Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems

Abstract

We study the existence and asymptotic behavior of solutions having positive and sign-changing components to the singularly perturbed system of elliptic equations [Formula: see text] in a bounded domain [Formula: see text] in [Formula: see text], with [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text] is the unit ball we obtain solutions with a prescribed combination of positive and nonradial sign-changing components exhibiting two different types of asymptotic behavior as [Formula: see text]: solutions whose limit profile is a rescaling of a solution with positive and nonradial sign-changing components of the limit system [Formula: see text] and solutions whose limit profile is a solution of the uncoupled system, i.e. after rescaling and translation, the limit profile of the [Formula: see text]th component is a positive or a nonradial sign-changing solution to the equation [Formula: see text]