Infinitely many positive solutions of nonlinear Schrödinger equations

The paper deals with the equation $$-Delta u+a(x) u =|u|^{p-1}u $$ - Δ u + a ( x ) u = | u | p - 1 u , $$u in H^1({mathbb {R}}^N)$$ u ∈ H 1 ( R N ) , with $$Nge 2$$ N ≥ 2 , $$p> 1, p< {N+2over N-2}$$ p > 1 , p < N + 2 N - 2 if $$Nge 3$$ N ≥ 3 , $$ain L^{N/2}_{loc}({mathbb {R}}^N)$$ a ∈ L loc N / 2 ( R N ) , $$inf a> 0$$ inf a > 0 , $$lim _{|x| ightarrow infty } a(x)= a_infty $$ lim | x | → ∞ a ( x ) = a ∞ . Assuming that the potential a ( x ) satisfies $$lim _{|x| ightarrow infty }[a(x)-a_infty ] e^{eta |x|}= infty orall eta > 0$$ lim | x | → ∞ [ a ( x ) - a ∞ ] e η | x | = ∞ ∀ η > 0 , $$ lim _{ ho ightarrow infty } sup left{ a( ho heta _1) - a( ho heta _2) : heta _1, heta _2 in {mathbb {R}}^N, | heta _1|= | heta _2|=1 ight} e^{ ilde{eta } ho } = 0 quad ext{ for } ext{ some } ilde{eta }> 0$$ lim ρ → ∞ sup a ( ρ θ 1 ) - a ( ρ θ 2 ) : θ 1 , θ 2 ∈ R N , | θ 1 | = | θ 2 | = 1 e η ~ ρ = 0 for some η ~ > 0 and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a ( x ) has radial symmetry, or that $$N=2$$ N = 2 , or that $$|a(x)-a_infty |$$ | a ( x ) - a ∞ | is uniformly small in $${mathbb {R}}^N$$ R N , etc.