168 research outputs found
Equations involving fractional Laplacian operator: Compactness and application
In this paper, we consider the following problem involving fractional
Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u=
|u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0
\,\, {\rm on}\, \, \partial\Omega, \end{equation} where is a smooth
bounded domain in , ,
. We show that for any
sequence of solutions of \eqref{eq:0.1} corresponding to
, satisfying in the
Sobolev space defined in \eqref{eq:1.1a}, converges strongly in
provided that and . An application of this compactness
result is that problem \eqref{eq:0.1} possesses infinitely many solutions under
the same assumptions.Comment: 34 page
Multiple nodal solutions of nonlinear Choquard equations
In this paper, we consider the existence of multiple nodal solutions of the
nonlinear Choquard equation \begin{equation*} \ \ \ \ (P)\ \ \ \ \begin{cases}
-\Delta u+u=(|x|^{-1}\ast|u|^p)|u|^{p-2}u \ \ \ \text{in}\ \mathbb{R}^3, \ \ \
\ \\ u\in H^1(\mathbb{R}^3),\\ \end{cases} \end{equation*} where . We show that for any positive integer , problem has
at least a radially symmetrical solution changing sign exactly -times
Multiple solutions of semilinear elliptic systems
summary:We obtain in this paper a multiplicity result for strongly indefinite semilinear elliptic systems in bounded domains as well as in
Asymptotics of ground states for fractional H\'enon systems
We investigate the asymptotic behavior of positive ground states for H\'enon
type systems involving a fractional Laplacian on a bounded domain, when the
powers of the nonlinearity approach the Sobolev critical exponent.Comment: 18 page
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