THE EXISTENCE OF WEAK SOLUTION TO FRACTIONAL ANISOTROPIC EQUATIONS IN WITH EXPONENTIAL GROWTH

Tóm tắt

In this paper, we investigate the existence of a weak solution to a fractional anisotropic equation with a Choquard reaction and exponential growth. The variational methods are used to study above problem. Namely, Lions’s result and Mountain Pass Theorem are combined to get the existence of a weak solution to our equation. Under some suitable conditions on nonlinear reaction, studies show that the energy function to our problem satisfies the geometric condition of the Mountain Pass Theorem. Then there exists the Palais-Smale sequence with a positive level. Next, for that given condition on the nonlinear function, the mountain pass level is proved small enough, as a result, the fractional Trudinger-Moser inequality can be applied to get the compact of that Palais-Smale sequence. The weak limit of the Palais-Smale sequence is the weak solution of our problem. Some difficulties needed to be solved of estimating the mountain pass level, as well as the compact of the Palais-Smale sequence due to the Choquard reaction and exponential nonlinearity. In the future work, our results can be applied to study the multiplicity of weak solutions for the autonomous problem associated with our problem.