On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis–Nirenberg problem

Abstract

In this paper we study the asymptotic and qualitative properties of least energy radial signchanging solutions of the fractional Brezis-Nirenberg problem ruled by the s-Laplacian, in a ball of R n , when s ∈ (0, 1) and n > 6s. As usual, λ is the (positive) parameter in the linear part in u. We prove that for λ sufficiently small such solutions cannot vanish at the origin, we show that they change sign at most twice and their zeros coincide with the sign-changes. Moreover, when s is close to 1, such solutions change sign exactly once. Finally we prove that least energy nodal solutions which change sign exactly once have the limit profile of a "tower of bubbles", as λ → 0 + , i.e. the positive and negative parts concentrate at the same point with different concentration speeds.

Sign-changing bubble-tower solutions to fractional semilinear elliptic problems

“…In addition [19,Theorem 1.4] only grants that u ε does not vanish in a set of positive measure. Nevertheless, combining the results of [8] with a new argument based on energy and regularity estimates, we show that for any s ∈ (0, 1) least energy radial sign-changing solutions to (1.2) vanish only where a change of sign occurs (see Lemma 4.5, Lemma 5.2).…”

“…In this paper we consider slightly subcritical nonlinearities f (u) = |u| 2 * s −2−ε u, and we extend the results of [8] to all s ∈ (0, 1) without any extra assumption. The same proofs work also in the case of critical nonlinearities with minor modifications.…”

“…We point out that for 2s < n ≤ 6s, according to a classical result of Atkinson, Brezis, and Peletier (see [1]), radial sign-changing solutions in a ball may not exist when ε > 0 is close to zero, while they do exist for n > 6s (see [8,Theorem 3.7]).…”

“…At first glance the answer seems to be positive, but differently from the case of constantsign solutions, several difficulties arise when studying the qualitative properties of sign-changing solutions. Indeed, in view of the non-local interactions between the nodal components, we cannot take benefit from the fractional moving plane method (see [5]), and the strong maximum principle does not work properly (see [8,Sect. 1]).…”

“…Moreover, when s > 1 2 , we proved that they behave like a tower of two bubbles as ε → 0 + , namely, the positive and the negative part blowup and concentrate at the same point (which is the center of the ball) with different speeds. Nevertheless, we needed to assume that these solutions change sign exactly once to determine which one between the positive and the negative part blew-up faster (see [8,Sect.1]).…”