Profile and Existence of Sign-Changing Solutions to an Elliptic Subcritical Equation

Abstract

In this paper, we study the nonlinear elliptic problem involving nearly critical exponent (Pε) : Δ2 u = |u|(8/(n-4))-εu, in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 5. We characterize the low energy sign-changing solutions (uε) of (Pε). We prove that (uε) are close to two bubbles with different signs and they have to blow up either at two different points with the same speed or at a critical point of the Robin function. Furthermore, we construct families of each kind of these solutions and we prove that the bubble-tower solutions exist in our case.

Sign-changing bubble tower solutions for a Paneitz-type problem

“…In the subcritical case, namely q = N+4 N−4 − ε with ε > 0 small, when K is a constant, the asymptotic behaviour of solutions of (1.3) has been studied in [4]. On the other hand, Ayed and Ghoudi [2] proved that the low energy sign-changing solutions to (1.3) that are close to two bubbles with different signs and they have to blow up either at two different points with the same speed or at a critical point of the Robin function. Yessine and Rabeh [40] constructed a solution with the shape of a tower of sign-changing bubbles as ε goes to zero.…”

Multispike Solutions for a slightly subcritical elliptic problem with non-power nonlinearity

“…When the Laplacian operator in (1.3) is replaced by the biharmonic one, the existence of multispike and bubble towers changing sign solutions for the counterpart of (1.3) have been studied in [6] and [8] respectively. Going back to (Pε), the existence of multispike solutions has not been studied yet and the main purpose of this paper is to focus on this issue.…”

On a biharmonic equation involving slightly supercritical exponent

“…The concentration phenomena for second-order elliptic equations (P ε ) involving a nearly subcritical exponent ( ε ∈ (1 − p, 0) ) were studied in [11,15,17] for K ̸ ≡ 1 and [5,8] for K ≡ 1 only.…”