5,380 research outputs found
Poisson problems for semilinear Brinkman systems on Lipschitz domains in Rn
The purpose of this paper is to combine a layer potential analysis with the Schauder fixed point theorem to show the existence of solutions of the Poisson problem for a semilinear Brinkman system on bounded Lipschitz domains in Rn (n 65 2) with Dirichlet or Robin boundary conditions and data in L2-based Sobolev spaces. We also obtain an existence and uniqueness result for the Dirichlet problem for a special semilinear elliptic system, called the Darcy\u2013Forchheimer\u2013 Brinkman system
Locating the peaks of semilinear elliptic systems
We consider a system of weakly coupled singularly perturbed semilinear
elliptic equations. First, we obtain a Lipschitz regularity result for the
associated ground energy function as well as representation formulas
for the left and the right derivatives. Then, we show that the concentration
points of the solutions locate close to the critical points of in the
sense of subdifferential calculus.Comment: To appear on Advanced Nonlinear Studies, 21 page
Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball
In [40], it was shown that the following singularly perturbed Dirichlet problem
\ep^2 \Delta u - u+ |u|^{p-1} u=0, \ \mbox{in} \ \Om,\]
\[ u=0 \ \mbox{on} \ \partial \Om
has a nodal solution u_\ep which has the least energy among all nodal solutions. Moreover, it is shown that u_\ep has exactly one local maximum point P_1^\ep with a positive value
and one local minimum point P_2^\ep with a negative value and, as \ep \to 0,
\varphi (P_1^\ep, P_2^\ep) \to \max_{ (P_1, P_2) \in \Om \times \Om } \varphi (P_1, P_2),
where \varphi (P_1, P_2)= \min (\frac{|P_1-P_2}{2}, d(P_1, \partial \Om), d(P_2, \partial \Om)). The following question naturally arises: where is the {\bf nodal surface} \{ u_\ep (x)=0 \}? In this paper, we give an answer in the case of the unit ball \Om=B_1 (0).
In particular, we show that for \epsilon sufficiently small, P_1^\ep, P_2^\ep and the origin must lie on a line. Without loss of generality, we may assume that this line is the x_1-axis.
Then u_\ep must be even in x_j, j=2, ..., N, and odd in x_1.
As a consequence, we show that \{ u_\ep (x)=0 \} = \{ x \in B_1 (0) | x_1=0 \}. Our proof
is divided into two steps:
first, by using the method of moving planes, we show that P_1^\ep, P_2^\ep and the origin must lie on the x_1-axis and u_\ep must be even in x_j, j=2, ..., N. Then,
using the Liapunov-Schmidt reduction method, we prove
the uniqueness of u_\ep (which implies the odd symmetry of u_\ep in x_1). Similar results are also proved for the problem with Neumann boundary conditions
On least Energy Solutions to A Semilinear Elliptic Equation in A Strip
We consider the following semilinear elliptic equation on a strip: where . When such that
for , the least energy solution is trivial, i.e., doesn't depend
on , and for , the least energy solution is nontrivial. When , it is shown that there are two numbers
such that the least energy solution is trivial when , the least energy solution is nontrivial when ,
and the least energy solution does not exist when . A connection
with Delaunay surfaces in CMC theory is also made.Comment: typos corrected and uniqueness adde
Entire large solutions for semilinear elliptic equations
We analyze the semilinear elliptic equation , in
, with a particular emphasis put on the qualitative
study of entire large solutions, that is, solutions such that
. Assuming that satisfies the
Keller-Osserman growth assumption and that decays at infinity in a
suitable sense, we prove the existence of entire large solutions. We then
discuss the more delicate questions of asymptotic behavior at infinity,
uniqueness and symmetry of solutions.Comment: Journal of Differential Equations 2012, 28 page
Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions
Assuming is a ball in , we analyze the positive
solutions of the problem that branch out from the constant solution as grows from to
. The non-zero constant positive solution is the unique positive
solution for close to . We show that there exist arbitrarily many
positive solutions as (in particular, for supercritical exponents)
or as for any fixed value of , answering partially a
conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for
and so that a given number of solutions exist. The geometrical properties
of those solutions are studied and illustrated numerically. Our simulations
motivate additional conjectures. The structure of the least energy solutions
(among all or only among radial solutions) and other related problems are also
discussed.Comment: 37 pages, 24 figure
Asymptotic behaviour of a semilinear elliptic system with a large exponent
Consider the problem \begin{eqnarray*} -\Delta u &=& v^{\frac 2{N-2}},\quad
v>0\quad {in}\quad \Omega, -\Delta v &=& u^{p},\:\:\:\quad u>0\quad {in}\quad
\Omega, u&=&v\:\:=\:\:0 \quad {on}\quad \partial \Omega, \end{eqnarray*} where
is a bounded convex domain in with smooth boundary
We study the asymptotic behaviour of the least energy
solutions of this system as We show that the solution remain
bounded for large and have one or two peaks away form the boundary. When
one peak occurs we characterize its location.Comment: 16 pages, submmited for publicatio
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