Symmetry of solutions to semilinear elliptic equations via Morse index

نویسندگان
چکیده

منابع مشابه

Symmetry-Breaking for Positive Solutions of Semilinear Elliptic Equations

In a recent interesting paper, GIDAS, NI, and NIRENBERG [2] proved that positive solutions of the Dirichlet problem for second-order semi-linear elliptic equations on balls must themselves be spherically symmetric functions. Here we consider the bifurcation problem for such solutions. Specifically, we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solutio...

متن کامل

Solutions of Superlinear at Zero Elliptic Equations via Morse Theory

(1) { −∆u = f(u) in Ω, u = 0 on ∂Ω, where Ω ⊂ R is an open bounded domain with smooth boundary. We assume that f ∈ C(R,R) satisfies f(0) = 0, so the constant function u ≡ 0 is a trivial solution of (1). We are interested in the existence of nontrivial solutions when f is superlinear at zero, that is near zero it looks like O(u|u|ν−2) for some ν ∈ (1, 2). More precisely, we assume that f and its...

متن کامل

Multiple Nontrivial Solutions of Elliptic Semilinear Equations

We find multiple solutions for semilinear boundary value problems when the corresponding functional exhibits local splitting at zero.

متن کامل

Newton's Method and Morse Index for semilinear Elliptic PDEs

In this paper we primarily consider the family of elliptic PDEs ∆u+ f(u) = 0 on the square region Ω = (0, 1)× (0, 1) with zero Dirichlet boundary condition. Following our previous analysis and numerical approximations which relied on the variational characterization of solutions as critical points of an “action” functional, we consider Newton’s method on the gradient of that functional. We use ...

متن کامل

Singular Solutions for some Semilinear Elliptic Equations

We are concerned with the behavior of u near x = O. There are two distinct cases: 1) When p >= N / ( N -2) and (N ~ 3) it has been shown by BR~ZIS & V~RON [9] that u must be smooth at 0 (See also BARAS & PIERRE [1] for a different proof). In other words, isolated singularities are removable. 2) When 1-< p < N / ( N 2) there are solutions of (1) with a singularity at x ---0. Moreover all singula...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

ژورنال

عنوان ژورنال: Proceedings of the American Mathematical Society

سال: 2007

ISSN: 0002-9939,1088-6826

DOI: 10.1090/s0002-9939-07-08652-2