We consider the problem \[ -Δu=|u|^{p-2}u in Ω, u=0 on \partialΩ, \] where $Ω:=\{(y,z)\in\mathbb{R}^{m+1}\times\mathbb{R}^{N-m-1}: 0<a<|y|<b<\infty\}$, $0\leq m\leq N-1$ and $N\geq2$. Let $2_{N,m}^{\ast}:=2(N-m)/(N-m-2)$ if $m<N-2$ and $2_{N,m}^{\ast}:=\infty$ if $m=N-2$ or $N-1$. We show that $2_{N,m}^{\ast}$ is the true critical exponent for this problem, and that there exist nontrivial solutions if $2<p<2_{N,m}^{\ast}$ but there are no such solutions if $p\geq2_{N,m}^{\ast}$.
In this paper, we deal with the existence and multiplicity of solutions for the fractional elliptic problems involving critical and supercritical Sobolev exponent via variational arguments. By means of the truncation combining with the Moser iteration, we prove that the problems has at least three solutions.
Given a smooth Riemannian manifold (M,g)we investigate the existence of positive solutions to a singularly perturbed supercritical elliptic equation which concentrate at some submanifold of M. We obtain a posive answer for some manifolds, which include warped products.
We consider the supercritical elliptic problem -Δu = λe^u, λ> 0, in an exterior domain $Ω= \mathbb{R}^N \setminus D$ under zero Dirichlet condition, where D is smooth and bounded in \mathbb{R}^N, N greater or equal than 3. We prove that, for λsmall, this problem admits infinitely many regular solutions.
We prove the existence of a family of slow decay positive solutions of a supercritical elliptic equation with Hardy potential in the entire space and study stability and oscillation properties of these solutions. We also establish the existence of a continuum of stable slow decay positive solutions for the relevant exterior Dirichlet problem.
We construct positive solutions of the semilinear elliptic problem $Δu+ λu + u^p = 0$ with Dirichet boundary conditions, in a bounded smooth domain $Ω\subset \R^N$ $(N\geq 4)$, when the exponent $p$ is supercritical and close enough to $\frac{N+2}{N-2}$ and the parameter $λ\in\R$ is small enough. As $p\to \frac{N+2}{N-2}$, the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino, Dolbeault and Musso \cite{DDM} when $Ω$ is a ball and the solutions are radially sym…
We consider the semilinear elliptic boundary value problem \[ -Δu=\left\vert u\right\vert ^{p-2}u\text{ in }Ω,\text{\quad }u=0\text{ on }\partialΩ, \] in a bounded smooth domain $Ω$ of $\mathbb{R}^{N}$ for supercritical exponents $p>\frac{2N}{N-2}.$ Until recently, only few existence results were known. An approach which has been successfully applied to study this problem, consists in reducing it to a more general critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both. The aim of this pa…
We study radial solutions of the semilinear elliptic equation $Δu+f(u)=0$ under rather general growth conditions on $f$. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole-Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $Δu+u^p=0$ and $Δu+…
Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or the…
This paper is concerned with the convergence of the solution of general elliptic boundary value problems in cylindrical domain, when some directions of the domain go to infinity.
In this paper, we aim to investigate the following class of singularly perturbed elliptic problem $$ \left\{ \begin{array}{ll} \displaystyle -\varepsilon^2\triangle {u}+|x|^ηu =|x|^ηf(u)& \mbox{in}\,\, A, u=0 & \mbox{on}\,\, \partial A, \end{array} \right. $$ where $\varepsilon>0$, $η\in\mathbb{R}$, $A=\{x\in\R^{2N}:\,\,0<a<|x|<b\}$, $N\ge2$ and $f$ is a nonlinearity of $C^1$ class with supercritical growth. By a reduction argument, we show that there exists a nodal solution $u_\e$ with exactly two positive and two negative peaks, which concentrate on two different orthogonal spheres of dimens…
The main thrust of our current work is to exploit very specific characteristics of a given problem in order to acquire improved compactness for supercritical problems and to prove existence of new types of solutions. To this end, we shall develop a variational machinery in order to construct a new type of classical solutions for a large class of supercritical elliptic partial differential equations.\\ The issue of symmetry and symmetry breaking is challenging and fundamental in mathematics and physics. Symmetry breaking is the source of many interesting phenomena namely phase transitions, inst…
We consider the supercritical Lane-Emden problem $$(P_\eps)\qquad -Δv= |v|^{p_\eps-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} $$ where $\mathcal A$ is an annulus in $\rr^{2m},$ $m\ge2$ and $p_\eps={(m+1)+2\over(m+1)-2}-\eps$, $\eps>0.$ We prove the existence of positive and sign changing solutions of $(P_\eps)$ concentrating and blowing-up, as $\eps\to0$, on $(m-1)-$dimensional spheres. Using a reduction method (see Ruf-Srikanth (2010) J. Eur. Math. Soc. and Pacella-Srikanth (2012) arXiv:1210.0782)we transform problem $(P_\eps)$ into a nonhomogeneous problem in an…
In this work we obtain positive bounded solutions of various perturbations of \begin{equation} \left\{ \begin{array}{lcl} \hfill -Δu - γ\sum_{i,j=1}^N \frac{x_i x_j}{|x|^2} u_{x_i x_j} &=& u^p \qquad \mbox{ in } B_1, \\ \hfill u &=& 0 \hfill \mbox{ on } \partial B_1, \end{array}\right. \end{equation} where $B_1$ is the unit ball in $ \IR^N$ where $N \ge 3$, $ γ>0$ and $ 1<p<p_{N,γ}$ where \begin{equation*} p_{N,γ}:=\left\{ \begin{array}{lc} \frac{N+2+3 γ}{N-2-γ} & \qquad \mbox{ if } γ<N-2, \\ \infty & \qquad \mbox{ if } γ\ge N-2. \end{array}\right. \end{equation*} Note for $γ>0$ this allows fo…
In this paper we prove a kind of rotational symmetry for solutions of semilinear elliptic systems in some bounded cylindrical domains. The symmetry theorems obtained hold for low-Morse index solutions whenever the nonlinearities satisfy some convexity assumptions. These results extend and improve those obtained in \cite{DaPaSys, DaGlPa1, Pa, PaWe}.
We consider the supercritical problem -Δu = |u|^{p-2}u in Ω, u=0 on \partialΩ, where $Ω$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and $p\geq2^{*}:= 2N/(N-2).$ Bahri and Coron showed that if $Ω$ has nontrivial homology this problem has a positive solution for $p=2^{*}.$ However, this is not enough to guarantee existence in the supercritical case. For $p\geq 2(N-1)/(N-3)$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as $p$ increases. More precisely, we show…
We consider the elliptic quasilinear equation --$Δ$ m u = u p |$\nabla$u| q in R N with q $\ge$ m and p > 0, 1 < m < N. Our main result is a Liouville-type property, namely, all the positive C 1 solutions in R N are constant. We also give their asymptotic behaviour : all the solutions in an exterior domain R N \B r0 are bounded. The solutions in B r0 \ {0} can be extended as a continuous functions in B r0. The solutions in R N \ {0} has a finite limit l $\ge$ 0 as |x| $\rightarrow$ $\infty$. Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a p…
We study the generalized boundary value problem for (E)\; $-Δu+|u|^{q-1}u=0$ in a dihedral domain $\Gw$, when $q>1$ is supercritical. The value of the critical exponent can take only a finite number of values depending on the geometry of $\Gw$. When $\gm$ is a bounded Borel measure in a k-wedge, we give necessary and sufficient conditions in order it be the boundary value of a solution of (E). We also give conditions which ensure that a boundary compact subset is removable. These conditions are expressed in terms of Bessel capacities $B_{s,q'}$ in $\BBR^{N-k}$ where $s$ depends on the characte…
We deals with nonlinear elliptic Dirichlet problems of the form $${\rm div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }Ω,\qquad u\in H^{1,p}_0(Ω) $$ where $Ω$ is a bounded domain in $\mathbb{R}^n$, $n\ge 2$, $p> 1$ and $f$ has supercritical growth from the viewpoint of Sobolev embedding. Our aim is to show that there exist bounded contractible non star-shaped domains $Ω$, arbitrarily close to domains with nontrivial topology, such that the problem does not have nontrivial solutions. For example, we prove that if $n=2$, $1<p<2$, $f(u)=|u|^{q-2}u$ with $q>{2p\over 2-p}$ and $Ω=\{(ρ\cosθ,ρ\sinθ)\ :\ …
In this paper we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument.
For second-order elliptic or parabolic equations with subcritical or critical drifts, it is well-known that the Harnack inequality holds and their bounded weak solutions are Hölder continuous. We construct time-independent supercritical drifts in $L^{n-λ}(\mathbb{R}^n)$ with arbitrarily small $λ>0$ such that the Harnack inequality and the Hölder continuity fail in both the elliptic and the parabolic cases, thus confirming a conjecture by Seregin, Silvestre, Sverak and Zlatos. These results are sharp, and they also apply to a toy model of the axi-symmetric Navier-Stokes equations in space dimen…
We examine the equation given by \begin{equation} \label{eq_abstract} -Δu + a(x) \cdot \nabla u = u^p \qquad \mbox{in $ \IR^N$,} \end{equation} where $p>1$ and $ a(x)$ is a smooth vector field satisfying some decay conditions. We show that for $ p < p_c$, the Joseph-Lundgren exponent, that there is no positive stable solution of (\ref{eq_abstract}) provided one imposes a smallness condition on $a$ along with a divergence free condition. In the other direction we show that for $ N \ge 4$ and $ p > \frac{N-1}{N-3}$ there exists a positive solution of (\ref{eq_abstract}) provided $a$ satisfies a …
We study a weakly coupled supercritical elliptic system of the form \begin{equation*} \begin{cases} -Δu = |x_2|^γ\left(μ_{1}|u|^{p-2}u+λα|u|^{α-2}|v|^βu \right) & \text{in }Ω,\\ -Δv = |x_2|^γ\left(μ_{2}|v|^{p-2}v+λβ|u|^α|v|^{β-2}v \right) & \text{in }Ω,\\ u=v=0 & \text{on }\partialΩ, \end{cases} \end{equation*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq 3$, $γ\geq 0$, $μ_{1},μ_{2}>0$, $λ\in\mathbb{R}$, $α, β>1$, $α+β= p$, and $p\geq 2^{*}:=\frac{2N}{N-2}$. We assume that $Ω$ is invariant under the action of a group $G$ of linear isometries, $\mathbb{R}^{N}$ is the sum $F\…
In this paper we investigate the behavior and the existence of positive and non-radially symmetric solutions to nonlinear exponential elliptic model problems defined on a solid torus $\bar{T}$ of $\mathbb{R}^3$, when data are invariant under the group $G=O(2)\times I \subset O(3)$. The model problems of interest are stated below: ${ll} {\bf(P_1)} & \displaystyle Δ\upsilon+γ=f(x)e^\upsilon, \upsilon>0\quad \mathrm{on} \quad T, \quad\upsilon |_{_{\partial T}}=0.$ and ${ll}\bf{(P_2)} & \displaystyle Δ\upsilon+a+fe^\upsilon=0, \upsilon>0\quad \mathrm{on}\quad T, [1.3ex] &\displaystyle \frac{\parti…
We prove Hölder continuity of weak solutions of the uniformly elliptic and parabolic equations %$Δu-\frac{A}{|x|^{2+β}}u=0,\,\,(β\geq 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01} \partial_{i} (a_{ij}(x) \partial_{j}u(x)) - \frac{A}{|x|^{2+β}} u(x) =0\quad (A>0,\quadβ\geq 0), \end{equation} \begin{equation}\label{02} \partial_{i} (a_{ij}(x,t) \partial_{j}u(x,t)) - \frac{A}{|x|^{2+β}} u(x,t)-\partial_{t}u(x,t) =0\quad (A>0,\quadβ\geq 0), \end{equation} with critical or supercritical 0-order term coefficients which are beyond De Giorgi-Nash-Moser's Theory. …
In this paper we are interested in positive classical solutions of \begin{equation} \label{eqx} \left\{\begin{array}{ll} -Δu = a(x) u^{p-1} & \mbox{ in } Ω, \\ u>0 & \mbox{ in } Ω, \\ u= 0 & \mbox{ on } \pOm, \end {array}\right. \end{equation} where $Ω$ is a bounded annular domain (not necessarily an annulus) in $\IR^N$ $(N \ge3)$ and $ a(x)$ is a nonnegative continuous function. We show the existence of a classical positive solution for a range of supercritical values of $p$ when the problem enjoys certain mild symmetry and monotonicity conditions. As a consequence of our results, we shall sh…
Given an isoparametric function $f$ on the $n$-dimensional round sphere, we consider functions of the form $u=w\circ f$ to reduce the semilinear elliptic problem \[ -Δ_{g_0}u+λu=λ | u\ | ^{p-1}u\qquad\text{ on }\mathbb{S}^n \] with $λ>0$ and $1<p$, into a singular ODE in $[0,π]$ of the form $w'' + \frac{h(r)}{\sin r} w' + \fracλ{\ell^2}\ (| w|^{p-1}w - w\ )=0$, where $h$ is an strictly decreasing function having exactly one zero in this interval and $\ell$ is a geometric constant. Using a double shooting method, together with a result for oscillating solutions to this kind of ODE, we obtain a …
We show that in some suitable torus-like domains D some supercritical elliptic problems have an arbitrary large number of sign-changing solutions with alternate positive and negative layers which concentrate at different rates along a k-dimensional submanifold of the boundary of D as p approaches 2*_{N,K} from below.
This paper is concerned with the structure of solutions to the elliptic problem for an Henon type equation with a forcing term. Under suitable assumptions on the forcing term, we give a complete classification of the existence/nonexistence of solutions to the problem.
We give new criteria for the existence of weak solutions to an equation with a super linear source term \begin{align*}-Δu = u^q ~~\text{in}~Ω,~~u=σ~~\text{on }~\partialΩ\end{align*}where $Ω$ is a either a bounded smooth domain or $\mathbb{R}\_+^{N}$, $q\textgreater{}1$ and $σ\in \mathfrak{M}^+(\partialΩ)$ is a nonnegative Radon measure on $\partialΩ$. One of the criteria we obtain is expressed in terms of some Bessel capacities on $\partialΩ$. We also give a sufficient condition for the existence of weak solutions to equation with source mixed terms. \begin{align*} -Δu = |u|^{q\_1-1}u|\nabla u…
Positive normalized solutions to a singular elliptic equation with a $L^2$-supercritical nonlinearity
College of Data Science, Jiaxing University（嘉兴大学数据科学学院）；School of Mathematics and Statistics, Northeast Normal University（东北师范大学数学与统计学院）；Departamento de Matemática, Universidade de Brasília（巴西利亚大学数学系）
We consider the supercritical problem \[ -Δv=|v|^{p-2}v in Θ_ε, v=0 on \partialΘ_ε, \] where $Θ$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $p>2^{\ast}:=2N/(N-2)$, and $Θ_ε$ is obtained by deleting the $ε$-neighborhood of some sphere which is embedded in $Θ$. In some particular situations we show that, for $ε>0$ small enough, this problem has a positive solution $v_ε$ and that these solutions concentrate and blow up along the sphere as $ε$ tends to 0. Our approach is to reduce this problem to a critical problem of the form \[ -Δu=Q(x)|u|^{4/(n-2)}u in Ω_ε, u=0 on \partialΩ_ε, \]…
Let $(M,g)$ and $(K,κ)$ be two Riemannian manifolds of dimensions $m$ and $k ,$ respectively. Let $ω\in C^2(N),$ $ω> 0.$ The warped product $ M\times _ωK$ is the $ (m+k)$-dimensional product manifold $M\times K$ furnished with metric $ g+ω^2 κ.$ We prove that the supercritical problem $$-Δ_{g+ω^2 κ}u+h u=u^{ {m+2\over m-2} \pm\varepsilon},\ u>0,\ \hbox{in}\ (M\times _ωK,g+ω^2 κ)$$ has a solution which concentrate along a $k$-dimensional minimal submanifold $Γ$ of $M\times _ωN$ as the real parameter $\varepsilon$ goes to zero, provided the function $h$ and the sectional curvatures along $Γ$ sat…
We examine the Hénon equation $ -Δu =|x|^αu^p$ in $ Ω\subset \mathbb{R}^N$ with $u=0$ on $ \partial Ω$ where $ 0 < α$. We show there exists a sequence $ \{p_k\}_k \subset [ \frac{N+2}{N-2}, p_α(N)]$ with $p_1 < p_2 <p_3 < ...$, $ p_k \nearrow p_α(N)$ such that for any $ \frac{N+2}{N-2} \le p < p_α(N)$, which avoids $ \{p_k\}_k $, there exists a positive classical solution of the Hénon equation, provided $ Ω$ is a sufficiently small perturbation of the unit ball. We also examine the Lane-Emden-Fowler equation in the case of an exterior domain; ie. $ -Δu = u^p$ in $ Ω$, an exterior domain, with …
We study the problem% \[ -Δv+λv=| v| ^{p-2}v\text{ in }Ω,\text{\qquad}v=0\text{ on $\partialΩ$},\text{ }% \] for $λ\in\mathbb{R}$ and supercritical exponents $p,$ in domains of the form% \[ Ω:=\{(y,z)\in\mathbb{R}^{N-m-1}\times\mathbb{R}^{m+1}:(y,| z| )\inΘ\}, \] where $m\geq1,$ $N-m\geq3,$ and $Θ$ is a bounded domain in $\mathbb{R}% ^{N-m}$ whose closure is contained in $\mathbb{R}^{N-m-1}\times(0,\infty)$. Under some symmetry assumptions on $Θ$, we show that this problem has infinitely many solutions for every $λ$ in an interval which contains $[0,\infty)$ and $p>2$ up to some number which i…
We study the problem \begin{eqnarray*} (-Δ)^s u &=& u^p - u^q \quad\text{in }\quad \mathbb{R}^N, u &\in& \dot{H}^s(\mathbb{R}^N)\cap L^{q+1}(\mathbb{R}^N), u&>0& \quad\text{in}\quad\mathbb{R}^N, \end{eqnarray*} where $s\in(0,1)$ is a fixed parameter, $(-Δ)^s$ is the fractional laplacian in $\mathbb{R}^N$, $q>p\geq \frac{N+2s}{N-2s}$ and $N>2s$. For every $s\in(0,1)$, we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions a…
We consider a parabolic-elliptic Keller-Segel type system, which is related to a simplified model of chemotaxis. Concerning the maximal range of existence of solutions, there are essentially two kinds of results: either global existence in time for general subcritical ($\|ρ_0\|_1<8π$) initial data, or blow--up in finite time for suitably chosen supercritical ($\|ρ_0\|_1>8π$) initial data with concentration around finitely many points. As a matter of fact there are no results claiming the existence of global solutions in the supercritical case. We solve this problem here and prove that, for a p…
In this paper we consider the following coupled gradient-type quasilinear elliptic system \begin{equation*} \left\{ \begin{array}{ll} - {\rm div} ( a(x, u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v) &\hbox{ in $Ω$,}\\[10pt] - {\rm div} ( b(x, v, \nabla v) ) + B_t(x, v, \nabla v) = G_v\left(x, u, v\right) &\hbox{ in $Ω$,}\\[10pt] u = v = 0 &\hbox{ on $\partialΩ$,} \end{array} \right. \end{equation*} where $Ω$ is an open bounded domain in $\mathbb{R}^N$, $N\ge 2$. We suppose that some $\mathcal{C}^{1}$-Carathéodory functions $A, B:Ω\times\mathbb{R}\times\mathbb{R}^N\rightarrow\mathbb{R}$ …
We study $H^1$ versus $C^1$ local minimizers for functionals defined on spaces of symmetric functions, namely functions that are invariant by the action of some subgroups of $\mathcal{O}(N)$. These functionals, in many cases, are associated with some elliptic partial differential equations that may have supercritical growth. So we also prove some results on classical regularity for symmetric weak solutions for a general class of semilinear elliptic equations with possibly supercritical growth. We then apply these results to prove the existence of a large number of classical positive symmetric …
Let $Ω$ be a bounded domain in $\R^n$, $n\ge 3$ with smooth boundary $\partialΩ$ and a small hole. We give the first example of sign-changing {\it bubbling} solutions to the nonlinear elliptic problem $$ -Δu=|u|^{{n+2\over n-2} +\ve -1 } u \, \, \mbox{ in } Ω, \quad \quad u=0 \mbox{ on } \partial Ω, $$ where $\ve$ is a small positive parameter. The basic cell in the construction is the sign-changing nodal solution to the critical Yamabe problem $$ -Δw = |w|^{\frac{4}{n-2}} w, \ \ w \in {\mathcal D}^{1,2} (\R^n) $$ which has large number ($3n$) of kernels.
Given $α>0$, we establish the following two supercritical Moser-Trudinger inequalities \[ \sup\limits_{u \in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( (α_n + |x|^α) |u|^{\frac{n}{n-1}} \big) dx < +\infty \] and \[ \sup\limits_{u\in W^{1,n}_{0,{\rm rad}}(B): \int_B |\nabla u|^n dx \leq 1} \int_B \exp\big( α_n |u|^{\frac{n}{n-1} + |x|^α} \big) dx < +\infty, \] where $W^{1,n}_{0,{\rm rad}}(B)$ is the usual Sobolev spaces of radially symmetric functions on $B$ in $\mathbb R^n$ with $n\geq 2$. Without restricting to the class of functions $W^{1,n}_{0,{\rm rad}}(B)$, …
We show uniqueness of cylindrical blowups for mean curvature flow in all dimension and all codimension. Cylindrical singularities are known to be the most important; they are the most prevalent in any codimension. Mean curvature flow in higher codimension is a nonlinear parabolic system where many of the methods used for hypersurfaces do not apply and uniqueness of cylindrical blowups remained a major open problem. Our results imply regularity of the singular set for the system.
We construct positive singular solutions for the problem $-Δu=λ\exp (e^u)$ in $B_1\subset \mathbb{R}^n$ ($n\geq 3$), $u=0$ on $\partial B_1$, having a prescribed behaviour around the origin. Our study extends the one in Y. Miyamoto [Y. Miyamoto, A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth. J. Differential Equations \textbf{264} (2018), 2684--2707] for such nonlinearities. Our approach is then carried out to elliptic equations featuring iterated exponentials.
The general self-adjoint elliptic boundary value problems are considered in a domain $G\subset \Bbb R^{n+1}$ with finitely many cylindrical ends. The coefficients are stabilizing (as $x\to\infty$, $x\in G$) so slowly that we can only describe some ``structure'' of solutions far from the origin. This problem may be understood as a model of ``generalized branching waveguide.'' We introduce a notion of the energy flow through the cross-sections of the cylindrical ends and define outgoing and incoming ``waves.'' An augmented scattering matrix is introduced. Analyzing the spectrum of this matrix on…
