Existence of nontrivial solution for a nonlocal problem with subcritical nonlinearity

摘要

In this paper, we consider the following new nonlocal Dirichlet boundary value problem: $$ extstyleegin{cases} -(a-bint_{Omega} ert abla u ert ^{2},dx)Delta u=lambda u+g(x,u),& xin Omega, u=0,& xinpartialOmega, end{cases} $$ (0.1) where a and b are positive, ?? is a positive parameter, (0leqlambda alambda_{1}), (lambda_{1}) is the first eigenvalue of operator a????. Under appropriate assumptions on the function g which is of subcritical growth, we obtain a nontrivial solution.KeywordsNonlocal problem??Nontrivial solution??Subcritical nonlinearity??MSC35B33??35B38??35B09??1 Introduction and main resultIn this paper, we consider the following new nonlocal Dirichlet boundary value problem: $$ extstyleegin{cases} -(a-bint_{Omega} ert abla u ert ^{2},dx)Delta u=lambda u+g(x,u),& xin Omega, u=0,& xinpartialOmega, end{cases} $$ (1.1) where a and b are positive, ?? is a positive parameter.The search for a nontrivial solution of problem (1.1) is a new subject and of great significance. We put forward a new nonlocal term (a-bint _{Omega}|abla u|^{2},dx), which is different from the well known nonlocal term (a+bint_{Omega}|abla u|^{2},dx) and presents a lot of interesting difficulties.