Bifurcation results for a fractional elliptic equation with critical exponent in R^n

By Serena Dipierro, María Medina, I. Peral, Enrico Valdinoci. 2014

47 citations recorded by the source corpus.

In this paper we study some nonlinear elliptic equations in $\R^n$ obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is $$ (-\Delta)^s u = \epsilon\,h\,u^q + u^p \ {{in}}\R^n,$$ where $s\in(0,1)$, $n>4s$, $\epsilon>0$ is a small parameter, $p=\frac{n+2s}{n-2s}$, $0<q<p$ and $h$ is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov-Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case $0<q<1$ is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a9bbde7dea1b258e970bf9407ce0985b6ed0873d

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In Applied Mathematics

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Citations in the corpus

Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay (2015, methodology): “ann data on the boundary. This is therefore a nonlinear nonlocal operator “based” on the half-Laplacian or Dirichlet to Neumann map. Their papers also use the Lyapunov-Schmidt procedure. The articles [13,17] also use the Lyapunov-Schmidt method, here for some semilinear fractional elliptic equations”
Fractional elliptic problems with critical growth in the whole of $\R^n$ (2015, methodology): “1As a technical observation, we stress that papers like [20] and [26] deal with a bifurcation method from a ground state solution that is already known to be non-degenerate, while in the present monograph we find: (i) solutions that bifurcate from zero, (ii) different solutions obtained by a contrad”