Multiple bump solutions to logarithmic scalar field equations

Abstract

We investigate multi-bump positive and nodal solutions to the logarithmic scalar field equation $$ -\Delta u + V(y) u = u\log |u|,\quad u\in H^1(\mathbb R^N), $$ where $N\geq2$ and the potential $V$ is bounded radially symmetric, which is a class of important Schrödinger equations in mathematical physics. The main difficulties to apply Lyapunov-Schmidt reduction to logarithmic scalar equations are caused by the non-smooth property and sub-linear growth of the logarithmic nonlinearity. To overcome these difficulties, we develop a new approach to carry out the Lyapunov-Schmidt reduction, which can be used to construct not only positive but also nodal solutions to the logarithmic equations. Finally, both infinitely many positive and infinitely many nodal solutions with an arbitrarily large number of bumps are constructed for scalar field equations when the potential satisfies different proper decay conditions.