Research On The Existence Of Solutions For A Class Of Kirchhoff-type Equations With Fast Increasing Weights

Variational method is one of the important methods in nonlinear functional analysis.Its basic idea is to transform the solution problem of differential equation into the critical point problem of the correnponding functional.In this paper,a class of Kirchhoff type equations with fast increasing weight is studied by using variational method.Kirchhoff equation has a wide range of applications,such as earthquake prediction,imaging,and weak signal detection and recognition.Firstly,we consider the following weighted Kirchhoff-type problem with concave-convex nonlinear term in R3,(?) and obtain the existence results of two non-negative and non-trivial solutions by using the Mountain pass lemma and the Ekeland’s variational principle,where a,b are positive constants,h(x,u)=f(x)|u|q-2u+g(x)|u|p-2u,f and g satisfy given condition.Secondly,we consider the following weighted Kirchhoff-type problem with critical growth (?) where f has exponential subcritical or exponential critical growth in the sense of Trudinger-Moser inequality.By using constrained variational methods,quantitative deformation lemma,and Miranda’s theorem,we prove the existence of least energy sign-changing solution,and the sign-change solution has exactly two nodal domains.