- On the intense worth of Nehari manifold for nonlocal singular Schr{ö}dinger-Kirchhoff equations in RN
Authors: Deepak Kumar Mahanta, Tuhina Mukherjee, Abhishek Sarkar
Summary: This text investigates the existence, non-existence, and multiplicity of weak options for a parameter-dependent nonlocal Schrödinger-Kirchhoff sort drawback on RN involving singular non-linearity. By performing high quality evaluation primarily based on Nehari submanifolds and fibre maps, our objective is to indicate the issue has at the least two optimistic options even when λ lies past the extremal parameter λ∗.
2. Nehari manifold method for fractional Kirchhoff issues with extremal worth of the parameter
Authors: P. K. Mishra, V. M. Tripathi
Summary: On this work we research the next nonlocal drawback
{M(∥u∥2X)(−Δ)su=λf(x)|u|γ−2u+g(x)|u|p−2uin Ω,u=0on RN∖Ω,
the place Ω⊂RN is open and bounded with clean boundary, N>2s,s∈(0,1),M(t)=a+btθ−1,t≥0 with θ>1,a≥0 and b>0. The exponents fulfill 1<γ<2<2θ<p<2∗s=2N/(N−2s) (when a≠0) and a pair of<γ<2θ<p<2∗s (when a=0). The parameter λ concerned in the issue is actual and optimistic. The issue into account has nonlocal behaviour as a result of presence of nonlocal fractional Laplacian operator in addition to the nonlocal Kirchhoff time period M(∥u∥2X), the place ∥u∥2X=∬R2N|u(x)−u(y)|2|x−y|N+2sdxdy. The load capabilities f,g:Ω→R are steady, f is optimistic whereas g is allowed to alter signal. On this paper an extremal worth of the parameter, a threshold to use Nehari manifold technique, is characterised variationally for each degenerate and non-degenerate Kirchhoff instances to indicate an existence of at the least two optimistic options even when λ crosses the extremal parameter worth by executing high quality evaluation primarily based on fibering maps and Nehari manifold.
