Lanthanide-doped photon-avalanche(PA)upconversion(UC)nanoparticles(NPs),characterized by highly nonlinear optical response,have recently attracted tremendous interest for applications in many frontier areas such as su...Lanthanide-doped photon-avalanche(PA)upconversion(UC)nanoparticles(NPs),characterized by highly nonlinear optical response,have recently attracted tremendous interest for applications in many frontier areas such as super-resolution imaging[1],dynamic photoswitching[2],ultrasensitive optical sensing[3],and high-density optical memory and computing[4].Specifically,the large nonlinearities(N)of PA have fueled the development of low-cost,single-beam super-resolution imaging techniques,offering a√N-fold improvement in spatial resolution[5].Although PA NPs with N plateauing 60s have been developed through energy transfer engineering based on core/shell architecture[6],further enhancement remains challenging.展开更多
In this paper,we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and a steep potential well{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)...In this paper,we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and a steep potential well{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)ϕ=u^(2),in R^(3),where s∈(3/4,1),t∈(0,1),q∈(1,2),p∈(4,2_(s)^(*)),2_(s)^(*):=6/3-2s is the fractional critical exponent in dimension 3,V_(λ)(x)=λV(x)+1 withλ>0.Under the case of steep potential well,we obtain the existence of the sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma.Furthermore,we prove that the energy of ground state sign-changing solution is strictly more than twice of the energy of the ground state solution.Our results improve the recent results in the literature.展开更多
基金the National Natural Science Foundation of China(Nos.12474418,U22A20398,22135008)the Natural Science Foundation of Fujian Province(No.2024J010038).
文摘Lanthanide-doped photon-avalanche(PA)upconversion(UC)nanoparticles(NPs),characterized by highly nonlinear optical response,have recently attracted tremendous interest for applications in many frontier areas such as super-resolution imaging[1],dynamic photoswitching[2],ultrasensitive optical sensing[3],and high-density optical memory and computing[4].Specifically,the large nonlinearities(N)of PA have fueled the development of low-cost,single-beam super-resolution imaging techniques,offering a√N-fold improvement in spatial resolution[5].Although PA NPs with N plateauing 60s have been developed through energy transfer engineering based on core/shell architecture[6],further enhancement remains challenging.
基金supported by the Natural Science Foundation of Sichuan(No.2023NSFSC0073)。
文摘In this paper,we investigate the following fractional Schrödinger-Poisson system with concave-convex nonlinearities and a steep potential well{(-Δ)^(s)u+V_(λ)(x)u+ϕu=f(x)|u|^(q-2)u+|u|^(p-2)u,in R^(3),(-Δ)^(t)ϕ=u^(2),in R^(3),where s∈(3/4,1),t∈(0,1),q∈(1,2),p∈(4,2_(s)^(*)),2_(s)^(*):=6/3-2s is the fractional critical exponent in dimension 3,V_(λ)(x)=λV(x)+1 withλ>0.Under the case of steep potential well,we obtain the existence of the sign-changing solutions for the above system by using the constraint variational method and the quantitative deformation lemma.Furthermore,we prove that the energy of ground state sign-changing solution is strictly more than twice of the energy of the ground state solution.Our results improve the recent results in the literature.
