Multi-bump solutions for logarithmic Schrödinger equations
The paper addresses the stationary version of the nonlinear Schrödinger equation for real function \(u(x)\) with the log nonlinearity in the \(N\)-dimensional space, with 1-periodic coefficient functions, \(V(x)\) and \(Q(x)\): \[ -\Delta u +V(x)u=Q(x)u\ln(u^2). \] This equation with constant coefficients \(V\) and \(Q\) gives rise to the well-known exact solutions in the form of \textit{gaussons}. This paper produces a proof of the existence of infinity many multi-peak solutions of the equation with the \(x\)-periodic coefficients. The solutions are looked for as stationary points of the respective Hamiltonian. The logarithmic singularity in the Hamiltonian, corresponding to the log nonlinearity of the equation, makes it necessary to address the fixed-point problem admitting its lack of smoothness.
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