Result type
journal article in Web of Science database
Description
In this paper, we study the following logarithmic Schrödinger equation
$−\Delta u+λa(x)u=u\logu^2 $ in V
on a connected locally finite graph $G=(V,E)$, where $\Delta$ denotes the graph Laplacian, λ>0 is a constant, and a(x)≥0 represents the potential. Using variational techniques in combination with the Nehari manifold method based on directional derivative, we can prove that, there exists a constant $λ_0>0$ such that for all $λ≥λ_0$, the above problem admits a least energy sign-changing solution $u_λ$. Moreover, as λ→+∞, we prove that the solution $u_λ$ converges to a least energy sign-changing solution of the following Dirichlet problem $−\Delta u=ulogu^2 $ in Ω, u(x)=0 on ∂Ω, where Ω={x∈V:a(x)=0} is the potential well.
$−\Delta u+λa(x)u=u\logu^2 $ in V
on a connected locally finite graph $G=(V,E)$, where $\Delta$ denotes the graph Laplacian, λ>0 is a constant, and a(x)≥0 represents the potential. Using variational techniques in combination with the Nehari manifold method based on directional derivative, we can prove that, there exists a constant $λ_0>0$ such that for all $λ≥λ_0$, the above problem admits a least energy sign-changing solution $u_λ$. Moreover, as λ→+∞, we prove that the solution $u_λ$ converges to a least energy sign-changing solution of the following Dirichlet problem $−\Delta u=ulogu^2 $ in Ω, u(x)=0 on ∂Ω, where Ω={x∈V:a(x)=0} is the potential well.
