Peak solutions for logarithmic scalar field systems

We are concerned with a class of important Schrdinger equations in mathematical physics with logarithmic nonlinearities: - 2 u +V (y)u = u log|u|, u > 0, in H 1 (R N ), where N 3, is a small positive parameter, and V (y) denotes the potential function.The main difficulties to apply Lyapunov-Schmidt reduction to logarithmic scalar equations are caused by the non-smooth property and sublinear growth of the logarithmic non-linearity.Our method is fundamentally based on a new type of innerouter decomposition, setting it apart from conventional gluing techniques that usually require distinct constructions for the inner and outer problems.Rather than this traditional separation, we incorporate the minimization operator for the outer problem directly with the operator related to the fixed-point theorem, enhancing the reduction framework to be applicable.We prove the existence of positive multipeak solutions under certain assumptions on V (y).Finally, we also use the local Pohozaev identities to obtain the non-degenerate of positive multipeak solutions.