We investigate the bi-harmonic problem{Δ^(2)u-α▽·(f(▽u))-βΔ_(p)u=g(x,u) in Ω,δu/δn=0,δ(Δu)/δn=0 on δΩ,where Δ^(2)u=Δ(Δu),Δ_(p)u=div(|▽u|^(p-2)▽u)with p>2.Ω is a bounded smooth domain in R^...We investigate the bi-harmonic problem{Δ^(2)u-α▽·(f(▽u))-βΔ_(p)u=g(x,u) in Ω,δu/δn=0,δ(Δu)/δn=0 on δΩ,where Δ^(2)u=Δ(Δu),Δ_(p)u=div(|▽u|^(p-2)▽u)with p>2.Ω is a bounded smooth domain in R^(N),N≥1.By using a special function space with the constraint ∫_(Ω)udx=0,under suitable assumptions on f and g(x,u),we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem.Recent results from the literature are extended.展开更多
In this paper, we establish fountain theorems over cones and apply it to the quasilinear elliptic problem{-△Pu=λ｜u｜q-2u＋μ｜u｜ y-2u,x∈Ω,u=0,x∈δΩ to show that problem （1） possesses infinitely many solution...In this paper, we establish fountain theorems over cones and apply it to the quasilinear elliptic problem{-△Pu=λ｜u｜q-2u＋μ｜u｜ y-2u,x∈Ω,u=0,x∈δΩ to show that problem （1） possesses infinitely many solutions, where 1 〈 p 〈 N, 1 〈 q 〈 P 〈 γ, Ω∩→ R^N is a smooth bounded domain and λ, μ∈ R.展开更多
In this work,we use the variant fountain theorem to study the existence of nontrivial solutions for the superquadratic fractional difference boundary value problem:{T△^(v)_(t-1)(t△^(v)_(v-1)x(t))=f(x(t+v-1)),t∈[0,T...In this work,we use the variant fountain theorem to study the existence of nontrivial solutions for the superquadratic fractional difference boundary value problem:{T△^(v)_(t-1)(t△^(v)_(v-1)x(t))=f(x(t+v-1)),t∈[0,T]N_(0),x(v-2)=[tΔ^(v)_(v-1)x(t)]_(t=T=0.The existence of nontrivial solutions is obtained in the case of super quadratic growth of the nonlinear term f by change of fountain theorem.展开更多
目的研究降低酒精润版液中酒精浓度的方法。方法首先将赛飞扬FS12-100润版液用纯净水配制成质量分数为3%的原液,测试表面张力;然后将不同质量分数的酒精和表面活性剂6501分别添加到原液中,测试表面张力;最后将不同质量分数的酒精和表面...目的研究降低酒精润版液中酒精浓度的方法。方法首先将赛飞扬FS12-100润版液用纯净水配制成质量分数为3%的原液,测试表面张力;然后将不同质量分数的酒精和表面活性剂6501分别添加到原液中,测试表面张力;最后将不同质量分数的酒精和表面活性剂6501同时添加到原液中,测试表面张力。结果在原液中添加质量分数为0.1%的表面活性剂6501和5%的酒精,表面张力最低为23.37 m N/m,可将酒精的质量分数由12%降低到5%。结论表面活性剂6501与酒精混合配制成新型的低酒精润版液是可行的。展开更多
文摘We investigate the bi-harmonic problem{Δ^(2)u-α▽·(f(▽u))-βΔ_(p)u=g(x,u) in Ω,δu/δn=0,δ(Δu)/δn=0 on δΩ,where Δ^(2)u=Δ(Δu),Δ_(p)u=div(|▽u|^(p-2)▽u)with p>2.Ω is a bounded smooth domain in R^(N),N≥1.By using a special function space with the constraint ∫_(Ω)udx=0,under suitable assumptions on f and g(x,u),we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem.Recent results from the literature are extended.
基金supported by ARC grant of Australiasupported by National Natural Sciences Foundations of China (10961016 and 10631030)NSF of Jiangxi(2009GZS0011)
文摘In this paper, we establish fountain theorems over cones and apply it to the quasilinear elliptic problem{-△Pu=λ｜u｜q-2u＋μ｜u｜ y-2u,x∈Ω,u=0,x∈δΩ to show that problem （1） possesses infinitely many solutions, where 1 〈 p 〈 N, 1 〈 q 〈 P 〈 γ, Ω∩→ R^N is a smooth bounded domain and λ, μ∈ R.
文摘In this work,we use the variant fountain theorem to study the existence of nontrivial solutions for the superquadratic fractional difference boundary value problem:{T△^(v)_(t-1)(t△^(v)_(v-1)x(t))=f(x(t+v-1)),t∈[0,T]N_(0),x(v-2)=[tΔ^(v)_(v-1)x(t)]_(t=T=0.The existence of nontrivial solutions is obtained in the case of super quadratic growth of the nonlinear term f by change of fountain theorem.
文摘目的研究降低酒精润版液中酒精浓度的方法。方法首先将赛飞扬FS12-100润版液用纯净水配制成质量分数为3%的原液,测试表面张力;然后将不同质量分数的酒精和表面活性剂6501分别添加到原液中,测试表面张力;最后将不同质量分数的酒精和表面活性剂6501同时添加到原液中,测试表面张力。结果在原液中添加质量分数为0.1%的表面活性剂6501和5%的酒精,表面张力最低为23.37 m N/m,可将酒精的质量分数由12%降低到5%。结论表面活性剂6501与酒精混合配制成新型的低酒精润版液是可行的。
