The Stationary Wave Solution Of Semilinear Sch(o|")dinger Equations With Biharmonic Diffusion

In this paper, we main concerned with the stationary wave solution of semilinear Schodinger equations with biharmonic diffusion in RN: where z:R×RNâ†' C, â–³2= △△A denotes iterated N-dimensional Laplacian, here we just consider the case N≥ 5; r is a real positive constant, f(x,｜z｜)z is a given function. The above equation main arise from plasma physics, fluid mechanics. Set z(x, t)= u(x)eit, where u is a real function, (0.1) can be reduced to the following equation: k is a real positive constant; f(x,｜u｜)u is a given real function. Our paper mainly discussed the existence and non-existence of the solution about equation (0.2) for different given real function f(x,｜u｜)u. For the subcritical exponent problem, we obtain solution by Mountain-Pass theorem. And for the loss of compactness due to global space and critical exponent, by Pohozaev Identity and Lions Lemma, we deduce some existence or non-existence results of above equation.