Branch switching at a corank-4 bifurcation point of semi-linear elliptic problems with symmetry

Branch switching of the problem at a corank-4 bifurcation point is investigated by exploitingsymmetry and other properties of the operator. Here f :RR isa smooth odd function. The singularity of the corank-4 bifurcationpoint is decomposed into various subspaces via symmetries suchthat all solution branches can be followed by continuation methodsand their modifications. At the same time, a modified Lyapunov-Schmidtmethod is used to determine solution branches having littlesymmetry on a slightly enlarged system in appropriate subspaces.We find altogether 40 different solution branches. Among themwe have 13 solutions, such that none of these solutions canbe generated by conjugation or hidden symmetries from any othersolutions.     

Persistence of the bifurcation structure for a semilinear elliptic problem on thin domains

We study a semilinear elliptic problem on thin domains with a bifurcation parameter. It is shown that the set of solutions is upper semicontinuous as the thickness of a domain tends to 0, and that solution branches including bifurcation points persist near those of a one-dimensional limiting equation.     

A global bifurcation result for a semilinear elliptic equation

We consider the problem

     

Non-existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry

We give two sufficient conditions for a branch consisting of non-trivial solutions of an abstract equation in a Banach space not to have a (secondary) bifurcation point when the equation has a certain symmetry. When the nonlinearity f is of Allen-Cahn type (for instance f(u)=u−u³), we apply these results to an unbounded branch consisting of non-radially symmetric solutions of the Neumann problem on a disk D⊂R²

     

Existence and bifurcation of positive solutions of a semilinear elliptic problem on {\Bbb R}^N

Existence of positive solutions of a semilinear elliptic problem on , arising from population genetics is obtained. We use an approximation procedure, by accumulation of branches of solutions of the problem stated on balls of sufficiently large radia. Received January 19, 1996     

Symmetry breaking for a class of semilinear elliptic problems and the bifurcation diagram for a 1-dimensional problem

We study the behaviour of the positive solutions to the Dirichlet problem IR ⁿ in the unit ball in IR ^R wherep<(N+2)/(N−2) ifN≥3 and λ varies over IR. For a special class of functionsg viz.,g(x)=u ₀ ^p (x) whereu ₀ is the unique positive solution at λ=0, we prove that for certain λ’s nonradial solutions bifurcate from radially symmetric positive solutions. WhenN=1, we obtain the complete bifurcation diagram for the positive solution curve.     

Uniqueness and bifurcation for semilinear elliptic equations on closed surfaces

Motivated by a question recently posed by H. Brezis we provide uniqueness and bifurcation results for some semilinear elliptic equations on two dimensional closed surfaces with exponential nonlinearities.     

On bifurcation for semilinear elliptic Dirichlet problems on geodesic balls

We study bifurcation from a branch of trivial solutions of semilinear elliptic Dirichlet boundary value problems on a geodesic ball, whose radius is used as the bifurcation parameter. In the proof of our main theorem we obtain in addition a special case of an index theorem due to S. Smale.     

Critical point theorems and applications to a semilinear elliptic problem

The objective of this article is to establish the existence of critical points for functionals of classC ²defined on real Hilbert spaces. The argument is based on the infinite dimensional Morse theory introduced by Gromoll-Meyer [13]. The abstract results are applied to study the existence of nonzero solutions for a class of semilinear elliptic problems where the nonlinearity possesses a superlinear growth on a direction of the real line.This research was partially supported by CNPq/Brazil     

Multiplicity and bifurcation of positive solutions for nonhomogeneous semilinear elliptic problems

In this paper we consider the following problem

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Parametrization of a solution at bifurcation points

We suggest algorithms for finding bifurcation points numerically. The suggested approach is based on the method of extension of a solution with respect to the best parameter. We prove theorems and analyze specific features and properties of algorithms, whose efficiency is illustrated by numerical examples.     

Local and global bifurcation results for a semilinear boundary value problem

We investigate the local and global nature of the bifurcation diagrams which can occur for a semilinear elliptic boundary value problem with Neumann boundary conditions involving sign-changing coefficients. It is shown that closed loops of positive and negative solutions occur naturally for such problems and properties of these loops are investigated.