A bifurcation problem for the equation in a bounded domain in with mixed boundary conditions, given nonnegative functions and a small perturbation is considered. The existence of a global bifurcation between two given simple eigenvalues of the Laplacian is proved under some assumptions about the supports of the functions . These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to .
Let L = -Δ + V be a Schrödinger operator in and be the Hardy type space associated to L. We investigate the bilinear operators T⁺ and T¯ defined by , where T₁ and T₂ are Calderón-Zygmund operators related to L. Under some general conditions, we prove that either T⁺ or T¯ is bounded from to for 1 < p,q < ∞ with 1/p + 1/q = 1. Several examples satisfying these conditions are given. We also give a counterexample for which the classical Hardy space estimate fails.
We prove bilinear virial identities for the nonlinear Schrödinger equation, which are extensions of the Morawetz interaction inequalities. We recover and extend known bilinear improvements to Strichartz inequalities and provide applications to various nonlinear problems, most notably on domains with boundaries.
We consider a nonlinear Laplace equation Δu = f(x,u) in two variables. Following the methods of B. Braaksma [Br] and J. Ecalle used for some nonlinear ordinary differential equations we construct first a formal power series solution and then we prove the convergence of the series in the same class as the function f in x.