Boundary concentration phenomena for an anisotropic Neumann problem in $\mathbb{R}^{2}$

This paper is concerned with the analysis of solutions to the anisotropic Neumann problem

where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{2}$, $\lambda>0$ is a small parameter, $0<p<2$, $a(x)$ is a positive smooth function over $\overline{\Omega}$ and $\nu$ denotes the outer unit normal vector to $\partial\Omega$. This problem is the Euler-Lagrange equation for the energy functional

which is well defined because the critical Moser-Trudinger inequality implies the validity of the Sobolev-Orlicz compact subcritical embedding

We are interested in the existence of solutions of equation (1.1) that exhibit the boundary concentration phenomenon as the parameter $\lambda$ tends to zero. This work is strongly stimulated by some extensive research involving the isotropic case $a(x)\equiv 1$ in equation (1.1):

where $\mathcal{D}$ is a smooth bounded domain in $\mathbb{R}^{N}$ with $N\geq 2$. In the case of $p=1$, this scalar equation is equivalent to an elliptic system representing the stationary Keller-Segel chemotaxis system with linear sensitivity:

because the first equation in system (1.4) implies

and hence $\psi=\lambda e^{\upsilon}$ for some positive constant $\lambda$. Steady states of system (1.4), namely its solutions, are of basic importance for a better understanding of global dynamics to the following Keller-Segel system with $\tau\geq 0$:

which describes chemotactic feature of cellular slime molds sensitive to the gradient of a chemical substance secreted by themselves (see [15]). The one-dimensional form of system (1.4) was first studied by Schaaf [19]. In higher dimensions $N\geq 2$ Biler [2] established the existence of non-constant radially symmetric solution to (1.4) when the domain $\mathcal{D}$ is a ball. In the general two-dimensional case, Wang-Wei [22], independently of Senba-Suzuki [20], proved that for any $\mu\in(0,1/|\Omega|+\mu_{1})\setminus\{4\pi m|m=1,2,\ldots\}$ (where $\mu_{1}$ denotes the first positive eigenvalue of $-\Delta$ with Neumann boundary condition), system (1.4) has a non-constant solution such that $\int_{\mathcal{D}}\psi=\mu|\Omega|$. Meanwhile, if space dimension is $N=2$, it is known that as infinite time blow-up solutions of the parabolic-elliptic system (1.5) from chemotaxis, steady states of (1.4) produce a significant concentration phenomenon in mathematical biology referred as ‘chemotactic collapse’, namely the blow-up for the quantity $\psi$ in (1.4) takes place as a finite sum of Dirac measure at points with masses equal to $8\pi$ or $4\pi$, respectively, depending on whether the blow-up points lie inside the domain or on the boundary. By analyzing the asymptotic behavior of families of solutions to equation $(\ref{1.3})|_{p=1}$ Senba-Suzuki [20, 21] exhibited this phenomenon for the term $\lambda e^{\upsilon}$ in $(\ref{1.3})|_{p=1}$ with positive, uniformly bounded mass $\lambda\int_{\mathcal{D}}e^{\upsilon}$ as $\lambda$ tends to zero. More precisely, if $\upsilon_{\lambda}$ is a family of solutions of $(\ref{1.3})$ under $p=1$ and $N=2$, such that

then there exist non-negative integers $k$, $l$ with $k+l\geq 1$ for which $L=4\pi(k+2l)$. Moreover, once $\lambda$ tends to zero, this family of solutions concentrate at $l$ different points $\xi_{1},\ldots,\xi_{l}$ inside the domain $\mathcal{D}$ and $k$ different points $\xi_{l+1},\ldots,\xi_{k+l}$ on the boundary $\partial\mathcal{D}$. In particular, far away from these concentration points the asymptotic profile of $\upsilon_{\lambda}$ is uniformly described as

In addition, these concentration points or blow-up points $\xi=(\xi_{1},\ldots,\xi_{k+l})$ are nothing but critical points of a functional

where $c_{i}=8\pi$ for $i=1,\ldots,l$, but $c_{i}=4\pi$ for $i=l+1,\ldots,k+l$, $G(x,y)$ denotes the Green’s function of the problem

and $H(x,y)$ its regular part defined as

Reciprocally, in the spirit of the Lyapunov-Schmidt finite-dimensional reduction method del Pino-Wei [12] constructed a family of mixed interior and boundary bubbling solutions for equation $(\ref{1.3})|_{p=1,N=2}$ with exactly the asymptotic profile above. Successively, when $N=2$ and $p$ is between $0$ and $2$, Deng [13] used a reductional argument to build solutions for equation (1.3) with bubbling profiles at points inside $\mathcal{D}$ and on the boundary $\partial\mathcal{D}$, which recovered the result in [12] when $p=1$. In general, such bubbling solutions are called solutions concentrating on $0$-dimensional sets with uniformly bounded mass.

Clearly, a natural question is to ask whether there exist a family of solutions of equation (1.3) concentrating on higher dimensional subsets of $\overline{\mathcal{D}}$ with or without uniformly bounded mass as the parameter $\lambda$ tends to zero. The first result in this direction was obtained by Pistoia-Vaira [17] in the case that $p=1$ and the domain $\mathcal{D}$ is a ball with dimension $N\geq 2$. Based on a fixed-point argument, they constructed a family of uniformly unbounded mass radial solutions of $(\ref{1.3})|_{p=1}$ in the ball, which blow up on the entire boundary and hence produce the boundary concentration layer. Following closely the techniques of $(\ref{1.3})|_{p=1}$ in [17], Bonheure-Casteras-Noris [5] constructed a family of boundary layer solutions in the annulus blowing up simultaneously along both boundaries, a family of internal layer solutions in the unit ball blowing up on an interior sphere, and a family of solutions in the unit ball with an internal layer and a boundary layer blowing up simultaneously on an interior sphere and the boundary. Very recently, when the domain $\mathcal{D}$ is a unit disk (corresponding to a unit ball with dimension $N=2$), Bonheure-Casteras-Román [7] have successfully constructed a family of uniformly unbounded mass radial solutions of $(\ref{1.3})|_{p=1}$ which concentrate at the origin and blow up on the entire boundary. Additionally, some bifurcation analyses of radial solutions to $(\ref{1.3})|_{p=1}$ in a ball with dimension $N\geq 2$ were also performed by Bonheure et al. in [4, 6]. As for a general smooth two-dimensional domain $\mathcal{D}$, it is very worth mentioning that inspired by the novel result in [17], del Pino-Pistoia-Varia [11] applied an infinite-dimensional form of Lyapunov-Schmidt reduction to establish the existence of a family of solution $\upsilon_{\lambda}$ for equation $(\ref{1.3})|_{p=1,N=2}$ with unbounded mass $\lambda\int_{\mathcal{D}}e^{\upsilon_{\lambda}}$, which exhibit a sharp boundary layer and blow up along the entire $\partial\mathcal{D}$ as $\lambda$ tends to zero but remains suitably away from a sequence of critical small values where certain resonance phenomenon occurs. Finally, when the domain $\mathcal{D}$ has suitable rational symmetries in higher dimensions $N\geq 3$, Agudelo-Pistoia [1] constructed several families of layered solutions $\upsilon_{\lambda}$ of the stationary Keller-Segel chemotaxis equation $(\ref{1.3})|_{p=1}$ with uniformly bounded mass $\lambda\int_{\mathcal{D}}e^{\upsilon_{\lambda}}$, which exhibit three different types of chemoattractant concentration along suitable $(N-2)$-dimensional minimal submanifolds of the boundary.

Problem (1.1) is seemingly similar to equation (1.3). Our original motivation in equation (1.3) is based on the fact that except for $p=1$, nothing is known about the existence or the boundary concentration phenomenon for solutions of equation (1.3) in higher dimensions $N\geq 3$. For this aim our idea is to consider partially axially symmetric solutions of equation (1.3) when the domain $\mathcal{D}$ has some rotational symmetries, which implies that problem (1.1) can be viewed as a special case of equation (1.3) in higher dimensions $N\geq 3$. Indeed, take $n\in\{1,2\}$ as a fixed integer. Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{2}$ such that

Fix $k_{1},k_{n}\in\mathbb{N}$ with $k_{1}+k_{n}=N-2\geq 1$ and set

Then $\mathcal{D}$ is a smooth bounded domain in $\mathbb{R}^{N}$ which is invariant under the action of the group $\Upsilon:=\mathcal{O}(k_{1}+1)\times\mathcal{O}(k_{n}+1)$ on $R^{N}$ given by

Note that $\mathcal{O}(k_{i}+1)$ is the group of linear isometries of $\mathbb{R}^{k_{i}+1}$ and $\mathbb{S}^{k_{i}}$ is the unit sphere in $\mathbb{R}^{k_{i}+1}$. If we seek $\Upsilon$-invariant solutions of equation (1.3), i.e. solutions $\upsilon$ of the form

a direct calculus shows that equation (1.3) is transformed to

Thus if we take anisotropic coefficient

then equation (1.6) can be rewritten as problem (1.1). Hence by considering rotational symmetry of $\mathcal{D}$, a fruitful approach for seeking layered solutions of equation (1.3) with concentration along some $(N-2)$-dimensional minimal submanifolds of $\overline{\mathcal{D}}$ diffeomorphic to $\mathbb{S}^{k_{1}}\times\mathbb{S}^{k_{n}}$ is to reduce it to produce point-wise blow-up solutions of the anisotropic problem (1.1) in the domain $\Omega$ of dimension $2$. This approach, together with some Lyapunov-Schmidt finite-dimensional reduction arguments, has recently been taken to construct multi-layer positive solutions of equation (1.3) concentrating along some $(N-2)$-dimensional minimal submanifolds of $\partial\mathcal{D}$, which can be found in [1] only for the case $p=1$.

In this paper, our goal is to obtain the existence of boundary separated or clustered layer positive solutions for equation (1.3) in the higher-dimensional domain with some rotational symmetries, by constructing bubbling solutions for the anisotropic planar problem (1.1) with simple or non-simple boundary concentration points when $p$ is between $0$ and $2$. We try to use a new reductional argument to investigate the effect of anisotropic coefficient $a(x)$ on the existence of boundary concentrating solutions to problem (1.1). As a result, with the help of some suitable assumptions on anisotropic coefficient $a(x)$ we prove that there exist a family of positive solutions of problem (1.1) with an arbitrary number of mixed interior and boundary bubbles which concentrate at totally different strict local maximum or minimal boundary points of $a(x)$ restricted to $\partial\Omega$, or accumulate to the same strict local maximum boundary point of $a(x)$ over $\overline{\Omega}$ as $\lambda$ tends to zero. In particular, we recover and improve the result in [1] when $p=1$.

Before precisely stating our results, let us start with some notations. Let $\varepsilon$ be a positive parameter given by the relation

Clearly, $\lambda\rightarrow 0$ if and only if $\varepsilon\rightarrow 0$, and $\lambda=\varepsilon^{2}$ if $p=1$. Let

and $G_{a}(x,y)$ be the anisotropic Green’s function associated to the Neumann equation

for every $y\in\overline{\Omega}$. The regular part of $G_{a}(x,y)$ is defined depending on whether $y$ lies inside the domain or on its boundary as

In this way, $y\in\overline{\Omega}\mapsto H_{a}(\cdot,y)\in C\big{(}\Omega,C^{\alpha}(\overline{\Omega})\big{)}\cap C\big{(}\partial\Omega,C^{\alpha}(\overline{\Omega})\big{)}$ and $H_{a}(x,y)\in C^{\alpha}\big{(}\overline{\Omega}\times\Omega\big{)}\cap C^{\alpha}\big{(}\overline{\Omega}\times\partial\Omega\big{)}\cap C^{1}\big{(}\overline{\Omega}\times\Omega\setminus\{x=y\}\big{)}\cap C^{1}\big{(}\overline{\Omega}\times\partial\Omega\setminus\{x=y\}\big{)}$ for any $\alpha\in(0,1)$, and the corresponding Robin’s function $y\in\overline{\Omega}\mapsto H_{a}(y,y)$ belongs to $C^{1}(\Omega)\cap C^{1}(\partial\Omega)$ (see [1]). Moreover, by the maximum principle, for any $y\in\overline{\Omega}$, $G_{a}(\cdot,y)>0$ over $\overline{\Omega}$.

Our first result concerns the existence of solutions of problem (1.1) whose mixed interior and boundary bubbles are uniformly far away from each other and interior bubbles lie in the domain with distance to the boundary uniformly approaching zero.

Our next result concerns the existence of solutions of problem (1.1) with mixed interior and boundary bubbles which accumulate to the same boundary point.

Now we find that if $0<p<2$ and the domain $\mathcal{D}$ has some rational symmetries in higher dimensions $N\geq 3$ such that the corresponding anisotropic coefficient $a(x)$ given by (1.7) satisfies the assumptions in Theorem 1.2, then equation (1.3) has a family of positive solutions with arbitrarily many mixed interior and boundary layers which collapse to the same $(N-2)$-dimensional minimal submanifold of $\partial\mathcal{D}$ as $\lambda$ tends to zero. Meanwhile, we observe that the assumptions in Theorem $1.2$ contain the following two cases:
(C1)   $\xi_{*}\in\partial\Omega$ is a strict local maximum point of $a(x)$ restricted to $\partial\Omega$;
(C2)   $\xi_{*}\in\partial\Omega$ is a strict local maximum point of $a(x)$ restricted in $\Omega$ and satisfies $\partial_{\nu}a(\xi_{*})=\langle\nabla a(\xi_{*}),\,\nu(\xi_{*})\rangle=0$.
Arguing as in the proof of Theorem $1.2$, we readily prove that if (C1) holds, then problem (1.1) has positive solutions with arbitrarily many boundary bubbles which accumulate to $\xi_{*}$ along $\partial\Omega$; while if (C2) holds, then problem (1.1) has positive solutions with arbitrarily many interior bubbles which accumulate to $\xi_{*}$ along the neighborhood near the inner normal direction of $\partial\Omega$. As for the latter case, our result seems to close some gap which was left open in the literature [1] regarding such type of chemoattractant concentration from the stationary Keller-Segel system with linear chemotactical sensitivity function, namely involving the existence of solutions of equation $(\ref{1.3})|_{p=1,N\geq 3}$ with an arbitrary number of interior layers which simultaneously accumulate along a suitable $(N-2)$-dimensional minimal submanifold of $\partial\mathcal{D}$ as $\lambda$ tends to zero. Finally, it is necessary to point out that radial solutions of equation $(\ref{1.3})|_{p=1}$ with concentration on an arbitrary number of internal spheres were built by Bonheure-Casteras-Noris [6] when the domain $\mathcal{D}$ is a ball with dimension $N\geq 2$, but a remarkable fact is that, in opposition to our result or an analogous one given by Malchiodi-Ni-Wei [16] for a singularly perturbed elliptic Neumann problem on a ball, the layers of those solutions do not accumulate to the boundary of $\mathcal{D}$ as $\lambda$ tends to zero.

The proof of our results relies on a very well known Lyapunov-Schmidt finite-dimensional reduction procedure. In Section $2$ we provide a good approximation for the solution of problem (1.1) and estimate the scaling error created by this approximation. Then we rewrite problem (1.1) in terms of a linear operator $\mathcal{L}$ for which a solvability theory is performed through solving a linearized problem in Section $3$. In Section $4$ we solve an auxiliary nonlinear problem. In Section $5$ we reduce the problem of finding bubbling solutions of (1.1) to that of finding a critical point of a finite-dimensional function. In section $6$ we give an asymptotic expansion of the energy functional associated to the approximate solution. In Section $7$ we provide the detailed proof of Theorems $1.1$-$1.2$. Finally, we give some technical explanations in Appendix $8$.

Notation: In this paper the letters $C$ and $D$ will always denote a universal positive constant independent of $\lambda$, which could be changed from one line to another. The symbol $o(t)$ (respectively $O(t)$) will denote a quantity for which $\frac{o(t)}{|t|}$ tends to zero (respectively, $\frac{O(t)}{|t|}$ stays bounded ) as parameter $t$ goes to zero. Moreover, we will use the notation $o(1)$ (respectively $O(1)$) to stand for a quantity which tends to zero (respectively, which remains uniformly bounded) as $\lambda$ tends to zero.

The original cells for the construction of an approximate solution of problem (1.1) are based on the four-parameter family of functions

which exactly solve

Set

The configuration space for $m$ concentration points $\xi=(\xi_{1},\ldots,\xi_{m})$ we try to look for is the following

where $l\in\{0,\ldots,m\}$, $\varepsilon$ is sufficiently small and uniquely defined by $\lambda$ in (1.8), and $\kappa$ is given by

Let $m\in\mathbb{N}^{*}$ and $\xi=(\xi_{1},\ldots,\xi_{m})\in\mathcal{O}_{\varepsilon}$ be fixed. For numbers $\mu_{i}$, $i=1,\ldots,m$, yet to be determined, but we always assume

for some $C>0$, independent of $\varepsilon$. Define

and for each $i=1,\ldots,m$,

Here, $\omega^{j}_{\mu_{i}}$, $j=1,2,3,4$, are radial solutions of

with

and

and

and