We are concerned with the following Dirichlet problem: -△u(x) = f(x, u), x ∈ Ω. u ∈ H_0~1(Ω). (P) where f(x, t) ∈ C(Ω×R), f(x, t)/t is nondecreasing in t ∈ R and tends to an L~∝-function q(x) uniformly ...We are concerned with the following Dirichlet problem: -△u(x) = f(x, u), x ∈ Ω. u ∈ H_0~1(Ω). (P) where f(x, t) ∈ C(Ω×R), f(x, t)/t is nondecreasing in t ∈ R and tends to an L~∝-function q(x) uniformly in x ∈ Ω as t→+∝ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case. an Ambrosetti-Rabinowitz-type condition, that is. for some θ>2. M>0, 0<θF(x. s)≤ f(x, s)s, for all |s|≥M and x ∈ Ω, (AR) is no longer true, where F(x, s) = integral from n=0 to s f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable, conditions on f(x, t) and q(x). Our methods also work for the case where f(x, f) is superlinear in t at infinity. i.e., q(x) ≡∞.展开更多
By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε...By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε^(N)(-△)^(s)N/sμ+V(x)|μ|^(N/s-2μ)=Q(x)h(μ)in R^(N),where ε>0 is a parameter,s ∈(0,1),2≤p<+oo and N=ps.The nonlinear term h is a diferentiable function with exponential critical growth,the absorption potential V and the reaction potential Q are continuous functions.展开更多
This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forc-ing perturbations q+Aq+Vq(t,q)=f(t),where q=(q1,q2,..qN)∈R^(N),A is an ...This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forc-ing perturbations q+Aq+Vq(t,q)=f(t),where q=(q1,q2,..qN)∈R^(N),A is an antisymmetric constant N×N matrix,V(t,q)=-K(t,q)+W(t,q)with K,W ∈C^(1)(R,R^(N))and satisfying b1|q|^(2)≤K(t,q)≤b_(2)|q|^(2)for some positive constants b_(2)≥b_(1)>0 and external forcing term f∈C(R,R^(N))being small enough.Under some new weak superquadratic conditions for W,by using the mountain pass theorem,we obtain the existence of at least one nontrivial homoclinic solution.展开更多
This paper concerns the existence of multiple homoclinic orbits for the second-order Hamiltonian system-L（t）z＋Wz（t,z）=0,where L∈C（R,RN2）is a symmetric matrix-valued function and W（t,z）∈C1（R×RN,R）is a...This paper concerns the existence of multiple homoclinic orbits for the second-order Hamiltonian system-L（t）z＋Wz（t,z）=0,where L∈C（R,RN2）is a symmetric matrix-valued function and W（t,z）∈C1（R×RN,R）is a nonlinear term.Since there are no periodic assumptions on L（t）and W（t,z）in t,one should overcome difficulties for the lack of compactness of the Sobolev embedding.Moreover,the nonlinearity W（t,z）is asymptotically linear in z at infinity and the system is allowed to be resonant,which is a case that has never been considered before.By virtue of some generalized mountain pass theorem,multiple homoclinic orbits are obtained.展开更多
For the following elliptic problem {-△u-μu/｜x｜^2=｜u｜^2^＊（s）-2u/｜x｜^s＋h（x）, on R^N u∈D^1,2（R^N）, N≥3, 0≤μ〈μ^-=（N-2）^2/4, 0≤s〈2, where 2^＊（s）=2（N-s）/N-2 is the critical Sobolev-Hardy expon...For the following elliptic problem {-△u-μu/｜x｜^2=｜u｜^2^＊（s）-2u/｜x｜^s＋h（x）, on R^N u∈D^1,2（R^N）, N≥3, 0≤μ〈μ^-=（N-2）^2/4, 0≤s〈2, where 2^＊（s）=2（N-s）/N-2 is the critical Sobolev-Hardy exponent, h（x） ∈ （D^1,2（R^N））^＊, the dual space of （D^1,2（R^N））, with h（x）≥（≠）0. By Ekeland＇s variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if ｜｜h｜｜＊〈CN,sAs^N-s/4-2s（1-μ/μ）^1/2, CN,s=4-2s/N-2（N-2/N＋2-2s）^N＋2-2s/4-2s and As = inf u∈D^1,2（R^N）／{0}∫R^N（｜△↓u｜^2-μu^2/｜x｜^2）dx/（∫R^N｜u｜^2^＊（s）/｜x｜^sdx）^2/2^＊（s）.展开更多
This article considers the equation △2u = f（x,u）with boundary conditions either u｜aΩ = au/an｜aΩ = 0 or u｜aΩ = △u｜aΩ = 0, where f（x, t） is asymptotically linear with respect to t at infinity, and Ω is a ...This article considers the equation △2u = f（x,u）with boundary conditions either u｜aΩ = au/an｜aΩ = 0 or u｜aΩ = △u｜aΩ = 0, where f（x, t） is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in R^N, N 〉 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f（x, t）.展开更多
文摘We are concerned with the following Dirichlet problem: -△u(x) = f(x, u), x ∈ Ω. u ∈ H_0~1(Ω). (P) where f(x, t) ∈ C(Ω×R), f(x, t)/t is nondecreasing in t ∈ R and tends to an L~∝-function q(x) uniformly in x ∈ Ω as t→+∝ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case. an Ambrosetti-Rabinowitz-type condition, that is. for some θ>2. M>0, 0<θF(x. s)≤ f(x, s)s, for all |s|≥M and x ∈ Ω, (AR) is no longer true, where F(x, s) = integral from n=0 to s f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable, conditions on f(x, t) and q(x). Our methods also work for the case where f(x, f) is superlinear in t at infinity. i.e., q(x) ≡∞.
基金supported by National Natural Science Foundation of China(No.12171152)。
文摘By using the Ljusternik-Schnirelmann category and variational method,we s-tudy the existence,multiplicity and concentration of solutions to the fractional Schrodinger equation with potentials competition as follows,ε^(N)(-△)^(s)N/sμ+V(x)|μ|^(N/s-2μ)=Q(x)h(μ)in R^(N),where ε>0 is a parameter,s ∈(0,1),2≤p<+oo and N=ps.The nonlinear term h is a diferentiable function with exponential critical growth,the absorption potential V and the reaction potential Q are continuous functions.
基金supported by the National Natural Science Foundation of China(Grant No.12171253).
文摘This paper is concerned with the existence of nontrivial homoclinic solutions for a class of second order Hamiltonian systems with external forc-ing perturbations q+Aq+Vq(t,q)=f(t),where q=(q1,q2,..qN)∈R^(N),A is an antisymmetric constant N×N matrix,V(t,q)=-K(t,q)+W(t,q)with K,W ∈C^(1)(R,R^(N))and satisfying b1|q|^(2)≤K(t,q)≤b_(2)|q|^(2)for some positive constants b_(2)≥b_(1)>0 and external forcing term f∈C(R,R^(N))being small enough.Under some new weak superquadratic conditions for W,by using the mountain pass theorem,we obtain the existence of at least one nontrivial homoclinic solution.
文摘This paper concerns the existence of multiple homoclinic orbits for the second-order Hamiltonian system-L（t）z＋Wz（t,z）=0,where L∈C（R,RN2）is a symmetric matrix-valued function and W（t,z）∈C1（R×RN,R）is a nonlinear term.Since there are no periodic assumptions on L（t）and W（t,z）in t,one should overcome difficulties for the lack of compactness of the Sobolev embedding.Moreover,the nonlinearity W（t,z）is asymptotically linear in z at infinity and the system is allowed to be resonant,which is a case that has never been considered before.By virtue of some generalized mountain pass theorem,multiple homoclinic orbits are obtained.
文摘For the following elliptic problem {-△u-μu/｜x｜^2=｜u｜^2^＊（s）-2u/｜x｜^s＋h（x）, on R^N u∈D^1,2（R^N）, N≥3, 0≤μ〈μ^-=（N-2）^2/4, 0≤s〈2, where 2^＊（s）=2（N-s）/N-2 is the critical Sobolev-Hardy exponent, h（x） ∈ （D^1,2（R^N））^＊, the dual space of （D^1,2（R^N））, with h（x）≥（≠）0. By Ekeland＇s variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if ｜｜h｜｜＊〈CN,sAs^N-s/4-2s（1-μ/μ）^1/2, CN,s=4-2s/N-2（N-2/N＋2-2s）^N＋2-2s/4-2s and As = inf u∈D^1,2（R^N）／{0}∫R^N（｜△↓u｜^2-μu^2/｜x｜^2）dx/（∫R^N｜u｜^2^＊（s）/｜x｜^sdx）^2/2^＊（s）.
基金This work was supported by NSFC(10571174,10631030)and CAS(KJCX3-SYW-S03)
文摘This article considers the equation △2u = f（x,u）with boundary conditions either u｜aΩ = au/an｜aΩ = 0 or u｜aΩ = △u｜aΩ = 0, where f（x, t） is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in R^N, N 〉 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f（x, t）.
