Abstract
In this paper we prove some integral estimates on the minimal growth of the positive part \(u_+\) of subsolutions of quasilinear equations
on complete Riemannian manifolds M, in the non-trivial case \(u_+\not \equiv 0\). Here A satisfies the structural assumption \(|A(x,u,\nabla u)|^{p/(p-1)} \le k \langle A(x,u,\nabla u),\nabla u\rangle \) for some constant \(k>0\) and for \(p>1\) the same exponent appearing on the RHS of the equation, and V is a continuous positive function, possibly decaying at a controlled rate at infinity. We underline that the equation may be degenerate and that our arguments do not require any geometric assumption on M beyond completeness of the metric. From these results we also deduce a Liouville-type theorem for sufficiently slowly growing solutions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In the recent paper [4] (Lemma 8), the following theorem was established. Let M be a complete Riemannian manifold (without boundary), \(\lambda >0\) a constant and \(u\in C^2(M)\). If the superlevel set \(\Omega _+:= \{x\in M: u(x)>0\}\) is not empty and u satisfies
then for any fixed point \(x_0\in M\) we have
where \(u_+:= \max \{u,0\}\) is the positive part of u and \(B_R(x_0)\) is the geodesic ball of radius R centered at \(x_0\). Indeed, inspection of the proof also shows that there exists a constant \(C(\lambda )\), not depending on M or u, such that
and that the optimal value for \(C(\lambda )\) is not smaller than \(\frac{\log 2}{4}\sqrt{\lambda }\). This can be regarded as a sort of “gap” theorem for subsolutions of \(\Delta u = \lambda u\): if \(u\in C^2(M)\) satisfies
then either \(u\le 0\) or the positive part of u has to be sufficiently large in an integral sense (that is, its \(L^2\) norm on \(B_R(x_0)\) must grow at least exponentially with respect to R). In fact, the result from [4] is more general and also covers the case of weighted Laplacians and locally Lipschitz weak solutions of (1).
In this paper we generalize the above theorem by considering differential inequalities for a wider class of (possibly degenerate) quasilinear elliptic operators in divergence form, including the p-Laplace operator
and also replacing the constant \(\lambda \) by a positive continuous function V possibly decaying at infinity at a controlled rate, namely, not faster than a negative power \(r(x)^{-\mu }\), \(\mu >0\), of the distance \(r(x)=\textrm{dist}(x,o)\) from some fixed point \(o\in M\). More precisely, for a given pair of parameters \(\lambda >0\) and \(\mu \ge 0\) we shall assume that
These conditions are clearly satisfied, for instance, if
Also, in case \(\mu >0\) the triangle inequality implies that the validity of (\(V_{\lambda ,\mu }\)) does not depend on the choice of the reference base point \(o\in M\).
To give an example of our main result, we state it in the model case of the p-Laplace operator. To do so, we have to precise some terminology. For a function \(u\in W^{1,p}_\textrm{loc}(M)\), we denote by \(\Omega _+:= \{x\in M: u(x)>0\}\) its positivity set and for a given measurable function \(V\ge 0\) we say that u satisfies
if
where
(Note that if \(|\Omega _+|>0\) then the space \(D^+(\Omega _+)\) of test functions is non-trivial because it contains at least elements of the form \(\varphi = u_+\psi \), with \(0\le \psi \in C^\infty _c(M)\), so (4) is a meaningful condition.) In particular, (4) is always satisfied if
or even if
since \(\nabla u_+ = \textbf{1}_{\Omega _+} \nabla u\) almost everywhere on M. Note that (6) is a weaker condition than (5), as follows from work of Le, [7].
Theorem 1
Let M be a complete Riemannian manifold, \(p\in (1,+\infty )\), \(\mu \in [0,p]\), \(\lambda >0\), and \(V: M \rightarrow (0,+\infty )\) a continuous function satisfying (\(V_{\lambda ,\mu }\)).
Let \(u\in W^{1,p}_\textrm{loc}(M)\). If \(\Omega _+:= \{x\in M: u(x)>0\}\) is of positive measure and
then for any \(x_0\in M\) and \(q\in (p-1,+\infty )\) we have
where \(C_0\) and \(C_1\) are explicitely given by
where \(p'=\frac{p}{p-1}\) is the exponent conjugate to p. Moreover, in case \(\mu =p\) we have
whenever the limit on the LHS exists.
Remark 2
Note that the value \(C_1>p\) determined by (9) satisfies \(C_1<C_0+p\), hence (10) gives a stronger estimate than (8) when its LHS is well defined.
The constants appearing in (7) and (10) are sharp, that is, for each combination of values of p, \(\mu \), \(\lambda \) and q it is possible to find M and u for which the equality in (7) or (10) is attained. This is shown by explicit examples described at the end of Sect. 3. We don’t know whether the value of \(C_1>p\) in (9) is sharp or not for the validity of (8). It seems worth to underline that the case \(p=2\), \(q=2\), \(\mu =0\) in the above theorem implies that the optimal value for \(C(\lambda )\) in (3) is \(C(\lambda )=2\sqrt{\lambda }\).
Theorem 3
Let M be a complete Riemannian manifold, \(\mu \in [0,2]\), \(\lambda >0\) and \(V: M \rightarrow (0,+\infty )\) a continuous function satisfying (\(V_{\lambda ,\mu }\)).
Let \(u\in W^{1,2}_\textrm{loc}(M)\). If \(\Omega _+:= \{x\in M: u(x) > 0\}\) is of positive measure and
then for any \(x_0\in M\) and \(q\in (1,+\infty )\) we have
and in case \(\mu =2\)
provided the limit exists.
In full generality, in our main theorem we deal with differential inequalities involving quasilinear differential operators L formally defined by
where \(A: {\mathbb {R}}\times TM \rightarrow TM\) is a continuous function (or, more generally, a Carathéodory-type function as specified in Sect. 2) satisfying
for all \(x\in M\), \(s\in {\mathbb {R}}\), \(\xi \in T_x M\) with some constant \(k>0\). If these conditions are satisfied, we say that the differential operator L defined by (11) is weakly p-coercive with coercivity constant k. The p-Laplace operator falls in this class since it can be expressed as in (11) for the choice \(A(x,s,\xi ) = |\xi |^{p-2}\xi \), which fulfills (12) with \(k=1\). In analogy with what we did above, we say that a function \(u\in W^{1,p}_\textrm{loc}(M)\) satisfies
if
Theorem 4
Let M be a complete Riemannian manifold, \(p\in (1,+\infty )\), \(\mu \in [0,p]\) and \(\lambda >0\). Let L be a weakly p-coercive operator as in (11) with coercivity constant \(k>0\) and \(V: M \rightarrow (0,+\infty )\) a continuous function satisfying (\(V_{\lambda ,\mu }\)).
Let \(u\in W^{1,p}_\textrm{loc}(M)\). If \(\Omega _+:= \{ x\in M: u(x) > 0 \}\) is of positive measure and
then for any \(x_0\in M\) and \(q\in (p-1,+\infty )\) we have
where \(C_0>0\) and \(C_1>p\) are determined by
with \(p'=\frac{p}{p-1}\). Moreover, in case \(\mu =p\) we have
whenever the limit exists.
We point out that the RHS’s of (14) and (15) both converge to p from above as \(\lambda \rightarrow 0^+\). Hence, if \(u\in W^{1,p}_\textrm{loc}(M)\) satisfies
with \(|\Omega _+|\ne 0\) and V a continuous positive function decaying to 0 faster than \(r(x)^{-p}\) as \(x\rightarrow \infty \), then on arbitrary manifolds we couldn’t expect the possible validity of an estimate stronger than
In fact, we are able to prove a weaker growth estimate (with \(\liminf \) replaced by \(\limsup \)) holds more generally for any \(u\in W^{1,p}_\textrm{loc}(M)\) satisfying
for some measurable function \(f:M\rightarrow [0,+\infty ]\) such that \(f>0\) on a set \(E\subseteq \Omega _+\) of positive measure. Of course, by (16) we mean that
Note that if (16) holds with f as above then there exists \(\varphi \in D^+(\Omega _+)\) for which the LHS of (17) is strictly positive (this follows by considering a test function of the form \(\varphi = u_+\psi \) for some \(0\le \psi \in C^\infty _c(M)\) strictly positive on a portion of E of positive measure), and then it must also be \(A(x,u,\nabla u)\ne 0\) on a subset \(E_0\subseteq \Omega _+\) of positive measure.
Theorem 5
Let M be a complete Riemannian manifold, \(p\in (1,+\infty )\), L a weakly p-coercive operator as in (11) and \(u\in W^{1,p}_\textrm{loc}(M)\) such that \(\Omega _+:= \{x\in M: u(x)>0\}\) has positive measure. If u satisfies
and further
then for any \(q\in (p-1,+\infty )\)
In particular,
As said, (18) holds if u satisfies (16) for some measurable \(f:M\rightarrow [0,+\infty ]\) with f not a.e. vanishing on \(\Omega _+\). Alternatively, (18) is satisfied also when u is not constant on M and positive somewhere (so that \(|\Omega _+|>0\)) and A obeys the following mild non-degeneracy condition:
Theorem 5 is a consequence of the next Theorem 6, proved in the last part of the paper where we extend some arguments from [8] to general weakly p-coercive operators L of the form (11).
Theorem 6
Let M be a complete, non-compact Riemannian manifold, \(p\in (1,+\infty )\), L a weakly p-coercive operator as in (11) and \(u\in W^{1,p}_\textrm{loc}(M)\). If \(\{u>0\}\) has positive measure, u satisfies
and for some \(q>p-1\) it holds
then \(A(x,u,\nabla u) = 0\) almost everywhere on \(\{u>0\}\). In particular, if the structural condition (20) holds, then u is constant on M.
We remark that condition (22) amounts to saying that the function \(\varphi : (0,+\infty ) \rightarrow [0,+\infty ]\) given by
is not in \(L^1((r,+\infty ))\) for any \(r>0\). In fact, as proved in Lemma 19 below, in the assumptions of Theorem 6 there exists \(r_0\ge 0\) such that \(\varphi \) is finite a.e. on \((r_0,+\infty )\) and \(\varphi \in L^1((r,R))\) for any \(r_0<r<R<+\infty \), so that (22) is satisfied if and only if \(\varphi \) is not integrable in a neighborhood of \(+\infty \). Note that in general \(\varphi \) may be integrable at \(+\infty \) and still satisfy \(\varphi =+\infty \) on \((0,r_0)\) for some \(r_0>0\). For instance, for fixed \(n\in {\mathbb {N}}\) and \(p>n\), the function
satisfies \(\Delta _p u = 0\) on \(\Omega _+ = {\mathbb {R}}^n{\setminus }\overline{B_1}\), and for any \(q>p-1\)
(with \(C=|\partial B_1|\)) is integrable at \(+\infty \): indeed,
and (under the assumption \(p>n\)) we have \(-\frac{(n-1)(p-1)+q(p-n)}{(p-1)^2}<-1\) if and only if \(q>p-1\). This shows that the clause “\(\forall \,r>0\)” in (22) cannot in general be replaced by “for some \(r>0\)”.
Note that (22) is a condition about the growth of the integral of \(u_+^q\) on geodesic spheres \(\partial B_s\). This can be related to the growth of the integral of \(u_+^q\) on balls \(B_s\). More precisely, (22) is implied (see Proposition 1.3 in [8]) by the stronger condition
which in turn is satisfied, for instance, when
Since this last condition is exactly the negation of condition (19) above, Theorem 5 follows at once from Theorem 6.
As hinted at the beginning of this Introduction, our main Theorem 4 can be also interpreted as a “gap” theorem for functions \(u\in W^{1,p}_\textrm{loc}(M)\) satisfying
Namely, if u satisfies the above differential inequality, then either \(u\le 0\) a.e. on M or the positive part of u must grow sufficiently fast. As an easy consequence we have the following Liouville-type result (for its proof it is enough to apply Theorem 4 to both u and \(-u\)). For the sake of simplicity, we only state it in case V is a positive constant, but the interested reader can immediately generalize it to the case where V is a function satisfying (\(V_{\lambda ,\mu }\)) for some \(\lambda >0\) and \(\mu \in [0,p]\).
Theorem 7
Let M be a complete Riemannian manifold, \(p\in (1,+\infty )\), \(\lambda >0\) and L a weakly p-coercive operator as in (11) with coercivity constant \(k>0\). Let \(u\in W^{1,p}_\textrm{loc}(M)\) satisfy
If for some \(x_0\in M\) and \(q\in (p-1,+\infty )\)
for some constant \(C<\frac{p(q-p+1)^{1/p'}}{(p-1)^{1/p'}} \frac{\lambda ^{1/p}}{k}\), then \(u\equiv 0\).
We conclude this introduction with a few comments on some technical points. First, in all the results stated above, except for Theorem 6, M is not explicitely assumed to be non-compact. Indeed, if M is compact (without boundary) and u satisfies
for some measurable f, then necessarily \(f=0\) and \(A(x,u,\nabla u) = 0\) a.e. on \(\Omega _+\) (see Lemma 8 in Sect. 2). Hence, in the assumptions of Theorems 1, 3, 4 and 5, M is necessarily non-compact. Secondly, in all our results we do not make additional regularity assumptions on the subsolutions beside their belonging to the appropriate Sobolev class \(W^{1,p}_\textrm{loc}(M)\). Since we do not know if Sobolev subsolutions of possibly degenerate equations of the form
are always locally essentially upper bounded (that is, if they necessarily satisfy \(u_+\in L^\infty _\textrm{loc}(M)\)), in some of our arguments we have to follow more winding roads using approximation procedures.
The paper is organized as follows. In Sect. 2 we collect the notation and all relevant definitions. In Sect. 3 we prove the main Theorem 4 and we provide examples showing sharpness of the constants in the statements. Section 4 is devoted to the proof of Theorem 6, from which Theorem 5 can be easily deduced (see Corollary 22 and Remark 23).
Comparison results and the case \(p=1\) will appear in a forthcoming paper.
We recently learned that on arXiv has just appeared a paper by Bisterzo, Farina and Pigola [2] which is somehow related to our work, at least where L is the Laplace–Beltrami operator. However, even in the above overlapping case, the two papers are different in setting, scope and sharpness of the results.
2 Definitions and Notation
Throughout this paper, M will always be a connected Riemannian manifold withouth boundary. We denote by TM its tangent bundle and by \(\langle \, , \, \rangle \) its Riemannian metric. For any \(p\in (1,+\infty )\) we also denote by \(W^{1,p}_\textrm{loc}(M)\) the space of Sobolev functions u whose restrictions to any relatively compact set \(\Omega \subseteq M\) belong to \(W^{1,p}(\Omega )\). This is equivalent to requiring that \(u\circ \psi ^{-1} \in W^{1,p}_\textrm{loc}(\psi (U))\) for any local chart \(\psi : U \subseteq M \rightarrow {\mathbb {R}}^m\), where \(m=\dim M\). We also denote by \(W^{1,p}_c(M)\) the subspace of \(W^{1,p}_\textrm{loc}(M)\) consisting of functions with compact support.
We consider quasilinear differential operators L in divergence form weakly defined on functions \(u\in W^{1,p}_\textrm{loc}(M)\) by
Here \(A: {\mathbb {R}}\times TM \rightarrow TM\) is a function such that
and whose local representation \(\tilde{A}: \psi (U) \times {\mathbb {R}}\times {\mathbb {R}}^m \rightarrow {\mathbb {R}}^m\) in any chart \(\psi : U \subseteq M \rightarrow {\mathbb {R}}^m\) satisfies the Carathéodory conditions
-
\(\tilde{A}(y,\cdot ,\cdot )\) is continuous for a.e. \(y\in \psi (U)\)
-
\(\tilde{A}(\cdot ,s,v)\) is measurable for every \((s,v)\in {\mathbb {R}}\times {\mathbb {R}}^m\).
(The representation \(\tilde{A}\) is defined by
where \(y^1,\dots ,y^m\) are the coordinates induced by \(\psi \).) In particular, these conditions on \(\tilde{A}\) are satisfied whenever A is a continuous function of its arguments. Following terminology from [5, Definition 2.1], we say that A and the corresponding operator L given by (23) are weakly-p-coercive for some \(p\in (1,+\infty )\) if
for some constant \(k>0\) that we will call the coercivity constant of A. Note that the above conditions imply that
Indeed, this is clearly true when \(A(x,s,\xi )=0\); otherwise, by Cauchy–Schwarz inequality and (25) we have \(|A(x,s,\xi )|^p \le k^p |A(x,s,\xi )|^{p-1}|\xi |^{p-1}\), and then (26) follows dividing both sides by \(|A(x,s,\xi )|^{p-1}\). In particular, we have
On the other hand, in general we do not assume non-degeneracy of A, that is, we do not assume that \(A(x,s,\xi )\ne 0\) when \(\xi \ne 0\).
Let A be a weakly p-coercive function for some \(p\in (1,+\infty )\). For any given \(u\in W^{1,p}_\textrm{loc}(M)\) and any \(s_0\in {\mathbb {R}}\) we set
and for any non-negative measurable \(f:M\rightarrow [0,+\infty ]\) we say that u satisfies
if
where
We remark that our assumptions on A and u imply that \(|A(x,u,\nabla u)|\in L^{p'}_\textrm{loc}(M)\), with \(p'=\frac{p}{p-1}\) the exponent conjugate to p, and that \(\langle A(x,u,\nabla u),\nabla \varphi \rangle \) is measurable for each \(\varphi \in D^+(\Omega _{s_0})\) (see for instance [9, Lemma 2.4]). Hence, the LHS of (29) is well defined and finite for each \(\varphi \in D^+(\Omega _{s_0})\).
The next lemma justifies our focus on complete, non-compact manifolds in the introduction and in the following sections.
Lemma 8
Let M be a compact manifold without boundary, \(p\in (1,+\infty )\) and L a weakly p-coercive operator as in (23). If \(u\in W^{1,p}(M)\) satisfies
for some measurable \(f: M \rightarrow {\mathbb {R}}\) and some \(s_0\in {\mathbb {R}}\), then
Proof
Considering the test function \(\varphi = (u-s_0)_+ \in D^+(\Omega _{s_0})\) we have
and by the weak coercivity condition (25) we obtain
By non-negativity of f and of \(|\,\cdot \,|\), this immediately yields (30). \(\square \)
Lastly, we precise the following terminology. For an open interval \(I\subseteq {\mathbb {R}}\) we say that a function \(F: I \rightarrow {\mathbb {R}}\) is piecewise \(C^1\) if F is continuous on I and there exists a discrete (possibly empty) set \(E\subseteq I\) such that
If \(u\in W^{1,p}_\textrm{loc}(M)\) with \(u(M)\subseteq I\) and \(F'\) is bounded on \(I\setminus E\), then by Stampacchia’s lemma the function \(v = F(u)\) is also in \(W^{1,p}_\textrm{loc}(M)\) and
see for instance Theorem 7.8 in [6]. (Here and in the following statements, “a.e.” always referes to the m-dimensional Riemannian volume measure of M.) Since \(\nabla u = 0\) a.e. on each level set of u, we can further write
3 Proof of the Main Theorem
The aim of this section is to prove the main Theorem 13 below, which is slightly more general than Theorem 4 from the Introduction. To do so, we have to collect some preliminary lemmas about functions u satisfying \(Lu\ge 0\) on some superlevel set \(\Omega _{s_0}:= \{x\in M: u(x)>s_0\}\), \(s_0\in {\mathbb {R}}\). Note that for the validity of the following lemmas it is not necessary to assume that \(|\Omega _{s_0}|>0\), that is, \(s_0\) may be a priori larger than or equal to \(\textrm{ess}\,\textrm{sup}_M u\) (in which case it is clearly true that \(Lu\ge 0\) on \(\Omega _{s_0}\) in the sense of (29), and the thesis of each lemma holds trivially).
Lemma 9
Let M be a Riemannian manifold, \(p>1\) and L a weakly p-coercive operator as in (23) with coercivity constant \(k>0\). Let \(u\in W^{1,p}_\textrm{loc}(M)\) satisfy
for some \(s_0\in {\mathbb {R}}\) and some measurable \(f: M \rightarrow {\mathbb {R}}\). Let F be a non-negative, non-decreasing, piecewise \(C^1\) function on \((0,+\infty )\). Then for every \(0\le \eta \in C^\infty _c(M)\)
where \(w:=(u-s_0)_+\), \(A_u:=A(x,u,\nabla u)\) and \(p'=\frac{p}{p-1}\).
Proof
Let \(0\le \eta \in C^\infty _c(M)\) be given and let
as in the statement. Let \(\lambda \in C^\infty ({\mathbb {R}})\) be such that
and for any \(\varepsilon >0\) define \(\lambda _\varepsilon \in C^\infty ({\mathbb {R}})\) by
Clearly we have
where \(\textbf{1}\) denotes the indicator function and \(\nearrow \) denotes monotone convergence from below. Let \(h>0\) be fixed and for any \(\varepsilon \in (0,h/2)\) let \(F_{\varepsilon ,h}: {\mathbb {R}}\rightarrow [0,+\infty )\) be given by
By our choice of \(\lambda \) and our assumptions on F, the function \(F_{\varepsilon ,h}\) is non-negative, non-decreasing, piecewise \(C^1\) on \({\mathbb {R}}\) (with an additional corner point at \(s=h\)) and globally Lipschitz, so \(F_{\varepsilon ,h}(w)\in W^{1,p}_\textrm{loc}(M)\) with
In particular we have
Set
We have \(0\le \varphi \in W^{1,p}_c(M)\) and by the choice of \(\lambda _\varepsilon \) we also have that \(\varphi \) vanish outside \(\{w>0\} \equiv \Omega _{s_0}\). So \(\varphi \) is an admissible test function for (29) and we have
By direct computation and using that \(\eta F(w)\lambda _\varepsilon '(w)\langle A_u,\nabla u\rangle \ge 0\) by our assumptions on \(\lambda _\varepsilon \), F, \(\eta \) and A, together with weak p-coercivity (25) of A and Cauchy–Schwarz inequality we have
We substitute into (36) and rearrange terms to get
Using non-negativity of F, \(F'\), f, \(\eta \), monotonicity of F and (35), by the monotone convergence theorem we get
and then we obtain (85). \(\square \)
We underline that the LHS of (32) can be further estimated from above via Young’s inequality in two different ways, both useful in what will follow.
(1) Suppose that \(F'>0\) on \((0,+\infty )\). By Hölder’s and Young’s inequalities with conjugate exponents p and \(p'\), for any \(\sigma >0\) we get
(2) If \(0\le \psi \in C^\infty _c(M)\), then applying (32) with \(\eta :=\psi ^p\in C^\infty _c(M)\) we get
and by Young’s inequality we have, again for any \(\sigma >0\),
By suitably choosing \(\sigma \) in (39) and rearranging terms we deduce the following
Lemma 10
In the assumptions of Lemma 9, if
then for any \(\varepsilon >0\) and for any \(0\le \eta \in C^\infty _c(M)\) we have
In particular, (40) holds under one of the following assumptions:
-
(a)
\(F(s) = O(s)\) as \(s\rightarrow +\infty \)
-
(b)
\(u_+\in L^r_\textrm{loc}(M)\) and \(F(s)=O(s^{r/p})\) as \(s\rightarrow +\infty \), for some \(r>p\)
-
(c)
\(u_+\in L^\infty _\textrm{loc}(M)\).
Proof
If \(\varepsilon >0\) is given then for \(\sigma = (\varepsilon p')^{-1/p'} k\) we have
and then from (39) we get
In the assumption (40) we can rearrange terms to obtain (41). In view of (32) and since \(f\ge 0\) on \(\Omega _{s_0}\), condition (40) is automatically satisfied if \(F(w)|A_u|\textbf{1}_{\Omega _{s_0}}\in L^1_\textrm{loc}(M)\). In particular this is always the case if \(F(w)\textbf{1}_{\Omega _{s_0}}\in L^p_\textrm{loc}(M)\), because then \(F(w)|A_u|\textbf{1}_{\Omega _{s_0}}\in L^1_\textrm{loc}(M)\) by Hölder inequality (recall that \(u\in W^{1,p}_\textrm{loc}(M)\), so \(|A_u|\le k^p|\nabla u|^{p-1} \in L^{p'}_\textrm{loc}(M)\)), and condition \(F(w)\textbf{1}_{\Omega _{s_0}}\in L^p_\textrm{loc}(M)\) is in turn satisfied in either one of the cases (a), (b) or (c). \(\square \)
A case that will be relevant for our subsequent discussion is where \(u_+\in L^q_\textrm{loc}(M)\) and \(F(s)=s^{q-p+1}\) for some \(q\in (p-1,+\infty )\). In this setting the desired inequality takes the form
where \(\gamma :=q-p+1\in (0,+\infty )\). Note that for \(p-1<q\le p\) we have \(0<\gamma \le 1\), hence \(F(s)=s^{q-p+1}=s^\gamma =O(s)\) and this scenario is covered by alternative (a) in Lemma 10, while for \(q>p\) (and without assuming \(u_+\in L^\infty _\textrm{loc}(M)\)) we cannot refer to (b) or (c).
Lemma 11
Let M be a Riemannian manifold, \(p\in (1,+\infty )\) and L a weakly p-coercive operator as in (23) with coercivity constant \(k>0\). Let \(u\in W^{1,p}_\textrm{loc}(M)\) satisfy
for some \(s_0\in {\mathbb {R}}\) and some measurable \(f: M \rightarrow {\mathbb {R}}\). Let \(w:=(u-s_0)_+\) and \(A_u:=A(x,u,\nabla u)\). Then for any \(q\in (p-1,+\infty )\) and for every \(0\le \eta \in C^\infty _c(M)\)
where \(\gamma :=q-p+1\). If \(u_+\in L^q_\textrm{loc}(M)\), this can be strengthened to
In particular, if \(u_+\in L^\infty _\textrm{loc}(M)\) then this holds for any \(q\in (p-1,+\infty )\).
Proof
Let \(0\le \eta \in C^\infty _c(M)\), \(q\in (p-1,+\infty )\) be given and set \(F(s)=s^\gamma \) for \(s>0\), where \(\gamma :=q-p+1\) as in the statement of the Lemma.
If \(p-1<q\le p\) then \(0<\gamma \le 1\) and by Lemma 10 we have the validity of (45) for any \(\varepsilon \in (0,1]\). (Note that in this case (44) and (45) coincide.)
If \(q>p\) then we proceed by approximating F from below with globally Lipschitz functions. For any \(h>0\) let \(F_h: (0,+\infty ) \rightarrow (0,+\infty )\) be defined by
Then \(F_h\) is piecewise smooth with a corner point at \(s=h\) and satisfies \(F_h(s) = O(s)\) as \(s\rightarrow +\infty \), so by Lemma 10 we have
By direct computation we have
We substitute the second estimate into the previous inequality to obtain
and then letting \(h\rightarrow +\infty \) we get, by the monotone convergence theorem,
proving (44).
If additionally \(u_+\in L^q_\textrm{loc}(M)\), then for any given \(0\le \eta \in C^\infty _c(M)\)
and from (44) applied for any \(\varepsilon \in (0,1)\) we deduce (since \(f\ge 0\)) that also
Since this holds for any \(0\le \eta \in C^\infty _c(M)\) we have that \(F'(w)|A_u|^{p'}\textbf{1}_{\Omega _{s_0}}\in L^1_\textrm{loc}(M)\), that is, the hypothesis (40) in Lemma 10 is satisfied, and then (45) directly follows from that lemma. \(\square \)
We briefly comment on the condition \(u_+\in L^\infty _\textrm{loc}(M)\). If the function A satisfies the additional coercivity condition
for some constant \(k_2>0\) (note that this is the case for the p-Laplacian \(L=\Delta _p\)) then subsolutions of \(Lu = 0\) on M are locally essentially bounded above, that is, condition \(u_+\in L^\infty _\textrm{loc}(M)\) is automatically satisfied for any \(u\in W^{1,p}_\textrm{loc}(M)\) satisfying
More generally, \(u_+\in L^\infty _\textrm{loc}(M)\) holds for functions \(u\in W^{1,p}_\textrm{loc}(M)\) such that, for some \(s_0\in {\mathbb {R}}\), the truncation \(w:= (u-s_0)_+\) satisfies \(Lw\ge 0\) weakly on M.
Proposition 12
Let M be a Riemannian manifold, \(p>1\) and L as in (23) a weakly p-coercive operator for which (46) holds. Let \(u\in W^{1,p}_\textrm{loc}(M)\) satisfy
for some \(s_0\in {\mathbb {R}}\). Then \(u_+ \in L^\infty _\textrm{loc}(M)\).
Sketch of proof
For \(p>\dim M\) the thesis holds because \(W^{1,p}_\textrm{loc}(M) \subseteq C(M)\) by (local) Sobolev embeddings, while for \(1<p\le \dim M\) the statement can be proved by Moser iteration technique, using the Caccioppoli-type inequality
obtained by (44) (with the choices \(\varepsilon =1/2\) and \(f=0\)) and (46), together with the fact that every point \(x\in M\) has a relatively compact neighborhood \(U\subseteq M\) on which a Sobolev inequality holds. In fact, the Moser technique can be used to prove that \((u-s_0)_+ \in L^\infty _\textrm{loc}(M)\), from which \(u_+\in L^\infty _\textrm{loc}(M)\) immediately follows. \(\square \)
Since the argument above is of local nature, clearly it also applies in case (46) is satisfied with \(k_2: M \rightarrow (0,+\infty )\) a continuous function possibly decaying to zero at infinity. However, in our analysis we are not assuming coercivity conditions of the form (46), and in fact we don’t know whether a function \(u\in W^{1,p}_\textrm{loc}(M)\) such that \(Lu\ge 0\) on some superlevel set \(\{u>s_0\}\), with L only satisfying assumptions (24)–(25) from Sect. 2, is necessarily locally upper bounded.
We are now ready to state and prove the main theorem of this section.
Theorem 13
Let M be a complete Riemannian manifold, \(p\in (1,+\infty )\) and L a weakly p-coercive operator as in (23) with coercivity constant \(k>0\). Let \(\lambda >0\), \(\mu \in [0,p]\) and \(V: M \rightarrow (0,+\infty )\) be a continuous function satisfying
Let \(u\in W^{1,p}_\textrm{loc}(M)\) satisfy, for some \(0\le s_0<\textrm{ess}\,\textrm{sup}_M u\),
Then for any \(x_0\in M\) and \(q\in (p-1,+\infty )\) we have
where \(C_0\) and \(C_1\) are determined by
Moreover, in case \(\mu =p\) we have
whenever the limit on the LHS exists.
Remark 14
Note that \(C_0+p>C_1>C_0\) always.
Proof
Let us set \(w:=(u-s_0)_+\) and \(A_u:=A(x,u,\nabla u)\). Let \(x_0\in M\) and \(q\in (p-1,+\infty )\) be given. For the sake of brevity, for any \(R>0\) we shall write \(B_R\) to denote the geodesic ball \(B_R(x_0)\). Without loss of generality we can assume \(w^q \in L^1_\textrm{loc}(M)\), since otherwise \(\int _{B_R} w^q = +\infty \) for each sufficiently large \(R>0\) and the conclusion is trivial. Note that under this assumption we also have \(w^{q-p}|A_u|^{p'} \textbf{1}_{\Omega _{s_0}}\in L^1_\textrm{loc}(M)\), as a consequence of (45) in Lemma 11. Let \(G,H:(0,+\infty ) \rightarrow [0,+\infty )\) be defined by
By the previous observation, the functions G and H are well defined, non-decreasing and absolutely continuous on any compact interval contained in \((0,+\infty )\). In particular, they are differentiable a.e. on \((0,+\infty )\).
Since \(s_0\ge 0\), we have \(u^{p-1}\ge w^{p-1}\) on \(\Omega _{s_0}\). Then by applying Lemma 9 with the choices \(F(s) = s^{q-p+1}\) and \(f = V w^{p-1}\) we have
for any \(0\le \eta \in C^\infty _c(M)\), where \(\gamma :=q-p+1>0\), and applying Young’s inequality as in (37) we have, for any \(\sigma >0\),
Let \(\varepsilon \in (0,\lambda )\) be given. By condition (49) and continuity and (strict) positivity of V, there exists \(R_0=R_0(x_0,\varepsilon )>0\) large enough so that
and
Indeed, for \(\mu =0\) this is clearly true since \(V\ge \lambda \) everywhere on M by assumption (49). In case \(\mu >0\), note that it is possible to first find \(r_0>0\) such that
since from (49) and the triangle inequality we have
and then for any \(R>r_0\) we get
From the assumption that V is continuous and strictly positive on M we have \(\inf _{B_{r_0}} V > 0\), so we can find \(R_0\ge r_0\) such that \(\inf _{B_{r_0}} V \ge (\lambda -\varepsilon )/R_0^\mu \). Then for any \(R>R_0\) the RHS in (59) is just \((\lambda -\varepsilon )/R^\mu \), and so (56)–(57) hold for such \(R_0\).
Let \(t>R_0\) be a value for which \(G'(t)\) and \(H'(t)\) both exist. For any \(0<\delta <t\) choose \(\eta _\delta \in C^\infty _c(M)\) satisfying
Since \(|\nabla \eta _\delta |\le (1+\delta ^{-1}) \textbf{1}_{B_R{\setminus } B_{R-\delta }}\) we have
and letting \(\delta \searrow 0\) we get
Similarly, we have
On the other hand, since \(\eta _\delta =0\) on \(M\setminus B_t\) and \(\eta _\delta \rightarrow \textbf{1}_{B_t}\) pointwise as \(\delta \rightarrow 0\), by the dominated convergence theorem and also using (57) we get
Thus, in view of (54)–(55) we have, for any \(\sigma >0\),
and using (57) to further estimate
we obtain
We apply the above reasoning to each value \(t>R_0\) for which G and H are simultaneously differentiable to deduce that for any \(\sigma : (0,+\infty ) \rightarrow (0,+\infty )\)
that is, multiplying everything by \(p[\sigma (t)]^{-p}\) and recalling that \(p+p'=pp'\),
for a.e. \(t>R_0\). We now consider separately the cases \(\mu \in [0,p)\) and \(\mu =p\).
Case \(\mu \in [0,p)\). Assume that \(\mu \in [0,p)\). Choosing
we get
Let \(\Phi : (0,+\infty ) \rightarrow [0,+\infty )\) be defined by
The function \(\Phi \) is absolutely continuous on each compact subset of \((0,+\infty )\) with
Then, in view of the previous inequality and since \(\mu c_2 t^{\mu -1} H(t) \ge 0\), we get
We have \(|\Omega _{s_0}|>0\) because \(s_0<\textrm{ess}\,\textrm{sup}_M u\), so there exists \(R_1>R_0\) such that \(G(R_1)>0\). Let \(R>R_1\) be given. By monotonicity of G and since \(c_2 t^\mu H(t)\ge 0\), we have \(\Phi (t) \ge G(t)\ge G(R_1)>0\) for all \(t\in [R_1,R]\). Since \([G(R_1),+\infty ) \ni s \mapsto \log s\) is Lipschitz, the function \(\log \Phi \) is absolutely continuous on \([R_1,R]\) with
Thus, integrating (63) and using that \(\Phi (R_1) \ge G(R_1)>0\) we get
Note that dividing both sides by \(R^{1-\frac{\mu }{p}}\), letting \(R\rightarrow +\infty \) and then \(\varepsilon \rightarrow 0^+\) we would obtain
which is (formally) weaker than (50) since \(\Phi (R)\ge G(R)\). To show that the same inequality holds with \(\log G(R)\) in place of \(\Phi (R)\), we proceed as follows. Let \(R>R_1\) and \(h>0\) be given. By inequality (45) in Lemma 11 applied with the choice \(\varepsilon =\frac{1}{2}\) and with a cut-off function \(0\le \eta \in C^\infty _c(M)\) satisfying
we get
and thus, choosing \(h=R^{\mu /p}\),
where in the last inequality we used monotonicity of G. Then, from (64) we get
Dividing both sides by \((R+R^{\mu /p})^{1-\frac{\mu }{p}}\) and then letting \(R\rightarrow +\infty \) we get
that is,
and letting \(\varepsilon \rightarrow 0^+\) we obtain (50).
Case \(\mu =p\). Assume now that \(\mu =p\). We first prove (51), and then (52) in the assumption that its LHS is well defined.
Proof of (51). Choosing
for a suitable constant \(c_4=c_{4,\varepsilon }\) to be suitably selected later, from (61) we get
for a.e. \(t>R_0\). In analogy with the previous case, we aim at using this to deduce an inequality of the form
with
for suitable constants \(c_5=c_{5,\varepsilon }\) and \(c_6=c_{6,\varepsilon }\). Computing \(\Phi '\) and rearranging terms we see that the desired inequality takes the form
so we want to choose \(c_4\), \(c_5\) and \(c_6\) matching the following relations:
Expressing everything in terms of \(c_5\) this amounts to
that is, raising everything to the power \(1/p'\) in the last relation, we choose \(c_5=c_{5,\varepsilon }\) as the unique value in \((p,+\infty )\) satisfying
and then we let \(c_4\) and \(c_6\) be defined accordingly by (71). Summarizing, for there choices of \(c_4\), \(c_5\) and \(c_6\) we have that (67) and (70) coincide, and each of them is equivalent to (68) for \(\Phi \) defined as in (69). Then choosing \(R_1>R_0\) such that \(G(R_1)>0\) and reasoning as in the previous case we see that
and then by applying (65) with \(h=R\) we obtain
Dividing both sides by \(\log (2R)\) and using that \(\log (2R)\sim \log R\) as \(R\rightarrow +\infty \) we get (after relabeling)
and then letting \(\varepsilon \rightarrow 0\) we get (51).
Proof of (52). Assume that
exists. From (51) we already know that \(\ell \ge C_1>p\). If \(\ell =+\infty \) then (52) is trivially satisfied, so let us assume that \(\ell <+\infty \). Let \(\varepsilon >0\) be as above and small enough so that \(\ell -\varepsilon > p\). Then there exists \(R_2>R_0\) such that
We recall, from the discussion preceding the treatment of case \(\mu <p\), that for each \(t>R_2\) such that \(G'(t)\) and \(H'(t)\) exist we have (60), that is,
for any \(\sigma >0\). Using the co-area formula twice together with (56) we get
where \(m=\dim M\) and \({\mathcal {H}}\) is the Hausdoff measure induced by the Riemannian structure. Substituting into the above inequality and multiplying both sides by \(p\sigma ^{-p}t^{-p}\) we get
and then choosing
this yields
Let \(\Psi : (R_2,+\infty ) \rightarrow [0,+\infty )\) be defined by
The function \(\Psi \) is absolutely continuous on each compact interval contained in \((R_2,+\infty )\) and inequality (73) can be restated as
Reasoning as in the previous cases, since \(\Psi \not \equiv 0\) we reach the conclusion
We now use this to deduce (52). Let \(R>R_2\) be given. Applying (65) with \(h=R\), integrating by parts and then using (72) twice we get
Since G is non-decreasing, we have \(G(R)\le G(2R)\) and then
as \(R\rightarrow +\infty \). By (72) we see that \(R^{-p+2\varepsilon } G(2R) > 2^p R^{2\varepsilon } \rightarrow +\infty \), so
as \(R\rightarrow +\infty \), and then
and then, using that \(\log R\sim \log (2R)\), after relabeling we get
Substituting this into (75) yields
and then letting \(\varepsilon \rightarrow 0^+\) we finally obtain (52). \(\square \)
Remark 15
As a byproduct of the previous proof (namely, inequality (66) above), we showed that if \(u\in W^{1,p}_\textrm{loc}(M)\) satisfies
with \(V: M \rightarrow (0,+\infty )\) continuous and matching (49) for some \(\lambda >0\) and \(\mu \in [0,p]\), then for each \(\varepsilon \in (0,\lambda )\) and \(R_0>0\) large enough (so that (56)–(57) are satisfied) and for each \(R_1>R_0\) such that
we have
where
do not depend on u. Inequality (76) only involves the integrals of \(w=(u-s_0)_+^q\) on geodesic balls, so it would still hold for functions \(u\in L^q_\textrm{loc}(M)\) that can be approximated pointwise and in \(L^q\) norm on balls B of arbitrary large radii by Sobolev functions \(\tilde{u}\in W^{1,p}_\textrm{loc}(B)\) satisfying
For instance, when \(L=\Delta \) is the Laplace–Beltrami operator and \(V\equiv 1\), a nontrivial result concerning local smooth monotone approximation of distributional \(L^1_\textrm{loc}\) subsolutions of \(\Delta u = u\) (namely, Theorem D in [3]) allows to extend the estimate
to distributional and not everywhere negative \(L^1_\textrm{loc}\) subsolutions of \(\Delta u = u\).
The following examples are aimed at showing the sharpness of the constant appearing in (50) and (52). Let M be a model surface, that is, a complete Riemannian manifold diffeomorphic to \({\mathbb {R}}^2\) and radially symmetric around some point \(o\in M\) so that in global polar coordinates \((r,\theta )\) centered at o the metric takes the form
for a smooth \(g:(0,+\infty ) \rightarrow (0,+\infty )\) satisfying \(g'(0^+)=1\) and \(g^{(2k)}(0^+)=0\) for each \(k\in \{0\}\cup {\mathbb {N}}\). Let \(v:[0,+\infty ) \rightarrow {\mathbb {R}}\) be smooth and such that
Then \(u:=v\circ r \in C^\infty (M)\), \(|\nabla u|\ne 0\) on \(M\setminus \{o\}\) and for any \(p>1\) we have
Case \(\mu \in [0,p)\). Let \(p>1\) and \(\mu \in [0,p)\) be given. Consider \(a,c\in {\mathbb {R}}\) satisfying
and set
Choose g and v satisfying the above requirements and such that
and
By (77) we have
where
Let \(s_0>e^c\). Since v is non-decreasing, the set \(\Omega _{s_0}:= \{u>s_0\}\) coincides with \(M{\setminus }\overline{B_{t_0}}\), where \(t_0 = [(\log s_0)/c]^{1/\beta } > 1\), so in particular \(\Omega _{s_0}\subseteq \Omega \). Also, for any \(q>p-1\) we have
where the asymptotic equivalence between the integrals holds because
(Recall that \(a+qc>a+(p-1)c>0\) due to our assumptions on a and c.) Integrating by parts yields
hence, rearranging terms and using that \(\beta \in (0,1]\), we get
for \(R\rightarrow +\infty \), with
Passing to logarithms, we obtain
as \(R\rightarrow +\infty \), that is, multiplying both sides by \(\beta R^{-\beta }\) and recalling that \(\beta =1-\frac{\mu }{p}\),
On the other hand, from (80) we clearly have
Since the p-Laplacian is weakly p-coercive with coercivity constant \(k=1\), to prove that estimate (50) is sharp it is enough to show that for any p and \(q>p-1\) there exist a and c satisfying (78) and such that
This can be done by picking any a and \(c>0\) such that
since this would yield
and then
as desired. For instance, a feasible choice for a and c would be the following:
Case \(\mu =p\). Let \(p>1\) be given, consider \(a,c\in {\mathbb {R}}\) satisfying (78) and choose g and v satisfying the general requirements and such that
and
By (77) we have
with
Let \(s_0>1\) be given. Then \(\Omega _{s_0}:= \{u>s_0\}\) is contained in \(\{u>1\} = M\setminus \overline{B_1}\) and for any \(q>p-1\) we have
that is,
and then again to prove sharpness of (52) we need to show that for any \(p>1\) and \(q>p-1\) we can choose a and c satisfying (78) and
but this is precisely what we did in the previous case.
4 The Case \(Lu\ge 0\)
In this section we are concerned with lower bounds on the growth of functions u satisfying the differential inequality \(Lu\ge 0\) on a non-empty superlevel set. The main result of this section is Theorem 20 below, corresponding to Theorem 6 from the Introduction. The starting point in this case is again Lemma 9. For ease of the reader we point out that in this case it takes the following form.
Lemma 16
Let M be a Riemannian manifold, \(p\in (1,+\infty )\) and L a weakly p-coercive operator as in (23). Let \(u\in W^{1,p}_\textrm{loc}(M)\) satisfy
for some \(s_0\in {\mathbb {R}}\). Then for any \(0\le \eta \in C^\infty _c(M)\) and for any non-negative, non-decreasing, piecewise \(C^1\) function on \((0,+\infty )\) we have
where \(w:=(u-s_0)_+\) and \(A_u:=A(x,u,\nabla u)\).
The main tool to prove Theorem 20 is the next proposition.
Proposition 17
Let M be a complete, non-compact Riemannian manifold, \(p\in (1,+\infty )\) and L a weakly p-coercive operator as in (23). Let \(u\in W^{1,p}_\textrm{loc}(M)\) satisfy
for some \(s_0\in {\mathbb {R}}\).
(a) For any \(q>p-1\) and for any \(x_0\in M\) and \(0<r<R\)
where \(w:=(u-s_0)_+\), \(A_u:=A(x,u,\nabla u)\) and \(\gamma :=q-p+1\).
(b) If \(u_+\in L^\infty _\textrm{loc}(M)\) and F is a non-negative, piecewise \(C^1\) function on \((0,+\infty )\) such that \(F'>0\) everywhere on \((0,+\infty )\), then
for every \(x_0\in M\) and \(0<r<R\), with w and \(A_u\) as above.
Remark 18
We remark that the exponents \(1-p\) and \(1/(1-p)\) appearing on the RHS’s of (87) and (88) are negative. With the agreement that \(0^a = +\infty \) and \((+\infty )^a = 0\) for any \(a\in (-\infty ,0)\), the inequalities make sense also in case one or more of the integrals on the RHS’s are either vanishing or diverging.
Proof
Let w and \(A_u\) be as in the statement. We first prove (b), since the proof of (a) relies on the same idea coupled with suitable approximation arguments.
Proof of (b). Suppose that \(u_+\in L^\infty _\textrm{loc}(M)\) and let F be as in the statement. The function F satisfies all the requirements in Lemma 16 and therefore
for any \(0\le \eta \in C^\infty _c(M)\). Note that both integrals are finite since \(F(w),F'(w)\in L^\infty (\Omega _{s_0})\) and \(|A_u|\textbf{1}_{\Omega _{s_0}}\in L^{p'}_\textrm{loc}(M)\). Applying Hölder inequality with conjugate exponents p and \(p'\) as in (37) we further obtain
where the middle integral is again finite since \([F(w)]^p/[F'(w)]^{p-1} \in L^\infty (\Omega _{s_0})\). Let \(x_0\in M\) be fixed and let us write \(B_s\) for the geodesic ball \(B_s(x_0)\), for any \(s>0\). Let \(G,H:(0,+\infty ) \rightarrow [0,+\infty )\) be defined by
Since \(F'(w)|A_u|^{p'}\textbf{1}_{\Omega _{s_0}}\in L^1_\textrm{loc}(M)\) and \([F(w)]^p/[F'(w)]^{p-1} \textbf{1}_{\Omega _{s_0}} \in L^\infty (M) \subseteq L^1_\textrm{loc}(M)\), the functions G and H are well defined, non-decreasing and absolutely continuous on any compact interval contained in \((0,+\infty )\). In particular, they are differentiable a.e. on \((0,+\infty )\). Let \(s>0\) be a value for which \(G'(s)\) and \(H'(s)\) both exist. For any \(\varepsilon >0\) choose \(\eta _\varepsilon \in C^\infty _c(M)\) satisfying
Then
and passing to limits as \(\varepsilon \rightarrow 0^+\) we get
Similarly, we obtain
and by dominated convergence theorem we also have
Then by (90) we deduce
Moreover, by the co-area formula we have
Let \(0<r<R\) be given. If \(G(r)=0\) then (88) is trivially satisfied. If \(G(r)>0\) then by monotonicity of G we have that \(G(s)\ge G(r)\) for all \(s\in [r,R]\). Since \(G'(s)\) is finite for a.e. \(s\in [r,R]\), from (92) and (93) we infer that \(\varphi (s)>0\) for a.e. \(s\in [r,R]\) and then
Since \(G(s)\ge G(r)>0\) for all \(s\in [r,R]\) and \([G(r),+\infty ) \ni t\mapsto t^{1/(1-p)}\) is Lipschitz, the function \(G^{1/(1-p)} \equiv G^{1-p'}\) is absolutely continuous on [r, R] with
Thus, integrating (94) we get (noting that \(p'/p=1/(p-1)\))
Discarding the term containing G(R) and raising everything to \(1-p\) we get
that is, (88).
Proof of (a). We observe that the argument developed above can be applied straightforwardly, without the assumption \(u_+\in L^\infty _\textrm{loc}(M)\), as long as we consider a piecewise \(C^1\) function \(F:(0,+\infty ) \rightarrow (0,+\infty )\) with \(F'>0\) such that
Indeed, if the conditions in (95) are satisfied then all the integrals appearing in (89) and (90) are finite and the functions G and H defined as in (91) are again finite-valued, non-decreasing and absolutely continuous on every compact interval contained in \((0,+\infty )\).
Case \(q\ge p\). Set \(\gamma := q-p+1 \ge 1\). For any \(h>0\) define \(F_h\) by
Note that \(F_h\) is positive and \(C^1\) on \((0,+\infty )\) with
We have \(F_h'>0\) everywhere on \((0,+\infty )\) and \(F_h'(w) \in L^\infty (\Omega _{s_0})\), therefore also \(F_h'(w)|A_u|^{p'}\textbf{1}_{\Omega _{s_0}} \in L^1_\textrm{loc}(M)\), due to (26) and \(u\in W^{1,p}_\textrm{loc}(M)\). Moreover,
so in particular
since \(\gamma \ge 1\) and \(u\in W^{1,p}_\textrm{loc}(M)\). Hence, conditions (95) are satisfied for \(F=F_h\) and we can repeat the argument in the proof of (a) up to obtaining
with
From (98) and recalling that \(\gamma =q-p+1\) we also have
hence
Reasoning again as in the proof of (a) we deduce that either \(G_h(r)=0\) or
In any case we get
and the conclusion follows by the monotone convergence theorem letting \(h\rightarrow +\infty \).
Case \(p-1<q<p\). Set \(\gamma := q-p+1\) as in the previous case and note that now \(\gamma \in (0,1)\). For any \(h>0\) let \(F_h\) be defined as in (96). We note that \(F_h\) is positive and \(C^1\) on \((0,+\infty )\) in this case too, with \(F_h'>0\) everywhere on \((0,+\infty )\). Then from Lemma 16 we get
From the expression (96) we see that \(F_h(w) \le C_{h,\gamma }(1+w)\), hence \(F_h(w)|A_u|\textbf{1}_{\Omega _{s_0}}\in L^1_\textrm{loc}(M)\) by Hölder inequality. By (99) this also yields
On the other hand, we have
where the inequality in the middle holds because \(w-h<(w-h)/\gamma \) on \(\{w>h\}\), since \(0<\gamma <1\) in this case. From this estimate we get
Hence, both conditions in (95) are satisfied. Setting again
we can repeat once more the general argument to get that either \(G_h(r)=0\) or
and in any case we get
We now let \(h\rightarrow +\infty \) in both sides of (101). By Fatou’s lemma we have
Concerning the RHS of (101), we aim at showing that
with
From (100) and recalling that \(\gamma +p-1=q\) we have
Since \(w\in W^{1,p}_\textrm{loc}(M)\), for a.e. \(s\in [r,R]\) we have \(w\in L^p(\partial B_s)\) by the co-area formula. Then, using the monotone convergence theorem on the first integral in (104) together with the fact that \(h^{\gamma -1} \rightarrow 0\) as \(h\rightarrow +\infty \) (due to \(\gamma <1\)) we get
If \(\varphi ^{1/(1-p)}\not \in L^1([r,R])\), then by (105) and Fatou’s lemma we have
so (103) holds with both sides equalling \(+\infty \). Suppose, instead, that \(\varphi ^{1/(1-p)}\in L^1([r,R])\). From the first line in (100) we also deduce the reversed estimate
where in the second inequality we exploited again the fact that \(0<\gamma <1\). Then, for every \(h>0\) we also have \(\varphi _h \ge \gamma ^p \, \varphi \) and therefore
Hence, if \(\varphi ^{1/(1-p)} \in L^1([r,R])\) then (103) follows by the dominated convergence theorem. In any case, from the continuity of \([0,+\infty ] \ni t \mapsto t^{1-p} \in [0,+\infty ]\) with the agreement that \(0^{1-p} = +\infty \) and \((+\infty )^{1-p}=0\) we get
By (101), (102) and (106) we obtain the desired conclusion. \(\square \)
From Proposition 17 we easily deduce the following lemma.
Lemma 19
Let M be a complete, non-compact Riemannian manifold, \(p\in (1,+\infty )\) and L a weakly p-coercive operator as in (23). Let \(u\in W^{1,p}_\textrm{loc}(M)\) satisfy
for some \(s_0\in {\mathbb {R}}\) and also suppose that
Then there exists \(r_0\ge 0\) such that for any \(q>p-1\)
In particular,
where \({\mathcal {H}}^{m-1}\) denotes the \((m-1)\)-dimensional Hausdorff measure. Moreover, if \(u_+\in L^\infty _\textrm{loc}(M)\) then also
Proof
Choose \(r_0\ge 0\) such that \(|B_r\cap E_0|>0\) for every \(r>r_0\), where \(E_0\) is as in (108). Then, for every \(r>r_0\)
and then applying Proposition 17.(a) we see that the RHS of (87) must be strictly positive for any \(R>r\), that is (since \(1-p < 0\)),
In particular, \(\left( \int _{\partial B_s} w^q\right) ^{1/(1-p)}\) must be finite for a.e. \(s>r\), hence for a.e. \(s>r_0\) by arbitrariness of \(r>r_0\), and therefore it must be \(\int _{\partial B_s} w^q>0\) for a.e. \(s>r_0\), yielding (110). If \(u_+\in L^\infty _\textrm{loc}(M)\), to prove (111) we start from the two-sided estimate
holding for each \(s>r_0\), from which we deduce
The function \(v(r):= {\mathcal {H}}^{m-1}(\partial B_r)\) satisfies
see Proposition 1.6 in [1], so we have
and by Proposition 17.(b) applied with the choice \(f\equiv 1\) and \(F(s) = 1 + s\) we get
Putting together all inequalities above we obtain (111). \(\square \)
We are now ready for the proof of the main result of this section.
Theorem 20
Let M be a complete, non-compact Riemannian manifold, \(p\in (1,+\infty )\) and L a weakly p-coercive operator as in (23). Let \(u\in W^{1,p}_\textrm{loc}(M)\) satisfy
for some \(s_0\in {\mathbb {R}}\) and suppose that for some \(x_0\in M\) and \(q>p-1\) it holds
Then \(A(x,u,\nabla u) = 0\) a.e. on \(\Omega _{s_0}\). Thus, if A satisfies the structural condition
then either \(u\equiv c\) a.e. on M for some constant \(c>s_0\), or \(u\le s_0\) a.e. on M.
Remark 21
Condition (113) can be stated, more briefly, as
with this notation meaning that the function \(\varphi : (0,+\infty ) \rightarrow [0,+\infty ]\) given by
is not in \(L^1((r,+\infty ))\) for any \(r>0\). The previous Lemma 19 implies that this is a meaningful condition, since in general only two cases are possible:
-
(i)
\(\varphi =+\infty \) a.e. on \((0,+\infty )\), and then \(\Omega _{s_0}\) has zero measure while condition (113) is obviously satisfied, or
-
(ii)
there exists \(r_0\ge 0\) such that \(\varphi <+\infty \) a.e. on \((r_0,+\infty )\) and \(\varphi \in L^1((r,R))\) for any \(r_0<r<R<+\infty \), so that (113) is satisfied if and only if \(\varphi \) is not integrable in a neighborhood of \(+\infty \).
Concerning case (ii), note that in general \(\varphi \) may be integrable at \(+\infty \) and still satisfy \(\varphi =+\infty \) on \((0,r_0)\) for some \(r_0>0\) (for instance, on \({\mathbb {R}}^n\) this may happen if u satisfies \(u\le s_0\) on \(B_{r_0}\) and \(u(x)\ge |x|^a\) as \(x\rightarrow \infty \) for some \(a>(p-n)/q\)), so the clause “\(\forall \,r>0\)” in (113) cannot in general be replaced by “for some \(r>0\)”.
Proof of Theorem 20
Suppose, by contradiction, that \(A_u:= A(x,u,\nabla u)\) is non-zero on a set \(E_0\subseteq \Omega _{s_0}\) of positive measure. Then reasoning as in the proof of Lemma 19 we see that there exists \(r>0\) such that
and by Proposition 17 this implies that
for all \(R>r\), with \(\gamma = q-p+1\). Since the RHS of this inequality is finite, letting \(R\rightarrow +\infty \) in the LHS we reach the desired contradiction. So, we conclude that \(A(x,u,\nabla u)=0\) a.e. on \(\Omega _{s_0}\).
If A satisfies the non-degeneracy condition (114) then we further deduce that \(\nabla u=0\) a.e. on \(\Omega _{s_0}\), and since the function \(w:= (u-s_0)_+ \in W^{1,p}_\textrm{loc}(M)\) has weak gradient \(\nabla w = \textbf{1}_{\Omega _{s_0}}\nabla u\) this yields \(\nabla w \equiv 0\) a.e. on M. By connectedness of M this implies that \(w = a\) a.e. on M for some constant \(a\ge 0\). If \(a>0\) then \(u = c:= s_0+a\) a.e. on M (and \(\Omega _{s_0}\) is of full measure), while if \(a=0\) then \(u\le s_0\) a.e. on M (and \(\Omega _{s_0}\) has zero measure). \(\square \)
As a consequence of Theorem 20 we have the following Liouville-type theorem.
Corollary 22
Let M be a complete, non-compact Riemannian manifold, \(p\in (1,+\infty )\) and L a weakly p-coercive operator as in (23). Let \(u\in W^{1,p}_\textrm{loc}(M)\) satisfy
for some \(s_0\in {\mathbb {R}}\) and suppose that for some \(x_0\in M\) and \(q>p-1\) it holds
Then \(A(x,u,\nabla u) = 0\), and if A satisfies the structural condition (114) then either \(u\equiv c\) a.e. on M for some \(c>s_0\) or \(u\le s_0\) a.e. on M.
Proof
The corollary is a direct consequence of Theorem 20 since (115) implies (113). For the details, see the proof of Proposition 1.3 in [8] (the parameter \(\delta \) there corresponds to \(p-1\) in our setting). \(\square \)
Remark 23
Note, in particular, that (115) holds if
or even if, for some \(n\in {\mathbb {N}}\)
where
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
References
Bianchini, B., Mari, L., Rigoli, M.: On some aspects of oscillation theory and geometry. Mem. AMS 225, 1056 (2013)
Bisterzo, A., Farina, A., Pigola, S.: \(L^p_{{\rm loc}}\) positivity preservation and Liouville-type theorems, available at arXiv:2304.00745
Bisterzo, A., Marini, L.: The \(L^\infty \)-positivity preserving property and stochastic completeness. Potential Anal. (2022). https://doi.org/10.1007/s11118-022-10041-w
Colombo, G., Mari, L., Rigoli, M.: Einstein-type structures, Besse’s conjecture, and a uniqueness result for a \(\varphi \)-CPE metric in its conformal class. J. Geom. Anal. 32(11), Paper No. 267 (2022)
D’Ambrosio, L., Mitidieri, E.: Quasilinear elliptic equations with critical potentials. Adv. Nonlinear Anal. 6(2), 147–164 (2017)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin (2001)
Le, V.K.: On some equivalent properties of sub- and supersolutions in second order quasilinear elliptic equations. Hiroshima Math. J. 28(2), 373–380 (1998)
Rigoli, M., Setti, A.G.: Liouville type theorems for \(\varphi \)-subharmonic functions. Rev. Mat. Iberoamericana 17(3), 471–520 (2001)
Rindler, F.: Calculus of Variations. Universitext. Springer, Cham (2018)
Acknowledgements
This research is part of the grant PID2021-124157NB-I00, funded by MCIN / AEI / 10.13039 / 501100011033/ “ERDF A way of making Europe”, Spain, and is also supported by Comunidad Autónoma de la Región de Murcia, Spain, within the framework of the Regional Programme in Promotion of the Scientific and Technical Research (Action Plan 2022), by Fundación Séneca, Regional Agency for Science and Technology, REF. 21899/PI/22. The second named author wants to express gratitute for the hospitality provided by the Department of Mathematics of the University of Murcia, where part of this work was done.
Funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Alías, L.J., Colombo, G. & Rigoli, M. Growth of Subsolutions of \(\Delta _p u = V|u|^{p-2}u\) and of a General Class of Quasilinear Equations. J Geom Anal 34, 44 (2024). https://doi.org/10.1007/s12220-023-01490-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-023-01490-9