This paper is dedicated to studying the following elliptic system of Hamiltonian type:■where N≥3,V,Q∈C(RN,R),V(x)is allowed to be sign-changing and inf Q>0,and F∈C1(R2,R)is superquadratic at both 0 and infinity...This paper is dedicated to studying the following elliptic system of Hamiltonian type:■where N≥3,V,Q∈C(RN,R),V(x)is allowed to be sign-changing and inf Q>0,and F∈C1(R2,R)is superquadratic at both 0 and infinity but subcritical.Instead of the reduction approach used in Ding et al.(2014),we develop a more direct approach—non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than those in Ding et al.(2014).We can find anε0>0 which is determined by terms of N,V,Q and F,and then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for allε∈(0,ε0].展开更多
We investigate the Kirchhoff type elliptic problem(a+b∫_(R^(N))[|∇u|^(2)+V(x)u^(2)]dx)[-Δu+V(x)u]=f(x,u),x∈R^(N),where both V and f are periodic in x,0 belongs to a spectral gap of−∆+V.Under suitable assumptions on...We investigate the Kirchhoff type elliptic problem(a+b∫_(R^(N))[|∇u|^(2)+V(x)u^(2)]dx)[-Δu+V(x)u]=f(x,u),x∈R^(N),where both V and f are periodic in x,0 belongs to a spectral gap of−∆+V.Under suitable assumptions on V and f with more general conditions,we prove the existence of ground state solutions and infinitely many geometrically distinct solutions.展开更多
We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1...We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1-periodic in each of x1, x2,..., x N and sup[σ(-△ + V0) ∩(-∞, 0)] < 0 < inf[σ(-△ +V0)∩(0, ∞)], V1∈ C(RN) and lim|x|→∞V1(x) = 0. Inspired by previous work of Li et al.(2006), Pankov(2005)and Szulkin and Weth(2009), we develop a more direct approach to generalize the main result of Szulkin and Weth(2009) by removing the "strictly increasing" condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N0 by using the diagonal method.展开更多
We survey recent results on ground and bound state solutions E:?→R^3 of the problem {▽(▽×E)+}λE=|E|^(P-2)E in Ω,v×E=0 on Ω on a bounded Lipschitz domain ??R^3,where?×denotes the curl operator in R...We survey recent results on ground and bound state solutions E:?→R^3 of the problem {▽(▽×E)+}λE=|E|^(P-2)E in Ω,v×E=0 on Ω on a bounded Lipschitz domain ??R^3,where?×denotes the curl operator in R^3.The equation describes the propagation of the time-harmonic electric field R{E(χ)e^(iwt)}in a nonlinear isotropic material ? withλ=-μεω~2≤0,where μ andεstand for the permeability and the linear part of the permittivity of the material.The nonlinear term|E|^(P-2)E with 2<p≤2*=6 comes from the nonlinear polarization and the boundary conditions are those for?surrounded by a perfect conductor.The problem has a variational structure;however the energy functional associated with the problem is strongly indefinite and does not satisfy the Palais-Smale condition.We show the underlying difficulties of the problem and enlist some open questions.展开更多
We consider the nonlinear difference equations of the form Lu=f（n,u）,n∈Z,where L is a Jacobi operator given by（Lu）（n）=a（n）u（n＋1）＋a（n-1）u（n-1）＋b（n）u（n） for n ∈Z,{a（n）} and {b（n）} are real val...We consider the nonlinear difference equations of the form Lu=f（n,u）,n∈Z,where L is a Jacobi operator given by（Lu）（n）=a（n）u（n＋1）＋a（n-1）u（n-1）＋b（n）u（n） for n ∈Z,{a（n）} and {b（n）} are real valued N-periodic sequences,and f（n,t） is superlinear on t.Inspired by previous work of Pankov[Discrete Contin.Dyn.Syst.,19,419-430（2007）]and Szulkin and Weth[J.Funct.Anal.,257,3802-3822（2009）],we develop a non-Nehari manifold method to find ground state solutions of Nehari-Pankov type under weaker conditions on f.Unlike the Nehari manifold method,the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold by using the diagonal method.展开更多
基金supported by National Natural Science Foundation of China(Grant No.11171351)
文摘This paper is dedicated to studying the following elliptic system of Hamiltonian type:■where N≥3,V,Q∈C(RN,R),V(x)is allowed to be sign-changing and inf Q>0,and F∈C1(R2,R)is superquadratic at both 0 and infinity but subcritical.Instead of the reduction approach used in Ding et al.(2014),we develop a more direct approach—non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than those in Ding et al.(2014).We can find anε0>0 which is determined by terms of N,V,Q and F,and then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for allε∈(0,ε0].
基金supported by Natural Science Foundation of Fujian Province(Nos.2022J01339 and 2020J01708)National Foundation Training Program of Jimei University(ZP2020057)。
文摘We investigate the Kirchhoff type elliptic problem(a+b∫_(R^(N))[|∇u|^(2)+V(x)u^(2)]dx)[-Δu+V(x)u]=f(x,u),x∈R^(N),where both V and f are periodic in x,0 belongs to a spectral gap of−∆+V.Under suitable assumptions on V and f with more general conditions,we prove the existence of ground state solutions and infinitely many geometrically distinct solutions.
基金supported by National Natural Science Foundation of China(Grant No.11171351)Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120162110021)
文摘We consider the semilinear Schrdinger equation-△u + V(x)u = f(x, u), x ∈ RN,u ∈ H 1(RN),where f is a superlinear, subcritical nonlinearity. We mainly study the case where V(x) = V0(x) + V1(x),V0∈ C(RN), V0(x) is 1-periodic in each of x1, x2,..., x N and sup[σ(-△ + V0) ∩(-∞, 0)] < 0 < inf[σ(-△ +V0)∩(0, ∞)], V1∈ C(RN) and lim|x|→∞V1(x) = 0. Inspired by previous work of Li et al.(2006), Pankov(2005)and Szulkin and Weth(2009), we develop a more direct approach to generalize the main result of Szulkin and Weth(2009) by removing the "strictly increasing" condition in the Nehari type assumption on f(x, t)/|t|. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold N0 by using the diagonal method.
基金supported by the National Science Centre of Poland (Grant No. 2013/09/B/ST1/01963)
文摘We survey recent results on ground and bound state solutions E:?→R^3 of the problem {▽(▽×E)+}λE=|E|^(P-2)E in Ω,v×E=0 on Ω on a bounded Lipschitz domain ??R^3,where?×denotes the curl operator in R^3.The equation describes the propagation of the time-harmonic electric field R{E(χ)e^(iwt)}in a nonlinear isotropic material ? withλ=-μεω~2≤0,where μ andεstand for the permeability and the linear part of the permittivity of the material.The nonlinear term|E|^(P-2)E with 2<p≤2*=6 comes from the nonlinear polarization and the boundary conditions are those for?surrounded by a perfect conductor.The problem has a variational structure;however the energy functional associated with the problem is strongly indefinite and does not satisfy the Palais-Smale condition.We show the underlying difficulties of the problem and enlist some open questions.
基金Supported by NSFC(Grant No.11571370)the Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20120162110021)of China
文摘We consider the nonlinear difference equations of the form Lu=f（n,u）,n∈Z,where L is a Jacobi operator given by（Lu）（n）=a（n）u（n＋1）＋a（n-1）u（n-1）＋b（n）u（n） for n ∈Z,{a（n）} and {b（n）} are real valued N-periodic sequences,and f（n,t） is superlinear on t.Inspired by previous work of Pankov[Discrete Contin.Dyn.Syst.,19,419-430（2007）]and Szulkin and Weth[J.Funct.Anal.,257,3802-3822（2009）],we develop a non-Nehari manifold method to find ground state solutions of Nehari-Pankov type under weaker conditions on f.Unlike the Nehari manifold method,the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold by using the diagonal method.
