1. Introduction
The aim of this paper is to study qualitative properties such as symmetry and monotonicity of positive weak solutions to the quasilinear elliptic problems involving Hardy potential:
where $\Omega$
is a $C^2$
-bounded domain in $\mathbb {R}^N$
, $0\in \Omega$
, $\vartheta \geq 0$
, $p\in (1,N)$
, $q\in (0,p)$
.
The interest in this class of problems in recent years is related to the possibility of modelling wide classes of elliptic problems. For instance, taking into account the results contained in [Reference Cianchi and Maz'ya4–Reference Cianchi and Maz'ya6] we know that if the real function $A: \mathbb {R}^+ \rightarrow \mathbb {R}^+$
is assumed to be of class $C^1(\mathbb {R}^+)$
and fulfills the following
we can consider problems in which the differential operator satisfies ellipticity and monotonicity conditions, not necessarily of power type. On the other hand, the choice $A(t)=t^{p-2}+bt^{q-2}$
, $b\geq 0$
leads to the well-known classical semilinear elliptic problem [Reference Damascelli and Pacella14, Reference Damascelli and Sciunzi15, Reference Esposito, Montoro and Sciunzi17] for $b=0$
, or to double-phase type problems (see [Reference Colombo and Mingione8–Reference Cupini, Marcellini and Mascolo12, Reference Esposito, Leonetti and Mingione19, Reference Marcellini24]). Such problems are therefore particular cases contained in the more general setting that we consider here. We shall borrow some ideas from [Reference Damascelli13, Reference Damascelli and Sciunzi15] but, contrary to the first feeling, the adaptation of such techniques is a quite delicate issue, once we want to deal with a real general class of operators. In the following we will also suppose that
We remark that hypothesis (1.3), that we prefer to state in this way, is equivalent to request that
The position of the origin with respect to the domain is connected to the presence of the Hardy potential and it is crucial already to prove existence results (see [Reference Chirillo, Montoro, Muglia and Sciunzi7] and the references therein). Regarding the nonlinear term $g \in C^{1}(\bar {\Omega } \times \mathbb {R})$
we will suppose that:
(g1) $g(x,\cdotp )$
is a locally Lipschitz continuous, uniformly with respect to $x$, i.e. for every $\Omega ' \subset \Omega$ and for every $M>0$, there is a positive constant $L=(M,\Omega ')$ such that for every $x \in \Omega '$ and every $u,v\in [0,M]$, we have that
\[ |g(x,u)-g(x,v)|\leq L|u-v|; \](g2) $g(\cdotp,u)$
is locally Lipschitz continuous, uniformly with respect to $u$.(g3) $g(\cdotp,u)$
is non-decreasing w.r.t. the $x_1$-direction in the set $\Omega _{\lambda }:=\{x=(x_1,\ldots,x_N)\in \Omega : x_1 < \lambda \text {with}\ \lambda <0\}$;(g4) $g(x, \cdotp )>0$
is positive and, more precisely, $g(x,u)>0$ in $\Omega '$ for every $\Omega ' \subset \Omega$ and for every $u>0$.
The monotonicity assumption on $g$
, with respect to the first variable, is necessary for the applicability of the moving plane method; this is well-known already in the case of non singular source terms. We refer to [Reference Canino, Montoro and Sciunzi3, Reference Merchàn, Montoro, Peral and Sciunzi25, Reference Merchàn, Montoro and Sciunzi26] for a discussion about such a condition. In this setting the notion of solution has to be understood in the weak sense as in the next definition
Definition 1.1 We say that $u$
is a weak solution to problem (1.1) if $u\in W_0^{1,p}(\Omega )$
, $g(\cdot,u(\cdot ))\in L^1(\Omega )$
and
Remark 1.2 We do not prefer to discuss here the wide literature regarding the $L^\infty$
-boundedness of the solutions, that is a well-known issue e.g. in the critical or sub-critical case. Once we reduce to deal with bounded far from the origin $W_0^{1,p}(\Omega )$
solutions, by [Reference DiBenedetto16, Reference Tolksdorf29], we get that $u\in C_{loc}^{1,\alpha }(\Omega {\setminus} \{0\})$
; furthermore, supposing that $\Omega$
is smooth, the $C_{loc}^{1,\alpha }(\bar {\Omega }{\setminus} \{0\})$
regularity follows by [Reference Lieberman23] while, from [Reference Gilbarg and Trudinger22], it also follows that $u\in C^2(\Omega {\setminus} (Z_u \cup \{0\}))$
, where
Moreover, arguing as in [Reference Chirillo, Montoro, Muglia and Sciunzi7], we have that $u_i\in W_{loc}^{1,2}(\Omega {\setminus} \{0\})$
for $p\in (1,3)$
and $u_i\in W_{loc}^{1,q}(\Omega {\setminus} \{0\})$
for every $q<\frac {p-1}{p-2}$
and $p\geq 3$
.
Our result will be obtained by means of the moving plane method; this technique is mostly used in this topic and it goes back to the seminal papers of Alexandrov [Reference Alexandrov1] and Serrin [Reference Serrin28] and the celebrated papers [Reference Berestycki and Nirenberg2, Reference Gidas, Ni and Nirenberg21].
This tool was adapted to the case of the $p$
-Laplacian operator in bounded domains firstly in [Reference Damascelli and Pacella14] for the case $1< p<2$
and, later on, in [Reference Damascelli and Sciunzi15] for the case of positive nonlinearities and $p>2$
. In this paper we will apply the moving plane technique by means of the achievement of a weak comparison principle in small domains and the strong comparison principle; this approach is more close to those introduced in [Reference Damascelli and Pacella14, Reference Damascelli and Sciunzi15] and goes back to the illuminating papers [Reference Berestycki and Nirenberg2, Reference Gidas, Ni and Nirenberg21].
Such a technique can be performed in general convex domains providing partial monotonicity results near the boundary. For simplicity of exposition and without loss of generality, we assume directly in all the paper that $\Omega$
is a strictly convex domain in the $x_1$
-direction and symmetric with respect to the hyperplane $\{x_1=0\}$
. In this setting, our main result is the following:
Theorem 1.3 Let $u\in C^1(\bar {\Omega } {\setminus} \{0\})$
be a weak solution to (1.1). Let $\Omega$
be strictly convex with respect to $x_1$
-direction and symmetric with respect to the hyperplane $\{x_1=0\}$
. Then it follows that $u$
is symmetric with respect to the hyperplane $\{x_1=0\}$
and non-decreasing in the $x_1$
-direction in $\Omega \cap \{x_1 <0\}$
.
Moreover, if $\Omega$
is a ball centred at the origin, then $u$
is radially symmetric with $\frac {\partial {u}}{\partial {r}}(r)<0$
for $r\neq 0$
.
To prove theorem (1.3) we have to face some difficulties, mainly related to the nonlinear degenerate nature of the operator. In particular, in order to carry on the procedure, we need to study the asymptotic behaviour of the solution near to zero. This task was also faced in [Reference Merchàn, Montoro, Peral and Sciunzi25]. However, the presence of the distorsion $A(\cdot )$
, causes that we can not repeat the same argument of [Reference Merchàn, Montoro, Peral and Sciunzi25]. Furthermore, when proving the weak comparison principle in small domains, we have to overcome the difficulty arising from many homogeneity problems, that the reader will appreciate during the reading. Finally, note that, taking into account the literature regarding the existence of the solutions, we choose a setting of assumptions that involves also cases when the nonlinearity is not Lipschitz continuous at zero.
The paper is organized as follows: in next section we introduce some preliminary tools and results such as summability properties of the second derivatives of the solutions. Furthermore we prove that the solution $u$
blow-up near the origin and we develop the proof of the weak comparison principle for small domains. The main result, namely theorem (1.3), is proved in the last section. Finally, for the readers convenience, we add the proofs of some standard results in the appendix.
2. Preliminaries and useful results
In this section, briefly, we enclose some definitions, results and remarks that it will be useful in the rest of the paper. From now on, in order to get a readable notation, generic numerical constant will be denoted by $c$
and will be allowed to vary within a single line or formula. Moreover we denote with $f^+:=\max \{f,0\}$
.
Remark 2.1 With respect to the setting (1.2)–(1.4), we observe that:
• From [Reference Cianchi and Maz'ya5, lemma 4.2], calling $\bar {c}:=(1+\min \{0,m_A\})>0$
, one has that
(2.1)\begin{equation} [A(|\xi|)\xi-A(|\eta|)\eta]\cdot [\xi-\eta] \geq \bar{c}|\xi-\eta|^2\int_0^1 A(|\eta+s(\xi-\eta)|)\,{\rm d}s, \end{equation}for every $\xi,\eta \in \mathbb {R}^N.$ Using (1.4), we get
\[ [A(|\xi|)\xi-A(|\eta|)\eta]\cdot [\xi-\eta] \geq c_2\bar{c}|\xi-\eta|^2\int_0^1 |\eta+s(\xi-\eta)|^{p-2}\,{\rm d}s \]and, arguing as in [Reference Damascelli13, lemma 2.1], one has that for every $\xi, \eta \in \mathbb {R}^N$
(2.2)\begin{equation} \left[A( |\xi|)\xi-A( |\eta|)\eta\right]\cdot [\xi-\eta] \geq c_2\bar{c} (|\xi|+|\eta|)^{p-2}|\xi-\eta|^2. \end{equation}• In [Reference Cianchi and Maz'ya5, proposition 4.1], it is already proved that
(2.3)\begin{equation} A(1)\min\{t^{m_A},t^{M_A}\}\leq A(t)\leq A(1)\max\{t^{m_A},t^{M_A}\}. \end{equation}Since $m_A>-1$, by (2.3) there exists $\eta \in [0,1)$ such that $m_A+\eta >0$; hence
(2.4)\begin{equation} \lim_{t\to 0}t^\eta A(t)=0 \end{equation}and, in particular $\lim _{t\to 0} tA(t)=0.$
Then, if $0< t< K$
there exists a constant $C_K:=C(K)>0$ such that
(2.5)\begin{equation} |tA(t)| \leq C_K. \end{equation}Moreover, $j(t):=tA(t)$
is a non-decreasing function on $[0,+\infty )$ since
\[ j'(t)= A(t)+tA'(t)= A(t)\left[1+\frac{tA'(t)}{A(t)}\right] \geq A(t)(1+m_A)\geq 0. \]• If $m_A \geq 0$
, by (1.2), one has
\[ \frac{tA'(t)}{A(t)} \geq m_A \Rightarrow A'(t)\geq \frac{m_AA(t)}{t}\geq 0. \]Then,(2.6)\begin{equation} \text{if}\ m_A \geq 0,\ A(t)\ \text{is a non-decreasing function on}\ (0,+\infty). \end{equation}• For $t \in \mathbb {R}^+$
and $t\geq K$, using (1.3) and (2.3), we get
\[ A(1)t^{m_A} \leq A(t) \leq c_1t^{p-2} \Rightarrow t^{m_A-p+2}\leq \frac{c_1}{A(1)}. \]Hence, we note that $(m_A-p+2)$ has to be necessarily negative, i.e.
(2.7)\begin{equation} m_A \leq p-2. \end{equation}Analogously, for $t \in \mathbb {R}^+$ and $t\geq K$, using (1.4) and (2.3), we get
\[ c_2t^{p-2} \leq A(t) \leq A(1)t^{M_A} \Rightarrow t^{p-2-M_A}\leq \frac{A(1)}{c_2}. \]Hence, we note that $(p-2-M_A)$ has to be necessarily negative, i.e.
(2.8)\begin{equation} M_A \geq p-2. \end{equation}
The following theorems are devoted to obtain some summability properties of the second derivatives and the gradient of solutions to (1.1).
Theorem 2.2 Assume that $\Omega$
is a bounded smooth domain and $1< p< N$
. Consider $u\in C^{1,\alpha }_{loc}(\bar {\Omega } {\setminus} \{0\})\cap C^2(\Omega {\setminus} (Z_u \cup \{0\}))$
a solution to (1.1), where $g(\cdot, u) \in W^{1,\infty }(\Omega )$
and $g(x, \cdot ) \in W^{1,\infty }(\mathbb {R})$
. We have
for any $E \Subset \Omega {\setminus} \{0\}$
and uniformly for any $y \in E$
, with
for $0 \leq \beta <1$
and $\gamma < (N-2)$
if $N \geq 3$
($\gamma =0$
if $N=2$
).
Moreover, if we also assume that $g$
is positive in $\Omega$
in the sense of $(g4)$
, we have that
where $\tilde {\Omega }$
is a compact such that $\tilde {\Omega } \subset \bar {\Omega } {\setminus} \{0\}$
.
Remark 2.3 As showed in [Reference Chirillo, Montoro, Muglia and Sciunzi7, remark 4.4], if $g$
is a positive function, the set $Z_u$
is such that $|Z_u|=0$
.
Theorem 2.4 Let $u\in C^{1,\alpha }(\Omega {\setminus} \{0\})\cap C^{2}(\Omega {\setminus} (Z_u \cup \{0\}))$
be a solution to (1.1) with $g(\cdot, u) \in W^{1,\infty }(\Omega )$
and $g(x, \cdot ) \in W^{1,\infty }(\mathbb {R})$
and $g(x,u)$
positive in the sense of $(g4)$
in $B_{2 \rho }(x_0) \subset \Omega {\setminus} \{0\}$
. Then
for any $y\in B_{\rho }(x_0)$
, with $\alpha :=\frac {p-1}{p-2}$
if $p>2$
and $\alpha :=\frac {m_A+1}{m_A}$
if $p\in (1,2)$
, $r\in (0,1)$
, $\gamma < N-2$
if $N \ge 3$
, $\gamma =0$
if $N=2$
and
If $\Omega$
is a smooth domain and $g$
is nonnegative in $\Omega$
where $\tilde {\Omega }$
is a compact such that $\tilde {\Omega } \subset \bar {\Omega } {\setminus} \{0\}$
and $y\in \tilde {\Omega }$
.
Remark 2.5 Even if we consider a solution to the problem (1.1), the proof of theorems 2.2 and 2.4 repeats verbatim the arguments exploited in [Reference Chirillo, Montoro, Muglia and Sciunzi7, theorems 4.2, 4.6, 4.8].
Assume that $\tilde {\Omega } \Subset \Omega {\setminus} \{0\}$
. We recall the definition of weighted Sobolev space $H^{1,2}_{\rho }(\tilde {\Omega })$
.
Definition 2.6 Let $\rho, \rho ^{-1} \in L^1(\Omega )$
. The space $H_{\rho }^{1,2}(\tilde {\Omega })$
is defined as the completion of $C^1(\tilde {\Omega })$
(or $C^{\infty }(\tilde {\Omega })$
) with the norm
where
For a discussion on this topic we refer the reader to [Reference Zhikov31].
We also recall that $H_{0,\rho }^{1,2}(\tilde {\Omega })$
is defined as the completion of $C_{c}^1(\tilde {\Omega })$
(or $C_c^{\infty }(\tilde {\Omega })$
) under the norm
Theorem 2.7 Weighted Sobolev inequality, [Reference Farina, Montoro and Sciunzi20]
Let $\rho$
be a weight function such that
with $1<\sigma < \frac {p-1}{p-2}$
if $p>2$
, $\gamma < N-2$
if $N\geq 3$
, $\gamma =0$
if $N=2$
. Assume, in the case $N \geq 3$
, without no loss generality that $\gamma >N-2\sigma,$
which implies $N\sigma -2N+2\sigma +\gamma >0$
. Then, for any $w \in H_{0, \rho }^{1,2}(\tilde {\Omega })$
, there exists a constant $C_{s}$
such that
for any $1 \leq q< 2^{*}(\sigma )$
where
In particular we prove the following Poincaré's inequality that we will use later considering the weight $\rho = A(|\nabla u|)$
. Note that this choice is possible thanks to theorem 2.4.
Corollary 2.8 Let us consider $w\in H_{0,\rho }^{1,2}(\tilde {\Omega })$
. Then
where $C_{pp}(\tilde {\Omega }) \rightarrow 0$
when $|\tilde {\Omega }| \rightarrow 0$
.
Proof. Choose $2< q<2^*(\sigma )$
. By Lebesgue's spaces embedding
then using theorem 2.7 we get
where $C_{pp}(\tilde {\Omega }):= |\tilde {\Omega }|^{\frac {q-2}{q}} C_s^2$
.
To state the next results we need some notations.
For a real number $\lambda$
we set
where $x_{\lambda }$
is the reflection through the hyperplane $T_{\lambda }:= \{x\in \mathbb {R}^N: x_1=\lambda \}.$
Also let us define $a:= \inf \limits _{x \in \Omega } x_1.$
Let us denote with $B_{\rho }$
the open ball with centre 0 and radius $\rho >0$
. Define $\phi _{\rho }(x)\in C_c^{1}(\Omega )$
, $\phi _{\rho } \geq 0$
such that
where $k$
is a positive constant.
Lemma 2.9 Let $u \in C^1(\bar {\Omega } {\setminus} \{0\})$
a solution to (1.1). Then, the set $\Omega {\setminus} Z_u$
does not contain any connected component $\mathcal {C}$
such that $\bar {\mathcal {C}}\subset \Omega$
.
Moreover, if we assume that $\Omega$
is a smooth bounded domain with connected boundary, it follows that $\Omega {\setminus} Z_u$
is connected.
Proof. We proceed by contradiction. Let us assume that such component exists, namely
By remark 2.3, we get that $|Z_u|=0$
.
For all $\varepsilon >0$
, let us define $J_{\varepsilon }: \mathbb {R}^+ \cup \{0\} \rightarrow \mathbb {R}$
by setting
We consider the function
We point out that $\text {supp }\psi _{\varepsilon } \subset \mathcal {C}$
and $\psi _{\varepsilon }\in W_0^{1,p}(\Omega ) \cap L^{\infty }(\Omega )$
. By using $\psi _{\varepsilon }$
as test function in problem (1.1), since
we obtain
Denoting $h_{\varepsilon }(t)= \frac {J_{\varepsilon }(t)}{t}$
, we get
and, by straightforward calculation, we see that $|\nabla (|\nabla u|)| \leq \left \Vert D ^2u\right \Vert$
. Therefore, the first term on the left-hand side of (2.12) can be estimated as
Choosing $\eta$
as in (2.4) and using theorem 2.2, one has
hence, by Lebesgue embedding space, $A(|\nabla u|)\left \Vert D ^2u\right \Vert \phi _{\rho } \in L^1(\mathcal {C}), \ \text { for all } \rho >0$
. Since
we get that $\lim \limits _{\varepsilon \rightarrow 0} |\nabla u||h_{\varepsilon }'(|\nabla u|)|=0 \text { a.e. in } \mathcal {C}$
and $|\nabla u||h_{\varepsilon }'(|\nabla u|)|\leq |\nabla u|\displaystyle \frac {2\varepsilon }{|\nabla u|^2} \leq 2.$
Then, since $A(|\nabla u|)|\nabla u| |h'_{\varepsilon }(|\nabla u|)|\left \Vert D ^2u\right \Vert \phi _{\rho } \leq 2 A(|\nabla u|)\left \Vert D ^2u\right \Vert \phi _{\rho } \in L^1(\mathcal {C})$
, using Dominated convergence theorem in (2.13), we obtain
From (2.15), passing to the limit in (2.12), we get
Since
using Dominated convergence theorem, we obtain
and in particular, by (2.11), one has
Using Hölder inequality and (2.5), we estimate the left-hand side of (2.17) as
Therefore, passing to the limit in (2.17) for $\rho \rightarrow 0$
, we get
On the other hand, since $\phi _{\rho }(x) \rightarrow 1$
for $\rho \rightarrow 0$
and $g(x,u) > 0$
, using again dominated convergence theorem, we get
which contradicts (2.18). If $\Omega$
is smooth, since the right-hand side of (1.1) is positive, by Höpf's Lemma (see [Reference Pucci and Serrin27, Reference Vazquez30]), a neighbourhood of the boundary belongs to a component $\mathcal {C}$
of $\Omega {\setminus} Z_u$
. The case $\Omega {\setminus} Z_u$
not connected would imply the existence of a second connected component $\mathcal {C}'$
with $\partial \mathcal {C}' \subset Z_u$
, but this would provide a contradiction with the first part of the proof. Thus $\Omega {\setminus} Z_u$
is connected.
Remark 2.10 We remark that the previous theorem still holds if we replace the assumption $g(x,u)> 0$
by the assumption $\frac {u^q}{|x|^p} + g(x,u)>0$
for every $x\in \Omega$
.
Lemma 2.11 Let $u\in W_0^{1,p}(\Omega )$
be a nonnegative weak solution to the problem (1.1). Then
Proof. Since $g(x,u)> 0$
, $u$
is a supersolution to $-\text {div}(A(|\nabla u|)\nabla u) = \vartheta \frac {u^q}{|x|^p}.$
By strong maximum principle (see [Reference Pucci and Serrin27, theorem 7.1.2]), if we set $\tilde {C}:= \inf \limits _{B_R} u >0$
, we get
If we consider
we have that (2.21) admits an unique radial non-increasing solution $w\in W_0^{1,p}(B_R)$
such that $w(r) \rightarrow +\infty$
for $r\rightarrow 0^+$
(see the appendix for details).
Since $u\geq w \text { on } \partial B_R$
, using $(w-u)^+ \in W_0^{1,p}(B_R)$
as function test in (2.20) and (2.21), we get
Using (2.2), we have that
then $(w-u)^+=0$
on $B_R$
, i.e. $w \leq u \text { on } B_R.$
Therefore, since $w(r)\rightarrow +\infty$
for $r\rightarrow 0$
, we have that $\lim \limits _{|x| \rightarrow 0} u(x)= +\infty.$
3. Symmetry and monotonicity results
3.1 Weak comparison principle
Let us first prove the following result:
Theorem 3.1 Weak comparison principle
Let $\lambda <0$
and $\tilde {\Omega }$
be a bounded domain such that $\tilde {\Omega } \Subset \Omega _{\lambda }$
. Assume that $u \in C^1(\bar {\Omega } {\setminus} \{0\})$
is a solution to (1.1) such that $u\leq u_{\lambda }$
on $\partial \tilde {\Omega }$
. Then there exists a positive constant $\delta =\delta (\lambda, p, q, g, dist(\tilde {\Omega }, \partial \Omega ))$
such that if $|\tilde {\Omega }| < \delta$
, then it holds
Proof. We have (in the weak sense) that $u$
and $u_{\lambda }$
satisfy
If $(u-u_{\lambda })^+\equiv 0$
in $\tilde {\Omega }$
, we have the claim. Then, we assume by contradiction that $(u-u_{\lambda })^+ \not \equiv 0$
in $\tilde {\Omega }$
and we want to use $\varphi := (u-u_{\lambda })^+\chi _{\tilde {\Omega }}$
as test function.
In order to do this, we notice that, by lemma 2.11, $\lim \limits _{|x| \rightarrow 0} u(x)= +\infty$
.
Then, since $u\in C^1(\bar {\Omega }{\setminus} \{0\})$
(see remark 1.2) and $0\notin \tilde {\Omega }$
, we have that $u\in L^{\infty }(\tilde {\Omega })$
; therefore $u_{\lambda } \in L^{\infty }(\tilde {\Omega })$
in supp$(u-u_{\lambda })^+$
.
By lemma 2.11, we have that
hence
From the assumption $u \leq u_{\lambda }$
on $\partial {\tilde {\Omega }}$
, it follows $\varphi := (u-u_{\lambda })^+\chi _{\tilde {\Omega }}\in W_0^{1,p}(\Omega )$
, then, we can choose $\varphi$
as test function in weak formulation of (3.1) and (3.2) obtaining
Then, if we set $\Omega ^+:= \text {supp}(u-u_{\lambda })^+ \cap \tilde {\Omega }$
and we subtract (3.4) and (3.5), we get
Noticing $|x|\geq |x_{\lambda }|$
, (3.6) becomes
Using monotonicity of $g(\cdot, u)$
, the locally Lipschitz continuity of $g(x,\cdot )$
and taking in account that for $\lambda <0$
one has that $|x| \geq C$
in $\Omega _{\lambda }$
for some positive constant $C$
, one has
We recall that $y^q$
is locally Lipschitz continuous in $(0,+\infty )$
and the solution $u$
is strictly positive in $\tilde {\Omega }$
and also in $\Omega ^+$
. Then, (3.7) becomes
Let us now consider separately the following cases.
• $1< p<2$
.• $p\geq 2$
, $m_A \geq 0$.• $p\geq 2$
, $m_A < 0$ and $A(t) \geq \tau (C)>0$ with $\tau$ positive constant that depends on $C\subset [0,+\infty )$ compact.• $p\geq 2$
, $m_A < 0$, $\exists \tilde {C} \subset [0,+\infty )$ compact such that $\inf \limits _{\tilde {C}} A(t)=0$.
Case: $\mathbf {1< p<2}$
.
Using (2.2) and classic Poincaré inequality, (3.8) becomes
Now, from (3.3), we infer that $|\nabla u_{\lambda }| \in L^{\infty }(\Omega ^+)$
.
Then, since $|\nabla u| \in L^{\infty }(\Omega ^+)$
, using also the fact that $2-p>0$
, $(|\nabla u|+|\nabla u_{\lambda }|)^{2-p}$
is bounded in $\Omega ^+$
. Thus, equation (3.9) becomes
which gives a contradiction if $cC_p^2(|\Omega ^+|) < 1$
.
This occurs if we consider $\delta := \delta (\lambda,q, p, g, \text {dist}(\tilde {\Omega }, \partial {\Omega }))$
sufficiently small such that $|\tilde {\Omega }| \leq \delta$
(that satisfies our assumption) and $C_p(|\Omega ^+|)=C_p(|\tilde {\Omega } \cap \text {supp}(u-u_{\lambda })^+|) < \sqrt {\frac {1}{c}}$
jointly.
Then, taking into account the boundary condition,
This shows that actually $(u-u_{\lambda })^+=0$
on $\tilde {\Omega }$
, that is
Case: $\mathbf {p\geq 2}$
and $\mathbf {m_A \geq 0}$
.
Since $m_A \geq 0$
, by (2.6) we get that $A(t)$
is a non-decreasing function; then we can use [Reference Cianchi and Maz'ya5, lemma 4.3], namely
Hence, (3.8) becomes
Moreover, exploiting theorem 2.4, we can use in the right-hand side of (3.11) the weighted Poincaré inequality (see corollary 2.8) with $\rho = A(|\nabla u|)$
. Therefore, we get
which gives a contradiction if $cC_{pp}^2(|\Omega ^+|) < \frac {1}{3}$
.
Arguing as in the case $1 < p <2$
, we obtain
Case: $\mathbf {p\geq 2}$
, $\mathbf {m_A < 0}$
and $\mathbf {A(t) \geq \tau (C)>0}$
with $\mathbf {\tau }$
positive constant that depends on $\mathbf {C\subset [0,+\infty )}$
compact.
Using (2.1) in the left-hand side of (3.8), one has
As we remarked in the first part of the proof, $|\nabla u|$
and $|\nabla u_{\lambda }|$
are in $L^{\infty }(\Omega ^+)$
; hence there exists $\tilde {K} \in \mathbb {R^+}$
such that
Therefore, if we consider the compact set $[0,\tilde {K}]$
, equation (3.13) becomes
Hence, substituting in (3.8) and using classic Poincaré inequality, we have
which gives a contradiction if $cC_p(|\Omega ^+|) < \bar {c}\tau (\tilde {K})$
.
Arguing as in the case $1 < p <2$
, we obtain
Case: $\mathbf {p\geq 2}$
, $\mathbf {m_A < 0}$
and there exists $\mathbf {\tilde {C} \subset [0,+\infty )}$
compact such that $\mathbf {\inf \limits _{\tilde {C}} A(t)=0}$
.
By (1.4), $A(t) \rightarrow +\infty$
for $t \rightarrow +\infty$
, then if there exists $\tilde {C} \subset [0,+\infty )$
compact set such that $\inf \limits _{\tilde {C}} A(t)=0$
, we have that necessarily $\liminf \limits _{t \rightarrow 0^+}A(t)=0$
.
Therefore, by (1.5), there exists $\delta >0$
such that $A(t)$
is a non-decreasing function on $I_0:=(0,\delta )$
.
Let us set
where $K$
is chosen as in (1.3).
Using the weighted Poincaré inequality in the right-hand side of (3.8) with $\rho =A(|\nabla u|)$
, we get
For the left-hand side of (3.8) we use (3.10) on the set $\Omega ^+_1$
since $A(t)$
is non-decreasing, while we use (2.2) on $\Omega ^+_2$
and $\Omega ^+_3$
. Then
Using (3.15) and (3.16) in (3.8), we get
and arguing as for $1< p<2$
we get that $u\leq u_{\lambda } \text { in } \tilde {\Omega }.$
3.2 Symmetry result
Now we can prove our symmetry and monotonicity result.
Proof of theorem 1.3. The proof follows via the moving planes technique [Reference Berestycki and Nirenberg2, Reference Damascelli and Pacella14, Reference Damascelli and Sciunzi15, Reference Esposito, Sciunzi and Trombetta18, Reference Gidas, Ni and Nirenberg21]. First we define
We start showing that
To prove this, we observe that, since $u(a)=0$
and $\Omega$
is smooth and strictly convex, by Höpf Lemma [Reference Pucci and Serrin27, theorem 5.5.1]
Moreover we note that, by the Höpf Lemma applied to a solution $u$
of problem (1.1), we know that
Hence, we can consider $a< \lambda < a+\varepsilon$
with $\varepsilon$
a small positive constant such that
So, by monotonicity, we have that
Now we define
We want to show that $u \leq u_{\lambda }$
in $\Omega _{\lambda }$
for every $\lambda \in (a, 0]$
, namely that:
Assume by contradiction that $\bar {\lambda } <0$
. We will prove that $u \leq u_{\bar {\lambda }+\tau }$
in $\Omega _{\bar {\lambda }+\tau }$
for any $0<\tau <\bar {\tau }$
with $\bar {\tau }$
small enough.
By continuity we have that $u\leq u_{\bar {\lambda }}$
in $\Omega _{\bar {\lambda }} {\setminus} 0_{\bar {\lambda }}$
.
Let us consider $Z_{u, \bar {\lambda }}^R:= \{x \in \Omega _{\bar {\lambda }}: \nabla u(x)= 0 \vee \nabla u_{\bar {\lambda }}(x) =0 \}$
and open set $A_{\bar {\lambda }} \subset \Omega _{\bar {\lambda }}$
such that
We note that by Höpf Lemma, we can assume that $A_{\bar {\lambda }} \Subset \Omega$
and since $|Z_{u, \bar {\lambda }}^R|=0$
, we can take $A_{\bar {\lambda }}$
of arbitrarily small measure. Since we are working in $\Omega _{\bar {\lambda }}$
, with $\bar {\lambda } < 0$
, the weight $\frac {1}{|x|^p}$
is not singular there. Moreover, in a neighbourhood of $0_{\lambda }$
, by lemma 2.11, we have that $u< u_{\lambda }$
. Since elsewhere $\frac {1}{|x_{\lambda }|^p}$
is not singular and $u, u_{\lambda }$
are bounded, we can use Strong Comparison Principle (see [Reference Pucci and Serrin27, theorem 2.5.2]) to get that, if $\mathcal {C}$
is a connected component of $\Omega _{\bar {\lambda }} {\setminus} Z_{u,\bar {\lambda }}^R$
, then
Actually, we prove that the latter case is not possible. In fact, supposing that $u \equiv u_{\bar {\lambda }}\ \text {in}\ \mathcal {C}$
, if we reflect $\mathcal {C}$
through the hyperplane $T_{\bar {\lambda }}$
, we obtain that
We note that $\bar {\mathcal {C}} \cap \partial \Omega \neq 0$
follows by Dirichlet datum. Then, (3.18) would imply the existence of a local symmetry phenomenon where $(\partial \mathcal {C} {\setminus} T_{\bar {\lambda }}) \cup R_{\bar {\lambda }}(\partial \mathcal {C} {\setminus} T_{\bar {\lambda }}) \subset Z_u$
; therefore $\Omega {\setminus} Z_{u}$
would be not connected, contradicting lemma 2.9.
Then
Now, let us consider a compact set $K \subset \Omega _{\bar {\lambda }} {\setminus} A_{\bar {\lambda }}$
. We get that:
• By lemma 2.11, in a neighbourhood of $0_\lambda$
, $u< u_{\lambda }$ for every $\lambda \in [\bar {\lambda }, \bar {\lambda }+\tau ]$ for any $0<\tau < \bar {\tau }$ with $\bar {\tau }>0$ small.By uniform continuity of $u$
we have that $u < u_{\bar {\lambda }+\tau }$ in $K$; it is equivalent to say that
(3.20)\begin{equation} w := (u-u_{\bar{\lambda}+\tau})^+{=}0 \text{ in } K \end{equation}then(3.21)\begin{equation} \text{supp}(w) \subset \Omega_{\bar{\lambda}+\tau} {\setminus} K. \end{equation}• Taking into account the zero Dirichlet boundary datum and using Höpf Lemma, it is easy to show that, for some $\bar {\delta }>0$
, there exists a tubular neighbourhood such that
(3.22)\begin{equation} u < u_{\bar{\lambda}+\tau} \text{ in } I_{\bar{\delta}}(\partial \Omega)\cap \Omega_{\bar{\lambda}+\tau} \end{equation}for any $0<\tau < \bar {\tau }$.
Instead, for the region near $\partial {\Omega } \cap T_{\bar {\lambda }+\tau }$
, we use the monotonicity properties of solutions obtained by Höpf Lemma. In fact, since $\Omega$ is smooth and strictly convex, if we consider $p \in \partial {\Omega } \cap T_{\bar {\lambda }+\tau }$, we get that
\[ \exists \bar{\delta} >0 \text{ such that } \frac{\partial u}{\partial x_1 }(x) > 0 \text{ for all } x\in I_{\bar{\delta}}(p). \]So we have obtained that(3.23)\begin{equation} u < u_{\bar{\lambda}+\tau} \text{ in } I_{\bar{\delta}}(\partial \Omega) \cap T_{\bar{\lambda}+\tau}. \end{equation}If we denote with $\mathcal {N}_{\bar {\lambda }+\tau }$ a neighbourhood of $\partial \Omega _{\bar {\lambda }+\tau } \cap \partial \Omega$, from (3.22) and (3.23), we obtain that
(3.24)\begin{equation} u < u_{\bar{\lambda}+\tau} \text{ in } \mathcal{N}_{\bar{\lambda}+\tau}. \end{equation}
Summarizing up, in particular we get that
Moreover, equation (3.21), using (3.24), becomes
Now , we choose $|K|$
big enough such that $|\Omega _{\bar {\lambda }+\tau } {\setminus} (K \cup \mathcal {N}_{\bar {\lambda }+\tau })|$
is sufficiently small so that theorem 3.1 works.
Since $K$
is a closed set, using (3.20) and (3.24), it follows
Therefore, by theorem 3.1 it follows that
for any $0<\tau <\bar {\tau }$
.
Exploiting also (3.25), we have that
for any $0<\tau <\bar {\tau }$
, but this gives a contradiction with the definition of $\bar {\lambda }$
. So we have that $\bar {\lambda }=0$
and so
If we perform the moving plane technique in the opposite direction, we obtain that
Then $u$
is symmetric with respect to the hyperplane $\{x_1=0\}$
.
Moreover, the fact that the solution is non-decreasing in the $x_1$
-direction in $\Omega \cap \{x_1 <0\}$
is implicit in the moving plane procedure. Finally, if $\Omega$
is a ball centred at origin, repeating this argument along any direction, it follows that $u$
is radially symmetric. The fact the $\frac {\partial {u}}{\partial {r}}<0$
for $r\neq 0$
, follows by Höpf boundary lemma which works in this case since the level sets are balls and therefore fulfill the interior sphere condition.
Acknowledgments
L. Montoro, L. Muglia and B. Sciunzi are partially supported by PRIN project 2017JPCAPN (Italy): Qualitative and quantitative aspects of nonlinear PDEs, and L. Montoro by Agencia Estatal de Investigación (Spain), project PDI2019-110712GB-100.
Appendix A.
In this appendix we study the existence and the asymptotic behaviour of solutions to the problem
Taking into account [Reference Cianchi and Maz'ya5], the problem (A.1) is the Euler equation of the minimization problem for the functional $J: W_0^{1,p}(B_R) \rightarrow \mathbb {R}$
defined as
where $G: [0,+\infty ) \rightarrow [0, +\infty )$
is defined as
By (1.4), $sA(s) \geq c_2s^{p-1}$
for $s>0$
. Integrating
Our goal is to apply Weierstrass Theorem to prove that the functional $J$
has a global minimum point.
We set $h(x):= \frac {C}{|x|^{p}} \in L^{q}(B_R)$
with $1\leq q<\frac {N}{p}$
.
• First, we prove that $J$
is coercive.Let us consider a minimizing sequence $w_m \in W_0^{1,p}(B_R)$
for $J$. Using (A.3) we get
\begin{align*} J(w_m) & = \int_{B_R} G(|\nabla w_m|)\,{\rm d}x -\int_{B_R}h(x) \ w_m \,{\rm d}x\\ & \geq \frac{c_2}{p} \int_{B_R} |\nabla w_m|^p\,{\rm d}x - \int_{B_R}h(x) |w_m| \,{\rm d}x\\ & \geq \frac{c_2}{p}\left\Vert w_m\right\Vert_{W_0^{1,p}(B_R)}^p-\left\Vert h(x)\right\Vert_{p^{*'}}\left\Vert w_m\right\Vert_{p^*} \\ & \geq \frac{c_2}{p}\left\Vert w_m\right\Vert_{W_0^{1,p}(B_R)}^p -c\left\Vert h(x)\right\Vert_{p^{*'}}\left\Vert w_m\right\Vert_{W_0^{1,p}(B_R)}. \end{align*}Since $p^{*'}< \frac {N}{p}$ and $p> 1$, if $\left \Vert w _m\right \Vert _{W_0^{1,p}(B_R)} \rightarrow +\infty$ we have $J(w_m) \rightarrow +\infty$.• To prove the weak lower semi-continuity of $J$
, we write the functional as
\[ J(w_m)= \int_{B_R} G(|\nabla w_m|)\,{\rm d}x -\int_{B_R}h(x) \ w_m \,{\rm d}x = J_1(w_m)-J_2(w_m). \]Let us consider $w_m \rightharpoonup w$ in $W_0^{1,p}(B_R)$. We take $s< p^*$ and $s'$, conjugate exponent of $s$, such that $s'<\frac {N}{p}$. By compact embedding, up to a subsequence, $w_m \rightarrow w$ in $L^s(B_R)$, hence
\[ |J_2(w_m)-J_2(w)| \leq \int_{B_R} |h(x)w_m-h(x)w|\,{\rm d}x \leq \left\Vert h(x)\right\Vert_{s'}\left\Vert w_m-w\right\Vert_{s} \rightarrow 0 \]for $m\rightarrow +\infty$. Therefore, we have proved that
(A.4)\begin{equation} \lim\limits_{m \rightarrow +\infty} \int_{B_R}h(x)w_m\,{\rm d}x = \int_{B_R}h(x)w\,{\rm d}x. \end{equation}While for the functional $J_1( w_m)$, we first prove the strict convexity.
For $w_m \neq z_m$
,
\begin{align*} & (J'_1(w_m)-J'_1(z_m))(w_m- z_m) \\ & = \int_{B_R} \left(A(|\nabla w_m|)\nabla w_m - A(|\nabla z_m|)\nabla z_m, \nabla (w_m-z_m)\right)\,{\rm d}x\\ & \geq c_2\bar{c} \int_{B_R} (|\nabla w_m|+|\nabla z_m|)^{p-2}|\nabla (w_m-z_m)|^2\,{\rm d}x > 0 \end{align*}then $J_1$ is strictly convex.
Now, let us consider $w_m \rightarrow w$
in $W_0^{1,p}(B_R)$. Using Lagrange Theorem, (A.2), (2.5), Hölder inequality and (1.3), for $\xi \in \left \{\min \{|\nabla w_m|, |\nabla w|\}, \max \{|\nabla w_m|,\right. \left. |\nabla w|\}\right \}$, we have
\begin{align*} & |J_1(w_m)-J_1( w)| \leq \int_{B_R} |G(|\nabla w_m|)-G(|\nabla w|)|\,{\rm d}x \\ & = \int_{B_R}|G'(|\xi|)| \left||\nabla w_m|-|\nabla w|\right|\,{\rm d}x = \int_{B_R} |\xi|A(|\xi|) \left||\nabla w_m|-|\nabla w|\right|\,{\rm d}x \\ & \leq C_K \int_{B_R \cap \{|\xi| < K\}} \left||\nabla w_m|-|\nabla w|\right|\,{\rm d}x\\ & \quad+ c_1 \int_{B_R \cap \{|\xi| \geq K\}} \text{max}\{|\nabla w_m|^{p-1}, |\nabla w|^{p-1}\}\left||\nabla w_m|-|\nabla w|\right| \,{\rm d}x\\ & \leq C_K\left\Vert w_m-w\right\Vert_{W_0^{1,1}(B_R)} \\ & \quad+ c_1 \left(\int_{B_R}\text{max}\{|\nabla w_m|^p, |\nabla w|^p\}\,{\rm d}x \right)^{\frac{p-1}{p}} \left(\int_{B_R} |\nabla(w_m-w)|^p\,{\rm d}x\right)^{\frac{1}{p}}\\ & \leq C_K\left\Vert w_m-w\right\Vert_{W_0^{1,1}(B_R)} + c_1c \left\Vert w_m-w\right\Vert_{W_0^{1,p}(B_R)} \rightarrow 0, \text{ for } m\rightarrow +\infty. \end{align*}Then $J_1(w_m)$ is strong continuous. Since $J_1(w_m)$ is strictly convex, $J_1(w_m)$ is weakly lower semi-continuous.
Hence, using (A.4), we get
\begin{align*} J(w)& =J_1(w)-J_2(w) \leq \liminf\limits_{m \rightarrow +\infty} J_1(w_m)-\lim\limits_{m \rightarrow +\infty} J_2(w_m) \\ & \leq \liminf\limits_{m \rightarrow +\infty} \ ( J_1( w_m)-J_2(w_m)) = \liminf\limits_{m \rightarrow +\infty} J(w_m). \end{align*}Thus $J$ has a global maximum point $w$ that is a weak solution to the problem (A.1). In particular, since $J_1$ is strictly convex and $J_2$ is linear, the functional $J$ is strictly convex. Then, the solution $w$ is unique.
Moreover we point out that the solution $w$
is radial.
In fact, recalling the definition of $x_{\lambda }$
and $u_{\lambda }$
, since $|x|=|x_{\lambda =0}|$
, if we consider $w$
solution to (A.1) and $w_{0}$
, we get
Then, by (2.2) we obtain
and $w=w_0$
. Repeating this argument along any direction, it follows that $w$
is radially symmetric. The radiality can be obtained also by the uniqueness of solution.
Let us now study the asymptotic behaviour of $w$
near the origin.
First, we observe that $w$
is non-increasing with respect to $r$
.
In fact, for $\lambda <0$
, if we consider $w$
and $w_{\lambda }$
that respectively satisfy the following
we get, using the fact that $|x|>|x_\lambda |$
and (2.2), that
Then, $(w-w_{\lambda })^+=0$
, i.e. $w \leq w_{\lambda }$
$\forall {\lambda }<0$
.
If we choose $\varphi = \varphi (|x|)$
as test function in weak formulation of (A.1), we get
Passing in radial coordinates, for $r=|x|$
, we get
Since $w$
is radial, we have proved that $w$
is non-increasing with respect to $r$
. Then $w'$
is negative and $|w'(r)|=-w'(r)$
.
We rewrite (A.5) as
Then
From (A.5), since $Cr^{N-1-p}\geq 0$
, we have that $(-A(|w'|)w'r^{N-1})$
is a non-decreasing non negative function, hence
If $\alpha =0$
, we can apply de l'Hospital Theorem and we obtain
that we can rewrite as
Since $A \in C^{1}(\mathbb {R}^+)$
, $c \neq 0$
and (2.4) holds, we have to consider only the cases $0<|w'|< \tau$
with $\tau$
enough small and $|w'| \geq K$
.
Using (A.8), we get
then, there exists $\tilde {c}>0$
such that
If $|w'| \geq K$
, using (1.3) in (A.9) one has
Let us consider $\varepsilon >0$
. Integrating on $[\varepsilon, R]$
the following
we get
then $w(\varepsilon ) \rightarrow +\infty$
if $\varepsilon \rightarrow 0^+$
.
If $0 <|w'| < \tau$
, using (2.3) in (A.9) we get
Let us consider $\varepsilon >0$
. Integrating on $[\varepsilon, R]$
, one has that
and, using (2.7), we obtain that $\frac {p-1}{m_A+1}\geq 1$
and then $w(\varepsilon ) \rightarrow +\infty$
if $\varepsilon \rightarrow 0^+$
.
If $\alpha >0$
, arguing in (A.6) as in (A.8), we consider only the cases $0<|w'|< \tau$
with $\tau$
enough small and $|w'| \geq K$
.
From (A.6), there exists $\tilde {\alpha } >0$
such that
If $|w'| \geq K$
, using (1.3) in (A.10) one has
Integrating on $[\varepsilon, R]$
the following
since $\frac {N-1}{p-1}>1$
we obtain $w(\varepsilon ) \rightarrow +\infty$
if $\varepsilon \rightarrow 0^+$
.
If $0<|w'|<\tau$
, using (2.3) in (A.10) we get
Considering $\varepsilon >0$
. Integrating on $[\varepsilon, R]$
, one has
and, using (2.7), we obtain that $\frac {N-1}{m_A+1}\geq 1$
and then $w(\varepsilon ) \rightarrow +\infty$
if $\varepsilon \rightarrow 0^+$
.
