Abstract
We study local and global properties of positive solutions of \(-\Delta u=u^p+M\left| \nabla u\right| ^q\) in a domain \(\Omega \) of \(\mathbb {R}^N\), in the range \(\min \{p,q\}>1\) and \(M\in \mathbb {R}\). We prove a priori estimates and existence or non-existence of ground states for the same equation.
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1 Introduction
This article is concerned with local and global properties of positive solutions of the following type of equations
in \(\Omega {\setminus }\{0\}\) where \(\Omega \) is an open subset of \(\mathbb {R}^N\) containing 0, p and q are exponents larger than 1 and \(M,M'\) are real parameters. If \(M'\le 0\) 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 \(M'>0\) which will be assumed in all the article, and by homothety (1.1) becomes
If \(M=0\) (1.2) is called Lane–Emden equation
It turns out that it plays an important role in modelling meteorological or astrophysical phenomena [13, 15], 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 \(N\ge 3\), This equations exhibits two main critical exponents \(p=\frac{N}{N-2}\) and \(p=\frac{N+2}{N-2}\) which play a key role in the description of the set of positive solutions which can be summarized by the following overview:
-
1.
If \(1<p\le \frac{N}{N-2}\), there exists no positive solution if \(\Omega \) 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 \(1<p<\frac{N+2}{N-2}\), there exists no ground state, i.e. positive solution in \(\mathbb {R}^N\). Furthermore any positive solution u in a ball \(B_R=B_R(a)\) satisfies
$$\begin{aligned} u(x)\le c(R-|x-a|)^{-\frac{2}{p-1}}, \end{aligned}$$(1.4)where \(c=c(N,p)>0\), see [19].
-
3.
If \(p=\frac{N+2}{N-2}\) all the positive solutions in \(\mathbb {R}^N\) are radial with respect to some point a and endow the following form
$$\begin{aligned} u(x):=u_\lambda (x)=\frac{(N(N-2)\lambda )^{\frac{N-2}{4}}}{\left( \lambda +|x-a|^{2}\right) ^{\frac{N-2}{2}}}. \end{aligned}$$(1.5)All the positive solutions in \(\mathbb {R}^N{\setminus }\{0\}\) are radial, see [12].
-
4.
If \(p>\frac{N+2}{N-2}\) 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 \(T_k\) where \(k>0\) with
$$\begin{aligned} T_k[v](x)=k^{\frac{2}{p-1}}v(kx). \end{aligned}$$(1.6)
Indeed, the first significant non-radial results deals with the case \(1<p\le \frac{N}{N-2}\). They are based upon the Brezis–Lions lemma [11] which yields an estimate of solutions in the Lorentz space \(L^{\frac{N}{N-2},\infty }\), implying in turn the local integrability of \(u^q\). 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 \(1<p<\frac{N+2}{N-2}\); 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 \(p=\frac{N+2}{N-2}\), 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 \(M\ne 0\) 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 [9] where we study local and global aspects of positive solutions of
where \(p\ge 0\), \(0\le q<2\), mostly in the superlinear case \(p+q-1>0\). Therein we prove the existence of a critical line of exponents
The subcritical range corresponds to the fact that (p, q) is below \((\mathfrak {L})\). In this region Serrin’s celebrated results [24] can be applied and we prove [9, Theorem A] that positive solutions of (1.7) in the punctured ball \(B_2{\setminus }\{0\}\) satisfy, for some constant \(c>0\) depending on the solution,
When (p, q) is above \((\mathfrak {L})\), i.e. in the supercritical range, we introduced two methods for obtaining a priori estimate of solutions: The pointwise Bernstein method and the integral Bernstein method. The first one is based upon the change of unknown \(u=v^{-\beta }\), and then to show that \(\left| \nabla v\right| \) satisfies an inequality of Keller–Osserman type. When (p, q) lies above \((\mathfrak {L})\) and verifies
-
(i)
either \(1\le p<\frac{N+3}{N-1}\) and \(p+q-1<\frac{4}{N-1}\),
-
(ii)
or \(0\le p<1\) and \(p+q-1<\frac{(p+1)^2}{p(N-1)}\),
we prove that any positive solution of (1.7) in a domain \(\Omega \subset \mathbb {R}^N\) satisfies
for some positive \(c^*\) and a depending on N, p and q [9, Theorem B]. As a consequence we prove that any positive solution of (1.7) in \(\mathbb {R}^N\) is constant. With the second method we combine the change of unknown \(u=v^{-\beta }\) with integration and cut-off functions. We show the existence of a quadratic polynomial G in two variables such that for any \((p,q)\in \mathbb {R}_+\times [0,2)\) satisfying \(G(p,q)<0\) any positive solution of (1.7) in \(\mathbb {R}^N\) is constant [9, Theorem C]. The polynomial G is not simple but it is worth noting that if \(0\le p<\frac{N+2}{N-2}\), there holds \(G(p,0)<0\), which recovers Gidas and Spruck result [19].
For Eq. (1.2) we first observe that the equation is invariant under the scaling transformation (1.6) for any \(k>0\) if and only if q is critical with respect to p, i.e.
In general the transformation \(T_k\) exchanges (1.2) with
hence if \(q<\frac{2p}{p+1}\), the limit equation when \(k\rightarrow 0\) is (1.3). We say that the exponent p is dominant. We can also consider the transformation
when \(q\ne 2\), which is the same as \(T_k\) if \(q=\frac{2p}{p+1}\), and more generally transforms (1.2) into
Hence if \(q>\frac{2p}{p+1}\), the limit equation when \(k\rightarrow 0\) is the Riccati equation
It is also important to notice that the value of the coefficient M (and not only its sign) plays a fundamental role, only if \(q=\frac{2p}{p+1}\). If \(q\ne \frac{2p}{p+1}\) the transformation
allows to transform (1.2) into
Equation (1.2) has been essentially studied in the radial case when \(M<0\) in connection with the parabolic equation
see [14, 16, 17, 25, 27, 30, 31]. The studies mainly deal with the case \(q\ne \frac{2p}{p+1}\), although not complete when \(q>\frac{2p}{p+1}\). When \(q=\frac{2p}{p+1}\) 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 the nonradial case, any nonnegative nontrivial solution is positive since \(p,q>1\). We first observe, using a standard averaging method applied to positive supersolutions of (1.3), that if \(M\ge 0\), \(1<p\le \frac{N}{N-2}\) when \(N\ge 3\), any \(p>1\) if \(N=1,2\), then for any \(q>0\) there exists no positive solution in an exterior domain. When \(0<q<\frac{2p}{p+1}\) the equation endows some character of the pure Emden–Fowler equation (1.3) by the transformation \(T_k\). In [23] it is proved that if \(0<q<\frac{2p}{p+1}\), \(1<p<\frac{N+2}{N-2}\) and \(M\in \mathbb {R}\), any positive solution of (1.3) in an open domain satisfies
Note that this does not imply the non-existence of ground state. In [1] Alarcón, García-Melián and Quass study the equation
in an exterior domain of \(\mathbb {R}^N\) emphasizing the fact that positive solutions are super harmonic functions. They prove that if \(1<q\le \frac{N}{N-1}\) and if f is positive on \((0,\infty )\) and satisfies
for some \(p> \frac{N}{N-2}\), 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 [9]. Our first nonradial result dealing with the case \(q>\frac{2p}{p+1}\) is the following:
Theorem A
Let \(N\ge 1\), \(p>1\) and \(q>\frac{2p}{p+1}\). Then for any \(M>0\), any solution of (1.2) in a domain \(\Omega \subset \mathbb {R}^N\) satisfies
As a consequence, any ground state has at most a linear growth at infinity:
Our proof relies on a direct Bernstein method combined with Keller–Osserman’s estimate applied to \(\left| \nabla u\right| ^2\). It is important to notice that the result holds for any \(p>1\), showing that, in some sense, the presence of the gradient term has a regularizing effect. In the case \(q<\frac{2p}{p+1}\) we prove a non-existence result
Theorem A \('\) Let \(N\ge 1\), \(p>1\), \(1<q<\frac{2p}{p+1}\) and \(M>0\). Then there exists a constant \(c_{N,p,q}>0\) such that there is no positive solution of (1.2) in \(\mathbb {R}^N\) satisfying
When q is critical with respect to p the situation is more delicate since the value of M 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 \(N\ge 2\), \(p>1\) if \(N=2\) or \(1<p\le \frac{N}{N-2}\) if \(N=3\), \(q=\frac{2p}{p+1}\) and \(M>-\mu ^*\) where
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 \(M\le -\mu ^*\) there exists positive singular solutions. The constant \(\mu ^*\) will play an important role in the description developed in [10] 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 \(N\ge 1\), \(p>1\) and \(q=\frac{2p}{p+1}\).
-
1.
If \(\Omega \) is a domain with a compact boundary satisfying the Wiener criterion and \( M\ge -\mu ^*(2)\) there exists no positive supersolution of (1.2) in \(\Omega \) satisfying
$$\begin{aligned} \displaystyle \lim _{\mathrm{dist}\,(x,\partial \Omega )\rightarrow 0}u(x)=\infty . \end{aligned}$$(1.25) -
2.
If G is a bounded convex domain, \(\Omega =\overline{G}^c\) and \( M< -\mu ^*(1)\) there exists a positive solution of (1.2) in \(\Omega \) satisfying (1.25).
We show in [10] that the inequality \( M< -\mu ^*(1)\) 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 \(\Omega \subset \mathbb {R}^N\), \(N\ge 1\), be a domain, \(p>1\), \(q=\frac{2p}{p+1}\). For any
and any \(\nu >0\) such that \((1-\nu )M>M_\dag \), there exists a positive constant \(c_{N,p,\nu }\) such that any solution u in \(\Omega \) satisfies
Consequently there exists no nontrivial solution of (1.2) in \(\mathbb {R}^N\).
The next result, based upon an elaborate Bernstein method, complements Theorem C under a less restrictive assumption on M but a more restrictive assumption on p.
Theorem D
Let \(1<p<\frac{N+3}{N-1}\), \(N\ge 2\), \(1<q<\frac{N+2}{N}\) and \(\Omega \subset \mathbb {R}^N\) be a domain. Then there exist \(a>0\) and \(c_{N,p,q}>0\) such that for any \(M>0\), any positive solution u in \(\Omega \) satisfies
Hence there exists no nontrivial nonnegative solution of (1.2) in \(\mathbb {R}^N\).
It is remarkable that the constants a and \(c_{N,p,q}\) do not depend on \(M>0\), a fact which is clear when \(q\ne \frac{2p}{p+1}\) by using the transformation \(T_k\), but much more delicate to highlight when \(q=\frac{2p}{p+1}\) since (1.2) is invariant. When \(\left| M\right| \) is small, we use an integral method to obtain the following result which contains, as a particular case, the estimates in [19] and [10]. The key point of this method is to prove that the solutions in a punctured domain satisfy a local Harnack inequality.
Theorem E
Let \(N\ge 3\), \(1<p<\frac{N+2}{N-2}\), \(q=\frac{2p}{p+1}\). Then there exists \(\epsilon _0>0\) depending on N and p such that for any M satisfying \(\left| M\right| \le \epsilon _0\), any positive solution u in \(B_R{\setminus }\{0\}\) satisfies
As a consequence there exists no positive solution of (1.2) in \(\mathbb {R}^N\), and any positive solution u in a domain \(\Omega \) satisfies
Note that under the assumptions of Theorem E, there exist ground states for \(\left| M\right| \) large enough when \(1<p<\frac{N}{N-2}\), or any \(p>1\) if \(N=1,2\).
If u is a radial solutions of (1.2) in \(\mathbb {R}^N\) it satisfies
on \((0,\infty )\). Using several type of Lyapounov type functions introduced by Leighton [20] and Anderson and Leighton [3], we prove some results dealing with the case \(M>0\) which complement the ones of [25] relative to the case \(M<0\).
Theorem F
-
1.
Let \(p>1\) and \(q>\frac{2p}{p+1}\). Then there exists no radial ground state u satisfying \(u(0)=1\) when \(M>0\) is too large.
-
2.
Let \(1<p<\frac{N+2}{N-2}\). If \(1<q\le p\) there exists no radial ground state for any \(M>0\). If \(q>p\) there exists no radial ground state for \(M>0\) small enough.
-
3.
Let \(N\ge 3\), \(p>\frac{N+2}{N-2}\) and \(q\ge \frac{2p}{p+1}\). 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 \(\mathbb {R}^N{\setminus }\{0\}\) in the case \(q=\frac{2p}{p+1}\) 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 \((M,X_{M})\) occurs only if \(p\ge \frac{N+1}{N-3}\) and \(M\ge 0\) and \(M<0\) where there exists a countable set of bifurcations from \((M_k,X_{M_k})\), \(k\ge 1\), when \(1<p<\frac{N+1}{N-3}\).
In a subsequent article [10] we present a fairly complete description of the positive radial solutions of (1.2) in \(\mathbb {R}^N{\setminus }\{0\}\) in the scaling invariant case \(q=\frac{2p}{p+1}\).
2 The direct Bernstein method
We begin with a simple property in the case \(M\ge 0\) 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 \(\mathbb {R}^N{\setminus }\overline{B}_R\), \(R\ge 0\) if one of the two conditions is satisfied:
-
(i)
\(M\ge 0\), \(q\ge 0\) and either \(N=1,2\) and \(p>1\) or \(N\ge 3\) and \(1<p\le \frac{N}{N-2}\).
-
(ii)
\(M> 0\), \(N\ge 3\), \(p\ge 1\) and \(1<q\le \frac{N}{N-1}\).
-
(i)
-
2.
If \(N\ge 3\), \(q\ge 1\), \(p>\frac{N}{N-2}\) and \(u(x)=u(r,\sigma )\) is a positive solution of (1.2) in \(\mathbb {R}^N{\setminus }\overline{B}_R\), \(R\ge 0\). Then there exists \(\rho \ge R\) such that
$$\begin{aligned} {\displaystyle \frac{1}{N\omega _N} }{\displaystyle \int _{S^{N-1}}^{}}u(r,\sigma )dS:=\overline{u}(r)\le c_0r^{-\frac{2}{p-1}}\qquad \text {for all }\;r>\rho , \end{aligned}$$(2.1)with \(c_0:=\left( \frac{2N}{p-1}\right) ^{\frac{1}{p-1}}\) and
$$\begin{aligned} \left| {\displaystyle \frac{1}{N\omega _N} }{\displaystyle \int _{S^{N-1}}^{}}u_r(r,\sigma )dS\right| :=\left| \overline{u}_r(r)\right| \le (N-2)c_0r^{-\frac{p+1}{p-1}}\quad \text {for all }\;r>\rho .\ \ \ \end{aligned}$$(2.2) -
3.
If \(M>0\), \(p\ge 0\), and \(q>\frac{N}{N-1}\) there holds for
$$\begin{aligned} \left| \overline{u}_r(r)\right| \le \left( {\displaystyle \frac{(q-1)(N-1)-1}{(q-1)M} }\right) ^{\frac{1}{q-1}}r^{-\frac{1}{q-1}}\qquad \text {for all }\;r>\rho , \end{aligned}$$(2.3)and
$$\begin{aligned} \overline{u}(r)\le \Bigl ({\displaystyle \frac{q-1}{2-q} }\Bigr )\left( {\displaystyle \frac{(q-1)(N-1)-1}{(q-1)M} }\right) ^{\frac{1}{q-1}}r^{\frac{q-2}{q-1}}\qquad \text {for all }\;r>\rho , \end{aligned}$$(2.4)Furthermore, if \(R=0\), inequalities (2.1), (2.2) and (2.3) hold with \(\rho =0\).
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. [5] and references therein). In this statement we denote by \((r,\sigma )\in \mathbb {R}_+\times S^{N-1}\) the spherical coordinates in \(\mathbb {R}^N\), by \(\omega _N\) the volume of the unit N-ball and thus \(N\omega _N\) is the (N-1)-volume of the unit sphere \(S^{N-1}\). Writing (1.2) in spherical coordinates and using Jensen formula, we get
It implies that \(r\mapsto w(r):= -r^{N-1}\overline{u}_r\) is increasing on \((R,\infty )\), thus it admits a limit \(\ell \in (-\infty ,\infty ]\). If \(\ell \le 0\), then \(\overline{u}_r(r)> 0\) on \((R,\infty )\). Hence \(\overline{u}(r)\ge \overline{u}(\rho ):=c>0\) for \(r\ge \rho >R\). then
which implies \( \overline{u}_r(r)\rightarrow -\infty \), thus \( \overline{u}(r)\rightarrow -\infty \) as \(r\rightarrow -\infty \), contradiction. Therefore \(\ell \in (0,\infty ]\) and either \(\overline{u}_r(r)<0\) on \((R,\infty )\) or there exists \(r_{\ell }>R\) such that \(\overline{u}_r(r_\ell )=0\), \(\overline{u}\) is increasing on \((R,r_\ell ,)\) and decreasing on \((r_\ell ,\infty )\). If \(\overline{u}_r(r)<0\) on \((R,\infty )\), then we have for \(r> 2R\)
which yields (2.1). If we are in the second case with \(r_\ell >R\), we apply the same inequality with \(r> 2r_\ell \) and again (2.1) for \(r> 2r_\ell \). Since \(\overline{u}\) is superharmonic, the function \(v(s)=\overline{u}(r)\) with \(s=r^{2-N}\) is concave on \((0,R^{2-N})\) and it tends to 0 when \(s\rightarrow 0\). Thus
This implies (2.1) and (2.2). Note that the case \(r_\ell >R\) cannot happen if \(R=0\), so in any case, if \(R=0\) then \(\rho =0\).
If \(M>0\), we have with \(w(r)=-r^{N-1}\overline{u}_r\)
We have seen that \(w(r)>0\) at infinity with limit \(\ell \in (0,\infty ]\), hence, on the maximal interval containing \(\infty \) where \(w>0\), we have \((w^{1-q})_r\le (1-q)Mr^{(N-1)(1-q)}\). We have for \(r>s>R\)
if \(q=\frac{N}{N-1}\) and
if \(q<\frac{N}{N-1}\), and both expressions which tend to \(-\infty \) when \(r\rightarrow \infty \), a contradiction. This proves 1-(ii). If \(q>\frac{N}{N-1}\), the above expression yields, when \(r\rightarrow \infty \),
This implies
and (2.3).\(\square \)
Remark
The previous is a particular case of a much more general one dealing with quasilinear operators proved in [5, Theorem 3.1].
2.1 Proof of Theorems A, A\('\) and C
The function u is at least \(C^{3+\alpha }\) for some \(\alpha \in (0,1)\) since \(p,q>1\). Hence \(z=\left| \nabla u\right| ^2\) is \(C^{2+\alpha }\). Since there holds by Bochner’s identity and Schwarz’s inequality
we obtain from (1.2),
Since for \(\delta >0\),
we obtain for any \(\nu \in (0,1)\), provided \(\delta \) is small enough,
where \(c_1=c_1(M,N,\nu )>0\).
2.1.1 Proof of Theorem A
We recall the following technical result proved in [9, 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 \(\overline{B}_R\) and \(C^1\) on the set \({\mathcal {U}}_+=\{x\in B_R:v(x)>0\}\). If v satisfies, for some real number a,
on each connected component of \({\mathcal {U}}_+\), then
Abridged proof
Assuming \(a>0\), we set \(W=v^\alpha \) for \(0<\alpha \le \frac{1}{a+1}\), this transforms (2.8) into
and then we apply Keller–Osserman inequality. \(\square \)
Proof of Theorem A
Suppose \(\frac{2p}{p+1}<q\). We set \(r=\frac{2p}{p-1}\), \(r'=\frac{r}{r-1}\), then, for any \(\epsilon >0\)
We fix \(\eta \in (0,1)\) and \(\epsilon \) so that \(\epsilon ^r=\frac{2(1-\eta )}{N(p-1)}\) and get
where \(c_2=\frac{p+1}{2}\left( \frac{N(p-1)}{2(1-\eta )}\right) ^{\frac{p+1}{p-1}}\). We perform the change of scale (1.6) in order to reduce (1.2) to the case \(M=1\) by setting \(u(x)=\alpha ^{\frac{2}{p-1}}v(\alpha x)\) with \(\alpha =M^{-\frac{p-1}{(p+1)q-2p}}\). Then the equation for \(z=\left| \nabla v\right| ^2\) is considered in \(\Omega _\alpha =\alpha \Omega \). Choosing now \(\eta =\frac{1}{2}\) we obtain
where \(c_3=c_3(N,p,q)>0\), hence
Put \(\tilde{z}=\left( z-\left( 4Nc_3\right) ^{\frac{1}{q}}\right) _+\), then
hence, from Lemma 2.2, we derive
where \(c_4=c_4(N,q,c_1)>0\) which implies
Then (1.21) and (1.22) follow.
Assume now that there exists a ground state u. Fix \(y\in \mathbb {R}^N\) and consider \(\{y_n\}\subset \mathbb {R}^N\) such that \(|y_n|=2n>|y|\). We apply (2.11) with \(\Omega _\alpha =B_n(y_n)\). Then
and letting \(n\rightarrow \infty \) we infer
Hence, by the definition of v and y we see that
which is exactly (1.22). \(\square \)
2.1.2 Proof of Theorem A\('\)
Suppose \(1<q<\frac{2p}{p+1}\). 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 with \(\alpha =M^{\frac{p-1}{2p-(p+1)q}}\). From (2.7) with \(\nu =\frac{1}{4}\) the function \(z=\left| \nabla v\right| ^2\) satisfies
By Hölder’s inequality,
Since \((p-1)q'=2p+\frac{2p-(p+1)q}{q-1}\) we derive
If \(\max v\le c_{N,p,q}:=(4^{q'-1}p^{q'}N^{q'})^{-\frac{q-1}{2p-(p+1)q}}\), we obtain
which implies that \(z=0\) by Lemma 2.2, hence v is constant and thus \(v=0\) from the equation. \(\square \)
Remark
If u is a positive ground state of (1.2) radial with respect to 0, it satisfies \(u_r(0)=0\) and it is a decreasing function of r. The previous theorem asserts that it must satisfy
2.1.3 Proof of Theorem C
Suppose \(\frac{2p}{p+1}=q\). For \(A>0\) we consider the expression
Now the function \(Z\mapsto \Phi _A(Z)=u^{p}+AZ^{q}-\sqrt{Np\,}\,u^{\frac{p-1}{2}}Z\) achieves its minimum at \(Z_0=\left( \frac{\sqrt{Np}}{qA }\right) ^{\frac{p+1}{p-1}}u^{\frac{p+1}{2}}\) and
Thus setting
we obtain that if \(A\ge M_\dag \), then \(\Phi _A(Z)\ge 0\) for all Z. Put \(M_\nu =(1-\nu ) M\) for \(\nu \in (0,1)\) such that \(M_\dag <M_\nu \), we derive from (2.7)
which yields
Using again Lemma 2.2 we obtain
which is equivalent to (1.27). \(\square \)
2.2 Proof of Theorems B and B\('\)
2.2.1 Proof of Theorem B
Since the result is known when \(M\ge 0\) from Proposition 2.1, we can assume that \(M=-m<0\) and \(N=1,2\) or \(N\ge 3\) with \(p<\frac{N}{N-2}\), u is a nonnegative supersolution of (1.2) in \(\overline{B}^c_R\) and we set \(u=v^b\) with \(b>1\). Then
Here again \(q=\frac{2p}{p+1}\), setting \(z=\left| \nabla v\right| ^2\) we obtain
where
Thus \(\Phi \) achieves it minimum for
and
In order to ensure the optimal choice, when \(N\ge 3\) we take \(1+b(p-1)=\frac{N}{N-2}\), hence \(b=\frac{2}{(N-2)(p-1)}\) which is larger than 1 because \(p<\frac{N}{N-2}\). Finally
Hence, if
we have for some \(\delta >0\),
and by Proposition 2.1 that is no positive solution in an exterior domain of \(\mathbb {R}^N\).
If \(N=2\) for a given \(b>1\) we have from (2.19) that if
then, for some \(\delta >0\),
The result follows from Proposition 2.1 by choosing b large enough. \(\square \)
2.2.2 Proof of Theorem B\('\)
1. We assume that such a supersolution u exists and we denote \(u=e^v\), then
where
Clearly, if \(M\ge 0\), then \(F(X)\ge 0\) for any \(X\ge 0\). Next we assume \(M<0\), then
Hence, if \(\left| M\right| \le \mu ^*(2)\), v is a positive superharmonic function in \(\Omega \) 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 \(\min _{\left| x\right| =R}v(x)\) for R large enough if \(\Omega \) is an exterior domain). Letting \(k\rightarrow \infty \) we derive a contradiction.
2. Let \(R>0\) such that \(\Omega ^c\subset B_R\) and let w be the solution of
with \(a=1-\left( \frac{\left| M\right| }{\mu ^*(2)}\right) ^{p+1}<0\), obtained by approximations. By the argument used in 1,
hence
Therefore \(v=e^{w}\) is nonnegative and satisfies
Next we extend v by zero in \(B_R^c\) and denote by \(\tilde{v}\) the new function. It is a nonnegative subsolution of (1.2) which tends to \(\infty \) on \(\partial \Omega \). For constructing a supersolution we recall that if \(M\le -\mu ^*(1)\) there exist two types of explicit solutions of
defined on \(\mathbb {R}\) by \(U_{j,M}(t)=\infty \) for \(t\le 0\) and \(U_{j,M}(t)=X_{j,M}t^{-\frac{2}{p-1}}\), j=1,2, for \(t>0\) where \(X_{1,M}\) and \(X_{2,M}\) are respectively the smaller and the larger positive root of
Since \(\Omega ^c\) is convex it is the intersection of all the closed half-spaces which contain it and we denote by \({\mathcal {H}}_\Omega \) the family of such hyperplanes which are touching \(\partial \Omega \). If \(H\in {\mathcal {H}}_\Omega \) let \(\mathbf{n}_H\) be the normal direction to H, inward with respect to \(\Omega \), \({\mathcal {H}}_+=\{x\in \mathbb {R}^N:\langle \mathbf{n}_H,x-\mathbf{n}_H\rangle >0\}\) and we define \(U_H\) in the direction \(\mathbf{n}_H\) by putting
Hence and set, for \(x\in \Omega :=\cap _{H\in {\mathcal {H}}_\Omega } {\mathcal {H}}_+\),
Then \(u_\Omega \) is a nonnegative supersolution of (1.2) in \(\Omega \) and
Next \(v_\Omega =\ln u_\Omega \) blows up on \(\partial \Omega \), is finite on \(\partial B_R\) and satisfies
By comparison with w since \(a<0\), \(v_\Omega \ge w\). Hence \(u_\Omega \ge v\) in \(B_R{\setminus }\Omega ^c\). Extending v by zero as \(\tilde{v}\) we obtain \(u_\Omega \ge \tilde{v}\) in \(\Omega ^c\). Hence \(u_\Omega \) is a supersolution in \(\Omega ^c\) where it dominates the subsolution \(\tilde{v}\). It follows by [29, Theorem 1-4-6] that there exists a solution u of (1.2) satisfying \(\tilde{v}\le u\le u_\Omega \), which ends the proof. \(\square \)
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 \(u=v^{-\beta }\) as in [7] and [19] and the study of the equation satisfied by \(\left| \nabla v\right| \). However we do not use integral techniques. Since u is a positive solution of (1.2) in \(B_R\), the function v is well defined and satisfies
in \(B_R\). We set
and derive
Combining Bochner’s formula and Schwarz identity we have classically
We explicit the different terms
Hence
3.1 Proof of Theorem D
We develop the term \((\Delta v)^2\) in (3.3) and get
Next we set \(z=v^{-k}Y\) where k is a real parameter. Then \(\nabla z=-kv^{-k-1}Y\nabla v+v^{-k}\nabla Y\),
From (3.2)
therefore
Replacing \(\langle \nabla z,\nabla v\rangle \) and \(\Delta z\) given by the above expressions in (3.4) and z by \(v^{-k}Y\), leads to
For \(\epsilon _1,\epsilon _2>0\),
Hence
We first choose \(\epsilon _2=\frac{M\left| \beta \right| ^{q-1}}{qN}\), then
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 [9, Theorem B] where the exponents therein are \(q=0\), and \(p\in \left( 1,\frac{N+3}{N-1}\right) \). We set
where
and
Then
The sign of \(\tilde{H}_{\epsilon _1,1}\) depends on its discriminant \({\mathcal {D}}_{\epsilon _1}\) which is a polynomial in its coefficients. Then if for \(\epsilon _1=0\) this discriminant is negative \({\mathcal {D}}_{0}\) is negative, the discriminant \({\mathcal {D}}_{\epsilon _1}\) of \(\tilde{H}_{\epsilon _1,1}\) shares this property for \(\epsilon _1>0\) small enough and therefore \(H_{\epsilon _1,1}\) is positive. The proof is similar as the one of [9, Theorem B] in case (i) but for the sake of completeness we recall the main steps. Firstly
Then
where \(\tilde{L}=(p-1)k^2+p(\lambda +2)^2>0\). Put
After some computations we get, if \(k\ne -2\),
We choose \(S>2\) such that \(\frac{k}{k+2}-\frac{S}{2}=0\), hence \(\beta =\frac{2-k}{2(p-1)}\). If \(p<\frac{N+3}{N-1}\) the coefficient of \(S^2\) in \({\mathcal {T}}(S)\) is negative. Hence \({\mathcal {T}}(S)<0\) provided S is large enough which is satisfied if \(k<-2\) with \(\left| k+2\right| \) small enough. We infer from this that \(\beta >0\), \({\mathcal {D}}_0<0\) and \(\tilde{H}_{\epsilon _1,1}>0\) if \(\epsilon _1\) is small enough. In particular \(\tilde{H}_{\epsilon _1,1}(t)\ge c_6 (t^2+1)\) for some \(c_6=c_6(N,p,q)>0\), which means
Secondly the positivity of \(H_{M,2}\) is ensured, as \(\beta \) and M are positive, by the positivity of
Replacing s by its value, we obtain, since \(1<q<\frac{N+2}{N}\) and \(\beta +\frac{2+k}{2}>0\), which can be assume by taking \(\left| k+2\right| \) small enough,
Then we deduce that
and \(c_7=c_7(N,p,q)>0\) is independent of M. Since \(S=1-\frac{2\beta (p-1)}{k+2}=1-\frac{2-k}{k+2}=\frac{2k}{k+2}>0\), we have
From this we infer the inequality
Then we derive from Lemma 2.2 that in the ball \(B_R\) there holds
From this it follows
Setting \(a=-\frac{k+2}{2\beta }>0\) we get that for any domain \(\Omega \subset \mathbb {R}^N\) any positive solution in \(\Omega \) satisfies
The non existence of any positive of (1.2) solution in \(\mathbb {R}^N\) follows classically. \(\square \)
Corollary 3.1
Let \(\Omega \) be a smooth domain in \(\mathbb {R}^N\), \(N \ge 2\) with a bounded boundary, \(1<p<\frac{N+3}{N-1}\), \(1<q<\frac{N+2}{N}\) and \(M>0\). If u is a positive solution of (1.2) in \(\Omega \) there exists \(d_0\) depending on \(\Omega \) and \(c_9 = c_9(N,p,q) > 0\) such that
Proof
It is similar to the one of [9, Corollary B-2]. \(\square \)
4 The integral method
4.1 Preliminary inequalities
We recall the next inequality [6, Lemma 3.1].
Lemma 4.1
Let \(\Omega \subset \mathbb {R}^N\) be a domain. Then for any positive \(u\in C^2(\Omega )\), any nonnegative \(\eta \in C^\infty _0(\Omega )\) and any real numbers m and d such that \(d\ne m+2\), the following inequality holds
where
and
It is noticeable that d is a free parameter which plays a role only in the coefficients of the integral terms. The following technical result is useful to deal with the multi-parameter constraints problems which occur in our construction. It was first used in [7] under a simpler form and extended in [6, Lemma 3.4].
Lemma 4.2
For any \(N\in \mathbb {N}\), \(N\ge 3\) and \(1<p<\frac{N+2}{N-2}\) there exist real numbers m and d verifying
4.2 Proof of Theorem E
Step 1: The integral estimates. Let \(\eta \in C^{\infty }_0(\Omega )\), \(\eta \ge 0\). We apply Lemma 4.1 to a positive solution \(u\in C^2(\Omega )\) of (1.2), firstly with \(q>1\) and then with \(q=\frac{2p}{p+1}\).
We multiply (1.2) by \(\eta u^{m+p}\) and integrate over \(\Omega \). Then
We set
so that there holds
and
Eliminating V between (4.4) and (4.5), we get
where
Also
We fix now \(q=\frac{2p}{p+1}\), then
hence
and
Next we assume that \(|M|\le 1\). From (4.7), (4.9), it follows that
From now we fix m and d according Lemma 4.2. Therefore \(A>0\) by (4.2)-(iv) and \(B>0\) by combining (4.2)-(ii) and (4.2)-(iii). Furthermore \(B_0>0\) by (4.2)-(ii). Hence, from (4.7), (4.8) and (4.10) we derive, since \(\frac{N-1}{N}<1\) and \(m\le 0\) from (4.2)-(ii)
Plugging these estimates into (4.6) we infer
Since A and \(B_0\) are positive, there exists \(\mu _1\in (0,1)\) such that for any \(\left| M\right| <\mu _1\),
Set \(A_2=\min \{A_1,B_1\}\), then, and whatever is the sign of S,
In the sequel we denote by \(c_j\) some positive constants depending on N and p. Then
On the other hand, we have
Since
we deduce
Thus we derive from (4.13)
From this point we can use the method developed in [7, p 599] for proving the Harnack inequality satisfied by positive solutions of (1.3) in \(\Omega \). We set \(\eta =\xi ^\lambda \) with \(\xi \in C^\infty _0(\Omega )\) with value in [0, 1] and \(\lambda >4\). For \(\epsilon \in (0,1)\) we have by the Hölder–Young inequality
and
Hence
Let us denote by \(c_{4}X\) the right-hand side of (4.18). Combining (4.5), (4.16) and (4.18) we also get
and we finally obtain
Finally we estimate the different terms in X, using that \(m+p>0\) from (4.2)-(iii). For \(\epsilon >0\)
and
At end we obtain
Step 2: The Harnack inequality. We suppose that \(\Omega =B_R{\setminus }\{0\}:=B_R^*\), fix \(y\in B_{\frac{R}{2}}^*\), set \(r=|y|\), then \(B_r(y)\subset B_R^*\). Let \(\xi \in C^\infty _0(B_r(y))\) such that \(0\le \xi \le 1\), \(\xi =1\) in \(B_{\frac{r}{2}}(y)\), \(\left| \nabla \xi \right| \le cr^{-1}\) and \(\left| \Delta \xi \right| \le cr^{-2}\). We choose \(\lambda >\max \left\{ 4,\frac{m+2p}{p+1}\right\} \), then
and hence
We write (1.2) under the form
with
Set \(\sigma =\frac{m+2p}{p-1}\), then \(\sigma >\frac{N}{2}\) by (4.2)-(iii) and
Next we estimate G. For \(\tau ,\omega ,\gamma >0\) and \(\theta >1\), we have with \(\theta '=\frac{\theta }{\theta -1}\),
We fix
Then \(\omega >0\) and \(\theta >1\) from (4.2)-(iii), \(\omega >0\). Then \(u^{\omega \theta '}\left| \nabla u\right| ^{\gamma \theta }=u^{p+m-1}\left| \nabla u\right| ^{2}\) and \(u^{-\omega \theta }\left| \nabla u\right| ^{((q-1)\tau -\gamma )\theta '}=u^{m-2}\left| \nabla u\right| ^{4}\), thus
This implies
with \(\tau >N\). Using the results of [28, Section 5], we infer that a Harnack inequality, uniform with respect to r, is satisfied. Hence there exists \(c_{13}>0\) depending on N, p such that for any \(r\in (0,\frac{R}{2}]\) and y such that \(\left| y\right| =r\) there holds
which implies
and actually \(c_{14}=c_{13}^{7}\) by a simple geometric construction. By (4.24)
where \(\omega _N\) is the volume of the unit N-ball. This implies
The proof follows. \(\square \)
Remark
Using standard rescaling techniques (see e.g. [29, Lemma 3.3.2]) the gradient estimate holds
And the next estimate for a solution u in a domain \(\Omega \) satisfying the interior sphere condition with radius R is valid
5 Radial ground states
We recall that if \(q\ne \frac{2p}{p+1}\) and \(M\ne 0\), (1.2) can be reduced to the case \(M=\pm 1\) by using the transformation (1.15). Since any ground state u of (1.2) radial with respect to 0 is decreasing (this is classical and straightforward), it achieves its maximum at 0 and the following equivalence holds if v is defined by (1.15)
Hence large or small values of M for u are exchanged into large or small values of v(0) for v and in the sequel we will essentially express our results using the function u.
5.1 Energy functions
We consider first the energy function
Then
Hence, if \(M\le 0\), H is decreasing, a property often used in [25]. This implies in particular that a radial ground state satisfies
A similar estimate holds in all the cases.
Proposition 5.1
Let \(M>0\), \(p,q>1\). If u is a radial ground state solution of (1.2), then the function H defined in (5.2) is decreasing and in particular (5.3) holds.
Proof
Let u be such a radial ground state. By Proposition 2.1 we must have \(q>\frac{N}{N-1}\) and
this implies the claim. \(\square \)
5.1.1 Exponential perturbations
As we have seen it in the introduction, if \(q<\frac{2p}{p+1}\) Eq. (1.2) can be seen as a perturbation of the Lane–Emden equation (1.3) while if \(q>\frac{2p}{p+1}\) it can be seen as a perturbation of the Ricatti equation (1.14). Two types of transformations can emphasize these aspects.
(1) For \(p>1\) set
then
with
and
If \(q>\frac{2p}{p+1}\) (resp. \(q<\frac{2p}{p+1}\)), then \(\omega >0\) (resp. \(\omega <0\)) system (5.7) is a perturbation of the Lane–Emden system
at \(\infty \) (resp. \(-\infty \)). The following energy type function introduced in [20] is natural with (5.8)
and it satisfies
where \(L=N-2-\frac{4}{p-1}=K-\frac{2}{p-1}\). Relation (5.10) will be used later on.
(2) For \(p,q>1\) set
then
where
Note that if \(q<\frac{2p}{p+1}\) this system at \(\infty \) endows the form
It is therefore autonomous and much easier to study.
5.1.2 Pohozaev–Pucci–Serrin type functions
Let \(\alpha ,\gamma ,\theta ,\kappa \) be real parameters with \(\alpha ,\kappa >0\). Set
This type of function has been introduced in [25] in their study of Eq. (1.2) with \(M=1\) with specific parameters. We use it here to embrace all the values of M. We define \({\mathcal {U}}\) by the identity
Then
5.2 Some known results in the case \(M<0\)
We recall the results of [14, 25] and [23] relative to the case \(M<0\).
Theorem 5.2
-
1.
Let \(N\ge 3\) and \(1<p\le \frac{N}{N-2}\).
-
2
Assume \(\frac{N}{N-2}<p<\frac{N+2}{N-2}\) and let \(\overline{q}\) be the unique root in \((\frac{2p}{p+1},p)\) of the quadratic equation
$$\begin{aligned} (N-1)(X-p)^2-(N+2-(N-2)p)((p+1)X-2p)X=0. \end{aligned}$$-
2-(i)
If \(\overline{q}\le q<p\) there exists no ground state for any \(M<0\) [25, Theorem C].
-
2-(ii)
If \(\frac{2p}{p+1}<q<\overline{q}\), there exists no ground state for |M|. It is an open question whether there could exist a finite number of M for which there exists a ground state [25, Theorem 4].
-
2-(iii)
If \(1<q<\frac{2p}{p+1}\), there exists a ground state for large |M| [14, Proposition 5.7] and no ground state when |M| is small [23].
-
2-(i)
-
3
Assume \(p>\frac{N+2}{N-2}\) and \(q>1\) and let \(Q_{N,p}=\frac{2(N-1)p}{2N+p+1}\in (\frac{2p}{p+1},p)\).
-
3-(i)
If \(Q_{N,p}<q<p\) there exists a ground state for |M| small.
-
3-(ii)
If \(1<q\le Q_{N,p}\) there exists a ground state for any \(M<0\) [25, Theorem A].
-
3-(i)
-
4
Assume \(p=\frac{N+2}{N-2}\). There exists at least one \(M<0\) such that there exists a ground state if and only if \(1<q<p\). More precisely:
Remark
It is interesting to quote that when \(M<0\) and \(q\ge \frac{2p}{p+1}\), there holds [25, Theorem 3],
5.3 The case \(M>0\)
The next result is a consequence of Theorem A.
Theorem 5.3
Let \(M>0\), \(p>1\) and \(q>\frac{2p}{p+1}\) then there exists no radial ground state satisfying \(u(0)=1\) when M is large.
Proof
Suppose that such a solution u exists. From Theorem A and Proposition 2.1 there holds
As a consequence, if \(r>R>0\),
with \(c'_{N,p}=\frac{p-1}{2}c_{N,p}\). Since \(u(r)\rightarrow 0\) when \(r\rightarrow \infty \), we take \(R=|M|^{\frac{p-1}{(p+1)q-2p}}\) and derive
and the conclusion follows. \(\square \)
Remark
If we use Proposition 5.1 we can make estimate (5.19) more precise.
5.3.1 The case \(M>0\), \(1<p\le \frac{N+2}{N-2}\)
It is a consequence of our general results that there is no radial ground state for large M or for small M when \(1<q\le \frac{2p}{p+1}\) and \(1<p<\frac{N+2}{N-2}\). Indeed, if \(1<q<\frac{2p}{p+1}\) is a consequence of the equivalence statement between a priori estimate and non-existence of ground state proved in [23], and if \(q= \frac{2p}{p+1}\) it follows from Theorems C and E. Actually in the radial case, the result is more general.
Theorem 5.4
Let \(M>0\) and \(1< p<\frac{N+2}{N-2}\). If \(1<q\le p\), there exists no radial ground state for any M. If \(q>p\) there exists no radial ground state for M small enough.
Proof
By Proposition 2.1, we may assume \(N\ge 3\) and
(i) Assume first \(q< \frac{2p}{p+1}\). We use the system (5.5). Then \(\omega \), defined by (5.7) is negative. Hence the Leighton function \({\mathcal {N}}\) defined by (5.9) is nonincreasing since \(L\le 0\) when \(p\le \frac{N+2}{N-2}\). Furthermore since \((x(t),y(t))\rightarrow (0,0)\) when \(t\rightarrow -\infty \) and \(e^{-\omega t}\rightarrow 0\), we get \({\mathcal {N}}(-\infty )=0\) it follows that \({\mathcal {N}}(t)<0\) for \(t\in \mathbb {R}\). Moreover, by Proposition 2.1,
This implies \(e^{-\omega t}x^{q+1}(t)=O(e^{2\frac{q(p+1)-2p}{(p-1)(q-1)}t})=o(1)\) as \(t\rightarrow \infty \) and \({\mathcal {N}}(\infty )=0\), contradiction.
(ii) Assume next \(\frac{2p}{p+1}\le q\le p\). We consider the function (5.15) with the parameters
already used by [25] when \(M=-1\), and we get with \({\mathcal {U}}\) defined by (5.16),
where
By our assumptions \(A\ge 0\), \(B\ge 0\) and \(C>0\). Hence \({\mathcal {U}}>0\). This implies
since \({\mathcal {Z}}(0)=0\). If u is a ground state, then \(u'(r)\rightarrow 0\) as \(r\rightarrow \infty \), thus \(u\left| u'\right| ^q\le u\left| u'\right| ^{\frac{2p}{p+1}}\). Hence, from Proposition 2.1, \(u'^2(r)=O(r^{-2\frac{p+1}{p-1}})\) as \(r\rightarrow \infty \). The other terms \(u^{p+1}(r)\), \(r^{-1}u(r)u'(r)\) and \(u\left| u'\right| ^{\frac{2p}{p+1}}\) satisfy the same bound, hence
Then \({\mathcal {Z}}(r)\rightarrow 0\) when \(r\rightarrow \infty \), contradiction.
(iii) Suppose \(q>p\) and u is a ground state. By Proposition 5.1 and (5.18), there holds
Then \(\chi =\frac{p+3}{2+q(p+1)}r\left| u'\right| ^{q-1}\le c_{N,p}\). Hence, if \(M\le M_{N,p}\) for some \(M_{N,p}>0\), \({\mathcal {U}}\) is positive as A is. We conclude as above. \(\square \)
5.3.2 The case \(M>0\), \(p>\frac{N+2}{N-2}\)
We recall that in Theorem C if \(q=\frac{2p}{p+1}\) and \(p>1\) there is no ground state whenever \(M>M_{N,p}\), see (1.26). In Theorem A\('\) if \(1<q<\frac{2p}{p+1}\) and \(p>1\) there is no ground state u such that \(u(0)=1\) if M is too large. In the next result we complement Theorem 5.3 for small value of M in assuming \(q>\frac{2p}{p+1}\).
Theorem 5.5
If \(p>\frac{N+2}{N-2}\) and \(q\ge \frac{2p}{p+1}\) then there exist radial ground states for \(M>0\) small enough.
Proof
First we consider the function \({\mathcal {Z}}\) with \(k=N\) and obtain
The function vanishes at the origin. We compute \({\mathcal {U}}\) from the identity \({\mathcal {Z}}'+\theta \left| u'\right| ^{q-1}{\mathcal {Z}}=r^{N-1}{\mathcal {U}}\) and get
If \(\gamma =0\) and \(\theta =-2M\), then
If u is a regular solution which vanishes at some \(r_0>0\), then \({\mathcal {Z}}(r_0)=2^{-1}r_0^2u'^N(r_0)>0\). As \(p>\frac{N+2}{N-2}\), by choosing \(\alpha =\frac{1}{2}\left( \frac{N}{p+1}+\frac{N-2}{2}\right) \) we have \(\frac{N}{p+1}<\alpha <\frac{N-2}{2}\). We define \(\ell >0\) by \((N-2)p-(N+2)=4(p+1)\ell \), then \(\frac{N-2}{2}-\alpha =\alpha -\frac{N}{p+1}=\ell \) and then
Assume first \(q<2\), we have from Hölder’s inequality and \(0<r\le r_0\) where u is positive
and
since \(q\ge \frac{2p}{p+1}\) and \(u\le u(0)=1\). Taking \(M\le \frac{\ell }{\alpha }=\frac{(N-2)p-N-2}{(N-2)p+3N-2}\), \({\mathcal {U}}\) is negative implying that \(r\mapsto e^{-2M\int _0^r|u'|^{q-1}ds}{\mathcal {Z}}(r)\) is nonincreasing. Since \({\mathcal {Z}}(0)=0\), \({\mathcal {Z}}(r)\le 0\), a contradiction.
If \(q=2\), then \({\mathcal {U}}\le -\ell (u'^2+u^{p+1})+M\alpha u'^2\) since \(u\le 1\) on \([0,r_0]\). We still infer that \({\mathcal {U}}\le 0\) if we choose \(M\le \frac{\ell }{\alpha }\).
Finally, if \(q>2\), we have from Theorem A, \(u'\le C_{N,p,q} M^{-\frac{p+1}{(p+1)q-2p}}\). Therefore, using again the decay of u from \(u(0)=1\),
Hence \({\mathcal {U}}\le -\left( \ell -\alpha C^{q-2}_{N,p,q}M^{\frac{2}{(p+1)q-2p}}\right) u'^2\). Taking
we conclude that \({\mathcal {U}}<0\) which ends the proof as in the previous cases. \(\square \)
6 Separable solutions
We denote by \((r,\sigma )\in \mathbb {R}_+\times S^{N-1}\) the spherical coordinates in \(\mathbb {R}^N\). Then Eq. (1.2) takes the form
where \(\Delta '\) is the Laplace–Beltrami operator on \(S^{N-1}\) and \(\nabla '\) the tangential gradient. If we look for separable nonnegative solutions of (1.2) i.e. solutions under the form \(u(r,\sigma )=\psi (r)\omega (\sigma )\), then \(q=\frac{2p}{p+1}\), \(\psi (r)=r^{-\frac{2}{p-1}}\), and \(\omega \) is a solution of
where K is defined in (5.6). Throughout this section we assume
6.1 Constant solutions
The constant function \(\omega =X\) is a solution of (6.2) if
For \(N=1, 2\) and \(p>1\) or \(N\ge 3\) and \(1<p<\frac{N}{N-2}\), we recall that \(\mu ^*=\mu ^*(N)\) has been defined in (1.24). The following result is easy to prove
Proposition 6.1
-
1.
Let \(M\ge 0\) then there exists a unique positive root \(X_M\) to (6.4) if and only if \(p>\frac{N}{N-2}\). Moreover the mapping \(M\mapsto X_M\) is continuous and decreasing from \([0,\infty )\) onto \((0,\left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}}]\).
-
2.
Let \(M<0\), \(N\ge 3\) and \(p\ge \frac{N}{N-2}\) then there exists a unique positive root \(X_M\) to (6.4) and the mapping \(M\mapsto X_M\) is continuous and decreasing from \((-\infty ,0]\) onto \([\left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}},\infty )\).
-
3.
Let \(M<0\), \(N=1,2\) and \(p>1\) or \(N\ge 3\) and \(1<p<\frac{N}{N-2}\) then there exists no positive root to (6.4) if \(-\mu ^*<M\le 0\). If \(M=M^*:=-\mu ^*\) there exists a unique positive root \(X_{M^*}=\left( \frac{2|K|}{p(p-1)}\right) ^{\frac{1}{p-1}}\). If \(M<-\mu ^*\) there exist two positive roots \(X_{1,M}<X_{2,M}\). The mapping \(M\mapsto X_{1,M}\) is continuous and increasing from \((-\infty ,\mu ^*]\) onto \((0,X_{M^*}]\). The mapping \(M\mapsto X_{2,M}\) is continuous and decreasing from \((-\infty ,\mu ^*]\) onto \([X_{M^*},\infty )\).
Abridged proof
Set
then \(f'_M(X)=(p-1)X^{p-2}+M\frac{p-1}{p+1}\left( {\displaystyle \frac{2}{p-1} } \right) ^{\frac{2p}{p+1}}X^{-\frac{2}{p+1}}\).
-
1.
If M is nonnegative, \(f_M\) is increasing from \(-\frac{2K}{p-1}=-\frac{2[(N-2)p-N]}{(p-1)^2}\) to \(\infty \); hence, if \(p>\frac{N}{N-2}\) there exists a unique \(X_M> 0\) such that \(f_M(X_M)=0\), while if \(1<p<\frac{N}{N-2}\), \(f_M\) admits no zero on \([0,\infty )\). Since \(f_M>f_{M'}\) for \(M>M'>0\), there holds \(X_M>X_{M'}\), By the implicit function theorem the mapping \(M\mapsto X_M\) is \(C^1\) and decreasing from \([0,\infty )\) onto \((0,\left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}}]\). Actually it can be proved that (see [10, Proposition 2.2])
$$\begin{aligned} X_M=\frac{p-1}{2}\left( \frac{K}{M}\right) ^{\frac{p+1}{p-1}}(1+o(1))\quad \text {as }\;M\rightarrow \infty . \end{aligned}$$(6.6) -
2.
If M is negative, \(f_M\) achieves it minimum on \([0,\infty )\) at \(X_0=\left( \frac{-M}{p+1}\right) ^\frac{p+1}{p(p-1)}\left( \frac{2}{p-1}\right) ^\frac{2}{p-1}\), and
$$\begin{aligned} f_M(X_0)= & {} -{\displaystyle \frac{p}{\left( p+1\right) ^{\frac{p+1}{p}}} }\left( {\displaystyle \frac{2}{p-1} }\right) ^{2}(-M)^{\frac{p+1}{p}}-{\displaystyle \frac{2K}{p-1} }\\= & {} -\left( {\displaystyle \frac{2}{p-1} }\right) ^{2}\left( {\displaystyle \frac{p}{\left( p+1\right) ^{\frac{p+1}{p}}} }(-M)^{\frac{p+1}{p}}+{\displaystyle \frac{(N-2)p-N}{2} }\right) . \end{aligned}$$Since \(K>0\), there exists a unique \(X_M>0\) such that \(f_M(X_M)=0\) and \(X_M>X_0\). The mapping \(M\mapsto X_M\) is \(C^1\) and decreasing from \((-\infty , 0]\) onto \([\left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}},\infty )\). The following estimate holds
$$\begin{aligned}&\max \left\{ \left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}}, \left( \frac{2}{p-1}\right) ^{\frac{2}{p-1}}|M|^{\frac{p+1}{p(p-1)}}\right\} \le X_M\nonumber \\&\quad \le 2^{\frac{2}{p-1}}\left( \left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}} +\left( \frac{2}{p-1}\right) ^{\frac{2}{p-1}}|M|^{\frac{p+1}{p(p-1)}}\right) . \end{aligned}$$(6.7) -
3.
If \(N=1,2\) and \(p>1\) or \(N\ge 3\) and \(1<p<\frac{N}{N-2}\), then \(f_M(0)>0\). Hence, if \(f_M(X_0)>0\) there exists no positive root to \(f_M(X)=0\). Equivalently, if \(-\mu ^*<M<0\). If \(f_M(X_0)=0\), \(X_0\) is a double root and this is possible only if \(M=-\mu ^*\), hence \(X_{-\mu ^*}=\left( \frac{2|K|}{p(p-1)}\right) ^{\frac{1}{p-1}}\). If \(f_M(X)<0\), or equivalently, if \(M<-\mu ^*\), the equation \(f_M(X)=0\) admits two positive roots \(X_{1,M}<X_0<X_{2,M}\). The monotonicity of the \(X_{j,M}\), j=1,2, and their range follows easily from the monotonicity of \(M\mapsto f_M(X)\) for \(M<0\). Actually the following asymptotics hold when \(M\rightarrow -\infty \),
$$\begin{aligned} X_{1,M}=\frac{p-1}{2}\left( \frac{K}{M}\right) ^{\frac{p+1}{p-1}}(1+o(1))\; \text {and }\;\, X_{2,M}=\left( \frac{2}{p-1}\right) ^\frac{2}{p-1}\left( -M\right) ^{\frac{p+1}{p(p-1)}} (1+o(1)). \end{aligned}$$(6.8)\(\square \)
6.2 Bifurcations
We set
If \(\eta \in C^\infty (S^{N-1})\) and if there exists a constant positive solution X to \(A(X)=0\) we have
Hence the problem is singular if
where \(\lambda _k=k(k+N-2)\) is the k-th nonzero eigenvalue of \(-\Delta '\) in \(H^1(S^{N-1})\). The following result follows classically from the standard bifurcation theorem from a simple eigenvalue (which can always be assumed if we consider functions depending only on the azimuthal angle on \(S^{N-1}\) reducing the eigenvalue problem to a simple Legendre type ordinary differential equation) see e.g. [26, Chapter 13] and identity (6.4).
Theorem 6.2
Let \(M_0\in \mathbb {R}\) and \(X_{M_0}\) be a constant solution of (6.2). If \(X_{M_0}\) satisfies for some \(k\in \mathbb {N}^*\),
there exists a continuous branch of nonconstant positive solutions \((M,\omega _M)\) of (6.2) bifurcating from the \(({M_0},X_{M_0})\).
Since \(M\left( {\displaystyle \frac{2}{p-1} }\right) ^{\frac{2p}{p+1}}X_M^{\frac{p-1}{p+1}}={\displaystyle \frac{2K}{p-1} }-X_M^{p-1}\) by (6.4) the following statements follow immediately from Proposition 6.1.
Lemma 6.3
Set \(\Phi (M)=M\left( \frac{2}{p-1}\right) ^{\frac{2p}{p+1}}X_M^{\frac{p-1}{p+1}}\) when \(X_M\) is uniquely determined, and \(\Phi _j(M)=M\left( \frac{2}{p-1}\right) ^{\frac{2p}{p+1}}X_{j,M}^{\frac{p-1}{p+1}}\), j=1,2, if there exist two equilibria. Then
-
1.
If \(N\ge 3\) and \(p>\frac{N}{N-2}\), the mapping \(M\mapsto \Phi (M)\) is continuous and increasing from \([0,\infty )\) onto \([0,\frac{2K}{p-1})\).
-
2.
If \(N\ge 3\) and \(p\ge \frac{N}{N-2}\), the mapping \(M\mapsto \Phi (M)\) is continuous and increasing from \((-\infty ,0]\) onto \((-\infty ,0]\).
-
3.
If \(N=1,2\) and \(p>1\) or \(N\ge 3\) and \(1<p<\frac{N}{N-2}\), the mapping \(M\mapsto \Phi _1(M)\) (resp \(M\mapsto \Phi _2(M))\) is continuous and decreasing (resp. increasing) from \((-\infty ,-\mu ^*]\) onto \([\frac{2K}{p-1},0)\) (resp. \((-\infty ,\frac{2K}{p-1}])\).
If we analyse the range \(R[\Phi ]\) of \(\Phi \) or \(R[\Phi _j]\) of \(\Phi _j\), we prove the following result.
Theorem 6.4
-
1
Let \(N\ge 3\) and \(p\ge \frac{N}{N-2}\).
-
1-(i)
There exists a continuous curve of bifurcation \((M,\omega _M)\) issued from \((M_0,X_{M_0})\) for some \(M_0=M_0(p)\ge 0\) if and only if \(p\ge \frac{N+1}{N-3}\) and \(k=1\). Furthermore \(M_0(\frac{N+1}{N-3})=0\).
-
1-(ii)
The bifurcation curve \(s\mapsto (M(s),\omega _{M(s)})\), is defined on \((-\epsilon _0,\epsilon _0)\) for some \(\epsilon _0>0\) and verifies \((M(0),\omega _{M(0)})=(M_0,X_{M_0})\).
-
1-(i)
-
2.
Let \(N\ge 3\) and \(p\ge \frac{N}{N-2}\).
-
2-(i)
For any \(k\ge 1\) there exist \(M_k<0\) and a continuous branch of bifurcation \((M,\omega _M)\) issued from \((M_k,X_{M_k})\), with the restriction that \(p< \frac{N+1}{N-3}\) if \(k=1\).
-
2-(ii)
The bifurcation curve \(s\mapsto (M(s),\omega _{M(s)})\), is defined on \((-\epsilon _0,\epsilon _0)\) for some \(\epsilon _0>0\) and verifies \((M(0),\omega _{M(0)})=(M_0,X_{M_0})\). Finally \(M_k\rightarrow -\infty \) when \(k\rightarrow \infty \).
-
2-(i)
-
3.
let \(N=1,2\) and \(p>1\), or \(N\ge 3\) and \(1<p<\frac{N}{N-2}\).
-
3-(i)
There exists no \(M<0\) such that \(\frac{2K}{p-1}<\Phi _1 (M)<0\), and a countable set of \(M_k<0\), \(k\ge 1\), such that \(\Phi _2(M_k)=\frac{p+1}{p(p-1)}\left( 2K-\lambda _k\right) \).
-
3-(ii)
There exist a countable branches of bifurcation of solutions \((M_k(s),\omega _{M_k(s)})\) issued from \((M_k, X_{2,M_k})\).
-
3-(i)
Proof
Assertion 1. Since from Lemma 6.3, \(R[\Phi ]=[0,\frac{2K}{p-1})\) for \(M\ge 0\), we have to see under what condition on \(p\ge \frac{N}{N-2}\) one can find \(k\ge 1\) such that
Since \(K<N\) and \(\lambda _k\ge 2N\) for \(k\ge 2\), the only possibility for this last inequality to hold is \(k=1\). The inequality \(\frac{2K}{p+1} <N-1\) always holds since \(p>1\), while the inequality \(N-1=\lambda _1\le 2K\) is equivalent to \(p\ge \frac{N+1}{N-3}\). Therefore \(M_0=0\) and \(X_{M_0}=\left( \frac{2K}{p-1}\right) ^{\frac{1}{p-1}}\). If we consider only functions on the sphere \(S^{N-1}\) which depend uniquely on the azimuthal angle \(\theta =\tan ^{-1}(x_N\lfloor _{S^{N-1}})\), the function \(\psi _1(\sigma )= x_N\lfloor _{S^{N-1}}\) is a eigenfunction of \(-\Delta '\) in \(H^1(S^{N-1})\) with multiplicity one. Hence the bifurcation branch is locally a regular curve \(s\mapsto (M(s),\omega _{M(s)})\) with \(0\le s<\epsilon '_0\) and we construct the bifurcating solution on \(S^{N-1}\) using the classical tangency condition [26, Theorem 13.5],
where \(\zeta _s\in H^{1}(S^{N-1})\), is orthogonal to \(\psi _1\) in \(H^{1}(S^{N-1})\) and satisfies \(\left\| \zeta _s\right\| _{C^1}=o(1)\) when \(s\rightarrow 0\). This implies the claim.
Assertion 2. Since \(R[\Phi ]=(-\infty ,0)\) for \(M< 0\), we have to find \(k\ge 1\) such that
As in Case 1, \(K<2N\), then inequality \(2K\le \lambda _k\) holds for all \(k\ge 2\), and if \(k=1\) this is possible only if \(p< \frac{N+1}{N-3}\). The construction of the bifurcating curve is the same as in Case 1.
Assertion 3. We have \(R[\Phi _1]=[\frac{2K}{p-1},0)\) for \(M\le -\mu ^*\). If we look for the existence of some \(k\ge 1\) such that
we get an impossibility since \(K<0\). Hence there exists no \(M_0<0\) such that \((M_0,X_{1,M_0})\) is a bifurcation point. We have also \(R[\Phi _2]=(-\infty ,\frac{2K}{p-1}]\) for \(M\le -\mu ^*\). Now the condition for the existence of a bifurcation branch issued from \((M_0,X_{2,M_0})\) for some \(M_0\le -\mu ^*\) is
which is always true for any \(k\ge 1\) and \(1<p<\frac{N}{N-2}\). \(\square \)
Remark
The exponent \(p=\frac{N+1}{N-3}\) is the Sobolev critical exponent on \(S^{N-1}\). If one consider the Lane–Emden equation with a Leray potential
with \(\lambda \in \mathbb {R}\), then the separable solutions \(u(r,\sigma )=r^{-\frac{N-3}{2}}\omega (\sigma )\) verify
It was observed in [7] that there exists a branch of bifurcation \((\lambda ,\omega _\lambda )\) with \(\lambda >0\) issued from \((0,\omega _0)\), where \( \omega _0\) is the constant explicit solution of (6.14).
Remark
In Theorem 6.4-1 and the above remark, we conjectured that on the bifurcating curve there holds locally \(M(s)<M_0\), and that for any \(p\ge \frac{N+1}{N-3}\) there exists \(M_0:=M_0(p)\) such that for \(M> M_0\) all the positive solutions to (6.2) are constant, furthermore \(M_0\) is defined by (6.11). When \(p=\frac{N+1}{N-3}\), then \(M=0\) and there exists infinitely many positive solutions to (6.2) [7, Proposition 5.1]. When \(\frac{N}{N-2}<p< \frac{N+1}{N-3}\), it is unclear if the branches of bifurcation \((M(s),\omega _{M(s)})\) issued from \((M_0,\omega _{M_0})\) with \(M_0<0\) are such that M(s) keeps a constant sign. If it is the case one could expect that if \(M\ge 0\) and \(\frac{N}{N-2}<p< \frac{N+1}{N-3}\), all the positive solutions to (6.2) are constant.
The following statement is an immediate consequence of Theorem 6.4.
Corollary 6.5
-
1.
If \(p>1\) and \(q=\frac{2p}{p+1}\) there always exist nonradial positive singular solutions of (1.2) in \(\mathbb {R}^N{\setminus }\{0\}\) under the form \(u(r,\sigma )=r^{-\frac{2}{p-1}}\omega (\sigma )\).
-
2.
If \(N\ge 4\) and \(p>\frac{N+1}{N-3}\), the functions are obtained by bifurcation from \(X_M\) with \(M>0\).
-
3.
If \(N\ge 3\) and \(\frac{N}{N-2}\le p<\frac{N+1}{N-3}\), the functions are obtained by bifurcation from \(X_M\) with \(M<0\).
-
4.
If \(N=1,2\) and \(p>1\) or \(N\ge 3\) and \(1<p<\frac{N}{N-2}\), the functions are obtained by bifurcation from \((M_k, X_{2,M_k})\) with \(M_k<-\mu ^*\) and \(k\ge 1\).
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This article has been prepared with the support of the collaboration programs ECOS C14E08 and FONDECYT Grant 1160540 for the three authors.
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Bidaut-Véron, MF., Garcia-Huidobro, M. & Véron, L. A priori estimates for elliptic equations with reaction terms involving the function and its gradient. Math. Ann. 378, 13–56 (2020). https://doi.org/10.1007/s00208-019-01872-x
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DOI: https://doi.org/10.1007/s00208-019-01872-x