∥

²

L²

.

Proof. (i) Similar to the proof of Lemma 7.1 in Wu [19].

(ii) For eachu

∈

M⁻_η,µ, letc

=

1

/ 

1

− σµη

^{2(pp−−21)−N2}



,t^c

=

t^c

(

^u

) >

^{0 andt1}

=

t¹

(

^u

) >

0 such thatt^cu

∈

M^c_η,0and t¹u

∈

M_η,0. For

σ ∈ (

⁰

,

¹

)

^{, we have}











R^N

h_ηt^cudx









≤ η

^−N2





t^cu





H¹

∥

h

∥

_L2

≤ ση

^−N2 2





t^cu





2 H¹

+ η

^−N2

2

σ ∥

h

∥

²

L²

.

Then by part (i) and²⁽_p^p₋⁻₂¹⁾

−

^N

2

>

⁰

,

sup

t≥0

I_η,µ

(

^tu

) ≥

I_η,µ



t^cu



≥

¹ 2





t^cu





2 H¹

−

¹

p



R^N

f_η



t^cu+



dx

− µη

^{2(pp−−21)}

 ση

^−N2 2





t^cu





2 H¹

+ η

^−N2

2

σ ∥

h

∥

²

L²



=

¹ 2c





t^cu





2 H¹

−

¹

p



R^N

f_η



t^cu+



p

dx

− µη

^{2(pp−−21)−N2} 2

σ ∥

_h

∥

²

L²

=

¹ cI_η,^c₀



t^cu



− µη

^{2(pp−−21)−N2} 2

σ ∥

h

∥

²

L²

=



1

− σ µη

^{2(pp−−21)−N2}



p p−2



1 2

−

¹

p

 

∥

u

∥

^p

H¹



R^Nf_η

|

u+

|

^pdx



_p−²₂

− µη

^{2(pp−−21)−N2} 2

σ ∥

_h

∥

²

L²

=



1

− σµη

^{2(pp−−21)−N2}



p p−2

I_fη,0



t¹u



− µη

^{2(pp−−21)−N2} 2

σ ∥

h

∥

²

L²

.

Moreover, byTheorem 2.3and [20, Lemma 2.4],

sup

t≥0

I_η,µ

(

^tu

) =

I_η,µ

(

^u

) .

Thus,

I_η,µ

(

^u

) ≥



1

− σ µη

^{2(pp−−21)−N2}



p p−2

I_fη,0



t¹u



− µη

^{2(pp−−21)−N2} 2

σ ∥

h

∥

²

L²

.

Moreover,

I_fη,hη

(

^tu

) ≤



1

+ σ µη

^{2(pp−−21)−N2}



2

∥

tu

∥

²

H¹

−

¹ p



R^N

f_η

|

tu+

|

^pdx

+ µη

^{2(pp−−21)−N2} 2

σ ∥

h

∥

²

L²

and so

I_fη,hη

(

^u

) ≤



1

+ σ µη

^{2(pp−−21)−N2}



p p−2

I_fη,0



t¹u



+ µη

^{2(pp−−21)−N2} 2

σ ∥

h

∥

²

L²

.

This completes the proof.

Let

w (

^x

)

be a unique radially symmetric positive solution of Eq.(3)ande

∈

S^N−1

= 

x

∈

_R^N

| |

x

| =

₁



. We denote

w

l

(

^x

) = w (

^x

+

le

) ,

^l

∈ (

⁰

, ∞ ) .

Then we have the following results.

Lemma 2.6. Suppose that

η, µ >

^0with

µη

^{2(pp−−21)−N2}

<

^Λ₂⁰

.

^Then (i) there exists t₀

>

^{0such that}

I_η,µ



u¹_η,µ

+

t

w

l

 <

^Iη,µ



u¹_η,µ



for all t

≥

t₀and e

∈

S^N−1

;

(ii) there exists l₁

>

⁰such that for l

>

^l1

,

sup

t≥0

I_η,µ



u¹_η,µ

+

t

w

l

 <

^Iη,µ



u¹_η,µ



+ α

^∞

= α

η,µ

+ α

^∞

,

where u¹_η,µis the local minimum inTheorem2.3.

Proof. (i) Sinceu¹_η,µis a positive solution of Eq.(2), using the fact that



R^N

∇

u_η,µ

∇ w

ldx

= − 

R^N

w

l∆^uη,µdxwe have I_η,µ



u¹_η,µ

+

t

w

l

 ≤

I_η,µ



u¹_η,µ



+

^t

2

2

∥ w

l

∥

²

H¹

+

t



R^N

f_η

w

l



u¹_η,µ



p−1

dx

−

^t

p

p



R^N

f_η

w

l^pdx

≤

I_η,µ



u¹_η,µ



+

^t

2

2

∥ w

l

∥

²

H¹

+

t



R^N

f_η

w

l



u¹_η,µ



p−1

dx

−

^t

p

p



B^N(0;1)

w

^pdx

.

Sincep

>

^{2 and}

w >

^{0 in}R^N, we can chooset₀

>

0 large enough such that (i) holds.

(ii) SinceI_η,µis continuous inH¹



R^N



, there existst₁

>

0 such that forl

>

⁰

,

I_η,µ



u¹_η,µ

+

t

w

l

 <

^Iη,µ



u¹_η,µ



+ α

^{∞ for allt}

<

^t1ande

∈

S^N−1

.

Using part (i) we know that forl

>

⁰

,

sup

t≥t0

I_η,µ



u¹_η,µ

+

t

w

l

 <

^Iη,µ



u¹_η,µ



+ α

^{∞ for alle}

∈

S^N−1

.

Thus, we only need to show that there existsl₁

>

0 such that forl

>

^l1

,

sup

t1≤t≤t0

I_η,µ



u¹_η,µ

+

t

w

l

 <

^Iη,µ



u¹_η,µ



+ α

^{∞ for alle}

∈

S^N−1

.

By Brown and Zhang [22] and Willem [23], we know that

sup

t>⁰I^∞

(

^t

w) =

I^∞

(w) = α

^∞

.

⁽⁷⁾

Forl

>

^{0 andt}1

≤

t

≤

t₀

,

I_η,µ



u¹_η,µ

+

t

w

l



≤

I_η,µ



u¹_η



+

I^∞

(

^t

w) +

¹ p



R^N



1

−

f_η



t^p

w

^pldx

−

¹ p



B^N(^le;1) f_η



t1wl 0



u¹_η,µ

+

s



p−1

− 

u¹_η,µ



p−1

−

s^p−1dsdx

≤

I_η,µ



u¹_η,µ



+

I^∞

(

^t

w) + (

^I

) − (

^II

) .

Using(7), we have

sup

t1≤t≤t0

I_η,µ



u¹_η,µ

+

t

w

l

 ≤

I_η,µ



u¹_η,µ



+

I^∞

(w) + (

^I

) − (

^II

) .

We recall the fact that for somec

>

⁰

w ( |

x

| ) |

x

|

^N−21exp

( |

x

| ) →

c as

|

x

| → ∞ .

(See [1,2,9,24]). In particular, there exists a constantC₀

>

0 such that

w (

^x

) ≤

C₀exp

( −|

x

| )

^{for allx}

∈

_R^N

.

Then by the Taylor expansion,



t1wl 0



u¹_η,µ

+

s



p−1

− 

u¹_η,µ



p−1

−

s^p−1ds

≥



t1wl

0

(

^p

−

1

)

^sp−2u1η,µ

− 

u¹_η,µ



p−1

ds

=

 (

^t1

w

l

)

^p−2

− 

u¹_η,µ



p−2



t₁

w

lu¹_η,µ

.

⁽⁸⁾