Existence Of Solutions For Kirchhoff-Choquard Type Equation With Exponential Critical Growth Terms On A Plannar

Equations with vanishing potential and expotential critical growth are an important class of nonlinear partial differential equations.In recent years,the solutions of equations have received much attention from scholars.In this paper,by proving a weighted Sobolev embedding theorem and a weighted Trudinger-Moser inequality,using mountain road theorem and variational method,we are concerned with the existence of solutions for the following Kirchhoff equation where a,b>0,V,Q ∈ C(R2),there exist constants γ,β>0 such that γ≤2<β and a0,b0>0 such that We shall assume the nonlinearity f∈C(R2)and f(s)=0 for all s≤0,f(s)>0 for all s>0,it also satisfies the following conditions:(f0)there exists α0>0 such that(f1)lims→0 f(s)/s2=0;(f2)there exists θ≥4 such that 0<θF(s)≤f(s)s for all s>0;(f3)there exists λ>0 and ν>2 such that F(s)≥λsν for all s ∈(0,1].This paper is divided into three chapters.In the first chapter,the research contant,research ideas and main conclusions are introduced.In the second chapter,in order to obtain the nontrivial weak solution of the equation,the working space is studied.And a weighted Lebesgue space is also introducted,where p E[1,∞),ω∈Lloc(R2)and ω is a positive function.Then a class of weighted Trudinger-Moser inequality is studied by a special decomposition.In the third chapter,we discuss the properties of the energy functional corresponding to the equation.Finally,we prove the existence of the solution of the equation by using the mountain road theorem in the critical point theory.