A priori estimates for elliptic equations with reaction terms involving the function and its gradient

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terms involving the function and its gradient

Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Veron

To cite this version:

Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Veron. A priori estimates for elliptic equations with reaction terms involving the function and its gradient. Mathematische Annalen, Springer Verlag, In press. �hal-01906697v2�

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terms involving the function and its gradient

Marie-Fran¸coise Bidaut-V´eron^∗, Marta Garcia-Huidobro ^†

Laurent V´eron ^‡

AbstractWe study local and global properties of positive solutions of−∆u=u^p+M|∇u|^qin a domain Ω ofR^N, in the range min{p, q}>1 andM ∈R. We prove a priori estimates and existence or non-existence of ground states for the same equation.

2010 Mathematics Subject Classification. 35J62, 35B08, 6804.

Key words. elliptic equations; Bernstein methods; ground states;

1 Introduction

This article is concerned with local and global properties of positive solutions of the following type of equations

−∆u=M⁰|u|^p−1u+M|∇u|^q, (1.1)

in Ω\ {0} where Ω is an open subset of R^N containing 0, p and q are exponents larger than 1 and M, M⁰ are real parameters. If M⁰ ≤0 the equation satisfies a comparison principle and a big part of the study can be carried via radial local supersolutions. This no longer the case when M⁰>0 which will be assumed in all the article, and by homothety (1.1) becomes

−∆u=|u|^p−1u+M|∇u|^q. (1.2)

If M = 0 (1.2) is called Lane-Emden equation

−∆u=|u|^p−1u. (1.3)

It turns out that it plays an important role in modelling meteorological or astrophysical phe- nomena [15], [13], this is the reason for which the first study, in the radial case, goes back to the end of nineteenth century and the beginning of the twentieth. A fairly complete presentation can be found in [18]. If N ≥ 3, This equations exhibits two main critical exponents p = _N^N₋₂ and p= ^N_N⁺²₋₂ which play a key role in the description of the set of positive solutions which can be summarized by the following overview:

1- If 1 < p ≤ _N^N₋₂, there exists no positive solution if Ω is the complement of a compact set.

Even in that case solution can be replaced by supersolution. This is easy to prove by studying the inequality satisfied by the spherical average of a solution of the equation.

2- If 1 < p < ^N_N⁺²₋₂, there exists noground state, i.e. positive solution in R^N. Furthermore any positive solution u in a ballBR=BR(a) satisfies

u(x)≤c(R− |x−a|)⁻^p−1² , (1.4)

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where c=c(N, p)>0, see [19].

3- Ifp= ^N_N⁺²₋₂ all the positive solutions inR^N are radial with respect to some pointaand endow the following form

u(x) :=u_λ(x) = (N(N−2)λ)^N−2⁴ (λ+|x−a|²)^N−2²

. (1.5)

All the positive solutions inR^N \ {0} are radial, see [12].

4- If p > ^N+2_N−2 there exist infinitely many positive ground states radial with respect to some points. They are obtained from one say v, radial for example with respect to 0 by the scaling transformation T_k wherek >0 with

Tk[v](x) =k^p−1² v(kx). (1.6)

Indeed, the first significant non-radial results deals with the case 1 < p ≤ _N−2^N . They are based upon the Brezis-Lions lemma [11] which yields an estimate of solutions in the Lorentz spaceL^N−2^N ^,∞, implying in turn the local integrability ofu^q. Then a bootstrapping method as in [21] leads easily to some a priori estimate. Note that this subcritical case can be interpreted using the famous Serrin’s results on quasilinear equations [24]. The first breakthrough in the study of Lane-Emden equation came in the treatment of the case 1 < p < ^N+2_N−2; it is due to Gidas and Spruck [19]. Their analysis is based upon differentiating the equation and then obtaining sharp enough local integral estimates on the term u^q−1 making possible the utilization of Harnack inequality as in [24]. The treatment of the critical case p = ^N+2_N−2, due to Caffarelli, Gidas and Spruck [12], was made possible thanks to a completely new approach based upon a combination of moving plane analysis and geometric measure theory. As for the supercritical case, not much is known and the existence of radial ground states is a consequence of Pohozaev’s identity [22], using a shooting method.

The study of (1.2) when M 6= 0 presents some similarities with the one of Lane-Emden equation in the cases 1 and 2, except that the proof are much more involved. Actually the approach we develop in this article is much indebted to our recent paper [6] where we study local and global aspects of positive solutions of

−∆u=u^p|∇u|^q, (1.7)

where p ≥ 0, 0 ≤q < 2, mostly in the superlinear case p+q −1 > 0. Therein we prove the existence of a critical line of exponents

(L) :={(p, q)∈R+×[0,2) : (N−2)p+ (N−1)q =N}. (1.8) The subcritical range corresponds to the fact that (p, q) is below (L). In this region Serrin’s celebrated results [24] can be applied and we prove [6, Theorem A] that positive solutions of (1.7) in the punctured ball B2\ {0}satisfy, for some constant c >0 depending on the solution, u(x) +|x| |∇u(x)| ≤c|x|^2−N for all x∈B1\ {0}. (1.9) When (p, q) is above (L), i.e. in the supercritical range, we introduced two methods for obtaining a priori estimate of solutions: Thepointwise Bernstein methodand the integral Bern- stein method. The first one is based upon the change of unknown u = v^−β, and then to show

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that|∇v|satisfies an inequality of Keller-Osserman type. When (p, q) lies above (L) and verifies (i) either 1≤p < ^N_N⁺³₋₁ and p+q−1< _N−1⁴ ,

(ii) or 0≤p <1 andp+q−1< _p(N^(p+1)₋₁₎² ,

we prove that any positive solution of (1.7) in a domain Ω⊂R^N satisfies

|∇u^a(x)| ≤c^∗(dist (x, ∂Ω))^−1−a

2−q

p+q−1 for all x∈Ω, (1.10)

for some positive c^∗ andadepending onN,pand q [6, Theorem B]. As a consequence we prove that any positive solution of (1.7) in R^N is constant. With the second method we combine the change of unknown u=v^−β with integration and cut-off functions. We show the existence of a quadratic polynomial G in two variables such that for any (p, q) ∈ R+×[0,2) satisfying G(p, q)<0 any positive solution of (1.7) inR^N is constant [6, Theorem C]. The polynomial G is not simple but it is worth noting that if 0≤p <^N_N⁺²₋₂, there holdsG(p,0)<0, which recovers Gidas and Spruck result [19].

For equation (1.2) we first observe that the equation is invariant under the scaling transformation (1.6) for anyk >0 if and only ifq is critical with respect top, i.e.

q = 2p p+ 1. In general the transformation Tk exchanges (1.2) with

−∆v=v^p+M k

2p−q(p+1)

p−1 |∇v|^q, (1.11)

hence if q < _p+1^2p , the limit equation when k → 0 is (1.3). We say that the exponent p is dominant. We can also consider the transformation

S_k[v](x) =k

2−q

q−1v(kx), (1.12)

when q6= 2, which is the same asTk ifq = _p+1^2p , and more generally transforms (1.2) into

−∆v=k

q−p(2−q)

q−1 v^p+M|∇v|^q. (1.13)

Hence if q > _p+1^2p , the limit equation when k→0 is the Riccati equation

−∆v=M|∇v|^q. (1.14)

It is also important to notice that the value of the coefficient M (and not only its sign) plays a fundamental role, only if q = _p+1^2p . Ifq 6= _p+1^2p the transformation

u(x) =av(y) with a=|M|⁻^(p+1)q−2p² and y=a^p−1² x (1.15) allows to transform (1.2) into

−∆v=|v|^p−1v± |∇v|^q. (1.16)

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The equation (1.2) has been essentially studied in theradial casewhenM <0 in connection with the parabolic equation

∂_tu−∆u+M|∇u|^q =|u|^p−1u, (1.17) see [14], [16], [17], [25], [27], [30], [31]. The studies mainly deal with the case q6= _p+1^2p , although not complete when q > _p+1^2p . When q = _p+1^2p the existence of a ground state is proved in dimension 1. Some partial results that we will improve, already exist in higher dimension. The case M >0 attracted less attention.

In thenonradial case, any nonnegative nontrivial solution is positive sincep, q >1. We first observe, using a standard averaging method applied to positive supersolutions of (1.3), that if M ≥0, 1 1 if N = 1,2, then for any q > 0 there exists no positive solution in an exterior domain. When 0< q < _p+1^2p the equation endows some character of the pure Emden-Fowler equation (1.3) by the transformation Tk. In [23] it is proved that if 0< q < _p+1^2p , 1< p < ^N_N⁺²₋₂ andM ∈R, any positive solution of (1.3) in an open domain satisfies

u(x) +|∇u(x)|^p+1² ≤cN,p,q,M

1 + (dist (x, ∂Ω))⁻^p−1²

for all x∈Ω. (1.18) Note that this does not imply the non-existence of ground state. In [1] Alarc´on, Garc´ıa-Meli´an and Quass study the equation

−∆u=|∇u|^q+f(u), (1.19)

in an exterior domain of R^N emphasizing the fact that positive solutions are super harmonic functions. They prove that if 1< q≤ _N−1^N and if f is positive on (0,∞) and satisfies

lim sup

s→0

s^−pf(s)>0, (1.20)

for somep >_N^N₋₂, then (1.19) admits no positive supersolution. The same authors also study in [2] existence and non-existence of positive solutions of (1.19) in a bounded domain with Dirichlet condition.

The techniques we developed in this paper are based upon a delicate extension of the ones already introduced in [6]. Our first nonradial result dealing with the caseq > _p+1^2p is the following:

Theorem A Let N ≥1, p > 1 and q > _p+1^2p . Then for any M >0, any solution of (1.2) in a domain Ω⊂R^N satisfies

|∇u(x)| ≤c_N,p,q M⁻

p+1

(p+1)q−2p + (Mdist (x, ∂Ω))⁻^q−1¹

for all x∈Ω. (1.21) As a consequence, any ground state has at most a linear growth at infinity:

|∇u(x)| ≤c_N,p,qM⁻

p+1

(p+1)q−2p for all x∈R^N. (1.22)

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Our proof relies on a direct Bernstein method combined with Keller-Osserman’s estimate applied to |∇u|². It is important to notice that the result holds for anyp >1, showing that, in some sense, the presence of the gradient term has a regularizing effect. In the case q < _p+1^2p we prove a non-existence result

Theorem A’ Let N ≥ 1, p > 1, 1 < q < _p+1^2p and M > 0. Then there exists a constant c_N,p,q >0 such that there is no positive solution of (1.2)in R^N satisfying

u(x)≤c_N,p,qM^2p−(p+1)q² for allx∈R^N. (1.23)

Whenq is critical with respect top the situation is more delicate since the value ofM plays a fundamental role. Our first statement is a particular case of a more general result in [1], but with a simpler proof which allows us to introduce techniques that we use later on.

Theorem B Let N ≥ 2, p > 1 if N = 2 or 1 −µ^∗ where

µ^∗:=µ^∗(N) = (p+ 1)

N −(N −2)p 2p

_p+1^p

. (1.24)

Then there exists no nontrivial nonnegative supersolution of (1.2)in an exterior domain.

In this range of values of p this result is optimal since for M ≤ −µ^∗ there exists positive singular solutions. The constant µ^∗ will play an important role in the description developed in [7] of radial solutions of (1.2). Using a variant of the method used in the proof of Theorem B we obtain results of existence and nonexistence of large solutions.

Theorem B’Let N ≥1, p >1 and q = _p+1^2p .

1- If Ω is a domain with a compact boundary satisfying the Wiener criterion and M ≥ −µ^∗(2) there exists no positive supersolution of (1.2)in Ω satisfying

dist^{(x,∂Ω)→0}lim u(x) =∞. (1.25)

2- If Gis a bounded convex domain,Ω =G^c andM <−µ^∗(1) there exists a positive solution of (1.2) in Ωsatisfying (1.25).

We show in [7] that the inequalityM <−µ^∗(1) is the necessary and sufficient condition for the existence of a radial large solution in the exterior of a ball.

Concerning ground states, we prove their nonexistence for any p > 1 provided M > 0 is large enough: indeed

Theorem C Let Ω⊂R^N, N ≥1, be a domain, p >1, q= _p+1^2p . For any M > M†:=

p−1 p+ 1

^p−1_p+1

N(p+ 1)² 4p

_p+1^p

, (1.26)

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and any ν > 0 such that (1−ν)M > M†, there exists a positive constant c_N,p,ν such that any solution u in Ω satisfies

|∇u(x)| ≤cN,p,ν((1−ν)M −M†)⁻

p+1

p−1(dist (x, ∂Ω))⁻

p+1

p−1 for allx∈Ω. (1.27) Consequently there exists no nontrivial solution of (1.2)in R^N.

The next result, based upon an elaborate Bernstein method, complements Theorem C under a less restrictive assumption on M but a more restrictive assumption onp.

Theorem D Let 1 0 and cN,p,q>0 such that for any M >0, any positive solution u in Ωsatisfies

|∇u^a(x)| ≤cN,p,q(dist (x, ∂Ω))⁻^p−1^2a ⁻¹ for all x∈Ω. (1.28) Hence there exists no nontrivial nonnegative solution of (1.2) in R^N.

It is remarkable that the constants a and c_N,p,q do not depend onM > 0, a fact which is clear when q6= _p+1^2p by using the transformationT_k, but much more delicate to highlight when q = _p+1^2p since (1.2) is invariant. When |M| is small, we use an integral method to obtain the following result which contains, as a particular case, the estimates in [19] and [7]. The key point of this method is to prove that the solutions in a punctured domain satisfy a local Harnack inequality.

Theorem E Let N ≥3, 10 depending on N and p such that for any M satisfying |M| ≤₀, any positive solution u in B_R\ {0} satisfies

u(x)≤cN,p|x|⁻^p−1² for all x∈B^R

2

\ {0}. (1.29)

As a consequence there exists no positive solution of (1.2)inR^N, and any positive solution uin a domain Ω satisfies

u(x) +|∇u(x)|^p+1² ≤c⁰_N,p(dist (x, ∂Ω))⁻^p−1² for all x∈Ω. (1.30)

Note that under the assumptions of Theorem E, there exist ground states for |M| large enough when 11 ifN = 1,2.

Ifu is a radial solutions of (1.2) in R^N it satisfies

−u⁰⁰− N−1

r u⁰ =|u|^p−1u+M u⁰

q, (1.31)

on (0,∞). Using several type of Lyapounov type functions introduced by Leighton [20] and Anderson and Leighton [3], we prove some results dealing with the caseM >0 which complement the ones of [25] relative to the caseM <0.

Theorem F 1- Let p >1 and q > _p+1^2p . Then there exists no radial ground state u satisfying u(0) = 1when M >0 is too large.

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2- Let 1 0. If q > p there exists no radial ground state for M >0 small enough.

3- Let N ≥ 3, p > ^N_N⁺²₋₂ and q ≥ _p+1^2p . Then there exist radial ground states for M > 0 small enough.

We end the article in proving the existence of non-radial positive singular solutions of (1.2) in R^N \ {0} in the case q = _p+1^2p obtained by bifurcation from radial explicit positive singular solutions. Our result shows that the situation is very contrasted according M > 0 where a bifurcation from (M, XM) occurs only if p≥ ^N+1_N−3 and M ≥0 and M <0 where there exists a countable set of bifurcations from (Mk, XM_k), k≥1, when 1< p < ^N_N⁺¹₋₃.

In a subsequent article [7] we present a fairly complete description of the positive radial solutions of (1.2) inR^N \ {0} in the scaling invariant caseq = _p+1^2p .

Acknowledgements This article has been prepared with the support of the collaboration programs ECOS C14E08 and FONDECYT grant 1160540 for the three authors.

2 The direct Bernstein method

We begin with a simple property in the case M ≥0 which is a consequence of the fact that the positive solutions of (1.2) are superharmonic.

Proposition 2.1 1- There exists no positive solution of (1.2) in R^N \B_R, R≥0 if one of the two conditions is satisfied:

(i) M ≥0, q≥0 and eitherN = 1,2 and p >1 or N ≥3 and 1< p≤ _N−2^N . (ii) M >0, N ≥3, p≥1 and 1< q≤ _N^N₋₁.

2- If N ≥ 3, q ≥ 1, p > _N−2^N and u(x) = u(r, σ) is a positive solution of (1.2) in R^N \BR, R≥0. Then there exists ρ≥R such that

1 N ω_N

Z

S^N−1

u(r, σ)dS:=u(r)≤c0r⁻^p−1² for all r > ρ, (2.1) with c0 :=

2N p−1

_p−1¹ and

1 N ω_N

Z

S^N−1

ur(r, σ)dS

:=|u_r(r)| ≤(N −2)c0r⁻

p+1

p−1 for all r > ρ. (2.2) 3- If M >0, p≥0, and q > _N^N₋₁ there holds for

|u_r(r)| ≤

(q−1)(N−1)−1 (q−1)M

_q−1¹

r⁻^q−1¹ for all r > ρ, (2.3) and

u(r)≤q−1 2−q

(q−1)(N−1)−1 (q−1)M

_q−1¹ r

q−2

q−1 for all r > ρ, (2.4) Furthermore, if R= 0, inequalities (2.1), (2.2)and (2.3) hold withρ= 0.

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Proof. Assertion 1-(i) is not difficult to obtain by integrating the inequality satisfied by the spherical average of the solution and using Jensen’s inequality. For the sake of completeness, we give a simple proof although the result is actually valid for much more general equations (see e.g. [8] and references therein). In this statement we denote by (r, σ)∈R+×S^N⁻¹the spherical coordinates in R^N, byωN the volume of the unit N-ball and thus N ωN is the (N-1)-volume of the unit sphere S^N−1. Writing (1.2) in spherical coordinates and using Jensen formula, we get

−r^1−N r^N⁻¹ur

r ≥u^p+M|u_r|^q. (2.5)

It implies thatr 7→w(r) :=−r^N⁻¹u_ris increasing on (R,∞), thus it admits a limit`∈(−∞,∞].

If `≤0, then ur(r)>0 on (R,∞). Hence u(r)≥u(ρ) :=c >0 forr ≥ρ > R. then r^N⁻¹u_r

r ≤ −c^pr^N−1 =⇒u_r(r)≤ ρ^N⁻¹

r^N−1u_r(ρ)− c^p N

r− ρ^N r^N−1

,

which implies u_r(r)→ −∞, thus u(r) → −∞asr → −∞, contradiction. Therefore `∈(0,∞]

and either u_r(r) <0 on (R,∞) or there exists r_` > R such that u_r(r_`) = 0, u is increasing on (R, r`,) and decreasing on (r`,∞). Ifur(r)<0 on (R,∞), then we have forr >2R

−r^N−1ur(r)≥ Z r

r 2

t^N−1u^p(t)dt≥ r^Nu^p(r)

2N =⇒ u^1−p

r≥ (p−1)r

2N =⇒u(r)≤

2N (p−1)r²

_p−1¹ , which yields (2.1). If we are in the second case withr_`> R, we apply the same inequality with r > 2r` and again (2.1) for r >2r`. Since u is superharmonic, the function v(s) = u(r) with s=r^2−N is concave on (0, R^2−N) and it tends to 0 whens→0. Thus

vs(s)≤ v

s =⇒ |u_r(r)| ≤(N −2)u(r)

r ≤(N −2)c0r⁻

p+1 p−1.

This implies (2.1) and (2.2). Note that the caser_` > Rcannot happen ifR= 0, so in any case, ifR= 0 then ρ= 0.

If M >0, we have withw(r) =−r^N⁻¹ur

wr≥M r^(1−q)(N⁻¹⁾|w|^q.

We have seen that w(r) > 0 at infinity with limit ` ∈ (0,∞], hence, on the maximal interval containing ∞ wherew >0, we have (w^1−q)r≤(1−q)M r^{(N−1)(1−q)}. We have for r > s > R

w^1−q(r)−w^1−q(s)≤Mln r

s

, ifq = _N−1^N and

w^1−q(r)−w^1−q(s)≤ M(q−1) (q−1)(N −1)−1

r^{1−(q−1)(N}⁻¹⁾−s1−(q−1)(N−1)

ifq < _N^N₋₁, and both expressions which tend to−∞ whenr→ ∞, a contradiction. This proves 1-(ii). If q > _N^N₋₁, the above expression yields, whenr → ∞,

`^1−q−w^1−q(s)≤ − (q−1)M

(q−1)(N−1)−1s^{1−(q−1)(N}⁻¹⁾.

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This implies

w(s)≤

(q−1)(N −1)−1 (q−1)M

_q−1¹

s^N−1−^q−1¹ ,

and (2.3).

Remark. The previous is a particular case of a much more general one dealing with quasilinear operators proved in [8, Theorem 3.1].

2.1 Proof of Theorems A, A’ and C

The function u is at least C^3+α for some α ∈ (0,1) since p, q > 1. Hence z = |∇u|² is C^2+α. Since there holds by Bochner’s identity and Schwarz’s inequality

−1

2∆z+ 1

N(∆u)²+h∇∆u,∇ui ≤0, (2.6) we obtain from (1.2),

−1

2∆z+|u|^2p N +2M

N |u|^p−1uz^q² +M²

N z^q−p|u|^p−1z−M q

2 z^q²⁻¹h∇z,∇ui ≤0.

Since for δ >0,

z^q²⁻¹|h∇z,∇ui| ≤

z⁻¹²∇z

z^q−1² |∇u|=

z⁻¹²∇z

z^q² ≤δz^q+ 1 4δ

|∇z|² z , we obtain for any ν ∈(0,1), provided δ is small enough,

−1

2∆z+ |u|^2p N +2M

N |u|^p−1uz^q² +M²(1−ν)²

N z^q−p|u|^p−1z≤c1

|∇z|²

z , (2.7)

where c₁=c₁(M, N, ν)>0.

2.1.1 Proof of Theorem A

We recall the following technical result proved in [6, Lemma 2.2] which will be used several times in the course of this article.

Lemma 2.2 Let S >1, R >0 and v be continuous and nonnegative in BR and C¹ on the set U₊={x∈BR:v(x)>0}. Ifv satisfies, for some real number a,

−∆v+v^S ≤a|∇v|²

v (2.8)

on each connected component of U₊, then

v(0)≤c_N,S,aR⁻^S−1² . (2.9)

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Abridged proof. Assuminga >0, we set W =v^α for 0< α≤ _a+1¹ , this transforms (2.8) into

−∆W + 1

αW^α(S−1)+1≤0, (2.10)

and then we apply Keller-Osserman inequality.

Proof of Theorem A. Suppose _p+1^2p < q. We set r= _p−1^2p ,r⁰ = _r−1^r , then, for any >0 p|u|^p−1z≤ ^r|u|^(p−1)r

r + z^r⁰

^r⁰r⁰ = (p−1)^r|u|^2p

2 + (p+ 1)z^p+1^2p 2^r⁰ . We fix η∈(0,1) and so that ^r = _N(p−1)^2(1−η) and get

p|u|^p−1z≤(1−η)|u|^2p

N +c₂z^p+1^2p , where c₂ = ^p+1₂ _N(p−1)

2(1−η)

^p+1_p−1

. We perform the change of scale (1.6) in order to reduce (1.2) to the case M = 1 by setting u(x) = α^p−1² v(αx) with α = M⁻

p−1

(p+1)q−2p. Then the equation for z=|∇v|² is considered in Ωα=αΩ. Choosing nowη= ¹₂ we obtain

c2z

2p p+1 ≤ 1

4Nz^q+c3, where c3=c3(N, p, q)>0, hence

−1

2∆z+v^2p 2N + 1

4Nz^q≤c3+c1

|∇z|² z . Put ˜z=

z−(4N c₃)¹^q

+, then

−1

2∆˜z+ 1

4Nz˜^q≤c1

|∇˜z|²

˜ z , hence, from Lemma 2.2, we derive

˜

z(y)≤c₄(dist (y, ∂Ω_α))^q−1² where c4=c4(N, q, c1)>0 which implies

|∇v(y)| ≤c⁰₄

1 + (dist (y, ∂Ω_α))⁻^q−1¹

∀y∈Ω_α. (2.11)

Then (1.21) and (1.22) follow.

Assume now that there exists a ground stateu. Fixy ∈R^N and consider{y_n} ⊂R^N such that |y_n|= 2n >|y|. We apply (2.11) with Ω_α=Bn(yn). Then

|∇v(y)| ≤c⁰₄

1 +|2n− |y||⁻^q−1¹ ,

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and letting n→ ∞ we infer

|∇v(y)| ≤c⁰₄ ∀y ∈R^N. (2.12) Hence, by the definition of v and y we see that

|∇u(x)| ≤c⁰₄M⁻

p+1

(p+1)q−2p ∀x∈R^N

which is exactly (1.22 ).

2.1.2 Proof of Theorem A’

Suppose 1< q < _p+1^2p . By scaling we reduce to the case M = 1 and we replace u by v defined by (1.6) as in the proof of Theorem A withα=M

p−1

2p−(p+1)q. From (2.7) withν = ¹₄ the function z=|∇v|² satisfies

−1

2∆z+v^2p N + 1

2Nz^q−pv^p−1z≤c₁|∇z|²

z . (2.13)

By H¨older’s inequality,

pv^p−1z≤ 1

4Nz^q+p(4N p)^q⁰⁻¹v^(p−1)q⁰. Since (p−1)q⁰ = 2p+^2p−(p+1)q_q−1 we derive

−1

2∆z+v^2p N

1−4^q⁰⁻¹p^q⁰N^q⁰v

2p−(p+1)q q−1

+ 1

4Nz^q ≤c1

|∇z|² z . If maxv≤c_N,p,q := (4^q⁰⁻¹p^q⁰N^q⁰)⁻

q−1

2p−(p+1)q, we obtain

−1

2∆z+ 1

4Nz^q≤c₁|∇z|² z ,

which implies that z= 0 by Lemma 2.2, hencev is constant and thus v= 0 from the equation.

Remark. If u is a positive ground state of (1.2) radial with respect to 0, it satisfies ur(0) = 0 and it is a decreasing function ofr. The previous theorem asserts that it must satisfy

u(0)> c_N,p,qM^2p−(p+1)q² . (2.14)

2.1.3 Proof of Theorem C

Suppose _p+1^2p =q. For A >0 we consider the expression (u^p+A|∇u|^q)²−N pu^p−1|∇u|²

=

u^p+A|∇u|^q−√

N p u^p−1² |∇u| u^p+A|∇u|^q+√

N p u^p−1² |∇u|

.

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Now the function Z 7→ ΦA(Z) = u^p +AZ^q −√

N p u^p−1² Z achieves its minimum at Z0 = ^√

N p qA

^p+1_p−1

u^p+1² and

Φ_A(Z0) =

"

1−p−1 p+ 1

N(p+ 1)² 4p

_p−1^p A⁻

p+1 p−1

# u^p. Thus setting

M†=

p−1 p+ 1

^p−1_p+1

N(p+ 1)² 4p

_p+1^p

, (2.15)

we obtain that if A ≥M†, then Φ_A(Z) ≥0 for all Z. Put M_ν = (1−ν)M for ν ∈ (0,1) such that M†< Mν, we derive from (2.7)

−1

2∆z+(u^p+M†z^q²)²

N −pu^p−1z+M_ν²−M_†²

N z^q≤c₁|∇z|²

z , (2.16)

which yields

−1

2∆z+M_ν²−M_†²

N z^q≤c₁|∇z|² z . Using again Lemma 2.2 we obtain

|∇u(x)| ≤c⁰₁((1−ν)M−M_†)⁻^q−1¹ (dist (x, ∂Ω))⁻^q−1¹ , (2.17)

which is equivalent to (1.27).

2.2 Proof of Theorems B and B’

2.2.1 Proof of Theorem B

Since the result is known whenM ≥0 from Proposition 2.1, we can assume thatM =−m <0 and N = 1,2 or N ≥3 withp < _N−2^N , u is a nonnegative supersolution of (1.2) in B^c_R and we set u=v^b withb >1. Then

−∆v≥(b−1)|∇v|² v +1

bv^1+b(p−1)−mb^q−1v^{(b−1)(q−1)}|∇v|^q. (2.18) Here again q= _p+1^2p , setting z=|∇v|² we obtain

−∆v≥ Φ(z) bv where

Φ(z) =b(b−1)z−mb

2p p+1v

2+b(p−1) p+1 z

p

p+1 +v^2+b(p−1). Thus Φ achieves it minimum for

z₀ =

mpb^q−1 (b−1)(p+ 1)

p+1

b^p−1v^2+b(p−1)

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and

Φ(z0) =v^2+b(p−1)

1− p^p (p+ 1)^p+1

b b−1

p

m^p+1

. (2.19)

In order to ensure the optimal choice, when N ≥ 3 we take 1 +b(p−1) = _N−2^N , hence b =

2

(N−2)(p−1) which is larger than 1 becausep < _N^N₋₂. Finally Φ(z₀) =v^N−2^N ⁺¹

1− 1

(p+ 1)^p+1

2p N −p(N−2)

p

m^p+1

. Hence, if

m <(p+ 1)

N −p(N −2) 2p

_p+1^p

=µ^∗(N), (2.20)

we have for some δ >0,

−∆v≥δv^N−2^N , (2.21)

and by Proposition 2.1 that is no positive solution in an exterior domain of R^N. IfN = 2 for a givenb >1 we have from (2.19) that if

m <(p+ 1)

b−1 bp

_p+1^p , then, for some δ >0,

−∆v ≥δv^1+b(p−1). (2.22)

The result follows from Proposition 2.1 by choosing blarge enough.

2.2.2 Proof of Theorem B’

1- We assume that such a supersolution u exists and we denoteu=e^v, then

−∆v≥F(|∇v|²), (2.23)

where

F(X) =X+e^(p−1)v+M e

p−1 p+1v

X

p p+1.

Clearly, ifM ≥0, thenF(X)≥0 for any X≥0. Next we assumeM <0, then F(X)≥F(X0) =e^(p−1)v 1−p^p

|M| p+ 1

p+1!

=e^(p−1)v 1− |M|

µ^∗(2) p+1!

.

Hence, if|M| ≤µ^∗(2),vis a positive superharmonic function in Ω which tends to infinity on the boundary. Such a function is larger than the harmonic function with boundary value k >0 for any k (and taking the value min

|x|=Rv(x) for R large enough if Ω is an exterior domain). Letting k→ ∞ we derive a contradiction.

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2- Let R >0 such that Ω^c⊂B_R and let wbe the solution of

−∆w−ae^(p−1)w = 0 inB_R∩Ω dist(x,∂BlimR)→0

w(x) =−∞

dist^{(x,∂Ω)→0}lim w(x) =∞,

(2.24)

with a= 1− _|M_|

µ^∗(2)

p+1

<0, obtained by approximations. By the argument used in 1, ae^(p−1)w ≤ |∇w|²+e^(p−1)w− |M|e

p−1

p+1w|∇w|^p+1^2p , hence

−∆w≤ |∇w|²+e^(p−1)w− |M|e

p−1

p+1w|∇w|^p+1^2p . Thereforev =e^w is nonnegative and satisfies

−∆v−v^p+|M| |∇v|^p+1^2p ≤0 inB_R∩Ω v= 0 on ∂BR

dist^{(x,∂Ω)→0}lim v(x) =∞.

(2.25)

Next we extendvby zero inB_R^c and denote by ˜vthe new function. It is a nonnegative subsolution of (1.2) which tends to∞on∂Ω. For constructing a supersolution we recall that if M ≤ −µ^∗(1) there exist two types of explicit solutions of

−u⁰⁰=u^p+M|u⁰|^p+1^2p (2.26)

defined on Rby Uj,M(t) =∞ for t≤0 and Uj,M(t) =Xj,Mt⁻^p−1² , j=1,2, fort >0 whereX1,M

and X_2,M are respectively the smaller and the larger positive root of X^p−1− |M|

2 p−1

_p+1² X

p−1

p+1 +2(p+ 1)

(p−1)² = 0. (2.27)

Since Ω^c is convex it is the intersection of all the closed half-spaces which contain it and we denote by H_Ω the family of such hyperplanes which are touching∂Ω. IfH∈ H_Ω let nH be the normal direction to H, inward with respect to Ω, H₊ ={x∈ R^N :hn_H, x−n_Hi >0} and we define U_H in the direction n_H by putting

U_H(x) =U_2,M(hn_H, x−n_Hi) =X_2,M(hn_H, x−n_Hi)⁻^p−1² for all x∈ H₊. Hence and set, for x∈Ω :=∩_H_∈H_ΩH₊,

uΩ(x) = inf

H∈H_ΩUH(x). (2.28)

Then u_Ω is a nonnegative supersolution of (1.2) in Ω and

uΩ(x)≥X2,M(distx,Ω))⁻^p−1² ∀x∈Ω.

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Nextv_Ω = lnu_Ω blows up on∂Ω, is finite on ∂B_R and satisfies

−∆v_Ω−ae^(p−1)v^Ω ≥0 inB_R∩Ω. (2.29)

By comparison withwsincea <0,vΩ ≥w. HenceuΩ ≥v inBR\Ω^c. Extendingvby zero as ˜v we obtain u_Ω ≥v˜in Ω^c. Hence u_Ω is a supersolution in Ω^c where it dominates the subsolution

˜

v. It follows by [29, Theorem 1-4-6] that there exists a solutionuof (1.2) satisfying ˜v≤u≤u_Ω,

which ends the proof.

3 The refined Bernstein method

The method is a combination of the one used in the previous proofs. It is based upon the replacement of the unknown by setting first u = v^−β as in [19] and [10] and the study of the equation satisfied by |∇v|. However we do not use integral techniques. Since u is a positive solution of (1.2) inBR, the functionv is well defined and satisfies

−∆v+ (1 +β)|∇v|² v + 1

βv^{1−β(p−1)}+M|β|^q−2βv^{(β+1)(1−q)}|∇v|^q= 0 (3.1) inB_R. We set

z=|∇v|² , s= 1−q−β(q−1) = (1−q)(β+ 1), σ= 1−β(p−1), and derive

∆v = (1 +β)z v + 1

βv^σ+M|β|^q−2βv^sz^q². (3.2) Combining Bochner’s formula and Schwarz identity we have classically

1

2∆z≥ 1

N(∆v)²+h∇∆v,∇vi.

We explicit the different terms (∆v)²= (1 +β)²z²

v² +M²β^2(q−1)v^2sz^q+ v^2σ

β² + 2M(1 +β)|β|^q−2βv^s−1z¹⁺^q² +2(1 +β)

β v^σ−1z+ 2M|β|^q−2v^s+σz^q²,

∇∆v= (1 +β)∇z

v −(1 +β)z

v² ∇v+σ

βv^σ−1∇v+M s|β|^q−2βv^s−1z^q²∇v +M q

2 |β|^q−2βv^sz^q²⁻¹∇z, h∇∆v,∇vi=

1 +β v +M q

2 |β|^q−2βv^sz^q²⁻¹

h∇z,∇vi −(1 +β)z² v² +σ

βv^σ−1z +M s|β|^q−2βv^s−1z^q²⁺¹. Hence

−1

2∆z+ 1

N(∆v)²+

1 +β v +M q

2 |β|^q−2βv^sz^q²⁻¹

h∇z,∇vi

−(1 +β)z² v² +σ

βv^σ−1z+M s|β|^q−2βv^s−1z^q²⁺¹≤0.

(3.3)

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3.1 Proof of Theorem D

We develop the term (∆v)² in (3.3) and get

−1 2∆z+

(1 +β)²

N −(1 +β) z²

v² +M²β^2(q−1)

N v^2sz^q+M

s+2(1 +β) N

|β|^q−2βv^s−1z¹⁺^q² + v^2σ

N β² +

1 +β v +M q

2 |β|^q−2βv^sz^q²⁻¹

h∇z,∇vi+N σ+ 2(1 +β)

N β v^σ−1z+ 2M|β|^q−2

N v^s+σz^q²

≤0.

(3.4) Next we set z=v^−kY wherek is a real parameter. Then∇z=−kv^−k−1Y∇v+v^−k∇Y,

h∇z,∇vi=−kv^−k−1Y z+v^−kh∇Y,∇vi=−kv^−2k−1Y²+v^−kh∇Y,∇vi, h∇z,∇vi

v =−kv^−2k−2Y²+v^−k−1h∇Y,∇vi,

M v^sz^q²⁻¹h∇z,∇vi=−kM v^s−^qk²^−k−1Y^q²⁺¹+M v^s−^qk² Y^q²⁻¹h∇Y,∇vi,

−∆z= div kv^−k−1Y∇v−v^−k∇Y

=kv^−k−1Y∆v−k(k+ 1)v^−k−2Y z+ 2kv^−k−1h∇Y,∇vi −v^−k∆Y

=kv^−k−1Y∆v−k(k+ 1)v^−2k−2Y²+ 2kv^−k−1h∇Y,∇vi −v^−k∆Y.

From (3.2)

∆v = (1 +β)v^−k−1Y + 1

βv^σ+M|β|^q−2βv^s−k^q²Y^q², therefore

−∆z=k(β−k)v^−2k−2Y²+ k

βv^σ−k−1Y +kM|β|^q−2βv^s−k^q²^−k−1Y^q²⁺¹ + 2kv^−k−1h∇Y,∇vi −v^−k∆Y.

Replacing h∇z,∇vi and ∆z given by the above expressions in (3.4) and z by v^−kY, leads to

−∆Y +

k(β−k)

2 +(1 +β)²

N −(k+ 1)(β+ 1)

v^−k−2Y²+v^2σ+k

N β² + M²β^2(q−1)

N v^2s+k−kqY^q + k+β+ 1

v +M q|β|^q−2β

2 v^s+k−k^q²Y^q²⁻¹

!

h∇Y,∇vi+2M|β|^q−2

N v^s+σ+k−k^q²Y^q² +

s+2(1 +β)

N −k(q−1) 2

M|β|^q−2βv^s−k^q²⁻¹Y¹⁺^q² + 1 β

k

2 +σ+2(1 +β) N

v^σ−1Y ≤0.

For 1, 2 >0,

1

v|h∇Y,∇vi| ≤1v^−k−2Y²+ 1 4₁

|∇Y|² Y , v^s+k−k^q²Y^q²⁻¹|h∇Y,∇vi| ≤₂v^2s−kq+kY^q+ 1

4₂

|∇Y|² Y .

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Hence

−∆Y +v^2σ+k

N β² +2M|β|^q−2

N v^s+σ+k−k^q²Y^q² + M²β^2(q−1)

N −M q2|β|^q−1 2

!

v^2s+k−kqY^q +

k(β−k)

2 +(1 +β)²

N −(k+ 1)(β+ 1)− |k+β+ 1|₁

v^−k−2Y² + 1

β k

2 +σ+2(1 +β) N

v^σ−1Y +

s+2(1 +β)

N − k(q−1) 2

M|β|^q−2βv^s−k^q²⁻¹Y¹⁺^q²

≤ |k+β+ 1|

₁ +M q|β|^q−1 2₂

!|∇Y|² 4Y .

(3.5) We first choose 2= ^M^|β|

q−1

qN , then

−∆Y + v^2σ+k N β² +

k(β−k)

2 +(1 +β)²

N −(k+ 1)(β+ 1)− |k+β+ 1|1

v^−k−2Y² + 1

β k

2 +σ+ 2(1 +β) N

v^σ−1Y + M²β^2(q−1)

2N v^2s+k−kqY^q+ 2M|β|^q−2

N v^s+σ+k−k^q²Y²^q +

s+2(1 +β)

N −k(q−1) 2

M|β|^q−2βv^s−k^q²⁻¹Y¹⁺^q²

≤

|k+β+ 1|

₁ +N q² 2

|∇Y|² 4Y .

(3.6) In order to show the sign of the terms on the left in (3.5), we separate the terms containing the coefficient M from the ones which do not contain it. Indeed these last terms are associated to the mere Lane-Emden equation (1.3) which is treated, as a particular case, in [6, Theorem B]

where the exponents therein areq = 0, andp∈ 1,^N_N⁺³₋₁

. We set H₁_,1= v^2σ+k

N β² +

k(β−k)

2 +(1 +β)²

N −(k+ 1)(β+ 1)− |k+β+ 1|₁

v^−k−2Y² + 1

β k

2 +σ+2(1 +β) N

v^σ−1Y

=v^2σ+kH˜₁_,1(v^{−1−k−σ}Y),

(3.7)

where

H˜1,1(t) =

k(β−k)

2 +(1 +β)²

N −(k+ 1)(β+ 1)− |k+β+ 1|1

t² + 1

β k

2 +σ+2(1 +β) N

t+ 1

N β²,

(3.8)

and

H_M,2 = M²β^2(q−1)

2N v^2s+k−kqY^q+2M|β|^q−2

N v^s+σ+k−k^q²Y^q² +

s+2(1 +β)

N − k(q−1) 2

M|β|^q−2βv^s−k²^q⁻¹Y¹⁺^q².

(3.9)

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Then

−∆Y +v^2σ+kH˜₁_,1(v^{−1−k−σ}Y) +H_M,2≤

|k+β+ 1|

₁ +N q² 2

|∇Y|² 4Y .

The sign of ˜H1,1 depends on its discriminantD₁ which is a polynomial in its coefficients. Then if for 1 = 0 this discriminant is negative D₀ is negative, the discriminant D₁ of ˜H1,1 shares this property for ₁ > 0 small enough and therefore H₁_,1 is positive. The proof is similar as the one of [6, Theorem B] in case (i) but for the sake of completeness we recall the main steps.

Firstly

D₀⁰ :=N²β²D₀= N k

2 +σN + 2(1 +β) 2

−4

N k(β−k)

2 + (1 +β)²−N(k+ 1)(β+ 1)

. Then

D₀⁰ =

N(p−1)

4 −1

(2σ+k)²+ 2(p−1)(2σ+k) + ˜L where ˜L= (p−1)k²+p(λ+ 2)² >0. Put

S = 2σ+k

k+ 2 = 1− 2β(p−1)

k+ 2 and T(S) =

(N −1)(p−1)

4 −1

S²+ (p−1)S+p.

After some computations we get, if k6=−2, D⁰₁:= (p−1)D⁰₀

(k+ 2)² = (p−1) k

k+ 2−S 2

2

+T(S). (3.10)

We choose S >2 such that _k+2^k −^S₂ = 0, hence β = _2(p−1)^2−k . If p < ^N_N⁺³₋₁ the coefficient of S² in T(S) is negative. HenceT(S)<0 providedS is large enough which is satisfied if k <−2 with

|k+ 2|small enough. We infer from this thatβ >0,D₀ <0 and ˜H1,1 >0 if1 is small enough.

In particular ˜H1,1(t)≥c6(t²+ 1) for some c6 =c6(N, p, q)>0, which means v^2σ+kH˜1,1(v^{−1−k−σ}Y)≥c6

v^−k−2Y²+v^2σ+k

. (3.11)

Secondly the positivity of HM,2 is ensured, asβ and M are positive, by the positivity of A:=s+2(1 +β)

N −k(q−1)

2 .

Replacing sby its value, we obtain, since 1< q < ^N+2_N and β+^2+k₂ >0, which can be assume by taking|k+ 2|small enough,

A= 21 +β

N −(q−1)

β+ 1 + k 2

>−k N Then we deduce that

−∆Y +c6 v^−k−2Y²+v^2σ+k

≤c7

|∇Y|²

Y , (3.12)

A priori estimates for elliptic equations with reaction terms involving the function and its gradient

HAL Id: hal-01906697

https://hal.archives-ouvertes.fr/hal-01906697v2

terms involving the function and its gradient

Marie-Françoise Bidaut-Véron, Marta Garcia-Huidobro, Laurent Veron

To cite this version:

terms involving the function and its gradient

Contents

1 Introduction

2 The direct Bernstein method

3 The refined Bernstein method