On some critical problems for the fractional Laplacian operator

By Begoña Barrios, Eduardo Colorado, A. de Pablo, U. Sánchez. 2011

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Abstract We study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional Laplacian: { ( − Δ ) α / 2 u = λ u q + u N + α N − α , u > 0 in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N is a smooth bounded domain, N ⩾ 1 , λ > 0 , 0 q N + α N − α , 0 α min { N , 2 } . For suitable conditions on α depending on q , we prove: In the case q 1 , there exist at least two solutions for every 0 λ Λ and some Λ > 0 , at least one if λ = Λ , no solution if λ > Λ . For q = 1 we show existence of at least one solution for 0 λ λ 1 and nonexistence for λ ⩾ λ 1 . When q > 1 the existence is shown for every λ > 0 . Also we prove that the solutions are bounded and regular.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/07fb943570dcbc1ff0f4cf641c76cea48dc95588

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Citations in the corpus

A critical fractional equation with concave-convex power nonlinearities (2013, background): “Some related problems involving this operator have been studied in [11,14,19,48] As in [7] the purpose of this paper is to study the existence of weak solutions for (P λ .”
Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian (2012, background): “We would just cite some very recent papers which analyze fractional elliptic equations involving the critical Sobolev exponent, [2, 6, 9, 16, 23, 24, 26].”
Fractional conformal Laplacians and fractional Yamabe problems (2010, methodology): “approximation and prove our Theorem 1.4. In the last section we adopt the calculation from [16] and prove our Theorem 1.5 by choosing a suitable test function. We ﬁnally mention the two related works [4, 41] on nonlinearities with critical exponents for the fractional Laplacian. 2 Conformal fraction”
Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem (2015, background): “we refer the reader to [3,4,9,10,39] and the references therein.”
A critical Kirchhoff type problem involving a nonlocal operator (2013, background): “Recently, some studies have been performed for critical problems in a non-local setting; we refer the interested readers to [3, 6, 17, 18, 19, 20].”
Borderline Variational Problems Involving Fractional Laplacians and Critical Singularities (2015, background): “−α|v k∇η|2dxdy = kα Z E y1−α|∇η|2|v k| 2dxdy, where E := Supp(|∇η|). Since α ∈ (0,2), y1−α is an A2-weight, and since E is bounded, we have that the embedding H1(E,y1−α) → L2(E,y1−α) is compact (See [3] and [23]). It follows from (3.9) 1 that kα Z E y1−α|v k∇η|2dxdy → 0 as k → ∞. Therefore, kα Z Rn+”