Abstract
We provide a classification result for positive solutions to \(-\Delta u=\frac{1}{u^\gamma }\) in the half space, under zero Dirichlet boundary condition.
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1 Introduction
We deal with the classification of positive solutions to the singular problem
where \(N\ge 1\), \(\gamma >1\), \(x\in {\mathbb {R}}^N_+\) is represented by \(x=(x',x_N)\), \(x'\in {\mathbb {R}}^{N-1}\) and \({\mathbb {R}}^N_+:=\{x\in {\mathbb {R}}^N:x_N>0\}\). This is a captivating problem itself but it also arises in the study of limiting scaling arguments at the boundary, in bounded domains, for solutions to
since the term \({1}/{u^\gamma }\) is the leading part near zero, see e.g. [2, 4]. Although the expert reader may guess that the decreasing nature of the nonlinearity is favourable for the application of maximum and comparison principles, we stress that here, solutions are not in the right Sobolev space in order to do that (in particular for \(\gamma >1\)). This causes a deep and challenging issue that we approach introducing a technique based on precise asymptotic estimates. The underlying idea is that the equation is almost harmonic far from the boundary once we deduce precise estimates.
Taking into account the nature of the problem, it follows that, the natural assumption that we shall adopt in all the paper is
Note that the continuity up to the boundary of the solutions can be proved as in [3]. Therefore the equation is understood in the classic meaning in the interior of the domain, or in the variational meaning as in the following:
We will classify all the locally bounded solutions according to the following hypothesis
(hp) There exists \({{\bar{\lambda }}}>0\) such that u is bounded on the set \(\Sigma _{{{\bar{\lambda }}}}\). We set \(\theta \in {\mathbb {R}}\) such that
where the strip \(\Sigma _{{{\bar{\lambda }}}}\) is defined in Sect. 2. Our main result is the following
Theorem 1
Let u be a solution of (\({\mathcal {P}}_{\gamma }\)) fulfilling \(\mathbf{(hp)}\). Then
Consequently either
or
where \(v(t)\in C^2({\mathbb {R}}_+)\cap C(\overline{{\mathbb {R}}_+})\) is the unique solution to
The starting crucial issue in our proof is the accurate study of the asymptotic behavior of the solutions up to the boundary as well as at infinity. Actually we shall show that every solution has at most linear growth far from the boundary, which is a sharp estimate. This analysis will allow us to exploit a celebrated result of Berestycki, Caffarelli and Nirenberg [1] to deduce that the solutions exhibit 1-D symmetry. Then, taking into account the non standard nature of the equation arising from the singular term, we will carry out a ODE analysis to complete our proof.
The paper is organized as follows: in Sect. 2 we provide the proofs of the asymptotic analysis and exploit it to prove the 1-D result. In Sect. 3 we carry out the ODE analysis. We conclude in Sect. 4 with the proof of our main result.
2 Asymptotic analysis and 1-D symmetry
In all the paper we shall use the notation given in the following
Definition 2
Given \(0<a<b\), we define the strip \(\Sigma _{(a,b)}\) as the set given by
We also set \(\Sigma _{(0,b)}:=\Sigma _{b}\).
In all this section we will use some ODEs arguments that are actually contained in the more general analysis of Sect. 3.
We start proving
Lemma 3
Under the assumption \(\mathbf{(hp)}\), it follows that
with \(C=C(\gamma , \theta )\) a positive constant.
Proof
Let us consider the 1-D solution \(w(x_N)\) of
given in (25). Note that \(w_\beta =\beta w\) solves \(-\Delta w_\beta =\beta ^{\gamma +1}/w_\beta ^\gamma \). Therefore, for \(\beta >1\), we have
Now, since
we take \(\beta \) large so that, for \(\beta \ge \theta /w({{\bar{\lambda }}})\) we deduce
Consequently \(u,w_\beta \) are well ordered on the boundary of the strip \(\Sigma _{{{\bar{\lambda }}}}\) (see (3)), namely
In order to prove a comparison principle, we have to take into account that we are working in the unbounded domain \({\mathbb {R}}^N_+\) and both \(u,w_\beta \) lose regularity at the boundary of the half-space \({\mathbb {R}}^N_+\). For this reason we start defining \(\phi _R(x'):{\mathbb {R}}^{N-1}\rightarrow {\mathbb {R}}\) such that
where we recall that a point \(x\in \mathbb {{\mathbb {R}}}^N_+\) is denoted by \(x=(x',x_N)\) with \(x'\in {\mathbb {R}}^{N-1}\) and where
Moreover, let us define the translated function (indeed still a supersolution to (4))
and let \(\varphi _R\) defined as
One can check, using a suitable argument based on the continuity of u and \(w_{\beta , \varepsilon }\), that \(\varphi _R\) is indeed a suitable function test to both problems (\({\mathcal {P}}_{\gamma }\)) and (4). Let us also define the cylinder
Then using \(\varphi _R\) in the weak formulations satisfied by u and by \(w_\varepsilon \), we obtain
and
Subtracting the last inequalities we obtain
We observe that there exists a positive constant \(\eta =\eta (\gamma ,\theta )\) such that
since u is bounded on \(\Sigma _{{{\bar{\lambda }}}}\) by \(\mathbf{(hp)}\). Moreover using Young inequality and (7), we also deduce that
Therefore, using (9) in (8), we get
For \(\delta \) small fixed we deduce that
For R large we have that \({C(\delta )}/{R^2}<\eta \) and therefore
By Fatou’s Lemma for \(R\rightarrow +\infty \), we obtain
Exploiting (6) we deduce that actually
Finally, by continuity, we have that \(u\le w_{\beta }\). Recalling (5) and (26) we get thesis. \(\square \)
Without assuming any a priori assumption, we prove the following
Lemma 4
There exists a constant \(C=C(\gamma )\) such that
Proof
Let us consider the first eigenfunction \(\varphi _1 \in C^2(\overline{B_1(0)})\) solution to
Setting
with \(C>0\) to be chosen, by a straightforward computations
Using (11), we obtain
For \(C=C(\gamma )\) small enough, we get \(\alpha (x)<1\) and therefore w is a subsolution to \(-\Delta w= w^{-\gamma }\) in \(B_1(0)\).
Let now \(x_0=(x_0', x_{0,N})\in {\mathbb {R}}^N_+\) and set
where \(R= x_{0,N}\). We have
Let u be a solution to (\({\mathcal {P}}_{\gamma }\)); we observe that
For \(\varepsilon >0\), we can use
as a test function in (13) obtaining that
Then \(u\ge w_{x_0, R}\) in \(B_R(x_0)\), hence
Recalling that \(R= x_{0,N}\), since \(x_0\) is arbitrary, we obtain the thesis. \(\square \)
Proposition 5
Under the assumption (hp), there exists a positive constant \(C=C(\gamma , \theta , N)\) such that
in the set \({\mathbb {R}}^N_+{\setminus } \Sigma _{{{\bar{\lambda }}}}\).
Proof
In what follows, without loss of generality, from \(\mathbf (hp)\), using the natural scaling for the problem (\({\mathcal {P}}_{\gamma }\))
we may assume that our solution u is indeed bounded in the strip \(\Sigma _{(0,2)}\).
Let \(x_0\in {\mathbb {R}}^+_N\), \(x_0=(x_0';x_{0,N})\), with \(x_{0,N}>2\) and let R>0, such that
Let \(u_R(x)=u(x_0+R(x-x_0))\); then
\(u_R>0\) in \(B_4(x_0)\). Since in Lemma 4 we showed that
in the whole \({\mathbb {R}}^N_+\), we infer that
where C is the positive constant given in Lemma 4. Therefore
We point out that, from the arbitrariness of \(x_0\), we deduce that
Consequently from (16), we deduce that
By Harnack inequality [5, Theorem 8.20] we have that
where \(C_H=C_H(\gamma ,N)\). Now let us consider, for \(N\ge 3\), the fundamental solution of the Laplace operator. So let us define
that fulfills
for all \(c,k\in {\mathbb {R}}\). Exploiting (17) with \(u_0=u(x_0)\), we infer that
hence
We new choose c amd k such that
Direct computation shows that the system (18) holds for
with \({\tilde{c}}_N=C^{-1}_H4^{N-2}/({2^{N-2}-1})\). Summarizing we have that
Using \((v_{c,k}-u-\varepsilon )^+\), for \(\varepsilon >0\), as test function in (20) (see also (18)), we get
namely \(v_{c,k}\le u+\varepsilon ,\) for all \(\varepsilon >0\). Therefore
Therefore
where in the last line we used (19). Finally by Lagrange theorem ve have
Therefore, since \(u\in L^{\infty }( \Sigma _{(0,2)})\), we deduce
for some constant \(C=C(\gamma ,{{\bar{\lambda }}}, \theta , N)\) that does not depend on R. Since \(x_0\) is arbitrary we obtain that
Scaling back, using (14) and (15) we obtain the thesis for \(N\ge 3\). The case \(N=2\) follows repeating the same argument but replacing the fundamental solutions with the logarithmic one. \(\square \)
It is straightforward to deduce the following
Corollary 6
Under the assumption (hp), u has linear growth, namely there exits \(c_1,c_2>0\) depending on \(\gamma , \theta , N\) such that
Proposition 7
Under the assumption (hp), there exists \(C=C(\gamma , \theta , N)\) such that the following hold
Proof
Let us start noticing that, without loss of generality, we may and do assume that the solution is bounded in the strip \(\Sigma _{2{{\bar{\lambda }}}}\). Let now \(P \in \Sigma _{{{\bar{\lambda }}}}\), with \(P=(x',x_N)\). Set \(R =x_N\) and let us define
Then \(u_R\) satisfies
Exploiting Lemma 4, we deduce that
with \(C=C(\gamma )\), i.e. \(1/u_\delta ^\gamma \in L^{\infty }(B_{1/2} \left( {P}/{R}\right) ).\) On the other hand Lemma 3 we also get
where \(C=C(\gamma , {{\bar{\lambda }}})\). By regularity estimates, see e.g. [5, Theorem 3.9]
Consequently we deduce
and hence
thus proving (i). Arguing now in the same way, let us define
By Proposition 5 we have that \(u_R\le C(\gamma ,\theta , N)\) and it satisfies
where \(h(x)\le C(\gamma )\) in \(B_{1/2}\left( {P}/{R}\right) \) (see Lemma 4). By regularity estimates \(|\nabla u_R|\le C\) in \(B_{1/4}\left( {P}/{R}\right) \) and therefore \(|\nabla u(x)|\le C\) in \(B_{R/4}\left( {P}\right) \). \(\square \)
We are now ready to prove the \(1-D\) symmetry result.
Theorem 8
Let u be a solution to (\({\mathcal {P}}_{\gamma }\)). Under the assumption \(\mathbf{(hp)}\)
Proof
Let \(\tau ,\sigma \in {\mathbb {R}}\), with \(\sigma >0\), chosen opportunely later. Define
and for \(i=1,\ldots , N-1\). Obviously \(-\Delta u_{\tau ,\sigma } =1/u^\gamma _{\tau ,\sigma }\) in \({\mathbb {R}}^N_+\). Setting \(z:=u-u_{\tau ,\sigma }\), we get
In the following we use [1, Lemma 2.1]. From Lemmas 3 and 4 we infer that there exist constants \(C_1, C_2\) such that
For \(\sigma >0\), by (23) there exists \(\rho >0\) and \({{\hat{\lambda }}}<{{\bar{\lambda }}}\) (actually think to \({{\hat{\lambda }}}\approx 0\)) such that \(u<\rho \) in \(\Sigma _{{{\hat{\lambda }}}}\) and \(u_{\tau ,\sigma }>2\rho \) in \(\Sigma _{{{\hat{\lambda }}}}\), for all \(\tau \in {\mathbb {R}}\). Defining the strip \(D:={\mathbb {R}}^N_+{\setminus } \Sigma _{{{\hat{\lambda }}}}\), \(z\le 0\) on \(\partial D\) holds. Moreover using Lagrange theorem jointly to Proposition 7, we also get that z is bounded in \({{\overline{D}}}\).
Setting
we observe that c(x) is continuous in \({{\overline{D}}}\) (indeed \(u,u_{\tau ,\sigma }\ge c>0\) in D, see (23)) and \(c(x)\le 0\) (in D) by its own definition. By (22) applying [1, Lemma 2.1] to the problem
we obtain \(z:=u-u_{\tau ,\sigma }\le 0\) in D. We point out that already in \(\Sigma _{{{\hat{\lambda }}}}\cup \{x_n={{\hat{\lambda }}}\}\), we have \(u-u_{\tau ,\sigma }\le 0\). Hence \(u\le u_{\tau ,\sigma }\) in \({\mathbb {R}}^N_+\).
Letting \(\sigma \rightarrow 0\) we obtain
By the arbitrariness of \(\tau \) we deduce that \(u=u(x_N)\). \(\square \)
3 ODE analysis
We start with the study of the one dimensional problem. We consider the following
It is straighforward to verify that the function
where
is a solution of (24).
A scaling argument. Let \(v\in C^2({\mathbb {R}}_+)\cap C(\overline{{\mathbb {R}}_+})\) be a solution of problem (24). Let
for a given \(\lambda >0\) and \(\alpha \in {\mathbb {R}}\). Then \(\sigma (0)=0\), \(\sigma (t)>0\) and for \(t>0\)
Choosing \(\alpha =-{2}/{(1+\gamma )}\), then \(\sigma \) satisfies (24) too. A similar computation showed that the same scaling works in the main problem (\({\mathcal {P}}_{\gamma }\)).
Let us define, by means of (25), the function
and notice that, since \(u(t)<w(t)\) for \(t>0\),
Since \(w(0)=u(0)=0\), \(w(t)>0\) and
then w is a supersolution for problem (24). Moreover \(w'(t)\rightarrow C_\gamma \) as \(t\rightarrow +\infty \).
Taking into account the supersolution w, let us fix \(t_0>0\) and consider the following problem
Proposition 9
Each solution of problem (28) is such that \(v(t)>w(t)\) for \(t\ge t_0\) and there exists (finite) \(\lim \nolimits _{t\rightarrow \infty }v'(t)\ge C_\gamma \).
Proof
A unique local solution for problem (28) there exists; indeed, it can be proved, that the solution is defined in the whole \([t_0, +\infty )\) since it is concave. Moreover
Since \(\left( v'(t_0)-w'(t_0)\right) '>0\), there exists \(\delta >0\) such that for all \(t\in [t_0,t_0+\delta )\), \(\left( v'(t)-w'(t)\right) '>0\). Actually \(\left( v'(t)-w'(t)\right) '>0\) for each \(t>t_0\); if not, denoting by \(\tau :=\sup \{t>t_0:\left( v'(t)-w'(t)\right) '>0\}\), it follows that
hence
Since \((v'-w')\) is continuous and (strictly) increasing on the interval \([t_0,\tau )\), and \(v(t_0)>w(t_0)\), therefore \(v(\tau )>w(\tau )\). This contradict (29). As a consequence, \(v(t)>w(t)\) for \(t\ge t_0\).
The solution of (28) is positive on \([t_0,+\infty )\), therefore \(v''(t)\) is negative on the same interval. This implies that \(v'(t)\) is decreasing and its limit there exists for \(t\rightarrow \infty \). Thus \(\lim \nolimits _{t\rightarrow \infty }v'(t)\ge \lim \nolimits _{t\rightarrow \infty }w'(t)= C_\gamma \). \(\square \)
Lemma 10
For any \(L\in {\mathbb {R}}_+\), there exists a solution \(\tilde{v}\) for the problem
Proof
We start proving that, choosing \( v'(t_0)>{v(t_0)}/{t_0}\) in (28), then there exists \(\tau _0\in (0,t_0]\) such that v(t) given in Proposition 9, can be extended as a solution of
Indeed each extension of v(t) for \(t<t_0\) is such that \(v''(t)\le 0\) and therefore the graph of v(t) lies below to the tangent line to v(t) in \((t_0,v(t_0))\). Since \( v'(t_0)>{v(t_0)}/{t_0}\), then a such \(\tau _0>0\) exists.
Let \(v_0(t)\) be a such solution, let us define \(\tilde{v}(t):=v_0(t+\tau _0)\). Then \(\tilde{v}(0)=0\) and verifies (30). \(\square \)
Theorem 11
Let \(M>0\) be fixed. Then there exists a solution to
and the solution is unique.
Proof
Let v be a solution of problem (30). Let
where \(L:= \lim \nolimits _{t\rightarrow +\infty }v'(t)\). By the scaling (27), we have
is a solution of (31) and since \(v'(t)\rightarrow L\) as \(t\rightarrow +\infty \), \(w'(t)\rightarrow M\) as \(t\rightarrow +\infty \).
About the uniqueness, let us consider (by contradiction) \(w_1,w_2\) two different solutions of (31). At first, let us assume that there exists \(t_0>0\), the smallest value for which \(w_1(t_0)=w_2(t_0)\). Taking into account the initial condition \(w_1(0)=w_2(0)=0\) and that \(w_1,w_2\) are continuous, by the weak comparison principle it follows that \(w_1(t)=w_2(t)\) on the interval \([0,t_0]\). Indeed, let us suppose without loss of generality, that \(w_1\le w_2\) in \([0,t_0]\); for any \(\varepsilon >0\), let \(\varphi :=(w_2-w_1-\varepsilon )\) be a test function for problems (30) and (31). So we have
Then \(w_1 = w_2+\varepsilon \) in \([0,t_0]\) for all \(\varepsilon >0\), therefore \(w_1=w_2\) in \([0,t_0]\). As a rule \(w_1(t_0)=w_2(t_0)\) and \(w_1'(t_0)=w_2'(t_0)\) then \(w_1= w_2\) in \({\mathbb {R}}^+\) by uniqueness for ODEs (note that \(w_1,w_2>0\) in \({\mathbb {R}}^+\) so that \(-w''=w^{-\gamma }\) is a regular ODE). Consequently, different solutions \(w_1\) and \(w_2\) do not cross.
From now on we may assume that \(w_1 < w_2\) for all \(t\in {\mathbb {R}}^+\). Notice that,
Since \(w_1'(t),w_2'(t)\rightarrow M\) as \(t\rightarrow +\infty \), then \(w_1'(t)-w_2'(t)>0\) for all \(t\in {\mathbb {R}}^+\) namely \(w_1-w_2\) should be increasing in \({\mathbb {R}}^+\) causing \(w_1=w_2\) in \({\mathbb {R}}^+\). \(\square \)
4 Conclusion: proof of Theorem 1
Once that Theorem 8 is in force and therefore we know that
we get that u is a positive solution to
with \(u(0)=0\). Therefore the ODEs analysis of Sect. 3 allows us to conclude that, either the solution is given by (25) or has linear growth and is completely classified by Theorem 11, taking into account the scaling in (27).
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Acknowledgements
L. Montoro 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.
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Montoro, L., Muglia, L. & Sciunzi, B. Classification of solutions to \(-\Delta u={u^{-\gamma }}\) in the half-space. Math. Ann. 389, 3163–3179 (2024). https://doi.org/10.1007/s00208-023-02717-4
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DOI: https://doi.org/10.1007/s00208-023-02717-4