non-nehari manifold method for fractional p-laplacian ...3,4 rr rr r31+ rr rr, rr rr4 →0, u∈n....

Research ArticleNon-Nehari Manifold Method for Fractional p-LaplacianEquation with a Sign-Changing Nonlinearity

Huxiao Luo1 Shengjun Li 23 andWenfeng He2

1Department of Mathematics Zhejiang Normal University Jinhua Zhejiang 321004 China2College of Information Sciences and Technology Hainan University Haikou 570228 China3School of Mathematics and Statistics Central South University Changsha Hunan 410083 China

Correspondence should be addressed to Shengjun Li shjli626126com

Received 15 April 2018 Accepted 10 June 2018 Published 18 July 2018

Academic Editor Xinguang Zhang

Copyright copy 2018 Huxiao Luo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider the following fractional p-Laplacian equation (minusΔ)120572119901119906 + 119881(119909)|119906|119901minus2119906 = 119891(119909 119906) minus Γ(119909)|119906|119902minus2119906 119909 isin R119873 where119873 ge 2119901lowast120572 gt 119902 gt 119901 ge 2 120572 isin (0 1) (minusΔ)120572119901 is the fractional 119901-Laplacian and Γ isin 119871infin(R119873) and Γ(119909) ge 0 for ae 119909 isin R119873 119891 has the subcriticalgrowth but higher than Γ(119909)|119906|119902minus2119906 however the nonlinearity119891(119909 119906)minusΓ(119909)|119906|119902minus2119906may change sign If119881 is coercive we investigatethe existence of ground state solutions for p-Laplacian equation

1 Introduction

Consider the following nonlinear Schrodinger equation withfractional 119901-Laplacian(minusΔ)120572119901 119906 + 119881 (119909) |119906|119901minus2 119906 = 119891 (119909 119906) minus Γ (119909) |119906|119902minus2 119906

119909 isin R119873 (1)

where 119873 ge 2 119901lowast120572 gt 119902 gt 119901 ge 2 120572 isin (0 1) and (minusΔ)120572119901 is thefractional 119901-Laplacian 119881(119909) Γ(119909) and 119891(119909 119906) R119873 timesR 997888rarrR satisfy the following assumptions

( 119881 ) 119881 isin 119862(R119873R) 1198810 fl inf119909isinR119873119881(119909) gt 0 there exists aconstant 1198890 gt 0 such that

lim|119910|997888rarrinfin

meas 119909 isin R119873 1003816100381610038161003816119909 minus 1199101003816100381610038161003816 le 1198890 119881 (119909) le 119872 = 0

forall119872 gt 0 (2)

where meas (sdot) denotes the Lebesgue measure in R119873

( Γ ) Γ isin 119871infin(R119873) Γ(119909) ge 0 for ae 119909 isin R119873

( 1198911 ) 119891(119909 119905) R119873 times R 997888rarr R is measurable continuousin 119905 isin R for ae 119909 isin R119873 and there are 119862 gt 0 and2 le 119901 lt 119902 lt 119903 lt 119901lowast120572 such that

1003816100381610038161003816119891 (119909 119905)1003816100381610038161003816 le 119862 (1 + |119905|119903minus1) for all 119905 isin R 119909 isin R119873 (3)

where 119901lowast120572 = 119873119901(119873 minus 119901120572)( 1198912 ) 119891(119909 119905) = 119900(|119905|119901minus1) as |119905| 997888rarr 0 uniformly in 119909 isin R119873( 1198913 ) 119865(119909 119905)|119905|119902 997888rarr infin uniformly in 119909 as |119905| 997888rarr infin where

119865(119909 119905) = int1199050119891(119909 120591)119889120591

( 1198914 ) 119905 997891997888rarr 119891(119909 119905)|119905|119902minus1 is nondecreasing on (minusinfin 0) cup(0infin)When 119901 = 2 (1) arises in the study of the nonlinear Frac-

tional Schrodinger equation

(minusΔ)120572 119906 + 119881 (119909) 119906 = 119891 (119909 119906) minus Γ (119909) |119906|119902minus2 119906119909 isin R

119873 (4)

Problems with this type have occurred in many differentfields such as continuum mechanics phase transition phe-nomena population dynamics and game theory as they are

HindawiJournal of Function SpacesVolume 2018 Article ID 7935706 5 pageshttpsdoiorg10115520187935706

2 Journal of Function Spaces

the typical outcome of stochastically stabilization of Levyprocesses see [1ndash4]

When 120572 = 1 and Γ = 0 (4) reduces to be the nonlinearSchrodinger equation

minusΔ119906 + 119881 (119909) 119906 = 119891 (119909 119906) 119909 isin R119873 (5)

Using the Nehari-type monotonicity condition Szulkin andWeth [5] obtained the existence of ground state solutionsfor (5) But in this paper the Nehari manifold is usuallynot smooth and the Nehari-type monotonicity condition forthe nonlinearity is not satisfied then the Nehari manifoldmethod is invalid In this paper we are aimed to obtainground state solutions for (1) by the so-called non-Neharimanifold method which is established by Tang [6 7] Unlikethe Nehari manifold method the main idea of our approachlies on finding a minimizing sequence for the energy func-tional outside the Nehari manifold by using the diagonalmethod

Now we are ready to state the main result of this paper

Theorem 1 Suppose that (119881) (Γ) and (1198911) minus (1198914) hold Then(1) has a nontrivial ground state solution

2 Preliminaries

In the paper we will denote 119900119899(1) by the infinitesimal as 119899 997888rarr+infin For the sake of simplicity the norm of the space 119871119901(R119873)will be denoted by sdot 119901 and integrals over the wholeR119873 willbe written int

We define the Gagliardo seminorm by

[119906]120572119901 = (int1003816100381610038161003816119906 (119909) minus 119906 (119910)10038161003816100381610038161199011003816100381610038161003816119909 minus 1199101003816100381610038161003816119873+120572119901 119889119909 119889119910)

1119901

(6)

where119906 R119873 997888rarr R is ameasurable functionThen fractionalSobolev space119882120572119901(R119873) is given by

119882120572119901 (R119873)= 119906 isin 119871119901 (R119873) 119906 is measurable and [119906]120572119901 lt infin (7)

endowed with the norm

119906120572119901 = ([119906]119901120572119901 + 119906119901119901)1119901 (8)

For the basic properties of fractional Sobolev spaces we referthe interested reader to [8] By condition (119881) we define thefractional Sobolev space with potential 119881(119909) by

119864 fl 119906 isin 119882120572119901 int119881 (119909) |119906|119901 119889119909 lt infin (9)


119906 fl ([119906]119901120572119901 + int119881 (119909) |119906|119901 119889119909)1119901 (10)

The energy functional 119869 119864 997888rarr R defined by

119869 (119906) = 1119901 int1003816100381610038161003816119906 (119909) minus 119906 (119910)10038161003816100381610038161199011003816100381610038161003816119909 minus 1199101003816100381610038161003816119873+120572119901 119889119909 119889119910

+ 1119901 int119881 (119909) |119906 (119909)|119901 119889119909minus int(119865 (119909 119906) minus 1119902Γ (119909) |119906|119902)119889119909

(11)

Under our hypotheses 119869 is well defined on 119864 It is well knownthat 119869 isin 1198621(119864R) and its derivative is given by

⟨1198691015840 (119906) V⟩= int

R2119873

1003816100381610038161003816119906 (119909) minus 119906 (119910)1003816100381610038161003816119901minus2 (119906 (119909) minus 119906 (119910)) (V (119909) minus V (119910))1003816100381610038161003816119909 minus 1199101003816100381610038161003816119873+120572119901 119889119909 119889119910+ int

R119873119881 (119909) |119906|119901minus2 119906V 119889119909

minus int (119891 (119909 119906) V minus Γ (119909) |119906|119902minus2 119906V) 119889119909

(12)

for 119906 V isin 119864 It is standard to verify that the weak solutions of(1) correspond to the critical points of 119869 Now we review themain embedding result for the space 119864Lemma 2 ([9 Lemma 1]) Under assumption (119881) the embed-ding 119864 997893rarr 119871119903(R119873) is compact for any 119903 isin [119901 119901lowast120572)

In the following lemma wewill show that 119869 hasMountainPass geometric structure

Lemma 3 Suppose that (119881) (Γ) and (1198911) minus (1198914) hold(i) There is 1205750 gt 0 such that 1205880 fl inf 119906=1205750119869(119906) gt 119869(0) = 0(ii) For any 119906 = 0 there exists 119905 gt 0 such that 119869(119905119906) lt 0

Proof (i) By (1198911) and (1198912) we have1003816100381610038161003816119891 (119909 119906)1003816100381610038161003816 le 120576 |119906|119901minus1 + 119862120576 |119906|119903minus1 |119865 (119909 119906)| le 120576119901 |119906|119901 + 119862120576119903 |119906|119903 (13)

By 119864 997893rarr 119871119904(R119873) for 119904 isin [119901 119901lowast120572) and (13) we have

119869 (119906) = 1119901 119906119901 minus int119865 (119909 119906) 119889119909 + 1119902 int Γ (119909) |119906|119902 119889119909ge 1119901 119906119901 minus 1205761198621 119906119901 minus 1198621205761198622 119906119903

(14)

By the arbitrariness of 120576 and 119901 lt 119903 we get the conclusion(ii) Fix 119906 = 0 by (1198913) we have119869 (119905119906)119905119902 = 1119901119905119902minus119901 119906119901

minus int(119865 (119909 119905119906)(119905119906)119902 119906119902 minus 1119902Γ (119909) |119906|119902)119889119909997888rarr minusinfin

(15)

as 119905 997888rarr +infin Thus there exists 119905 gt 0 such that 119869(119905119906) lt 0

Journal of Function Spaces 3

Now we define the Nehari manifold by

N fl 119906 isin 119864 0 ⟨1198691015840 (119906) 119906⟩ = 0 (16)

It is easy to prove that N is not empty And we have thefollowing lemma

Lemma 4 Suppose that (119881) (Γ) and (1198911) minus (1198914) hold Let 120579 isin[0 1) cup (1infin) and 119906 isin N then 119869(120579119906) lt 119869(119906)Proof By (1198914) we have

119891 (119909 119904) le 119891 (119909 119905)|119905|119902minus1 |119904|119902minus1 119904 lt 119905

119891 (119904) ge 119891 (119909 119905)|119905|119902minus1 |119904|119902minus1 119904 gt 119905

(17)

Then

119865 (119909 119905) minus 119865 (119909 120579119905) = int119905120579119905119891 (119909 119904) 119889119904 le 1 minus 120579119902119902 119891 (119909 119905) 119905

forall120579 ge 0 119905 isin R(18)

Let ℎ(120579) = 119901120579119902 minus 119902120579119901 then ℎ1015840(120579) = 119901119902(120579119902minus1 minus 120579119901minus1) By asimple calculation we have ℎ1015840(1) = 0 and ℎ(120579) gt ℎ(1) = 119901minus119902for all 120579 isin [0 1) cup (1infin) Thus

120579119901 minus 1119901 minus 120579119902 minus 1119902 = 119901 minus 119902 minus ℎ (120579)119901119902 lt 0 (19)

Let 119906 isin N it follows from (18) and (19) that

119869 (120579119906)= 119869 (119906) + (119869 (120579119906) minus 119869 (119906) minus 120579119902 minus 1119902 ⟨1198691015840 (119906) 119906⟩)= 119869 (119906) + [120579119901 minus 1119901 minus 120579119902 minus 1119902 ] 119906119901

+ int[120579119902 minus 1119902 119891 (119909 119906) 119906 minus 119865 (119909 120579119906) + 119865 (119909 119906)] 119889119909lt 119869 (119906)

(20)

Lemma 5 Suppose that (119881) (Γ) and (1198911) minus (1198914) hold let119898 flinfN119869 then there exist 119906119899 isin 119864 119888lowast isin (1205880 119898] satisfying

119869 (119906119899) 997888rarr 119888lowast(1 + 10038171003817100381710038171199061198991003817100381710038171003817) 100381710038171003817100381710038171198691015840 (119906119899)10038171003817100381710038171003817 997888rarr 0 (21)

Proof By (i) of Lemma 3 there exist 1205750 gt 0 and 1205880 gt 0 suchthat

119906 isin 119864 119906 = 1205750 997904rArr 119869 (119906) ge 1205880 (22)

Choose V119896 isin N such that

119898 le 119869 (V119896) lt 119898 + 1119896 119896 isin N (23)

Since 119869(119905V119896) lt 0 for large 119905 gt 0 Mountain Pass Lemmaimplies that there exists 119906119896119899119899isinN sub 119864 satisfying

119869 (119906119896119899) 997888rarr 119888119896100381710038171003817100381710038171198691015840 (119906119896119899)10038171003817100381710038171003817 (1 + 10038171003817100381710038171199061198961198991003817100381710038171003817) 997888rarr 0119896 isin N

(24)

where 119888119896 isin [1205880 sup119905ge0119869(119905V119896)] By Lemma 4 we have 119869(V119896) =sup119905ge0119869(119905V119896)Hence by (23) and (24) we have

119869 (119906119896119899) 997888rarr 119888119896 lt 119898 + 1119896 100381710038171003817100381710038171198691015840 (119906119896119899)10038171003817100381710038171003817 (1 + 10038171003817100381710038171199061198961198991003817100381710038171003817) 997888rarr 0119896 isin N

(25)

Now we can choose a sequence 119899119896 sub N such that

119869 (119906119896119899119896) lt 119898 + 1119896 100381710038171003817100381710038171198691015840 (119906119896119899119896)10038171003817100381710038171003817 (1 + 1003817100381710038171003817100381711990611989611989911989610038171003817100381710038171003817) lt 1119896

119896 isin N(26)

Let 119906119896 = 119906119896119899119896 119896 isin N Then going if necessary to a subse-quence we have

119869 (119906119899) 997888rarr 119888lowast isin [1205880 119898] 100381710038171003817100381710038171198691015840 (119906119899)10038171003817100381710038171003817 (1 + 10038171003817100381710038171199061198991003817100381710038171003817) 997888rarr 0 (27)

3 Proof of Theorem 1

Proof of Theorem 1 In view of Lemma 5 we find a Ceramisequence 119906119899 satisfying (21) By (18) we have1119902119891 (119909 119906119899) 119906119899 minus 119865 (119909 119906119899) ge 0

for all 119909 isin R119873 119906119899 isin 119864

(28)

Combining (21) and (28) for 119899 big enough we have119888lowast + 1 ge 119869 (119906119899) minus 1119902 ⟨1198691015840 (119906119899) 119906119899⟩

= ( 1119901 minus 1119902) 10038171003817100381710038171199061198991003817100381710038171003817119901

+ int[1119902119891 (119909 119906119899) 119906119899 minus 119865 (119909 119906119899)] 119889119909ge ( 1119901 minus 1119902) 10038171003817100381710038171199061198991003817100381710038171003817119901

(29)


It follows that 119906119899 is bounded Passing to a subsequence wehave 119906119899 1199060 in119864 By Lemma 2 we have 119906119899 997888rarr 1199060 in 119871119903(R119873)for 119903 isin [119901 119901lowast120572) Then by (13) and the Holder inequality wehave 1003816100381610038161003816100381610038161003816int (119891 (119909 119906119899) minus 119891 (119909 1199060)) (119906119899 minus 1199060) 119889119909

1003816100381610038161003816100381610038161003816le 120576 int (10038161003816100381610038161199061198991003816100381610038161003816119901minus1 + 100381610038161003816100381611990601003816100381610038161003816119901minus1) 1003816100381610038161003816119906119899 minus 11990601003816100381610038161003816 119889119909+ 119862120576 int(10038161003816100381610038161199061198991003816100381610038161003816119903minus1 + 100381610038161003816100381611990601003816100381610038161003816119903minus1) 1003816100381610038161003816119906119899 minus 11990601003816100381610038161003816 119889119909

le 120576 (10038171003817100381710038171199061198991003817100381710038171003817119901minus1119901 + 100381710038171003817100381711990601003817100381710038171003817119901minus1119901 ) 1003817100381710038171003817119906119899 minus 11990601003817100381710038171003817119901+ 119862120576 (10038171003817100381710038171199061198991003817100381710038171003817119903minus1119903 + 100381710038171003817100381711990601003817100381710038171003817119903minus1119903 ) 1003817100381710038171003817119906119899 minus 11990601003817100381710038171003817119903 997888rarr 0

(30)

and 1003816100381610038161003816100381610038161003816int Γ (119909) (10038161003816100381610038161199061198991003816100381610038161003816119902minus2 119906119899 minus 100381610038161003816100381611990601003816100381610038161003816119902minus2 1199060) (119906119899 minus 1199060) 1198891199091003816100381610038161003816100381610038161003816

le Γinfin int(10038161003816100381610038161199061198991003816100381610038161003816119902minus1 + 100381610038161003816100381611990601003816100381610038161003816119902minus1) 1003816100381610038161003816119906119899 minus 11990601003816100381610038161003816 119889119909le (10038171003817100381710038171199061198991003817100381710038171003817119902minus1119902 + 100381710038171003817100381711990601003817100381710038171003817119902minus1119902 ) 1003817100381710038171003817119906119899 minus 11990601003817100381710038171003817119902 997888rarr 0

(31)

It follows from (30) (31) and Simon inequality ((|119886|119901minus2119886 minus|119887|119901minus2119887)(119886 minus 119887) ge (12119901minus2)|119886 minus 119887|119901) that⟨1198691015840 (119906119899) minus 1198691015840 (1199060) 119906119899 minus 1199060⟩= int 11003816100381610038161003816119909 minus 1199101003816100381610038161003816119873+120572119901 [

1003816100381610038161003816119906119899 (119909) minus 119906119899 (119910)1003816100381610038161003816119901minus2

sdot (119906119899 (119909) minus 119906119899 (119910)) minus 10038161003816100381610038161199060 (119909) minus 1199060 (119910)1003816100381610038161003816119901minus2sdot (1199060 (119909) minus 1199060 (119910))] [119906119899 (119909) minus 119906119899 (119910) minus 1199060 (119909)+ 1199060 (119910)] 119889119909 119889119910 + int119881 (119909) [10038161003816100381610038161199061198991003816100381610038161003816119901minus2 119906119899minus 100381610038161003816100381611990601003816100381610038161003816119901minus2 119906] (119906119899 minus 1199060) 119889119909 minus int (119891 (119909 119906119899)minus 119891 (119909 1199060)) (119906119899 minus 1199060) 119889119909 + intΓ (119909) (10038161003816100381610038161199061198991003816100381610038161003816119902minus2 119906119899minus 100381610038161003816100381611990601003816100381610038161003816119902minus2 1199060) (119906119899 minus 1199060) 119889119909 ge 12119901minus2sdot int 1003816100381610038161003816(119906119899 (119909) minus 119906119899 (119910)) minus (1199060 (119909) minus 1199060 (119910))10038161003816100381610038161199011003816100381610038161003816119909 minus 1199101003816100381610038161003816119873+120572119901 119889119909 119889119910+ 12119901minus2 int119881 (119909) 1003816100381610038161003816119906119899 minus 11990601003816100381610038161003816119901 119889119909 + 119900119899 (1) = 12119901minus2 1003817100381710038171003817119906119899minus 11990601003817100381710038171003817119901 + 119900119899 (1)

(32)

On the other hand by ⟨1198691015840(119906119899) 119906119899 minus 1199060⟩ 997888rarr 0 and 119906119899 1199060we have

⟨1198691015840 (119906119899) minus 1198691015840 (1199060) 119906119899 minus 1199060⟩ 997888rarr 0 (33)

Combining (32) and (33) we have 119906119899 997888rarr 1199060 in 119864 Then by119869 isin 1198621(119864R) we have 1198691015840(1199060) = 0 By (28) Lemma 5 andFatoursquos lemma we have

119898 ge 119888lowast = lim119899997888rarrinfin

[119869 (119906119899) minus 1119902 ⟨1198691015840 (119906119899) 119906119899⟩]= ( 1119901 minus 1119902) lim

119899997888rarrinfin

10038171003817100381710038171199061198991003817100381710038171003817119901

+ lim119899997888rarrinfin

int[1119902119891 (119909 119906119899) 119906119899 minus 119865 (119909 119906119899)] 119889119909ge ( 1119901 minus 1119902) 100381710038171003817100381711990601003817100381710038171003817119901

+ int[1119902119891 (119909 1199060) 1199060 minus 119865 (119909 1199060)] 119889119909= 119869 (1199060) minus 1119902 ⟨1198691015840 (1199060) 1199060⟩ = 119869 (1199060)

(34)

This shows that 119869(1199060) le 119898 and so 119869(1199060) = 119898 = infN119869 gt0Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally and significantly in writ-ing this article All authors read and approved the finalmanuscript

Acknowledgments

This work is supported by the Hainan Natural ScienceFoundation (Grant nos 118MS002 and 117005) National Nat-ural Science Foundation of China (Grant nos 11461016 and11571370) China Postdoctoral Science Foundation FundedProject (Grant no 2017M612577) and Young Foundation ofHainan University (Grant no hdkyxj201718)

References

[1] D Applebaum ldquoLevy processesmdashfrom probability to financeand quantum groupsrdquo Notices of the American MathematicalSociety vol 51 no 11 pp 1336ndash1347 2004

[2] N Laskin ldquoFractional quantum mechanics and Levy pathintegralsrdquo Physics Letters A vol 268 no 4ndash6 pp 298ndash305 2000

[3] R Metzler and J Klafter ldquoThe restaurant at the end ofthe random walk recent developments in the description ofanomalous transport by fractional dynamicsrdquo Journal of PhysicsA Mathematical and General vol 37 no 31 pp R161ndashR2082004


[4] B Bieganowski ldquoSolutions of the fractional Schrodinger equa-tionwith a sign-changing nonlinearityrdquo Journal ofMathematicalAnalysis and Applications vol 450 no 1 pp 461ndash479 2017

[5] A Szulkin and TWeth ldquoGround state solutions for some indef-inite variational problemsrdquo Journal of Functional Analysis vol257 no 12 pp 3802ndash3822 2009

[6] X H Tang ldquoNon-Nehari manifold method for superlinearSchrodinger equationrdquo Taiwanese Journal of Mathematics vol18 no 6 pp 1957ndash1979 2014

[7] X H Tang ldquoNon-Nehari manifold method for asymptoticallyperiodic Schrodinger equationsrdquo Science China Mathematicsvol 58 no 4 pp 715ndash728 2015

[8] E Di Nezza G Palatucci and E Valdinoci ldquoHitchhikerrsquos guideto the fractional Sobolev spacesrdquo Bulletin des Sciences Mathe-matiques vol 136 no 5 pp 521ndash573 2012

[9] V Ambrosio ldquoMultiple solutions for a fractional p-Laplacianequation with sign-changing potentialrdquo Electronic Journal ofDifferential Equations vol 2016 article 151 12 pages 2016

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the typical outcome of stochastically stabilization of Levyprocesses see [1ndash4]

When 120572 = 1 and Γ = 0 (4) reduces to be the nonlinearSchrodinger equation

minusΔ119906 + 119881 (119909) 119906 = 119891 (119909 119906) 119909 isin R119873 (5)

Using the Nehari-type monotonicity condition Szulkin andWeth [5] obtained the existence of ground state solutionsfor (5) But in this paper the Nehari manifold is usuallynot smooth and the Nehari-type monotonicity condition forthe nonlinearity is not satisfied then the Nehari manifoldmethod is invalid In this paper we are aimed to obtainground state solutions for (1) by the so-called non-Neharimanifold method which is established by Tang [6 7] Unlikethe Nehari manifold method the main idea of our approachlies on finding a minimizing sequence for the energy func-tional outside the Nehari manifold by using the diagonalmethod

Now we are ready to state the main result of this paper

Theorem 1 Suppose that (119881) (Γ) and (1198911) minus (1198914) hold Then(1) has a nontrivial ground state solution

2 Preliminaries

In the paper we will denote 119900119899(1) by the infinitesimal as 119899 997888rarr+infin For the sake of simplicity the norm of the space 119871119901(R119873)will be denoted by sdot 119901 and integrals over the wholeR119873 willbe written int

We define the Gagliardo seminorm by

[119906]120572119901 = (int1003816100381610038161003816119906 (119909) minus 119906 (119910)10038161003816100381610038161199011003816100381610038161003816119909 minus 1199101003816100381610038161003816119873+120572119901 119889119909 119889119910)

1119901

(6)

where119906 R119873 997888rarr R is ameasurable functionThen fractionalSobolev space119882120572119901(R119873) is given by

119882120572119901 (R119873)= 119906 isin 119871119901 (R119873) 119906 is measurable and [119906]120572119901 lt infin (7)


119906120572119901 = ([119906]119901120572119901 + 119906119901119901)1119901 (8)

For the basic properties of fractional Sobolev spaces we referthe interested reader to [8] By condition (119881) we define thefractional Sobolev space with potential 119881(119909) by

119864 fl 119906 isin 119882120572119901 int119881 (119909) |119906|119901 119889119909 lt infin (9)


119906 fl ([119906]119901120572119901 + int119881 (119909) |119906|119901 119889119909)1119901 (10)

The energy functional 119869 119864 997888rarr R defined by

119869 (119906) = 1119901 int1003816100381610038161003816119906 (119909) minus 119906 (119910)10038161003816100381610038161199011003816100381610038161003816119909 minus 1199101003816100381610038161003816119873+120572119901 119889119909 119889119910

+ 1119901 int119881 (119909) |119906 (119909)|119901 119889119909minus int(119865 (119909 119906) minus 1119902Γ (119909) |119906|119902)119889119909

(11)

Under our hypotheses 119869 is well defined on 119864 It is well knownthat 119869 isin 1198621(119864R) and its derivative is given by

⟨1198691015840 (119906) V⟩= int

R2119873

1003816100381610038161003816119906 (119909) minus 119906 (119910)1003816100381610038161003816119901minus2 (119906 (119909) minus 119906 (119910)) (V (119909) minus V (119910))1003816100381610038161003816119909 minus 1199101003816100381610038161003816119873+120572119901 119889119909 119889119910+ int

R119873119881 (119909) |119906|119901minus2 119906V 119889119909

minus int (119891 (119909 119906) V minus Γ (119909) |119906|119902minus2 119906V) 119889119909

(12)

for 119906 V isin 119864 It is standard to verify that the weak solutions of(1) correspond to the critical points of 119869 Now we review themain embedding result for the space 119864Lemma 2 ([9 Lemma 1]) Under assumption (119881) the embed-ding 119864 997893rarr 119871119903(R119873) is compact for any 119903 isin [119901 119901lowast120572)

In the following lemma wewill show that 119869 hasMountainPass geometric structure

Lemma 3 Suppose that (119881) (Γ) and (1198911) minus (1198914) hold(i) There is 1205750 gt 0 such that 1205880 fl inf 119906=1205750119869(119906) gt 119869(0) = 0(ii) For any 119906 = 0 there exists 119905 gt 0 such that 119869(119905119906) lt 0

Proof (i) By (1198911) and (1198912) we have1003816100381610038161003816119891 (119909 119906)1003816100381610038161003816 le 120576 |119906|119901minus1 + 119862120576 |119906|119903minus1 |119865 (119909 119906)| le 120576119901 |119906|119901 + 119862120576119903 |119906|119903 (13)

By 119864 997893rarr 119871119904(R119873) for 119904 isin [119901 119901lowast120572) and (13) we have

119869 (119906) = 1119901 119906119901 minus int119865 (119909 119906) 119889119909 + 1119902 int Γ (119909) |119906|119902 119889119909ge 1119901 119906119901 minus 1205761198621 119906119901 minus 1198621205761198622 119906119903

(14)

By the arbitrariness of 120576 and 119901 lt 119903 we get the conclusion(ii) Fix 119906 = 0 by (1198913) we have119869 (119905119906)119905119902 = 1119901119905119902minus119901 119906119901

minus int(119865 (119909 119905119906)(119905119906)119902 119906119902 minus 1119902Γ (119909) |119906|119902)119889119909997888rarr minusinfin

(15)

as 119905 997888rarr +infin Thus there exists 119905 gt 0 such that 119869(119905119906) lt 0



N fl 119906 isin 119864 0 ⟨1198691015840 (119906) 119906⟩ = 0 (16)



119891 (119909 119904) le 119891 (119909 119905)|119905|119902minus1 |119904|119902minus1 119904 lt 119905

119891 (119904) ge 119891 (119909 119905)|119905|119902minus1 |119904|119902minus1 119904 gt 119905

(17)

Then


forall120579 ge 0 119905 isin R(18)






(20)


119869 (119906119899) 997888rarr 119888lowast(1 + 10038171003817100381710038171199061198991003817100381710038171003817) 100381710038171003817100381710038171198691015840 (119906119899)10038171003817100381710038171003817 997888rarr 0 (21)


119906 isin 119864 119906 = 1205750 997904rArr 119869 (119906) ge 1205880 (22)


119898 le 119869 (V119896) lt 119898 + 1119896 119896 isin N (23)


119869 (119906119896119899) 997888rarr 119888119896100381710038171003817100381710038171198691015840 (119906119896119899)10038171003817100381710038171003817 (1 + 10038171003817100381710038171199061198961198991003817100381710038171003817) 997888rarr 0119896 isin N

(24)



(25)


119869 (119906119896119899119896) lt 119898 + 1119896 100381710038171003817100381710038171198691015840 (119906119896119899119896)10038171003817100381710038171003817 (1 + 1003817100381710038171003817100381711990611989611989911989610038171003817100381710038171003817) lt 1119896

119896 isin N(26)






(28)


= ( 1119901 minus 1119902) 10038171003817100381710038171199061198991003817100381710038171003817119901


(29)





(30)



(31)


1003816100381610038161003816119906119899 (119909) minus 119906119899 (119910)1003816100381610038161003816119901minus2


(32)


⟨1198691015840 (119906119899) minus 1198691015840 (1199060) 119906119899 minus 1199060⟩ 997888rarr 0 (33)



[119869 (119906119899) minus 1119902 ⟨1198691015840 (119906119899) 119906119899⟩]= ( 1119901 minus 1119902) lim


10038171003817100381710038171199061198991003817100381710038171003817119901



+ int[1119902119891 (119909 1199060) 1199060 minus 119865 (119909 1199060)] 119889119909= 119869 (1199060) minus 1119902 ⟨1198691015840 (1199060) 1199060⟩ = 119869 (1199060)

(34)







Acknowledgments


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Volume 2018










N fl 119906 isin 119864 0 ⟨1198691015840 (119906) 119906⟩ = 0 (16)



119891 (119909 119904) le 119891 (119909 119905)|119905|119902minus1 |119904|119902minus1 119904 lt 119905

119891 (119904) ge 119891 (119909 119905)|119905|119902minus1 |119904|119902minus1 119904 gt 119905

(17)

Then


forall120579 ge 0 119905 isin R(18)






(20)


119869 (119906119899) 997888rarr 119888lowast(1 + 10038171003817100381710038171199061198991003817100381710038171003817) 100381710038171003817100381710038171198691015840 (119906119899)10038171003817100381710038171003817 997888rarr 0 (21)


119906 isin 119864 119906 = 1205750 997904rArr 119869 (119906) ge 1205880 (22)


119898 le 119869 (V119896) lt 119898 + 1119896 119896 isin N (23)


119869 (119906119896119899) 997888rarr 119888119896100381710038171003817100381710038171198691015840 (119906119896119899)10038171003817100381710038171003817 (1 + 10038171003817100381710038171199061198961198991003817100381710038171003817) 997888rarr 0119896 isin N

(24)



(25)


119869 (119906119896119899119896) lt 119898 + 1119896 100381710038171003817100381710038171198691015840 (119906119896119899119896)10038171003817100381710038171003817 (1 + 1003817100381710038171003817100381711990611989611989911989610038171003817100381710038171003817) lt 1119896

119896 isin N(26)






(28)


= ( 1119901 minus 1119902) 10038171003817100381710038171199061198991003817100381710038171003817119901


(29)





(30)



(31)


1003816100381610038161003816119906119899 (119909) minus 119906119899 (119910)1003816100381610038161003816119901minus2


(32)


⟨1198691015840 (119906119899) minus 1198691015840 (1199060) 119906119899 minus 1199060⟩ 997888rarr 0 (33)



[119869 (119906119899) minus 1119902 ⟨1198691015840 (119906119899) 119906119899⟩]= ( 1119901 minus 1119902) lim


10038171003817100381710038171199061198991003817100381710038171003817119901



+ int[1119902119891 (119909 1199060) 1199060 minus 119865 (119909 1199060)] 119889119909= 119869 (1199060) minus 1119902 ⟨1198691015840 (1199060) 1199060⟩ = 119869 (1199060)

(34)







Acknowledgments


References


















Journal of












Journal of







Volume 2018






Volume 2018












(30)



(31)


1003816100381610038161003816119906119899 (119909) minus 119906119899 (119910)1003816100381610038161003816119901minus2


(32)


⟨1198691015840 (119906119899) minus 1198691015840 (1199060) 119906119899 minus 1199060⟩ 997888rarr 0 (33)



[119869 (119906119899) minus 1119902 ⟨1198691015840 (119906119899) 119906119899⟩]= ( 1119901 minus 1119902) lim


10038171003817100381710038171199061198991003817100381710038171003817119901



+ int[1119902119891 (119909 1199060) 1199060 minus 119865 (119909 1199060)] 119889119909= 119869 (1199060) minus 1119902 ⟨1198691015840 (1199060) 1199060⟩ = 119869 (1199060)

(34)







Acknowledgments


References


















Journal of












Journal of







Volume 2018






Volume 2018






















Journal of












Journal of







Volume 2018






Volume 2018








non-nehari manifold method for fractional p-laplacian ...3,4 rr rr r31+ rr rr, rr rr4 →0, u∈n....

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