Stable solutions for weighted quasilinear Schrödinger equations in half-space with nonlinear boundary value conditions

In this paper, we study the following quasilinear Schrödinger equation in half-space{−Δu−Δ(|u|2α)|u|2α−2u=‖x‖θ|u|q−1uin R+n ∂u∂η=‖x′‖γ|u|p−1ug2(u)on ∂R+n , where α>12 is a parameter, q>3α−1+α2α, 1−2, γ>0 and g(s)=1+2αs2(2α−1). We establish a Liouville type theorem for the class of stable bounded sign-changing solutions under suitable assumptions on θ, q, p, α and n. In order to prove our results, first we obtain several important results concerning the a priori estimates of bounded stable solutions. Employing these results and Pohozaev identity, the nonexistence of nontrivial bounded solutions which are stable outside a compact set is proved.