Quantum mechanics

Function (biology)

Mathematical physics

Biology

Statistics

Boundary value problem

Applied mathematics

Energy (signal processing)

Bounded function

Sign (mathematics)

Lipschitz continuity

Evolutionary biology

Physics

Nonlinear system

Lipschitz domain

Mathematics

Domain (mathematical analysis)

Mathematical analysis

Logarithm

p-Laplacian

Nehari manifold

Nonlinear Analysis

In this paper, we study the existence of least energy solutions to the following fractional Kirchhoff problem with logarithmic nonlinearity M([u]s,pp)(−Δ)psu=h(x)|u|θp−2uln|u|+λ|u|q−2ux∈Ω,u=0x∈RN∖Ω,where s∈(0,1), 1<p<N∕s, Ω⊂RN is a bounded domain with Lipschitz boundary, M([u]s,pp)=[u]s,p(θ−1)p with θ≥1 and [u]s,p is the Gagliardo seminorm of u, h∈C(Ω¯) may change sign, λ>0 is a parameter, q∈(1,ps∗) and (−Δ)ps is the fractional p−Laplacian. When θp<q<ps∗ and h is a positive function on Ω, the existence of least energy solutions is obtained by restricting the discussion on Nehari manifold. When 1<q<θp and h is a sign-changing function on Ω, two local least energy solutions are obtained by using the Nehari manifold approach.

Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity