Ground State Solu tions for a Semilinear Ellip tic Equation Involving Concave-Convex Non linearities

KHAZAEEKOHPARO.and KHADEM LOO S.

DepartmentofBasic Sciences,BabolUniversity ofTechnology,47148-71167,

Babol,Iran.

Ground State Solu tions for a Semilinear Ellip tic Equation Involving Concave-Convex Non linearities

KHAZAEEKOHPARO.?and KHADEM LOO S.

DepartmentofBasic Sciences,BabolUniversity ofTechnology,47148-71167,

Babol,Iran.

Received 7 June 2012;A ccep ted 18Decem ber 2012

.This w ork is devoted to the existence and m u ltip licity p roperties of the ground statesolutionsof thesem ilinearboundary valuep roblem-Δu=λa(x)u|u|q-2+ b(x)u|u|2?-2in a bounded dom ain coup led w ith Dirich let boundary cond ition.Here 2?is the critical Sobolev exponent,and the term ground state refers tom inim izers of the corresponding energy w ithin the setofnontrivialpositive solutions.Using the Neharim anifold m ethod w e p rove thatone can find an intervalΛsuch that there existat least tw o positive solu tions of the p roblem forλ∈Λ.

Sem ilinear ellip tic equations;Neharim anifold;concave-convex non linearities.

1 In troduction

We consider the follow ing sem ilinear ellip tic equation:

w here??RN(N≥3)isa sm ooth bounded dom ain,λ＞0,1≤q＜2,and 2?=2N/(N-2) is the critical Sobolev exponentand thew eight functions a,b are satisfying the follow ing cond itions:

(B)b+=m ax{b,0}6≡0 and b∈C(?).

Tsung-fang Wu[1]has investigated the follow ing equation:

If thew eight functions a≡b≡1,Am brosetti-Brezis-Ceram i[2]stud ied Eq.(1.2).They established that thereexistsλ0＞0 such thatEq.(1.2)attainsat least tw o positive solu tions forλ∈(0,λ0),has a positive solu tion forλ=λ0and no positive solu tion exists forλ＞λ0. Wu[3]found that if thew eight functions a changes sign inant for the em bedd ing of,b≡1 andλis su fficiently sm all in Eq.(1.2),then Eq.(1.2)has at least tw o positive solu tions.

The energy functional correspond ing to Eq.(1.1)is defined as follow s:

and then Jλisw ell defined on .It isw ell-know n that the solu tions of Eq.(1.1)are the critical pointsof the functional Jλ.

We define the follow ing constan ts:

Ou rm ain resu lt is the follow ing.

Theorem1.1.Assume that the conditions(A)and(B)hold;then thereexistsan intervalΛsuch that forλ∈Λ,Eq.(1.1)hasat least two positive solutions.

We om it dx in the integration for convenience.This paper is organized as follow s.In Section 2,w e give som e p ropertiesof the Neharim anifold.In Sections 3 and 4 w e p rove Theorem 1.1.

2 The Neharim anifold

is of in terest.So,u∈M≥if and on ly if

It has to be considered that M≥contains every nonzero solu tion of Eq.(1.1).Fu rtherm ore,w e have the follow ing resu lt.

Lemma2.1.Theenergy functional Jλis coerciveand bounded below on M≥.

Proof.If u∈M≥,then by(1.3),(2.1)and the H¨older and Young inequalities,w e have

Thus,Jλis coercive and bounded below on Mλ.

Proof.See[2,Theorem 2.3].

LetΛ=(0,λ0)w hereλ0is the sam e as in(1.4),then w e have the follow ing resu lt.

and so

Sim ilarly,using(1.3),(2.3),and the H¨older and Young inequalities,w e have

Hence

w hich is a contrad iction.This com p letes the p roof.

We consider the functionψu:R+→R defined by

The follow ing resu ltexp lains the behavior of the graph ofψu.

Lemma2.4.For sufficiently smallλ,ψuis strictly increasing on(0,tmax(u))and strictly decreasing on(tmax(u),∞)w ith limt→∞ψu(t)=-∞,where

Proof.Clearly tu∈Mλif and on ly if

M oreover,

and so it is easy to see that,if tu∈Mλ,then

Remark2.1.Note that ifλ∈Λ,then

M oreover,w e have the follow ing lemm a.

Proof.See[5,Lemm a 2.6].

3 The existence of a ground state

are non-em p ty and by Lemm a 2.1w em ay define

Then w e have the follow ing resu lt.

Theorem3.1.Ifλ∈Λthen

and so

M oreover,by(1.3)w e have

This im p lies

By(2.3)and(3.1),w e have

for som e positive constan t d0.

and so

Then w e have the follow ing resu lts.

Proposition3.1.Ifλ∈Λ,then

Proof.See[6,Proposition 9].

(ii)uλisapositive solution ofEq.(1.1).

(iii)‖uλ‖→0 asλ→0+.

Proof.ByProposition3.1(i),thereisaminimizingsequenceunforJλonMλsuchthat

such that

Thus,w e have

First,weclaimthatuλisanonzerosolutionofEq.(1.1).By(3.3)and(3.4),itiseasytosee that uλis a solution of Eq.(1.1).From uλ∈Mλand(2.2),w e deduce that

Let n→∞in(3.6),by(3.3),(3.5)andαλ＜0,w e get

Firstw e show that Jλ(uλ)=αλ.It su ffices to recall that un,uλ∈Mλ;by(3.7)and using w eakly low er sem icontinuity of Jλw e get

This im p lies that Jλ(uλ)=αλandnl→im∞‖un‖2=‖uλ‖2.Letνn=un-uλ;then by Bre′zis-Lieb lemm a[7]w e have

and so‖uλ‖→0 asλ→0+.

4 Proof of Theorem 1.1

Since

and

References

[1]Wu T.F.,M u ltip licity resu lt for a sem ilinear ellip tic equation involving sign-changing w eight function.R.M.J.,39(3)(2009),995-1011.

[2]Am brosetti A.,Brezis H.and Ceram iG.,Com binedeffects of concave and convex nonlinearities in som e ellip tic p roblem s.J.Funct.Anal.,122(1994),519-543.

[3]Wu T.F.,On sem ilinear ellip tic equations involving concave-convex non linearities and sign-changing w eight function.J.M ath.Anal.Appl.,318(2006),253-270.

[4]Brow n K.J.,Zhang Y.,The Neharim anifold for a sem ilinear ellip tic equation w ith a signchanging w eight function.Differential Equations,193(2003),481-499.

[5]Wu T.F.,M u ltip le positive solu tions for a class of concave-convex ellip tic p roblem s in RNinvolving sign-changing w eight.J.Funct.Anal.,258(2010),99-131.

[6]Wu T.F.,On sem ilinear ellip tic equations involving concave-convex non linearities and sign-changing w eight function.J.M ath.Anal.Appl.,318(2006),253-270.

[7]Br′ezisH.,Lieb E.,A relation betw een pointw ise convergenceof functionsand convergence of functionals.Proc.Amer.M ath.Soc.,88(1983),486-490.

[8]Trud inger N.S.,On Harnack type inequalities and their app lication to quasilinear ellip tic equations.Comm.Pure Appl.M ath.,20(1967),721-747.

10.4208/jpde.v26.n1.2 March 2013

?Correspond ing au thor.Email addresses:kolsoomkhazaee@yahoo.com(O.Khazaee Kohpar),s.khademloo@ nit.ac.ir(S.Khadem loo)

AMSSubjectClassifications:35J25,35J20,35J61

ChineseLibraryClassifications:O175.8,O175.25