Existence And Multiplicity Of Solutions For Several Nonlocal Elliptic Problems

In this paper,by means of the nonlinear functional analysis and critical point theory,we establish the existence and multiplicity of solutions for several nonlocal elliptic problems involving critical exponential nonlinearities.First,we take account to the nonhomogeneous Kirchhoff type elliptic problem as follows where ? is a smooth bounded domain in R2,m:R1?R1 represents the Kirchhoff function,f:?×R?R is a continuous function and satisfies the critical expo-nential growth behaves as e?u2(?>0)at infinity.? is a small positive parameter,h(x)?(W01,2(?))*,h(x)?0 and h(x)(?)0.Using the Ekeland variational principle and Trudinger-Moser inequality in bounded domain,we obtain the existence and multiplicity of solutions for above problem.Next,we consider a class of nonlocal N-Kirchhoff elliptic type equation with the singular nonlinear term as follows ,where N?2,0??<N,?u?N=?RN(|?u|N+V(x)|u|N)dx,?Nu=div(1?u|N-2?u)is the N-Lapiacian,m:R+?R-stands for a Kirchhoff function,V:RN?R is a continuous potential function,?:RN×R?R is a continuous function,and behaves like e?|u|N/N-1 when |u|?? for some ?>0,h(x)?(W01,N(RN))*,h(x)?0 and h(x)(?)0,?>0 is a parameter.Together with variational methods and sin-gular Trudinger-Moser inequality in the whole RN,there are at least two nontrivial solutions for above equation when ? is small enough.Later,we discuss the following fractional elliptic problem where ? is smooth bounded domain in RN(N?1),0<s<1,(-?)N/ss denotes the fractional N/s-Laplacian operator which,up to a normalization constant,is defined as ,for ? ? C0?(RN).Here B?(x)denotes the ball in RN centered at x with radius?>0.f(x,u)is continuous in ?×R and has critical exponential growth in sense of Trudinger-Moser,which behaves like e?|u|N/(N-s)as |u|?? for some ?>0.By applying the fractional Trudinger-Moser type inequality and mountain pass theorem,we obtain the existence of a nontrivial solution for the above equation.Finally,we are concerned with the existence of solutions for the following fractional Kirchhoff type problem where ? is a smooth bounded domain in RN(N?1),a,b>0,0<s<1,?u?=(??R2N|u(x)-u(y)|N/s/|x-y|2Ndxdy)s/N,f satisfies the critical exponential growth at infinity.There are Kirchhoff function and fractional N/s-Laplacian operator,which increase the difficulty in mathematical research.Similarly,together with the frac-tional Trudinger-Moser inequality and mountain pass theorem,we gain the related result.