A global bifurcation result for a semilinear elliptic equation

We consider the problem

     

On bifurcation braid monodromy of elliptic fibrations

We propose to study a new kind of monodromy homomorphism for families of regular elliptic fibrations of a given differentiable fibration type to get a hold on topological properties of moduli stacks of elliptic surfaces.In specific cases, including the most significant one, when all singular fibres are nodal irreducible rational curves, we compute the corresponding monodromy group, a subgroup of the mapping class group of the fibration base punctured at the singular values of the fibration.We study a tentative algebraic characterisation and give implications for the group of diffeomorphisms compatible with the fibration.     

Hamiltonian identities for elliptic partial differential equations

New identities for elliptic partial differential equations are obtained. Several applications are discussed. In particular, Young's law for the contact angles in triple junction formation is proven rigorously. Structure of level curves of saddle solutions to Allen-Cahn equation are also carefully analyzed.     

A detailed analysis on local bifurcation from infinity for nonlinear elliptic problems

In this paper we analyze the local side of the bifurcation from infinity at the first eigenvalue of several elliptic operators. We underline that the key to decide the local behavior of the bifurcation lies in the sign of certain integral involving all the values of the nonlinearity.     

A finite element method for computing the bifurcation function for semilinear elliptic BVPs

The method of alternative problems can be used to show that a semilinear elliptic boundary value problem (Lu + g(x, u) = 0 with g_u(x, u) bounded below) is equivalent to a finite-dimensional problem ( , ), in the sense that their solution sets, which are not necessarily singletons, are in a one-to-one correspondence. This correspondence is based on a map σ from low-frequency to high-frequency Fourier components of solutions. A numerical method is presented for approximating σ and hence also solutions of the BVP. The method uses finite element approximations and avoids the use of eigenfunction expansions. Existence, uniqueness, and error estimates for the approximations of σ and solutions u are derived.     

Remarks on bifurcation for elliptic operators with odd nonlinearity

On estimating the eigenvalues for a class of semilinear elliptic operators, we obtain bifurcation and comparison results concerning the eigenvalues of some related linear problem.     

Prediction of homoclinic bifurcation: the elliptic averaging method

A criterion to predict bifurcation of homoclinic orbits in strongly nonlinear autonomous oscillators is presented. The averaging method combined formally with the Jacobian elliptic functions is applied to determine an approximation of limit cycles near homoclinicity. We then introduce a criterion for predicting homoclinic orbits, based on the collision between the bifurcating limit cycle and the saddle equillibrium. In particular, we show that this criterion leads to the same results as the standard Melnikov technique. Explicit applications of this criterion to quadratic nonlinearities are included.     

On Hamiltonian elliptic systems with periodic or non-periodic potentials

This paper is concerned with the following Hamiltonian elliptic system

     

On ground state solutions for superlinear Hamiltonian elliptic systems

In this paper, we study the following Hamiltonian elliptic systems $$\left\{\begin{array}{ll}-\Delta u+V(x)u= g(x,v),\quad {\rm in }\, \mathbb{R}^N,\\-\Delta v+V(x)v= f(x,u),\quad {\rm in } \, \mathbb{R}^N.\end{array}\right.$$ where ${V(x)\in C(\mathbb R^N), f(x,t), g(x,t)\in C(\mathbb{R}^N\times \mathbb{R})}$ are superlinear in t at infinity. Without Ambrosetti–Rabinowtitz condition, the existences of ground state solutions are obtained via the combination of generalized linking theorem and monotonicity method.