In this paper we study the asymptotic and qualitative properties of least energy radial sign-changing solutions of the fractional Brezis--Nirenberg problem ruled by the s-Laplacian, in a ball of $R^n$, when $s in (0,1)$ and $n > 6s$. As usual, $lambda$ is the (positive) parameter in the linear part in $u$. We prove that for $lambda$ 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 $lambda o 0^+$, i.e. the positive and negative parts concentrate at the same point with different concentration speeds.
On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis–Nirenberg problem / Cora, G.; Iacopetti, A.. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 176:(2018), pp. 226-271. [10.1016/j.na.2018.07.001]
On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis–Nirenberg problem
Iacopetti, A.
2018
Abstract
In this paper we study the asymptotic and qualitative properties of least energy radial sign-changing solutions of the fractional Brezis--Nirenberg problem ruled by the s-Laplacian, in a ball of $R^n$, when $s in (0,1)$ and $n > 6s$. As usual, $lambda$ is the (positive) parameter in the linear part in $u$. We prove that for $lambda$ 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 $lambda o 0^+$, i.e. the positive and negative parts concentrate at the same point with different concentration speeds.File | Dimensione | Formato | |
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