Abstract
Given \(p>1\) and f Lipschitz, under appropriate assumptions on the smoothness of the bounded domain \(\varOmega \subset \mathbb {R}^{N}, \ N\ge 1\), we give a precise description of the asymptotic behaviour of the gradient of the unique solution of
In particular, we show that there exists a corrector function S, finite sum of singular terms, such that
Moreover, we prove that
where \(\nu \) is the outward unit normal to \(\partial \varOmega \).
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1 Introduction
This paper is devoted to the study of semilinear elliptic problems with explosive boundary conditions; more precisely, we are interested in the qualitative behaviour of solutions of
where \(\varOmega \) is a bounded smooth domain of \(\mathbb {R}^N\), with \(N\ge 1\), \(g\in C^1(\mathbb {R})\) is such that
and f is a Lipschitz continuous function. Here, solutions are meant in the classical sense, i.e. \(C^2(\varOmega )\) functions which satisfy the differential equation above pointwise and such that
In the literature, solutions that blow up at the boundary of the domain are known as large solutions. Looking naively at (1.1), one naturally wonders under which assumptions on g the existence of a large solution is assured, if the monotonicity assumption on g implies uniqueness of solution and how such a solution behaves near the boundary.
In the seminal works by Keller and Osserman (see [19, 28]), it is proved that the necessary and sufficient condition for the existence of a large solution for problem (1.1) is the following:
This growth condition at infinity, known as Keller–Osserman condition, arises solving the one-dimensional problem
We stress that, in fact, \(\phi (s)=\psi ^{-1}(s)\) solves problem (1.4). We refer the interested reader to [14] (see also the references cited therein) for existence issues with no monotonicity assumptions on g.
Uniqueness is not a trivial task in the sense that it is not known if the monotonicity of g is a sufficient condition for it; we refer to [26], where it is proved that if g is convex then (1.1) admits a unique large solution, and to [15] (see also [5]), where it is shown that assumptions of the type
imply uniqueness of large solution. It is worthy to mention that the special case \(g(s)=|s|^{p-1}s\) with \(p>1\) satisfies the latter condition.
Let us point out now that the function \(\phi \) defined in (1.4) is strongly related to the boundary behaviour of solutions of (1.1). In [4, 5], it has been proved that the behaviour of u is, in some sense, one dimensional near the boundary, i.e. it holds that
Moreover, if g is such that
then
namely the first-order term in the asymptotic of u near the boundary only depends on the corresponding ODE (1.4) and in particular is not affected by the geometry of the domain. In [7], the authors improve (1.6); assuming in addition to (1.5) that
they prove that
where the positive constant c depends on the mean curvature of the boundary of \(\varOmega \). After this first clue, the influence of the geometry of \(\partial \varOmega \) in the expansion of u has been studied in [21, 29] under different assumptions on g. The most general result in this direction has been proved in [8]; in order to state it, we need to define
and to assume that
where
Assuming (1.7), together with (1.3) and (1.5), it follows that
where H is a smooth function whose restriction to \(\partial \varOmega \) coincides with the mean curvature of the domain; moreover, it is worth stressing that (1.7) implies
The relation above, together with (1.8), tells us that the second-order contribution to the explosion of u is affected by the geometry of the domain through the mean curvature of \(\partial \varOmega \).
More recently in [12] (see also [10]), by means of an interesting application of the contraction theorem, all the singular terms of the asymptotic of u have been implicitly calculated in the special case \(\varOmega =B\).
For power-type nonlinearities, it is also possible to obtain the first asymptotic of the gradient of the solution by means of scaling arguments. In particular, in [4, 6] (see also [30]) it is proved that if
it holds true that
where \(\nu \) is the unit normal to \(\partial \varOmega \) (recall that \(\nu (\bar{x})=-\nabla d(\bar{x})\) for \(\bar{x}\in \partial \varOmega \)) and \(\tau \in \mathbb {S}^{N-1}\) is such that \(\tau (\bar{x})\cdot \nu (\bar{x})=0\) for every \(\bar{x}\in \partial \varOmega \). However, a general result for the second-order term in the expansion of \(\nabla u\) in the same spirit of (1.8) is not available in the literature (see anyway [3] for a partial result in convex domains).
Our aim is to complete the picture of the asymptotic behaviour of the gradient of solutions of problem (1.10) in the case \(g(s)=|s|^{p-1}s\), with \(p>1\) and \(\varOmega \) smooth enough. Thus, the problem we deal with is
where \(f\in W^{1,\infty }(\varOmega )\). It is easy to verify that with such a choice of g, assumptions (1.3)–(1.7) are satisfied. It is also worth to recall that in this case problem (1.10) has a unique large solution and that the function \(\phi \) defined in (1.4) has the following explicit form:
The result that we present in this paper will describe not only the second-order behaviour of the gradient of the large solution of (1.10), but also the complete asymptotic expansion of all the singular terms of u and \(\nabla u\), for every arbitrary sufficiently smooth domain and every \(p>1\). As a by-product of this expansion, we will be able to provide the expected second-order asymptotic for the normal and tangential components of \(\nabla u\) with respect to \(\partial \varOmega \). Indeed, we will prove
where \(c(\alpha ,N)\) is a precise constant that depends only on \(\alpha \) and N (see Corollary 1.4 for more details). More in general, we will be able to prove (see Theorem 1.3 for the precise statement) that there exists a unique explicit function S, sum of \([\alpha ] +1\) singular terms where \(\alpha \) is as in (1.11), such that
Let us say that the formula above expresses the leitmotiv of the paper, that is, try to find an explicit simple corrector function that describes the explosive behaviour of u.
Moreover, using a scaling argument and the previously obtained information on z, in Theorem 2.9 we prove that the function z satisfies the following boundary conditions:
The latter condition is a weak form of expressing the fact that the normal derivative of z is zero at the boundary of \(\varOmega \).
Finally, we consider a more general class of nonlinearities that will be easily treated with an extension of our method.
Before stating precisely our main results, we need to give some notation.
1.1 Notation
We shall often work in tubular neighbourhoods of \(\partial \varOmega \) of the type
We recall that \(\varOmega \) is always at least of class \(C^2\). Hence, the function dist\((\cdot ,\partial \varOmega )\) distance from the boundary is well defined and twice differentiable near \(\partial \varOmega \). More precisely, the following theorem, proved in [20], gives the relation between the regularity of the boundary and the regularity of the distance function.
Theorem 1.1
(Theorem 3 in [20]) Let \(\varOmega \) be a domain of class \(C^{\gamma }\) with \(\gamma \ge 2\). Then,
Thanks to the previous theorem, we can define the following smooth versions of the distance function.
Definition 1.2
Let \(\varOmega \) be a domain of class \(C^{\gamma }\) with \(\gamma \ge 2\), and let \(\bar{\delta }>0\) be given by (1.13). Then, we define the regularized distance as a function \(d\in C^{\gamma }(\varOmega )\) such that \(d(x)=\text{ dist }(x,\partial \varOmega )\) for every x that belongs to \(\varOmega _{\bar{\delta }}\). We moreover denote \(d_n(x):=d(x)+\frac{1}{n}\).
It is worthy to stress that \(d_n(\cdot )\) inherits from dist\((\cdot ,\partial \varOmega )\) the following important properties
where \(\nu \) is the outward normal to \(\partial \varOmega \) and \(H(\bar{x})\) is the mean curvature at \(\bar{x}\in \partial \varOmega \).
Finally, unless otherwise specified, we indicate with C a constant that depends only on the data of the problem and that can vary line to line also in the proof on the same theorem.
1.2 Main results
The ansatz that guides our approach is that it is possible to give an explicit description of the explosive behaviour of the large solution u and of its gradient \(\nabla u\) by means of a finite sum of singular terms. Inspired by (1.6), (1.8) and (1.9), we conjecture that
where \(\sigma _{k}\) with \(k=0,1,\ldots \) are smooth functions, and define the following regularized function
Hence, the first question we want to answer is:
Can we find \(\sigma _{k}\) with \(k=0,1,\ldots \) such that z and \(|\nabla z|\) belong to \(L^{\mathcal {1}}(\varOmega )\)?
Of course, the functions \(\sigma _{1}, \dots , \sigma _{k}\) shall take into account several characteristics of the problem, among others the geometry of the domain. Notice moreover that the definition (1.14) suggests that we need \([\alpha ]+2\) terms for having \(z\in W^{1,\infty }(\varOmega )\). Indeed, we have the following result.
Theorem 1.3
Let us assume \(p>1\) and fix \(\alpha :=\frac{2}{p-1}\). Let \(\varOmega \) be a bounded domain of class \(C^{[\alpha ]+5}\) with \([\alpha ]\) the integer part of \(\alpha \), let f belong to \(W^{1,\infty }(\varOmega )\), and let u be the unique large solution of (1.10). Let us define the following functions
Then, \(\sigma _{k}\in C(\overline{\varOmega })^{[\alpha ]+5-k}\) with \(k=0,\ldots ,[\alpha ]+1\), and the function \(S\in C^4(\varOmega )\), defined as
is such that
Moreover, it also holds true
Remark 1.4
Let us stress that the higher the value of \(\alpha \) (i.e. the closer p is to 1), the higher the number of singular terms is and the higher the regularity of \(\varOmega \) has to be.
Moreover, if we split the above estimate along normal and tangential directions we get a very precise estimate of all the singular terms in the expansion of the gradient. More specifically, we have that
while
\(\forall \tau \in \mathbb {S}^{N-1}\) such that \(\tau \cdot \nu =0\).
From (1.18) and (1.19), we easily obtain the second-order asymptotic of the gradient (1.12) mentioned in Introduction.
The core of Theorem 1.3 is a Bernstein-type estimate for \(|\nabla z|\). This type of technique, already used in the framework of large solutions for quasilinear problem in [22] (see also [23]), has been originally developed in [24, 25], and it allows to obtain \(L^{\mathcal {1}}(\varOmega )\)-estimates for solutions of a vast class of boundary value problems. Of course, we do not know a priori the boundary condition (if any) satisfied by \(u-S\); thus, it is not possible to obtain Bernstein estimates directly for z and \(|\nabla z|\). We overcome this obstacle arguing by approximation and considering the following regularized corrector function
where \(\sigma _{0},\ldots ,\sigma _{[\alpha ]+1}\) are the functions defined in (1.15), and the following approximated problem
Moreover, we define \(z_n(x):=u_n(x)-S_n(x)\) that solves
where
Let us stress that the choice of the Neumann boundary condition in (1.21) and in turn in (1.22) is not the only possible, but it is the most convenient for our scope; indeed, Neumann problems are particularly suited for the implementation of the previously mentioned Bernstein estimates. Observe at this point that \(u_n\) converges, at least in \(C^2_\mathrm{loc} (\varOmega )\) (see Proposition 2.6) to the unique large solution to (1.10), and this in turn implies that \(z_n\rightarrow z:=u-S\) in \(C^2_\mathrm{loc}(\varOmega )\) where \(S=\displaystyle \lim _{n\rightarrow \infty }S_n\). Hence, once a uniform estimate (with respect to n) in \(W^{1,\infty }(\varOmega )\) is obtained for the solution \(z_n\) of (1.22), it can be inherited by z as n diverges.
The proof of Theorem 1.3 is divided into the following main steps:
-
we prove that there exists a constant \(\bar{C}=\bar{C}(\sigma _{0},\ldots , \sigma _{[\alpha ]+1},f)\) such that \(d_n|\widetilde{f}_n|+d_n^2|\nabla \widetilde{f}_n|\le \bar{C}d^{1+[\alpha ]-\alpha }\) for every \(n\in \mathbb {N}\);
-
we show that for every \(n\in \mathbb {N}\) problem (1.21) admits a solution \(u_n\) and we describe the first-order behaviour of \(u_n\) near the boundary;
-
through a Bernstein-type estimate, we show that there exists a positive constant \(B=B(\sigma _{0},\ldots , \sigma _{[\alpha ]+1},f)\) such that \(\Vert z_n\Vert _{W^{1,\infty }(\varOmega )}\le B\) for every \(n\in \mathbb {N}\). This implies that \(\Vert z\Vert _{W^{1,\infty }(\varOmega )}\le B\).
Hence, Theorem 1.3 tells us that \(z\in C^2(\varOmega )\) satisfies
where
and \(\displaystyle \widetilde{f}=\lim _{n\rightarrow \infty }\widetilde{f}_n\). Note that so far we do not have any information on the boundary behaviour of z, apart from the fact that is globally Lipschitz continuous. Thus, it is natural to wonder if z satisfies some boundary condition; and indeed using scaling arguments, in the same spirit of [4, 30], we prove (1.17).
Let us now consider a class of nonlinearities for which Theorem 2.9 can be generalized with minor modifications. Let us thus focus on the following problem
where \(p>1\), \(h\in C^4(\bar{\varOmega })\) is such that for \(0<A<B\)
and \(r\in C^1(\bar{\varOmega }\times \mathbb {R})\) satisfies
In Theorem 2.7 of [5], it is proved that for any bounded domain \(\varOmega \) of class \(C^2\), under the assumptions (1.26) and (1.27), problem (1.25) admits a positive large solution; moreover, every large solution u of (1.25) has the following asymptotic behaviour near \(\partial \varOmega \)
Now, we make additional growth assumptions on the function r(x, s) in order to be able to implement the Bernstein technique as in Theorem 1.3. We require that
where \(r_x:=\nabla _x r\) and \(r_s:=\frac{\partial r}{\partial s}\). As a first consequence of (1.29), we deduce that for \(1<q<p\) the function \(g(x,s):=h(x)|s|^{p-1}s-r(x,s)\) satisfies
Indeed,
Thus, using (1.28) and (1.30) we can take advantage of (the proof of) Theorem 2 of [15] to infer that problem (1.25) admits a unique large solution.
We stress here that we obtain the asymptotic expansion of large solutions and their gradient via an approximation procedure; thus, in the absence of a uniqueness result, our method gives a description only of the large solution obtained by the approximating scheme, i.e. the minimal large solution.
We can state our last result.
Theorem 1.5
Let us assume \(p>1\), fix \(\alpha :=\frac{2}{p-1}\), and let \(\varOmega \) be a bounded domain of class \(C^{[\alpha ]+5}\). Assume moreover that (1.26), (1.27) and (1.29) hold true. Then, there exist functions \(\sigma _{h,k}=\sigma _{h,k}(p,\varOmega ,h)\) (see (2.21) for the precise definition) with \(\sigma _{h,k}\in C(\overline{\varOmega })^{[\alpha ]+5-k}\)\(k=0,\ldots ,[\alpha ]+1\), such that, defining \(S_h\) as
it results \(u -S_h \in W^{1,\infty }(\varOmega )\) and \(z(\bar{x}):=u(\bar{x})-S_h(\bar{x})=0\) for every \(\bar{x}\in \partial \varOmega \). If moreover we assume
we also have that
Let us stress that the functions \(\sigma _{h,k}\) do not depend on the function r, due to assumptions (1.29). Indeed, these growth conditions imply that the contribution of the perturbation r(x, s) does not affect the asymptotic behaviour prescribed by \(h(x)|s|^{p-1}s\).
On the other hand, the presence of the weight h requires some modifications in the definition of the corrector function \(S_h\). This in turn yields to even more involved formulas for \(\sigma _{h,0},\ldots ,\sigma _{h,[\alpha ]+1}\) than (1.15). Notice that
namely the first-order behaviour does not see the influence of the weight that comes into play from the second one onward. As a last comment to Theorem 1.5, notice that, in order to recover the Neumann boundary conditions for z, the additional growth assumption (1.32) is needed (see Remark 2.2).
Unfortunately, we are not able to treat problem (1.1) with g that satisfies just (1.2) and (1.3). The main obstacles in considering a general g(s) [that satisfies anyway (1.2) and (1.3)] are, on the one side, that the simple structure of the corrector function S is lost and, on the other, that it becomes much harder to manipulate terms as \(g(z+S)-g(S)\).
2 Gradient bound
2.1 The choice of \(S_n\)
In this first section, we determine the regularity of the functions \(\sigma _{k}\), \(k=0,\ldots ,[\alpha ]+1\), defined in (1.15) and we show that \(\widetilde{f}_n\), defined in (1.23), is such that
We prove a slightly more general result that emphasizes the relationship between the number of elements of \(S_n\) and the required regularity of \(\varOmega \).
Proposition 2.1
Let us take a natural number \(M\in [0,[\alpha ]+1]\), \(\varOmega \) a bounded open domain of class \(C^{M+4}\), \(\sigma _{k}\) as in (1.15) with \(k=0,\ldots , M\) and \(S_n\) as in (1.20). Then, we have that \(\sigma _{k}\in C(\bar{\varOmega })^{M+3-k}\) with \(k=0,\ldots ,M\) and that there exists \(n_0=n_0(\sigma _{0},\ldots ,\sigma _{M})\) such that for every \(n>n_0\)
where \(C=C(N,\alpha ,\partial \varOmega )\).
Proof
Note at first that the positive root of \((k-\alpha )(k-\alpha -1)-(2+\alpha )(\alpha +1)=0\) (seen as an equation in the variable k) is bigger then \([\alpha ]+1\): indeed denoting by \(k_i\)\(i=1,2\) the two roots, we have that
Thus, the denominator in (1.15) is always different from zero. As far as the regularity of the terms involved in (2.1) is concerned, Theorem 1.1 assures us that \(d_n\in C^{M+4}(\varOmega )\) (see also [16, 20]); moreover, as it is clear from the formulas in (1.15), the evaluation of \(\sigma _{k}\) involves derivatives of \(d_n\) of order \(k+1\). Hence, the regularity of \(\sigma _{k}\) is \(M+4-(k+1)=M+3-k\), i.e. \(\sigma _{k}\in C^{M+3-k}(\bar{\varOmega })\) with \(k=1,\ldots ,M\).
Let us show now that such a choice of \(\sigma _{k}\) implies that (2.1) holds true. Thanks to the proved regularity property, we are allowed to compute both the gradient and the Laplacian of \(S_n(x)\). Recalling that by definition \(\nabla d_n(x)=\nabla d (x)\) and \(\Delta d_n(x)=\Delta d(x)\), we have that
Ordering the previous expression according to the power of the distance function and working in \(\varOmega _{\bar{\delta }}\), in order to use that \(|\nabla d|^2=1\), we obtain
where \(r=r(\sigma _{M-1},\sigma _{M})\in C^1(\bar{\varOmega })\).
Now, let us focus on the nonlinear term \(|S_n|^{p-1}S_n\). Since any \(\sigma _{k}\) is bounded, there exists \(\delta _0=\delta _0(\sigma _{0},\ldots ,\sigma _{M})\), with \(\delta _0<\bar{\delta }\), \(n_0=n_0(\delta _0)\) and a function \(R=R(\sigma _{0},\ldots ,\sigma _{M})\in C^1(\bar{\varOmega })\) such that
Now, it becomes clear that the choice of \(\sigma _{0},\ldots , \sigma _{M}\) in (1.15) is the unique for which
and moreover
with \(C=C(\sigma _{0},\ldots ,\sigma _{M})\). The estimate in \(\varOmega {\setminus }\varOmega _{\delta _0}\) is straightforward thanks to the regularity of \(\sigma _{k}\). \(\square \)
Remark 2.2
For the proof of Theorem 1.3, we take \(M=[\alpha ]+1\), so that (2.1) becomes
Since \(f\in W^{1,\infty }(\varOmega )\), recalling the definition (1.23) of \(\widetilde{f}_n\), it follows
We anticipate that for the implementation of the Bernstein technique we only need that the quantity on the left-hand side above is bounded and not infinitesimal near \(\partial \varOmega \). Despite this fact, we need anyway to use \([\alpha ]+2\) terms in the definition of \(S_n\) because, unless \(\alpha \in \mathbb {N}\), \([\alpha ]-\alpha <0\). Anyway the extra information that \(d_n|\widetilde{f}_n|+d_n^2|\nabla \widetilde{f}_n|\) goes to zero has x approaches the boundary is used in the second part of Theorem 2.9.
Remark 2.3
For the sake of completeness, we explicitly compute the expression for \(\sigma _{2}\):
Of course, \(\sigma _{0}\) and \(\sigma _{1} (x)\) coincide with the ones already known in the literature.
2.2 Existence and preliminary estimates for \(u_n\)
In this section, we find suitable sub- and super-solutions for problem (1.21) in order to prove both existence and some preliminary estimates on the solutions \(u_n\) of (1.21).
We first observe that
while
where \(\psi _n\in C(\partial \varOmega )\) is uniformly bounded (with respect to n). More precisely,
Such a bound is crucial in order to prove that problem (1.21) admits a pair of sub- and super-solutions.
Proposition 2.4
Let \(p>1\), \(f\in W^{1,\infty }(\varOmega )\), \(S_n\) as (1.20) and T as in (2.3). Hence, problem (1.21) admits a pair of (classical) sub- and super-solutions.
Proof
Case \(\alpha >1\), sub-solution. We prove that it is possible to chose \(M_1\) and \(M_2\) positive constants such that \(w_n:=\sigma _{0}d_n^{-\alpha }-M_1d_n^{1-\alpha }-M_2\) is a sub-solution for (1.21). Fix at first
and observe that this choice of \(M_1\) implies that
Moreover, using the monotonicity of the function \(s\rightarrow |s|^{p-1}s\), let us fix \(\delta _0=\delta _0(M_1)<\bar{\delta }\) and \(n_0=n_0(\delta _0)\) so that
On the other hand, an easy computation shows that
Recalling that
we deduce that
where the last inequality holds true thanks to the choice of \(M_1\). Now, taking
and using once more the monotonicity of \(s\rightarrow |s|^{p-1}s\), it follows that
and we conclude that \(w_n\) is a sub-solution of problem (1.21).
Case \(\alpha >1\), super-solution. Let us show now that it is possible to take \(N_1\ge M_1\) such that \(v_n:=\sigma _{0}d_n^{-\alpha }+N_1d_n^{1-\alpha }\) turns out to be a super-solution for (1.21). As far as the boundary condition is concerned, we have
where the last inequality follows from the previous choice of \(N_1\).
Since \(v_n\) is positive, thanks to the convexity of the function \((1+s)^p\) with \(p>1\), we have that
As in the previous case, it follows that
where we have used that \(|\nabla d |^2=1\) in \(\varOmega _{\bar{\delta }}\). Thanks, once again, to the choice of \(N_1\), we can conclude that \(-\Delta v_n+v_n^p\ge f\) in \(\varOmega _{\bar{\delta }}\).
Finally, we have
where the last inequality comes from the fact that in \(\varOmega {\setminus }\varOmega _{\bar{\delta }}\)\(d_n\ge \bar{\delta }\) and that in \(-\Delta v_n\) only linear powers of \(N_1\) appear. So by increasing, if necessary, the value of \(N_1\), we have \(-\Delta v_n+v_n^p\ge f\) in \(\varOmega {\setminus }\varOmega _{\bar{\delta }}\). It is then possible to conclude that \(v_n\) is a super-solution of (1.21) in \(\varOmega \) and that \(v_n\ge w_n\).
For the range \(0<\alpha \le 1\), the proof is similar and we just stress the main differences.
Case \(\alpha =1\). Note that, with this choice of \(\alpha \), we have \(p=2\), \(\sigma _{0}=\sqrt{2}\). We claim that \(w_n:=\sqrt{2}d_n^{-1}+M_3\log {d_n}-M_4\) and \(v_n:=\sqrt{2}d_n^{-1}-N_3\log {d_n}+N_4\), with \(M_3,M_4,N_3,N_4>0\), are a sub- and a super-solution for (1.21). Let us choose \(M_3\ge T\) in order to obtain
Then, we fix \(\delta _0=\delta _0(M_3)<\min \{\bar{\delta },1\}\) (so that \(\log (\delta _0)<0\)) and \(n_0=n_0(\delta _0)\) such that
Hence, it follows that
Now, we fix
that implies \(-\Delta w_n+|w_n|^{p-1}w_n-f\le 0\) in \(\varOmega {\setminus }\varOmega _{\delta _0}\) and, in turn, that \(w_n\) is a sub-solution of problem (1.21).
For the super-solution \(v_n\), we consider \(N_3\ge T\). Thus, exactly as in the previous case, we get
Noticing that \(v_n\) is positive and using the convexity of the function \((1+s)^3\), we obtain
Moreover, we fix \(\delta _0\le \bar{\delta }\) and \(n_0=n_0(\delta _0)\) so that
At this point, choosing
it follows that \(-\Delta v_n+v_n^p-f\ge 0\) in \(\varOmega {\setminus }\varOmega _{\delta _0}\) and we conclude.
Case \(\alpha <1\). Finally, we consider \(w_n=\sigma _{0}d_n^{-\alpha }-M_5+M_6d_n^{-\alpha +1}\) and \( v_n=\sigma _{0}d_n^{-\alpha }+N_5-N_6d_n^{-\alpha +1} \) with \(M_5,M_6,N_5,N_6>0\). Let us fix \(M_6\ge \frac{T}{1-\alpha }\), in order to have
Moreover, taking \(M_5>0\) it is possible to select \(\delta _0\le \bar{\delta }\) and \(n_0=n_0(\delta _0)\) such that
and that
Finally, increasing the value of \(M_5\) so that
it follows
As far as \(v_n\) is concerned, let us fix as before \(N_6\ge \frac{\Vert \psi _n\Vert _{\infty }}{1-\alpha }\) in order to get
Take now \(N_5>0\) and fix \(\delta _0<\bar{\delta }\) and \(n_0=n_0(\delta _0)\) such that
and that
As in the previous cases, by increasing if necessary the value of \(N_5\), we have \(-\Delta v_n+|v_n|^{p-1}v_n-f\ge 0\) in \(\varOmega {\setminus }\varOmega _{\delta _0}\) and that \(v_n\) is a super-solution of (1.21).
The ordered sub- and super-solutions obtained in the previous proposition allow us to prove existence of a solution for problem (1.21) and, on the other hand, give us a control on the behaviour of \(u_n\) (and in turn of \(z_n\)) near \(\partial \varOmega \), which is crucial in order to prove the results of the next section.
Theorem 2.5
Let \(p>1\), \(f\in W^{1,\infty }(\varOmega )\), \(S_n\) as in (1.20). Then, problem (1.21) has a unique classical solution \(u_n\) for every \(n\in \mathbb {N}\). Moreover,
where
Proof
The proof of the existence and uniqueness is standard, and we sketch it here for the convenience of the reader. In Proposition 2.4, for every \(\alpha >0\) we have constructed a pair of ordered sub- and super-solutions for problem (1.21)
Let us set \(v_n^0=v_n\), \(C:=\max {\{||v_n||_{L^{\mathcal {1}}(\varOmega )},||w_n||_{L^{\mathcal {1}}(\varOmega )}\}}\), \(m>pC^{p-1}\), and let us define \(v_n^i\) for \(i=1,2,\ldots \) as the solutions of
The choice of m allows us to say that the function \(s\rightarrow ms -|s|^{p-1}s\) is increasing in \([-C,C]\) so that we can apply the standard procedure of the sub- and super-solution method for existence of solutions (see for instance [17]). We claim that \(v_n^{i-1}\ge v_n^i\) for every \(i=1,2,\ldots \). Indeed, for \(i=1\) we have that the function \(w:=v_n^1-v_n^0\) satisfies
Hopf’s lemma and the strong maximum principle assure us that \(w\le 0\), which implies \(v_0\ge v_1\), and we can conclude the proof of the claim by induction. Similarly, we can prove that \(w_n\le v_n^i\) for every \(i=1,2,\ldots \) Then, we have that \(v_n^i\searrow u_n\) a.e in \(\varOmega \) as \(i\rightarrow \infty \) and that
moreover, by compactness and regularity arguments (see, respectively, [1, 2]) it is possible to prove that \(u_n\in C^2(\varOmega )\cap C^1(\overline{\varOmega })\) solves (1.21). The uniqueness follows by Theorem 3.6 of [16].
In order to prove (2.7), we first consider the case \(\alpha >1\). We have that
where \(M_1, \ M_2\) and \(N_1\) are the constants given by Proposition 2.4. Subtracting \(S_n\), we get
with b and B bounded functions. Thanks to the choice of \(M_1\) and \(N_1\), it follows that there exists a positive constant \(C=C(\alpha ,N,\partial \varOmega ,f)\) such that
The case \(\alpha \le 1\) follows similarly using the respective sub- and super-solutions, and for brevity we omit the proof.
We close this section proving the following proposition.
Proposition 2.6
The sequence \(u_n\) of solutions of problem (1.21) converges in \(C^{2}_\mathrm{loc}(\varOmega )\) to the solution of problem (1.10).
Proof
Let us define \(\psi _n:=u_n-u_{n+1}\), which satisfies
The Neumann boundary condition assures us that the maximum of \(\psi _n\) cannot be reached on \(\partial \varOmega \). So let it be \(x_0\in \varOmega \) the maximum point for \(\psi _n\). This implies that \(-\Delta \psi _n(x_0)\ge 0\), and then we obtain from the equation the following information:
But, since \(s\rightarrow |s|^{p-1}s\) is increasing, it has to be \(\psi _n(x_0)=u_n(x_0)-u_{n+1}(x_0)\le 0\). Being \(x_0\) the maximum point, it follows that \(u_n\le u_{n+1}\) in \(\varOmega \). So we know that the sequence \(u_n\) is increasing and converges pointwise to some function u. Moreover, we know, thanks to Theorem 2.5, that each \(u_n\) is between the relative sub- and super-solutions \(w_n\le u_n\le v_n\). Thus, we have that for any \(\omega \) compactly contained in \(\varOmega \) there exists \(c=c(\omega ,\alpha , N)\) such that
where v is the limit as n diverges of the super-solutions \(v_n\). On the other hand, we also have that
where w is the limit of the sub-solutions. Thus, using standard compactness and interior elliptic regularity arguments, we have that for every \(\omega \subset \subset \varOmega \)\(u_n\rightarrow u\) in \(C^2(\omega )\). Thus, we can pass to the limit with respect to n in (1.21), and moreover we also obtain
\(\square \)
2.3 Estimates of \(z_n\) and \(|\nabla z_n|\) in \(L^{\mathcal {1}}(\varOmega )\)
Now, we are ready to prove the uniform estimate in \(W^{1,\infty }(\varOmega )\) for \(z_n:=u_n-S_n\), where \(u_n\) are the solutions of problem (1.21) and \(S_n\) has been defined in (1.20). Note that thanks to Proposition 2.6 we already know that for every \(\omega \) compactly contained in \(\varOmega \) we have that
Thus, the main concern here is to obtain a Lipschitz control in \(\varOmega _{\delta _0}\) for some \(\delta _0>0\) small enough.
Let us start with the bound in \(L^{\infty }(\varOmega _{\delta _0})\).
Theorem 2.7
Let \(z_n\) be as above. Then, there exists \(\delta _0=\delta _0(\sigma _{0},\ldots ,\sigma _{[\alpha ]+1})\) and a constant \(C=C(p,N,\partial \varOmega ,f,\delta _0)\) such that
Proof
We build barriers in a neighbourhood of \(\partial \varOmega \) through sub- and super-solutions method.
Bound from above. Let us fix \(\delta _0=\delta _0 (\sigma _{0},\ldots ,\sigma _{[\alpha ]+1})\) and \(n_0=n_0(\delta _0)\) such that \(S_n\ge 0\) in \(\varOmega _{\delta _0}\) and \(n>n_0\). By definition of \(S_n\) there exists \(K=K(\delta _0)\) such that \(pS_n^{p-1}d_n^2\ge K\) in \(\varOmega _{\delta _0}\). Choose now \(B>\frac{\delta _0\widetilde{C}}{k}\), where \(\widetilde{C}\) is the constant given by (2.2). Taking advantage of the convexity of \(s\rightarrow (1+s)^p\), we have
where we have used (2.2) and the choice of B. Then, if we define \(\bar{w}=B+\max _{\{x\in \varOmega : d(x)= \delta _0\}}|z_n|\) it easily follows that \(z_n\le \bar{w}\) in \(\varOmega \).
Bound from below. For any positive fixed constant \(B>0\), thanks to (2.7) there exist \(\delta _0=\delta (\sigma _{0},\ldots ,\sigma _{[\alpha ]+1},B)\), \(n_0=n_0(\delta _0)\) and a constant \(C=C(\delta _0)\) such that
Thus, it follows that
At this point, reasoning exactly as in the first part, it follows that \(\underline{w}=-B-\max _{\{x\in \varOmega : d(x)= \delta _0\}}|z_n|\) controls \(z_n\) from below in \(\varOmega \) and the proof is concluded. \(\square \)
We can now state and prove our main result.
Theorem 2.8
Let \(z_n\) be the functions defined in (1.22). Then, there exists \(\delta _0=\delta _0(\sigma _{0},\ldots ,\sigma _{[\alpha ]+1})\) and \(C=C(\alpha ,N,\partial \varOmega ,f,\delta _0)\) such that
Proof
We divide the proof in two steps.
-
Step 1. Inequality satisfied by \(|\nabla z_n|^2\).
-
Step 2. Application of maximum principle to \(w_n:=|\nabla z_n|^2e^{\lambda d_n}\) in \(\varOmega _{\delta _0}\).
Step 1. Thanks to (2.6), there exist \(\delta _0<\bar{\delta }\) and \(n_0=n_0(\delta _0)\) such that
where the positive constants \(C_1, \ C_2\) and \(C_3\) depend only on \(\alpha ,N, \partial \varOmega \). Moreover, from the definition of \(S_n\) and thanks to the boundary condition on \(z_n\) it follows that
where \(\varepsilon (s)\) is defined in (2.7). All the computations performed from now on are meant on \(\varOmega _{\delta _0}\) and with \(n>n_0\). At first, let us recover the equation satisfied by \(|\nabla z_n|^2\) (see [24] and reference therein). In order to do it, it is useful to recall that
Hence, through Schwarz inequality, we get
Now, we consider separately each one of the terms on the right-hand side above.
First term. We rewrite it as
Note that in the right-hand side above the first term is the coercive one, while the other has to be absorbed. Thanks to (2.9), the coercive term of (2.11) becomes
Recalling (2.7), Theorem 2.7 and (2.9), the last term of (2.11) can be controlled as follows:
where we have used both (2.9) and (2.10). Then, we get
Second term. We apply Young’s inequality and use (2.2) to obtain
Third term. Using the easy inequality \((a-b)^2\ge \frac{a^2}{4}-b^2\), we obtain
Moreover, using the fact that the function \((1+s)^p-1\) is convex for \(p>1\) and has strictly positive derivative in zero and recalling (2.2) we have
Hence, gathering together the inequalities above, we have that
Using Young’s inequality we get, for \(\epsilon >0\), that
So up to a decrease in \(\delta _0\) and an increase in \(n_0\), we finally obtain
Step 2. As in [22] let us consider now \(w_n:=|\nabla z_n|^2e^{\lambda d_n}\) with \(\lambda >2\Vert \Delta d\Vert _{L^{\mathcal {1}}(\varOmega )}\). Its boundary behaviour is described in Lemma 2.4 of [22] (using in turn an idea of [25]). For the convenience of the reader, we report here the computations. Notice at first that the boundary condition \( \frac{\partial z_n}{\partial \nu }=\nabla z_n\cdot \nu =0 \) implies that there exists a function \(\mu \in L^{\infty }(\partial \varOmega )\) such that
To get convinced of this fact, just observe that the regular function \(\nabla z_n\cdot d_n\) takes the value 0 on \(\varOmega \) (recall that \(\nu =-\nabla d_n\)). Then, its gradient evaluated on the boundary cannot have any tangential component, otherwise the condition for \(\nabla v_n\cdot d_n\) would be violated. Hence, we have
But the left-hand side above is zero, so that
and as a consequence
Hence, we can take \(\lambda \) large enough to have
Taking into account (2.12), it follows that \(w_n\) satisfies
that is,
Hence, up to a decrease in \(\delta _0\) and an increase in \(n_0\), we get
Coupling equation (2.15) together with the boundary condition (2.14), we take advantage of the maximum principle to conclude that
Being the last term above uniformly bounded thanks to Proposition 2.6, the theorem is proved. \(\square \)
2.4 Boundary behaviour of z
Thanks to the results of the previous sections, we deduce that z solves
and that moreover
where \(\displaystyle S=\lim \nolimits _{n\rightarrow \infty }S_n\) and \(\displaystyle \widetilde{f}=\lim \nolimits _{n\rightarrow \infty }\widetilde{f}_n\). It is worth to stress that so far we do not know if z satisfies some sort of boundary conditions: indeed, the only information we have is that z solves an equation and that it is Lipschitz up to the boundary. Anyway, this is enough to define the function z on \(\partial \varOmega \) and to try to deduce its boundary value by means of a scaling argument.
Indeed, we are going to prove that the fact that z solves (2.16) with (2.17) implies that z satisfies both Dirichlet and Neumann boundary conditions. Here, we follow the approach of [4] (see also [30]) in order to deduce the following result.
Theorem 2.9
Under the assumptions of Theorem 1.3, it follows that
Proof
Behaviour of z on \(\partial \varOmega \). Let us consider a point \(x_0\in \partial \varOmega \) and let us identify it as the origin \(\mathcal {O}_{\eta }\) of a new system of coordinates \((\eta _1,\ldots ,\eta _N)=(\eta _1,\eta ')\) such that \(e_{\eta _1}=\nabla d(x_0)=-\nu (x_0)\), where \(e_{\eta _1}\) is the versor of the \(\eta _1\)-axis. The equation for z remains unchanged by such a transformation being the Laplacian invariant under rotation and translation. In order to perform a blow-up near the origin \(\mathcal {O}_{\eta }\), let us consider, for \(\delta _0,\delta >0\) and \(0<\sigma <\frac{1}{2}\), the set
that, through the change of variable \(\xi =\frac{\eta }{\delta }\), becomes
We take \(\delta _0\) small enough in order to have \(U_\delta \subset \varOmega \); this is always possible due to the smoothness of \(\varOmega \). Moreover, notice that \(U_{\delta }\) collapses to \(x_0\) as \(\delta \) goes to zero; meanwhile, thanks to the choice of \(\sigma \), \(W_{\delta }\) converges to the half space \(\mathbb {R}^N_+\). Now, we can define
that inherits the following properties from z
From this information and from the fact that \(W_{\delta }\nearrow \mathbb {R}^N_+\), we can infer that there exists a sequence \(\delta _n\rightarrow 0\) such that \(\{v_{\delta _n}\}\rightarrow v\) in \(C_\mathrm{loc}^{0,\iota }(\mathbb {R}^N_+)\) with \(\iota \in (0,1)\) as \(n\rightarrow \infty \). Moreover, thanks to the second estimate above we have that \(v\equiv \) const and hence the theorem is proved if we show that \(v\equiv 0\).
Choosing \(\delta _0\) small enough, the equation satisfied by \(v_{\delta _n}\) becomes
It is important to stress that in \(W_{\delta _n}\) the lower-order term above and the datum are singular only at the origin \(\mathcal {O}_{\eta }\). In order to pass to the limit in the equation above, note that
Hence,
and for the datum
Passing to the limit w.r.t n, we deduce that v satisfies
that admits \(v\equiv 0\) as the unique constant solution. Being the previous argument independent of the considered sequence, we deduce that, if \(\delta _n\rightarrow 0\) as n diverges and for any sequence \(\{\eta _n\}\subset \varOmega \) such that \(\eta _n\in U_{\delta _n}\), it follows that
Behaviour of \(\frac{\partial z}{\partial \nu }\). In order to prove that z satisfies also the Neumann boundary condition, we prove that there exist \(1<\beta _1,\beta _2<2\) and \(A_1,A_2>0\) such that
Indeed, thanks to the previous step, (2.19) implies that
Using (2.17) and by the definition of S, there exists \(\delta _0\le \bar{\delta }\) such that
Moreover, for any small \(\epsilon >0\) fixed, decreasing the value of \(\delta _0\) if needed, it holds true that
This implies that, for every \(\delta \le \delta _0\), we have a comparison principle for sub- and super-solutions in \(C^2(\varOmega _{\tilde{\delta }})\cap C(\bar{\varOmega }_{\tilde{\delta }})\) for the problem
for any \(g\in C(\partial \varOmega _{\tilde{\delta }})\). Let us prove that there exist \(\beta _1, A_1\) positive constant such that \(w:=-A_1d^{\beta _1}\) is a sub-solution of the problem above. Noticing that \(-1<[\alpha ]-\alpha \le 0\), it is possible to choose \(\beta _1, \beta _2\) in such a way that \(1<\beta _1,\beta _2<2+[\alpha ]-\alpha \le 2\) and that
Let us take moreover \(\tilde{\delta }\le \delta _0\) such that
where \(\bar{C}\) is given by (2.17), and finally fix \(A_i:=\max \{\tilde{\delta }^{-\beta _i}\Vert z\Vert _{L^{\mathcal {1}}(\varOmega )},1\}\) for \(i=1,2\). With this choice, let us consider \(w=-A_1d^{\beta _1}\). Simple computations show that
Thanks to the inequality above, the choice of \(A_1\) and (2.20) we infer that \(w\le z\) in \(\varOmega _{\tilde{\delta }}\) (actually in all \(\varOmega \) thanks to the choice of \(A_1\)). In the very same way we prove that \(z\le v:=A_2d^{\beta _2}\) and thus (2.19) is proved. \(\square \)
Let us now give the proof of Theorem 1.3.
Proof of Theorem 1.3
Thanks to Theorems 2.7 and 2.8, we have a uniform Lipschitz bound for the sequence \(z_n=u_n-S_n\) in \(\varOmega _{\delta _0}\), while Proposition 2.6 assures the interior regularity. Thus, we can deduce that there exists a constant \(C=C(\alpha ,N,\partial \varOmega ,f)\) such that
and passing to the limit with respect to n
Moreover, (1.17) is deduced from Theorem 2.9. \(\square \)
2.5 Generalizations
In this last section, we give for brevity the sketch of the proof of Theorem 1.5, just stressing the main differences with respect to Theorem 1.3.
Sketch of the proof of Theorem 1.5
Let us give at first the compete expression of \(\sigma _{h,0},\ldots ,\sigma _{h,[\alpha ]+1}\)
where
A tedious computation shows that with such a choice, there exists a positive constant \(\widetilde{C}_h=\widetilde{C}_h(\alpha ,N,\partial \varOmega ,h,r)\) such that
where \(S_{h,n}(x) =\sum _{k=0}^{[\alpha ]+1}\sigma _{h,k} (x) \ \left( \sqrt{h(x)}d_n(x)\right) ^{k-\alpha } \). Hence, we can define the approximated problems
For the sake of clarity, we give some details of the construction of the sub-solution in the case \(\alpha >1\). Let us consider the function
with
Notice that thanks to (1.29)
Then, there exist \(\delta _0=\delta _0(M_{h,1},r)\) and \(n_0=n_0(\delta _0)\) such that
Up to an increase in the value of \(M_{h,1}\) and taking the value of \(M_{h,2}\) large enough, we deduce [following the same arguments that have led to (2.4) and (2.5)] that \(w_{h,n}\) is a sub-solution of (2.22).
Once that sub- and super-solutions are obtained, we proceed as in Proposition 2.4, Theorem 2.5 and Proposition 2.6 in order to deduce that the solution \(u_{h,n}\) of (2.22) converges (as \(n\rightarrow \infty \)) in \(C^2_\mathrm{loc}(\varOmega )\) to \(u_h\), unique solution of (1.25). Moreover, the following estimate is satisfied
where
Let us now define \(z_{h,n}:=u_{h,n}-S_{h,n}\), that solves
where \(\widetilde{f}_n:=\Delta S_{h,n}-|S_{h,n}|^{p-1}S_{h,n}\). Concerning the \(L^{\mathcal {1}}(\varOmega )\) estimate for \(z_{h,n}\), we adapt the proof of Theorem 2.7 as follows. Let us fix a positive constant \(B>0\), and let \(\delta _0=\delta _0 (S_{h,n})\) and \(n_0=n_0(\delta _0)\) be such that
where we have used the first condition of (1.29). Thus, we can continue as in the proof of Theorem 2.7 to conclude that
Let us now have a closer glance to the perturbed version of Theorem 2.8, for which the growth conditions (1.29) are especially designed. Exactly as in the previous section, we obtain that there exist \(\delta _0\) and \(n_0\) such that
The main concern of course is the third term on the right-hand side above; we have
Let us focus on the last three terms on the right-hand side above. Using assumption (1.29) and estimate (2.23), we get that for \(d(x)\rightarrow 0\) and \(n\rightarrow \infty \)
Thus, up to a decrease in the value of \(\delta _0\) and an increase in \(n_0\), we obtain
At this point, it is easy to deduce the counter of (2.12), i.e. there exist some \(\delta _0\) and \(n_0=n_0(\delta _0)\) such that
From now on, the proof follows closely Theorem 2.8.
Hence, we infer that there exists \(z_h\in C^2(\varOmega )\), such that \(z_{h,n}\rightarrow z_h\) in \(C^2_\mathrm{loc}(\varOmega )\), that solves
and that moreover
As far as the boundary conditions of \(z_h\) are concerned, fixing \(\bar{x}\in \partial \varOmega \) and following the same notation of Theorem 2.9, let us set \(v_{h,\delta }(\xi ):=z_{h}(\delta \xi )\) in \(W_{\delta }\) that inherits the following properties from \(z_h\)
Thus, the limit function \(v_{h}\) has to be a constant. Moreover, using the first assumption in (1.29) we have that
and thus \(v_h\) solves
whose unique constant solution is zero. From this, we infer that \(z(\bar{x}):=u(\bar{x})-S_h(\bar{x})=0\) for every \(\bar{x}\in \partial \varOmega \).
In order to recover the Neumann boundary condition, using the same notation of the second part of Theorem 2.9, we can infer that there exist \(1<\beta _{h,1}<2\), \(\gamma _{h,1}>0\), \(\tilde{\delta }\) and \(A_{h,1}>0\) such that
where the last inequality is implied by assumption (1.29). The rest of the proof closely follows Theorem 2.9. \(\square \)
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Buccheri, S. Gradient behaviour for large solutions to semilinear elliptic problems. Annali di Matematica 198, 1013–1040 (2019). https://doi.org/10.1007/s10231-018-0808-y
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DOI: https://doi.org/10.1007/s10231-018-0808-y