Advances in Differential Equations

Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent

Giovanni Molica Bisci and Raffaella Servadei

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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.

Article information

Source
Adv. Differential Equations Volume 20, Number 7/8 (2015), 635-660.

Dates
First available in Project Euclid: 8 May 2015

Permanent link to this document
https://projecteuclid.org/euclid.ade/1431115711

Mathematical Reviews number (MathSciNet)
MR3344613

Zentralblatt MATH identifier
1322.49021

Subjects
Primary: 49J35: Minimax problems 35A15: Variational methods 35S15: Boundary value problems for pseudodifferential operators 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 45G05: Singular nonlinear integral equations

Citation

Molica Bisci, Giovanni; Servadei, Raffaella. Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent. Adv. Differential Equations 20 (2015), no. 7/8, 635--660. https://projecteuclid.org/euclid.ade/1431115711.


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