Tuesday, April 12, 2022 - 16:00 in ZOOM - Video Conference
Various structures for semilinear equations in $\mathbb R^n$ driven by fractional Laplacian
A talk in the Nonlocal Equations: Analysis and Numerics series by
Alexander I. Nazarov from St.~Petersburg Dept of Steklov Institute and St.~Petersburg State University, St.~Petersburg, Russia
| Abstract: |
We study bounded solutions to the fractional equation
\begin{equation}
(-\Delta)^s u+u-|u|^{q-2}u=0
\end{equation}
in $\mathbb R^n$ for $n\ge2$ and subcritical exponent $q>2$. Applying the variational approach based on concentration arguments and symmetry considerations which was introduced by Lerman, Naryshkin and Nazarov [1] we construct several types of solutions with various structures (radial, rectangular, triangular, hexagonal, quasi-periodic, breather type, etc.). The main difference from the local case is that we construct solutions to equation (1) in the whole space from solutions to similar equations with various fractional Laplacians in domains.
The talk is based on the joint work with Alexandra Shcheglova [2].
[1] Lerman L.M., Naryshkin P.E., Nazarov A.I. Abundance of entire solutions to nonlinear elliptic equations by the variational method. Nonlin. Analysis, { 190} (2020), paper N111590.
[2] A.I. Nazarov, A.P. Shcheglova, Solutions with various structures for semilinear equations in $\mathbb R^n$ driven by fractional Laplacian, Preprint available at $\href{https://arxiv.org/abs/2111.07301}{https://arxiv.org/abs/2111.07301}.$ $28p$.
Within the CRC this talk is associated to the project(s): A7 |
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