Saddle point solutions for non-local elliptic operators

By A. Fiscella. 2012

51 citations recorded by the source corpus.

The paper deals with equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a solution for them using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, one can derive an existence theorem for the fractional Laplacian, finding solutions of the equation $$ \begin{cases} (-\Delta)^s u=f(x,u) & {\mbox{in }} \Omega,\\ u=0 & {\mbox{in }} \mathbb{R}^n\setminus \Omega, \end{cases} $$ where the nonlinear term $f$ satisfies a linear growth condition.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9236d62668455ced94c21d830ada9b013f6812a1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

A bifurcation result for non-local fractional equations (2015, background): “In [4] [7] [8] [11] [19] [20] [21] [23] [24] [25] [26] [27] [30] (see also the references therein) the authors studied non-local fractional Laplacian equations with superlinear and subcritical or critical nonlinearities, while in [12] [13] the asymptotically linear case was exploited.”
Kirchhoff–Hardy Fractional Problems with Lack of Compactness (2017, background): “When Ω is bounded, p = 2 and K(x) = |x|−N−2s, then Z(Ω) is equivalent to the Hilbert space defined in [13].”
Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent (2015, background): “Recently, a lot of interest has been devoted to this kind of problems and to their concrete applications; see, for instance the seminal papers [9, 10, 11] and [2, 3, 8, 12, 15, 16, 17, 23, 31, 40] as well as the references therein.”