ON THE CLASSIFICATION OF STABLE SOLUTION TO BIHARMONIC PROBLEMS IN LARGE DIMENSIONS

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1 ON THE CLASSIFICATION OF STABLE SOLUTION TO BIHARMONIC PROBLEMS IN LARGE DIMENSIONS JUNCHENG WEI, XINGWANG XU, AND WEN YANG Abstract. We give a ew boud o the expoet for oexistece of stable solutios to the biharmoic problem where p >, 0. u = u p, u > 0 i R. Itroductio Of cocer is the followig biharmoic equatio u = u p, u > 0 i R (.) where 5 ad p >. Set Λ u (ϕ) := ϕ dx p u p ϕ dx, ϕ H (R ). (.) The Morse idex of a classical solutio to (.), id(u) is defied as the maximal dimesio of all subspaces of H (R ) such that Λ u (ϕ) < 0 i H (R ) \ {0}. We say u is a stable solutio to (.) if Λ u (ϕ) 0 for ay test fuctio ϕ H (R ), i.e., the Morse idex is zero. I the first part of the paper, we obtai the followig classificatio result o stable solutios of (.). Theorem.. Let 0 ad < p < + 8p. The equatio (.) has o stable solutios. 8 I the above theorem, p stads for the smallest real root which is greater tha of the followig algebraic equatio 5( )x 6 + 4( )x 5 ( )x 4 + ( )x 3 ( )x + 4( )x 3( ) = 0. Some remarks are i order. Let us recall that for the secod order problem u + u p = 0 u > 0 i R, p >, (.3) Faria [4] gave a complete classificatio of all fiite Morse idex solutios. The mai result of [4] is that o stable solutio exists to (.3) if either 0, p > or, p < p JL. Here p JL deotes the well-kow Joseph-Ludgre expoet ([9]). O the other had, stable radial solutio exists for p p JL. For the fourth order case, the oexistece of positive solutios to (.) is show if p < +4 p > +4 +4, ad all etire solutios are classified if p =. See [3] ad [6]. Whe, radially symmetric solutios to (.) are completely classified i [5], [6]

2 JUNCHENG WEI, XINGWANG XU, AND WEN YANG ad []. The radial solutios are show to be stable if ad oly if p p JL ad 3, where p JL stads for the correspodig Joseph-Ludgre expoet (see[5], [6]). I the geeral oradial case, Wei ad Ye [7] showed the oexistece of stable or fiite Morse idex solutios whe either 8, p > or 9, p 8. I dimesios 9, a perturbatio argumet is used to show the oexistece of stable solutios for p < 8 + ɛ for some ɛ > 0. However, o explicit value of ɛ is give. The proof of [7] follows a earlier idea of Cowa-Esposito-Ghoussoub [3] i which a similar problem i a bouded domai was studied. Theorem. gives a explicit value o ɛ for 0. I the secod order case, the proof of Faria uses basically the Moser iteratios: amely multiply the equatio (.3) by the power of u, like u q, q >. Moser iteratio works because of the followig simple idetity u q 4q ( u) = R (q + ) u q+, u C0(R ). R I the fourth order case, such equality does ot hold, ad i fact we have u q ( 4q u) = R (q + ) u q+ q(q ) u R R q 3 u 4, u C0(R ). The additioal term u q 3 u 4 makes the Moser iteratio argumet difficult R to use. I [7], they used istead the ew test fuctio u ad showed that u is bouded. Thus the expoet R 8 is obtaied. I this paper, we use the Moser iteratio for the fourth order problem ad give a cotrol o the term u q 3 u 4 (Lemma.3). As a result, we obtai a better expoet R 8 + ɛ where ɛ is explicitly give. As far as we kow, this seems to be the first result for Moser iteratio for a fourth order problem. I the secod part of this paper, we show that the same idea ca be used to establish the regularity of extremal solutios to u = λ(u + ) p, λ > 0 i u > 0, i u = u = 0, o (.4) where is a smooth ad bouded covex domai i R. For problem (.4), it is kow ([]) that for p > +4 there exists a critical value λ > 0 depedig o p > ad such that If λ (0, λ ), equatio (.4) has a miimal ad classical solutio which is stable; If λ = λ, a uique weak solutio, called the extremal solutio u exists for equatio (.4); No weak solutio of equatio (.4) exists wheever λ > λ. The regularity of the extremal solutio of problem (.4) at λ = λ has bee studied i [3] ad i [7], where they showed that the extremal solutio is bouded provided 8 or p < 8 + ε, 9 (ε very small). Here, we also give a explicit boud for the expoet p i large dimesios ad our secod result is the followig. Theorem.. The extremal solutio u of (.4) whe λ = λ is bouded provided that 0 ad < p < + 8p, where p is defied as above.

3 CLASSIFICATION OF STABLE SOLUTIONS 3 As +, the value ɛ is asymptotically 8 8/3 ad thus the upper boud ( 8) 3/ for p has the followig expasio /3 ( 8) + O( ). (.5) 3/ ( 8) O the other had, for radial solutios, the Joseph-Ludgre expoet ([9]) has the followig asymptotic expasio ( 8) + O( ). (.6) 3/ ( 8) I this paper, we have oly cosidered the fourth order problems with power-like oliearity. Other kids of oliearity, such as expoetial ad egative powers, also appear i may applicatios. See [3]. However, our techique here yields o improvemets of results of [3] i the case of expoetial ad egative oliearities. This paper is orgaized as follows. We prove Theorem. ad Theorem. respectively i sectio ad sectio 3. Some techical iequalities are give i the appedix.. Proof of Theorem. I this sectio, we prove Theorem. through a series of Lemmas. First of all, we have followig. Lemma.. For ay ϕ C0(R 4 ), ϕ 0, γ > ad ε > 0 arbitrary small umber, we have ( (u γ ϕ γ )) (( u γ ϕ γ ) + ε u 4 ϕ γ u γ 4 + Cu γ 4 (ϕ γ ) ), (.) ( (u γ ϕ γ )) (( u γ ϕ γ ) ε u 4 ϕ γ u γ 4 Cu γ 4 (ϕ γ ) ), (.) ((u γ ) ij ) ϕ γ ((u γ ϕ γ ) ij ) + ε u 4 u γ 4 ϕ γ R + C u γ 4 (ϕ γ ), (.3) R where C is a positive umber oly depeds o γ, ε ad 4 (ϕ γ ) is defied by 4 (ϕ γ ) = ϕ γ ϕ γ 4 + ϕ γ ( ϕ γ ) + ϕ γ. I the followig, uless otherwise, the costat C i this sectio always deotes a positive umber which may chage term by term but oly depeds o γ, ε. Proof. Sice ϕ is compactly supported, we ca use itegratio by parts without cosiderig the boudary terms. First, by direct calculatios, we get ( (u γ ϕ γ )) =[( u γ )ϕ γ ] + 4 u γ ϕ γ ϕ γ u γ + 4 u γ ϕ γ u γ ϕ γ + 4( u γ ϕ γ ) + u γ u γ ϕ γ ϕ γ + u γ ( ϕ γ ). (.4) We ow eed to deal with the third ad the fifth term o the right had side of the above equality up to the itegratio both sides.

4 4 JUNCHENG WEI, XINGWANG XU, AND WEN YANG For the third term, we have u γ u γ ϕ γ ϕ γ = (u γ ) i (u γ ) ij (ϕ γ ) j ϕ γ R R (u γ ) i (u γ ) j (ϕ γ ) ij ϕ γ (u γ ) i (u γ ) j (ϕ γ ) j (ϕ γ ) i, where f i = f x i ad f ij = f x j x i. (Here ad i the sequel, we use the Eistei summatio covetio: a idex occurrig twice i a product is to be summed from up to the space dimesio, e.g., u i v i = i= u iv i, i (u i u j ϕ j ) = i,j i(u i u j ϕ j ).) The first term o the right had side of the previous equatio ca be estimated as (u γ ) i (u γ ) ij (ϕ γ ) j ϕ γ = j ((u γ ) i (u γ ) i (ϕ γ ) j ϕ γ ) ((u γ ) i ) (ϕ γ ) jj ϕ γ R ((u γ ) i ) (ϕ γ ) j (ϕ γ ) j. R Combiig these two equalities, we get u γ u γ ϕ γ ϕ γ = j ((u γ ) i (u γ ) i (ϕ γ ) j ϕ γ ) R R (u γ ) i (u γ ) j (ϕ γ ) ij ϕ γ (u γ ) i (u γ ) j (ϕ γ ) j (ϕ γ ) i R R + ((u γ ) i ) (ϕ γ ) jj ϕ γ + ((u γ ) i ) (ϕ γ ) j (ϕ γ ) j. Rewritig the above equality we have 4 u γ u γ ϕ γ ϕ γ = u γ ϕ γ ϕ γ + u γ ϕ γ R 4 (u γ ) i (u γ ) j (ϕ γ ) ij ϕ γ 4 (< u γ, ϕ γ >). (.5) For the fifth term o the right had side of Equatio (.4) we have u γ u γ ϕ γ ϕ γ = u γ < u γ, ( ϕ γ ) > ϕ γ R R < u γ, ϕ γ > u γ ϕ γ u γ ϕ γ ϕ γ. (.6) R R Combiig Equatios (.4), (.5) ad (.6), oe obtais ( (u γ ϕ γ )) ( u γ ) ϕ γ R R = u γ ϕ γ 4 ϕ γ ( ϕ γ ( u γ, u γ )) R R + u γ ϕ γ (ϕ γ ) u γ ( ϕ γ ). (.7) Now by the Youg equality, for ay ε > 0, there exists a costat C = C(γ, ε) such that u γ ϕ γ ε 4 uγ 4 u γ ϕ γ + C ϕ γ 4 u γ ϕ γ

5 CLASSIFICATION OF STABLE SOLUTIONS 5 ad ϕ γ ( ϕ γ ( u γ, u γ )) ε 8 uγ 4 u γ ϕ γ + Cu γ ϕ γ. Thus by the equatio (.7), together with the above two estimates, oe gets: ( (u γ ϕ γ )) ( u γ ) ϕ γ ε u γ 4 u γ ϕ γ + 6C u γ 4 ϕ γ. The estimates (.) ad (.) follow from this easily. Next we observe that u γ ϕ γ = [ uγ < u γ, u γ >]ϕ γ. Thus up to the itegratio by parts, with the help of equatio (.5) ad the estimates we just proved, the estimate (.3) also follows by oticig the idetity ( (u γ ϕ γ )) = R (u γ ϕ γ ). The proof of Lemma. is thus completed. R Let us retur to the equatio u = u p, u > 0 i R. (.8) Fix q = γ > 0 ad γ >. Let ϕ C0 (R ). Multiplyig (.8) by u q ϕ γ ad itegratio by parts, we obtai u (u q ϕ γ ) = R u p+q ϕ γ. R (.9) For the left had side of (.9), we have the followig lemma. Lemma.. For ay ϕ C0 (R ) with ϕ 0, for ay ε > 0 ad γ with q defied above, there exists a positive costat C depeds o γ, ε such that R γ q u (uq ϕ γ ) ( u γ ϕ γ ) Cu γ 4 (ϕ γ ) R R (γ (γ ) + ε)u γ 4 u 4 ϕ γ. (.0) R Proof. First, by direct computatios, we obtai u (u γ ϕ γ ) = u((γ )u γ uϕ γ + (γ )u γ u (ϕ γ ) + (γ )(γ )u γ 3 u ϕ γ + u γ ϕ γ ), ( u γ ϕ γ ) =γ u γ ( u) ϕ γ + γ (γ ) u γ 4 u 4 ϕ γ + (γ )γ u γ 3 u uϕ γ. Combiig the above two idetities, we get γ q u (uq ϕ γ ) =( u γ ϕ γ ) + γ u γ u u ϕ γ + γ q uγ u ϕ γ For the term u γ u u ϕ γ, we have γ (γ ) u γ 4 u 4 ϕ γ. (.) u γ u u ϕ γ = i (u γ u i u j (ϕ γ ) j ) (γ )u γ 3 (u i ) u j (ϕ γ ) j We ca regroup the term u γ u i u ij (ϕ γ ) j as u γ u i u ij (ϕ γ ) j u γ u i u j (ϕ γ ) ij. u γ u i u ij (ϕ γ ) j = j (u γ (u i ) (ϕ γ ) j ) (γ )u γ 3 u j (u i ) (ϕ γ ) j u γ (u i ) (ϕ γ ) jj.

6 6 JUNCHENG WEI, XINGWANG XU, AND WEN YANG Therefore we get u γ u u ϕ γ = i (u γ u i u j (ϕ γ ) j ) j (u γ (u i ) (ϕ γ ) j ) (γ )u γ 3 (u i ) u j (ϕ γ ) j + u γ (u i ) (ϕ γ ) jj u γ u i u j (ϕ γ ) ij. (.) For the last three terms o the right had side of (.), applyig Youg s iequality, we get u γ 3 (u i ) u j (ϕ γ ε ) j 6γ (γ ) uγ 4 u 4 ϕ γ + Cu γ 4 (ϕ γ ), u γ (u i ) (ϕ γ ) jj ε 6γ uγ 4 u 4 ϕ γ + Cu γ 4 (ϕ γ ), u γ u i u j (ϕ γ ) ij ε 6γ uγ 4 u 4 ϕ γ + Cu γ 4 (ϕ γ ). By the above three iequalities ad (.), we have γ u γ u u ϕ γ ε u γ 4 u 4 ϕ γ C u γ 4 (ϕ γ ). (.3) R Similarly we get R γ q uγ u ϕ γ ε u γ 4 u 4 ϕ γ C u γ 4 (ϕ γ ). (.4) Iequality (.0) follows from (.), (.3) ad (.4). As a result of (.) ad (.0), we have R γ q u (uq ϕ γ ) ( (u γ ϕ γ )) Cu γ 4 (ϕ γ ) R R (γ (γ ) + ε)u γ 4 u 4 ϕ γ. (.5) R Next we estimate the most difficult term R u γ 4 u 4 ϕ γ i (.5). This is the key step i provig Theorem.. Lemma.3. If u is the classical solutio to the biharmoic equatio (.8), ad ϕ is defied as above, the for ay sufficietly small ε > 0, we have the followig iequality ( ε) u γ 4 u 4 ϕ γ R γ ( (u γ ϕ γ )) + Cu γ 4 (ϕ γ ) Proof. It is easy to see that R u γ 4 u 4 ϕ γ = γ 4 4 R (4γ 3 + p)(p + ) uγ+p ϕ γ. (.6) R u γ u γ 4 ϕ γ, (.7)

7 CLASSIFICATION OF STABLE SOLUTIONS 7 ad u γ u γ 4 ϕ γ = u γ u γ u γ u γ ϕ γ R R = u γ u γ u γ ϕ γ R = u γ u γ u γ ϕ γ + u γ ( u γ ) u γ ϕ γ R R + u γ u γ u γ ϕ γ, (.8) R where i the last step we used itegratio by parts. For the first term i the last part of the above equality, we have u γ u γ u γ ϕ γ = γ 3 ((γ )u γ 4 u 4 ϕ γ + u γ 3 u uϕ γ ). (.9) Substitutig (.9) ito (.8), ad combiig with (.7), we obtai u γ 4 u 4 ϕ γ = R R γ 3 u γ ( u γ ) u γ ϕ γ + u γ 3 ( u ) uϕ γ R + R γ 3 u γ ( u γ ) u γ ϕ γ. (.0) The first term o the right had side of (.0) ca be estimated as u γ ( u γ ) u γ = u γ ((u γ ) ij (u γ ) i (u γ ) j ) As a cosequece, we have R γ 3 u γ ( u γ ) u γ ϕ γ + = + γ(u γ ) ij (u γ ) ij + u γ γ (uγ ) i (u γ ) j (u γ ) i (u γ ) j = γ u γ + u γ γ uγ 4. R R γ u γ ϕ γ + R γ 4 u γ u γ 4 ϕ γ γ (u γ ϕ γ ) + Cu γ 4 (ϕ γ ) R + 4γ ε γ 4 u γ u γ 4 ϕ γ R R γ ( (uγ ϕ γ )) + R Cu γ 4 (ϕ γ ) R + 4γ ε γ 4 u γ u γ 4 ϕ γ, (.) where we used (.3) i the last step. For the secod term o the right had side of (.0), applyig the estimate (.3) from [7], i.e., ( u) p+ up+, ad the fact that u < 0 from Theorem 3. i

8 8 JUNCHENG WEI, XINGWANG XU, AND WEN YANG [6] or Theorem. i [8], we have u γ 3 ( u ) uϕ γ R = + R R + R p + uγ 3+ p+ γ + p+ p+ γ + p+ Usig the iequality u p+ u p+, we get R p+ γ + p+ R u γ + p+ γ + u uϕ γ R p+ ( u )ϕ γ p+ γ + u uϕ γ p+ γ + u u ϕ γ. (.) p+ γ + p+ u γ+p ϕ γ. (.3) O the other had, for the secod term o the right had side of (.), we have p+ u ϕ γ p+ = L uγ + ϕ γ = R {x ϕ γ >0} {x ϕ γ 0} p+ L uγ + ϕ γ p+ L uγ + ϕ γ, (.4) where the first equality follows from itegratio by parts ad L = γ +. As for the first term o the last part of (.4), usig the iequality u p+ p+ u p+ < 0, we have p+ u γ u ϕ γ p+ L {x ϕ γ >0} {x ϕ γ >0} L uγ + ϕ γ. (.5) Similar to the proof of Lemma., it is easy to get {x ϕ γ >0} p+ L uγ u ϕ γ ε u γ 4 u 4 ϕ γ + R Cu γ 4 (ϕ γ ). R (.6) By (.5) ad (.6), we have p+ {x ϕ γ >0} L uγ + ϕ γ ε u γ 4 u 4 ϕ γ + Cu γ 4 (ϕ γ ). (.7) Similarly, we also obtai p+ {x ϕ γ 0} L uγ + ϕ γ ε u γ 4 u 4 ϕ γ + Cu γ 4 (ϕ γ ). (.8) By (.4), (.7) ad (.8), we have u γ + p+ u ϕ γ ε u γ 4 u 4 ϕ γ + Cu γ 4 (ϕ γ ). (.9) R

9 CLASSIFICATION OF STABLE SOLUTIONS 9 Combiig (.), (.3) ad (.9), we get the followig iequality u γ 3 u uϕ γ ε u γ 4 u 4 ϕ γ + Cu γ 4 (ϕ γ ) R 4 R (4γ 3 + p)(p + ) uγ+p ϕ γ. (.30) Fially, we apply Youg s iequality to the third term o the right had side of (.0), ad get R γ 3 u γ ( u γ ) u γ ϕ γ = u γ 3 u u (ϕ γ ) R ε u γ 4 u 4 ϕ γ + Cu γ 4 (ϕ γ ). By (.0), (.), (.30) ad (.3), we fially obtai ( ε) u γ 4 u 4 ϕ γ R γ ( (u γ ϕ γ )) + Cu γ 4 (ϕ γ ) (.3) 4 R (4γ 3 + p)(p + ) uγ+p ϕ γ. By (.9), (.5) ad (.6), sice the umber ε is arbitrary small i those three places, we have for δ > 0 sufficietly small, the followig iequality holds ( 4(γ ) δ)( (u γ ϕ γ )) γ ( R R γ 8γ (γ ) (4γ 3 + p)(p + ) )up+γ ϕ γ C δ u γ 4 (ϕ γ ), (.3) R where C δ is a positive costat depeds o δ oly. Here, we eed 4(γ ) > 0, sice we have assumed that γ > i Lemma.. So γ is required be i (, 3 ). If we ca choose δ small eough to make 4(γ ) δ positive, by the stability property of fuctio u, we obtai (E pδ)u p+q ϕ γ R C δ u γ 4 (ϕ γ ), R (.33) where E is defied to be E = p( 4(γ ) ) γ q + 8γ (γ ) (4γ 3 + p)(p + ). (.34) Now we take ϕ = η m with m sufficietly large, ad choose η a cut-off fuctio satisfyig 0 η, η = for x < R ad η = 0 for x > R. By Youg s iequality agai, we have u γ 4 (ϕ γ ) C δ R 4 u γ η γm 4 C δ,ɛ R 4 θ R u η γm 4 θ + ɛcδ R u γ+p η γm, (.35)

10 0 JUNCHENG WEI, XINGWANG XU, AND WEN YANG where C δ,ɛ is a positive costat depeds o δ, ɛ, θ is a umber such that ( θ) + (γ + p )θ = γ so that 0 < θ < for < γ < γ + p. By (.33) ad (.35), we get (E pδ ɛc δ ) u p+γ η γm C δ,ɛ R 4 θ u η γm 4 θ. (.36) Sice θ is strictly less tha ad will be fixed for give γ, p, we ca choose m sufficietly large to make γm 4 θ > 0. O the other had, if E > 0, we ca fid small δ ad the small ɛ, such that E pδ ɛc δ > 0. Therefore, by the defiitio of fuctio η ad (.36), we obtai (E pδ ɛc δ ) u p+γ C δ,ɛ R 4 θ B R u. B R (.37) By (.0) of [7], we have B R u CR 8 p, as a result, the left had side of (.37) is less equal tha C δ,ɛ R 8 p 4 θ, which teds to 0 as R teds to, provided the power 8 p 4 θ is egative, which is equivalet to (p + γ ) > (p ) 4 accordig to the defiitio of θ. So, if (p + γ ) > (p ) 4 ad E pδ C δ ɛ > 0, we have u 0. Thus, we have proved the oexistece of stable solutio to (.8) if p satisfies the coditio (p + γ ) > (p ) 4 ad E > 0 (for δ, ɛ are arbitrary small). By Lemma 4. i the appedix, the power p ca be i the iterval ( 8, + 8p ). Combiig with Theorem. of [7], we have proved Theorem., i.e., for ay < p < + 8p, 0, equatio (.8) has o stable solutio. 3. Proof of Theorem. I this sectio, we preset the proof of Theorem.. We ote that it is eough to cosider stable solutios u λ to (.4) sice u = lim λ λ u λ. Now we give a uiform boud for the stable solutios to (.4) whe 0 < d < λ < λ, where d is a fixed positive costat from (0, λ ). First, we eed to aalyze the solutio ear the boudary. 3.. Regularity of the solutio o the boudary. I this subsectio, we establish the regularity of stable solutio of (3.) ad its derivative ear the boudary of the followig equatio: u = λ(u + ) p, λ > 0 i u > 0, i u = u = 0, o (3.) Theorem 3.. Let be a bouded, smooth, ad covex domai. The there exists a costat C (idepedet of λ, u) ad small positive umber ɛ, such that for stable solutios u to (3.) we have u(x) < C, x ɛ := {z : d(z, ) < ɛ}. (3.)

11 CLASSIFICATION OF STABLE SOLUTIONS Proof. This result is well-kow. See []. For the sake of completeess, we iclude a proof here. By Lemma 3.5 of [3], we see that, there exists a costat C idepedet of λ, u, such that ( + u) p dx C. (3.3) We write Equatio (3.) as u + v = 0, i v + λ( + u) p = 0, i u = v = 0, i. If we deote f (u, v) = v, f (u, v) = λ(u + ) p, we see that f v = > 0 ad f u = λp(u + )p > 0. Therefore, the covexity of, Lemma 5. of [4], ad the movig plae method ear (as i the appedix of [7]) imply that there exist t 0 > 0 ad α which depeds oly o the domai, such that u(x tν) ad v(x tν) are odecreasig for t [0, t 0 ], ν R satisfyig ν = ad (ν, (x)) α ad x. Therefore, we ca fid ρ, ɛ > 0 such that for ay x ɛ := {z : d(z, ) < ɛ} there exists a fixed-sized coe Γ x (with x as its vertex) with meas(γ x ) ρ, Γ x {z : d(z, ) < ɛ}, ad u(y) u(x) for ay y Γ x. The, for ay x ɛ, we have ( + u(x)) p ( + u) p ( + u) p C. meas(γ x ) Γ x ρ This implies that ( + u(x)) p C, therefore u(x) C. Remark: By classical elliptic regularity theory, u(x) ad its derivatives up to fourth order are bouded o the boudary by a costat idepedet of u. See [5] for more details. 3.. Proof of Theorem.. I the followig, we will use the idea i Sectio to prove Theorem.. First of all, multiplyig (.4) by (u + ) q ad itegratio by parts, we have λ(u + ) p+q = u(u + ) q = ( u) + (u + ) (u + ) q. (3.4) Settig v = u +, by direct calculatios, we get ( v γ ) = γ v γ ( v) + γ (γ ) v γ 4 v 4 + γ (γ )v γ 3 v v, (3.5) v v q = q( v) v q + q(q ) v vv q. (3.6)

12 JUNCHENG WEI, XINGWANG XU, AND WEN YANG From (3.4), (3.5) ad (3.6), we obtai ( q γ ( vγ ) q(γ ) v 4 v γ 4 ) + ( v) = λv p+q. (3.7) For the secod term i (3.7), we have v 4 v γ 4 = γ 4 v γ v γ 4 = γ 4 v γ v γ ( v γ ) = γ 4 ( ( vγ v γ v γ ) + ( vγ ) v γ v γ + vγ v γ v γ ) = γ 4 Simple calculatio yields γ 4 v γ v γ v γ = γ γ v γ ( v γ ) v γ + v γ v γ γ Substitutig (3.9) ito (3.8), we get v 4 v γ 4 = v γ 3 v v + γ 3 v γ 4 v 4 + γ v γ 3 v v. (3.8) v γ 3 v v. (3.9) v γ ( v γ ) v γ v v. (3.0) We ow estimate the secod term o the right had side of (3.0). From the proof of Lemma.3, together with the idetity vγ = v γ + < v γ, v γ >, the followig iequality holds γ 3 v γ ( v γ ) v γ + γ v 4 v γ 4 + γ ( v γ ) v γ γ ( v γ ) vγ. (3.) By (3.0) ad (3.), thaks to the covexity of the domai, we get v 4 v γ 4 v γ 3 v v+ γ ( v γ ) (γ ) v v. (3.) For the first term o the right had side of (3.), sice v = u +, we have v = u < 0 by maximal priciple, ad the iequality v < < 0 by Lemma 3. of [3]. Thus v γ 3 v v Moreover, we have λ p+ p + vγ 3+ v = λ p+ p + vγ 3+ v. + λ p+ γ + p+ λ p+ γ + p+ λ p+ v p+ p+ γ + (v v) p+ γ + v v.

13 CLASSIFICATION OF STABLE SOLUTIONS 3 For the secod term o the right had side of the above equality, usig the iequality v < < 0 agai, we have λ p+ v p+ Hece, we obtai λ p+ γ + p+ v γ 3 v v p+ γ + v v λ p+ γ + p+ v λ p+ γ + p+ λ p+ γ + p+ v γ+p. v γ+p, (3.3) where we used v = u + =, for the boudary term i (3.4), (3.) ad (3.3). By the remark after Theorem 3., we fid that there exists a costat C (the costat C appeared ow ad later i this sectio is idepedet of u), such that ( u u + ( u) + u ) C. (3.4) Combiig (3.7), (3.), (3.3) ad (3.4), we get ( 4(γ ) ) ( (u + ) γ ) 8λγ (γ ) + ( (4γ + p 3)(p + ) λγ q ) (u + ) p+q C. (3.5) If ( 4(γ ) ) > 0, p( 4(γ ) ) + 8γ (γ ) (4γ+p 3)(p+) γ q > 0 ad u is a stable solutio to the equatio (.4), we have (p( 4(γ ) 8γ (γ ) ) + (4γ + p 3)(p + ) γ γ ) (u + ) p+q C λ. This leads to u + L p+q. If p + q > (p ) 4, the classical regularity theory implies that u L (). Therefore we have established the boud of extremal solutios of (.4) if ad p( 4(γ ) ) + 8γ (γ ) (4γ + p 3)(p + ) γ q > 0 p < 8γ By Lemma 4. ad Theorem 3.8 of [7], we prove the extremal solutio u, the uique solutio of equatio (.4) (where λ = λ ) is bouded provided that () 8, p >, () 9 9, there exists ε > 0 such that for ay < p < (3) 0, < p < + 8p. (p is defied as before.) 8 + ε,

14 4 JUNCHENG WEI, XINGWANG XU, AND WEN YANG 4. appedix I this appedix, we study the followig iequalities p( 4(γ ) ) γ γ + 8γ (γ ) > 0, (4.) (4γ 3 + p)(p + ) p < 8γ + 4. (4.) 4 I order to get a better rage of the power p from (4.) ad (4.), it is ecessary for us to study the followig equatio (Lettig p = 8γ+ i (4.)): 8γ + 4 ( 4(γ ) ) γ 4 γ + 8γ (γ ) (4γ 3 + 8γ+ )( 8γ+ + ) = 0. (4.3) We ca oly cosider the behavior of (4.3) for γ (, 3 ). Through tedious computatios, we see the followig equatio which appeared i the itroductio is the simplified form of (4.3). As a cosequece, they have same roots i (, 3 ) : 5( )γ 6 + 4( )γ 5 ( )γ 4 + ( )γ 3 ( )γ + 4( )γ 3( ) = 0. (4.4) We deote the left had side of the equatio (4.3) by h(γ). Notice that if γ = the p = 8 ad γ = 4 8. Hece h( 4 8 ) = 8 8 [ ]. 8, I fact, if = 0, the h( 4 3 ) = 5 > 0. O the other had, it is also easy to see that h( 3 8γ+ ) < 0, while it is obvious that (4γ 3 + )( 8γ+ + ) > 0 ad (γ ) > 0 whe γ ( 8, 3 ). Therefore, by cotiuity, equatio (4.3) possesses a root i ( 8, 3 ). We deote the smallest root of (4.3) which is greater tha 8 by p. Oce we pick out a γ from the iterval ( 8, p ), h(γ) is of course positive. By cotiuity, we ca fid a small positive umber δ such that, the iequality p( 4(γ ) ) γ γ + 8γ (γ ) 8γ+ (4γ 3+p)(p+) > 0 holds whe p ( δ, 8γ+ ). So, we coclude that whe γ rus i the whole iterval ( 8, p ), the power p ca be i the whole iterval ( 8, + 8p ). We summarize the result as follows: Lemma 4.. Whe 0, we have p which satisfies (4.) ad (4.) ca rage i ( 8, + 8p ) ad this iterval is ot empty. Ackowledgmets: The first author was supported from a Earmarked grat ( O Elliptic Equatios with Negative Expoets ) from RGC of Hog Kog.

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