Existence of a nonnegative bound state for a higher-order logarithmic Schrödinger equation

RESUMO

In this paper, we study the existence of nonnegative bound states for the higher-order logarithmic Schrödinger equation $$ -\alpha\Delta u+V(x)u-\beta u\ln|u| = \gamma u(\ln|u|)^m, \qquad u\in H^1(\mathbb R^N), $$ where $\alpha,\beta,\gamma>0$, $m$ is an odd positive integer and $V$ is a positive bounded potential converging to a constant at infinity. The simultaneous presence of the first- and higher-order logarithmic terms leads to a nonsmooth variational structure and additional compactness difficulties. We introduce a family of power-law approximations and derive estimates that are uniform as the approximation exponent tends to the logarithmic limit. Under a structural condition on the autonomous nonlinearity, we obtain uniqueness, up to translations, of the nonnegative autonomous profile with connected positivity set and a corresponding energy gap below the two-profile threshold. The small-amplitude behavior exhibits a maximum-principle/compact-support dichotomy: the case $m=1$ yields positivity, whereas the higher odd orders fall into the compact-support regime for the autonomous profile. We then establish a profile decomposition for constrained Palais-Smale sequences and construct a barycenter-based min-max level below the splitting threshold. This prevents loss of mass through multiple profiles and allows us to pass to the logarithmic limit. Under the stated structural and energy conditions, we obtain a nontrivial nonnegative bound state of the original equation.