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

By Giovanni Molica Bisci, Raffaella Servadei. Advances in Differential Equations, 2015

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In the present paper we study the weak lower semicontinuity of the functional

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On Dirichlet problem for fractional p-Laplacian with singular non-linearity (2016, background): “The existence andmultiplicity results for non-local operators, like the fractional Laplacian with a combination of convex and concave type non-linearities like uq + λup, p, q > 0, is studied in [3, 5, 36, 37, 44, 45].”
Bifurcation and multiplicity results for critical fractional p‐Laplacian problems (2014, result): “er [7] (see, e.g., [10, 9, 1, 23, 19, 38, 11, 15, 20, 22, 21, 12, 36] and references therein). Consequently, many results known in the local case s = 1 have been extended to problem (1.4) (see, e.g., [34, 31, 35, 32, 33, 2, 5, 37, 17]). In particular, 2 Fiscella et al. [17] have recently obtained th”
Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials (2016, background): “Last but not least, fractional elliptic problems with critical growth, in a bounded domain, have been studied by some authors in the last years, see [6, 7, 18, 28, 39, 41] and references therein.”
A critical fractional equation with concave-convex power nonlinearities (2013, background): “Previous works related to the operator deﬁned in (1.2), or by a more general kernel, can be found in [16,23, 30,34,35,37,38,41–43 Problems similar to (P λ ) have been also studied in the local setting with different elliptic operators. far as we know, the ﬁrst example in this direction was given in…”