A concave—convex elliptic problem involving the fractional Laplacian

By Cristina Brandle, Eduardo Colorado, A. de Pablo. Proceedings of the Royal Society of Edinburgh. Section A Mathematics, 2010

552 citations recorded by the source corpus.

We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave—convex term. We completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result. We also prove an associated trace inequality and some Liouville-type results.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7423e49f9cc4f03cdec9732e06c74e71629554ce

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Citations in the corpus

A General Fractional Porous Medium Equation (2011, unknown): “ed by Ω ⊂ RN bounded. On the other hand, let v∈ Hσ/2 0 (Ω). Consider its σ-extension w= E(v), (σ= 2γ), and let webe the extension of wby zero outside the half-cylinder. Then we have the estimate, see [10], [19], µσ Z RN+1 + y1−σ|∇we|2 ≥ S(σ,N) Z RN |Tr(we)|N2−Nσ N−σ N . i.e., (5.5) µσ Z Ω y1−σ|∇w|2 ”
Nonexistence results for a class of fractional elliptic boundary value problems (2012, background): “+ . Moreover, w does not attain an interior maximum or minimum point in RN+1 + unless wis constant. Note also that we have a well deﬁned continuous trace map D1,2(RN+1 + ;t 1−2s) → Ds,2(RN) (see e.g. [5]), and for the sake of simplicity we denote the trace of a function in D1,2(RN+1 + ;t1−2s) with t”
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 .”
On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (2011, unknown): “For σ ∈ (1/2, 1) and 1 < p < n+2σ n−2σ , this result has been proved in [13].”
Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity (2013, background): “For related investigations on bounded domain, one can see [5, 9, 14, 18, 40] and references therein.”
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].”