## 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.

### MSC:

 35J60 Nonlinear elliptic equations 42B37 Harmonic analysis and PDEs 35J92 Quasilinear elliptic equations with $$p$$-Laplacian
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