Infinitely many positive solutions of nonlinear Schrödinger equations

Abstract

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 $$N\ge 2$$ N ≥ 2 , $$p> 1,\ p< {N+2\over N-2}$$ p > 1 , p < N + 2 N - 2 if $$N\ge 3$$ N ≥ 3 , $$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$ a ∈ L loc N / 2 ( R N ) , $$\inf a> 0$$ inf a > 0 , $$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$ lim | x | → ∞ a ( x ) = a ∞ . Assuming that the potential a(x) satisfies $$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$ lim | x | → ∞ [ a ( x ) - a ∞ ] e η | x | = ∞ ∀ η > 0 , $$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$ 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. ....

Multiplicity results for ground state solutions of a semilinear equation via abrupt changes in magnitude of the nonlinearity

“…The multiplicity problem has been studied for the following non-autonomous problem −∆u = f (x, u), u(x) → 0 as |x| → ∞ for f of the form f (x, u) = g(x, u) − a(x)u by [AT1,AT2,AW,CMP,CPS,CZ,DWY,HL,WY,MP21]. Under different assumptions on the nonnegative function g and the coefficient a, they have established existence of multiple ground state solutions.…”

Multiplicity results for bound state solutions of a semilinear equation

“…The multiplicity problem has been studied mostly for the non-autonomous case f (x, u) = g(x, u)−a(x)u, for different nonnegative functions g and coefficients a. See for example: Cao and Zhou '96 [5], Adachi and Tanaka'00 [1], Cerami, Hsu and Lin '10 [25], Wei and Yan '10 [35], Ao and Wei'14 [2], Del Pino, Wei and Yao'15 [18], Cerami and Molle '19 [7], Molle and Passaseo'21 [29]. Some progress has also been made for the autonomous case, for example Dávila, del Pino and Guerra'13 [16] proved that for f (u) = −u + u p + λu q with N = 3, 1 < q < 3, p < 5 near 5, if λ is large enough, then there exist at least three radial ground state solutions to this problem.…”

Normalized solutions to mass supercritical Schrodinger equations with negative potential

“…A lot of efforts have been done studying problem (1.2), (1.3) for a fixed frequency λ ∈ R; the literature in this direction is huge and we do not even make an attempt to summarize it here (see e.g. the recent papers [14,23] and references therein).…”