5,380 research outputs found

    Poisson problems for semilinear Brinkman systems on Lipschitz domains in Rn

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    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

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    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 Σ\Sigma 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 Σ\Sigma 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

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    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

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    We consider the following semilinear elliptic equation on a strip: {arraylΔuu+up=0 in RN1×(0,L),u>0,uν=0 on (RN1×(0,L))array \left\{{array}{l} \Delta u-u + u^p=0 \ {in} \ \R^{N-1} \times (0, L), u>0, \frac{\partial u}{\partial \nu}=0 \ {on} \ \partial (\R^{N-1} \times (0, L)) {array} \right. where 1<pN+2N2 1< p\leq \frac{N+2}{N-2}. When 1<p0 1<p 0 such that for LLL \leq L_{*}, the least energy solution is trivial, i.e., doesn't depend on xNx_N, and for L>LL >L_{*}, the least energy solution is nontrivial. When N4,p=N+2N2N \geq 4, p=\frac{N+2}{N-2}, it is shown that there are two numbers L<LL_{*}<L_{**} such that the least energy solution is trivial when LLL \leq L_{*}, the least energy solution is nontrivial when L(L,L]L \in (L_{*}, L_{**}], and the least energy solution does not exist when L>LL >L_{**}. A connection with Delaunay surfaces in CMC theory is also made.Comment: typos corrected and uniqueness adde

    Entire large solutions for semilinear elliptic equations

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    We analyze the semilinear elliptic equation Δu=ρ(x)f(u)\Delta u=\rho(x) f(u), u>0u>0 in RD{\mathbf R}^D (D3)(D\ge3), with a particular emphasis put on the qualitative study of entire large solutions, that is, solutions uu such that limx+u(x)=+\lim_{|x|\rightarrow +\infty}u(x)=+\infty. Assuming that ff satisfies the Keller-Osserman growth assumption and that ρ\rho 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

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    Assuming BRB_{R} is a ball in RN\mathbb R^{N}, we analyze the positive solutions of the problem {Δu+u=up2u, in BR,νu=0, on BR, \begin{cases} -\Delta u+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_{\nu}u=0,&\text{ on } \partial B_{R}, \end{cases} that branch out from the constant solution u=1u=1 as pp grows from 22 to ++\infty. The non-zero constant positive solution is the unique positive solution for pp close to 22. We show that there exist arbitrarily many positive solutions as pp\to\infty (in particular, for supercritical exponents) or as RR \to \infty for any fixed value of p>2p>2, answering partially a conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for pp and RR 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

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    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 Ω\Omega is a bounded convex domain in RN,\R^N, N>2,N>2, with smooth boundary Ω.\partial \Omega. We study the asymptotic behaviour of the least energy solutions of this system as p.p\to \infty. We show that the solution remain bounded for pp 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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