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Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent. (English) Zbl 1322.49021

Summary: In the present paper, we study the weak lower semicontinuity of the functional \[ \Phi_{\lambda, \gamma}(u):=\frac{1}{2} \int_{\mathbb{R}^n\times \mathbb{R}^n} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,\text{d}x\,\text{d}y -\frac{\lambda}{2}\int_{\Omega}|u(x)|^2\,\text{d}x-\frac{\gamma}{2} \Big(\int_{\Omega}|u(x)|^{2^*}\,\text{d}x\Big)^{2/2^*}, \] 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 \[ -(-\Delta)^s u(x):= \int_{\mathbb{R}^{n}}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,\text{d}y, \;\;x\in \mathbb{R}^n. \] These two existence results were obtained using, respectively, the direct method in the calculus of variations and critical points theory.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35A15 Variational methods applied to PDEs
35R11 Fractional partial differential equations
35R09 Integro-partial differential equations
35S15 Boundary value problems for PDEs with pseudodifferential operators
47G20 Integro-differential operators
45G05 Singular nonlinear integral equations
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Full Text: Euclid