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July/August 2015 Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent
Adv. Differential Equations 20(7/8): 635-660 (July/August 2015).

## Abstract

In the present paper, we study the weak lower semicontinuity of the functional \begin{align*} \Phi_{\lambda, \gamma}(u): & =\frac 1 2 \int_{\mathbb R ^n\times\mathbb R ^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dx\,dy -\frac \lambda 2 \int_\Omega |u(x)|^2 dx \\ & -\frac\gamma 2 \Big(\int_\Omega |u(x)|^{2^*} dx\Big)^{2/2^*}, \end{align*} where $\Omega$ is an open bounded subset of $\mathbb R ^n$, $n >2s$, $s\in (0,1)$, with continuous boundary, $\lambda$ and $\gamma$ are real parameters and $2^*:=2n/(n-2s)$ is the fractional critical Sobolev exponent. As a consequence of this regularity result for $\Phi_{\lambda, \gamma}$, we prove the existence of a nontrivial weak solution for two different nonlocal critical equations driven by the fractional Laplace operator $(-\Delta)^{s}$ which, up to normalization factors, may be defined as $$-(-\Delta)^s u(x):= \int_{\mathbb{R}^{n}}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \mathbb R ^n.$$ These two existence results were obtained using, respectively, the direct method in the calculus of variations and critical points theory.

## Citation

Giovanni Molica Bisci. Raffaella Servadei. "Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent." Adv. Differential Equations 20 (7/8) 635 - 660, July/August 2015.

## Information

Published: July/August 2015
First available in Project Euclid: 8 May 2015

zbMATH: 1322.49021
MathSciNet: MR3344613

Subjects:
Primary: 35A15, 35S15, 45G05, 47G20, 49J35  