GENERALIZED NEHARI MANIFOLD AND SEMILINEAR SCHRÖDINGER EQUATION WITH WEAK MONOTONICITY CONDITION ON THE NONLINEAR TERM.

We study the Schrödinger equations –Δu + V (x)u = f(x, u) in RN and –Δu–λu = f(x, u) in a bounded domain Ω ⊂ RN. We assume that f is superlinear but of subcritical growth and u ⟼ f(x, u)/|u| is nondecreasing. In RN we also assume that V and f are periodic in x1, . . ., xN. We show that these equations have a ground state and that there exist infinitely many solutions if f is odd in u. Our results generalize those by Szulkin and Weth [J. Funct. Anal. 257 (2009), 3802-3822], where u ⟼ f(x, u)/|u| was assumed to be strictly increasing. This seemingly small change forces us to go beyond methods of smooth analysis. [ABSTRACT FROM AUTHOR]