Symmetry breaking for an elliptic equation involving the fractional Laplacian. (English) Zbl 1463.35251

In this article, the author investigates the problem \[(-\Delta)^su+|x|^a|u|^{q-2}u=|x|^b|u|^{p-2}u\,,\tag{1.2}\] with \(a,b>0\), and \(u\) in the natural energy space, \(H^s_{q,a}(\mathbb{R}^n)\), and analogous problems in a ball \(B_R=\{x\in\mathbb{R}^n:|x|<R\}\) (for \(q=2\)).
The main result reads as follows:
Theorem 1.1. Let \(n\ge 2\), \(1/2<s<1\), \(2<p<2^\star=\frac{2n}{n-2s}\), \(0<a<n\) and \(b>\frac{ap}{2}\). If in addition, \[a(p-2-2ps)+4bs<2s(p-2)(n-1)\,,\tag{1.3}\] then for every \(R>0\) large enough, problem \[\begin{cases}(-\Delta)^su+|x|^au=|x|^bu^{p-1}\quad\text{in }B_R \\ u>0\text{ a.e. in }B_R,\quad u\equiv 0\;\text{ in }\mathbb{R}^n\setminus B_R\,,\end{cases}\tag{1.4}\] has a nontrivial radial weak solution and a nonradial one.
To prove this theorem the author states and proves the generalization of Strauss inequality for the space \(H^s_{q,a}(\mathbb{R}^n)\), proves the Hölder continuity of the functions in that space, outside the origin and uses the Arzelà-Ascoli theorem for the compactness arguments.


35J60 Nonlinear elliptic equations
42B37 Harmonic analysis and PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
Full Text: arXiv