164 research outputs found
Periodic solutions for critical fractional problems
We deal with the existence of -periodic solutions to the following
non-local critical problem \begin{equation*} \left\{\begin{array}{ll}
[(-\Delta_{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &\mbox{in}
(-\pi,\pi)^{N} \\ u(x+2\pi e_{i})=u(x) &\mbox{for all} x \in \mathbb{R}^{N},
\quad i=1, \dots, N, \end{array} \right. \end{equation*} where , , , is the fractional critical
Sobolev exponent, is a positive continuous function, and is a
superlinear -periodic (in ) continuous function with subcritical
growth. When , the existence of a nonconstant periodic solution is
obtained by applying the Linking Theorem, after transforming the above
non-local problem into a degenerate elliptic problem in the half-cylinder
, with a nonlinear Neumann boundary
condition, through a suitable variant of the extension method in periodic
setting. We also consider the case by using a careful procedure of limit.
As far as we know, all these results are new.Comment: Calculus of Variations and Partial Differential Equations (2018
The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions
We consider a large class of self-adjoint elliptic problem associated with
the second derivative acting on a space of vector-valued functions. We present
two different approaches to the study of the associated eigenvalues problems.
The first, more general one allows to replace a secular equation (which is
well-known in some special cases) by an abstract rank condition. The latter
seems to apply particularly well to a specific boundary condition, sometimes
dubbed "anti-Kirchhoff" in the literature, that arise in the theory of
differential operators on graphs; it also permits to discuss interesting and
more direct connections between the spectrum of the differential operator and
some graph theoretical quantities. In either case our results yield, among
other, some results on the symmetry of the spectrum
Perturbed nonlocal fourth order equations of Kirchhoff type with Navier boundary conditions
Abstract We investigate the existence of multiple solutions for perturbed nonlocal fourth-order equations of Kirchhoff type under Navier boundary conditions. We give some new criteria for guaranteeing that the perturbed fourth-order equations of Kirchhoff type have at least three weak solutions by using a variational method and some critical point theorems due to Ricceri. We extend and improve some recent results. Finally, by presenting two examples, we ensure the applicability of our results
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