This paper is devoted to the quasilinear equation ■where p > 2,Ω is a(bounded or unbounded) domain of R^N,w_1,w_2 are nonnegative continuous functions and f is an increasing function. We establish a Liouville typ...This paper is devoted to the quasilinear equation ■where p > 2,Ω is a(bounded or unbounded) domain of R^N,w_1,w_2 are nonnegative continuous functions and f is an increasing function. We establish a Liouville type theorem for nontrivial stable solutions of the equation under some mild assumptions on Ω,w_1, w_2 and f, which extends and unifies several results on this topic.展开更多
Let 0<α,β<n and f,g∈ C([0,∞)×[0,∞))be two nonnegative functions.We study nonnegative classical solutions of the system{(-△)^(α/2)u=f(u,v)in R^(n),(-△)^(β/2)v=g(u,v)in R^(n),and the corresponding eq...Let 0<α,β<n and f,g∈ C([0,∞)×[0,∞))be two nonnegative functions.We study nonnegative classical solutions of the system{(-△)^(α/2)u=f(u,v)in R^(n),(-△)^(β/2)v=g(u,v)in R^(n),and the corresponding equivalent integral system.We classify all such solutions when f(s,t)is nondecreasing in s and increasing in t,g(s,t)is increasing in s and nondecreasing in i,and f(μ^(n-α)s,μ^(n-β)t)/μ^(n-α),g(μ^(n-α)s,μ^(n-β)t)/μ^(n-β)are nonincreasing in μ>0 for all s,t≥0.The main technique we use is the method of moving spheres in integral forms.Since our assumptions are more general than those in the previous literature,some new ideas are introduced to overcome this difficulty.展开更多
We prove Liouville type theorems for stable and finite Morse index H_(loc)^(1)∩L_(loc)^(∞)solutions of the nonlinear Schrodinger equation -Δu+λ|x|^(a)u=|x|b|^(u)|^(p-1)u in R^(N),where N≥2,λ>0,a,b>-2 and p...We prove Liouville type theorems for stable and finite Morse index H_(loc)^(1)∩L_(loc)^(∞)solutions of the nonlinear Schrodinger equation -Δu+λ|x|^(a)u=|x|b|^(u)|^(p-1)u in R^(N),where N≥2,λ>0,a,b>-2 and p>1,Our analysis reveals that all stable solutions of the equation must be zero for all p>1,Furthermore,finite Morse index solutions must be zero if N≥3 an p≥(N+2+2b)/(N-2).The main tools we use are integral estimates,a Pohozaev type identity and a monotonicity formula.展开更多
基金supported by Vietnam National Foundation for Science and Technology Development(NAFOSTED)under Grant No.101.02-2017.307
文摘This paper is devoted to the quasilinear equation ■where p > 2,Ω is a(bounded or unbounded) domain of R^N,w_1,w_2 are nonnegative continuous functions and f is an increasing function. We establish a Liouville type theorem for nontrivial stable solutions of the equation under some mild assumptions on Ω,w_1, w_2 and f, which extends and unifies several results on this topic.
基金This research is funded by Vietnam National Foundation for Science and Technology Development(NAFOSTED)under grant number 101.02-2020.22.
文摘Let 0<α,β<n and f,g∈ C([0,∞)×[0,∞))be two nonnegative functions.We study nonnegative classical solutions of the system{(-△)^(α/2)u=f(u,v)in R^(n),(-△)^(β/2)v=g(u,v)in R^(n),and the corresponding equivalent integral system.We classify all such solutions when f(s,t)is nondecreasing in s and increasing in t,g(s,t)is increasing in s and nondecreasing in i,and f(μ^(n-α)s,μ^(n-β)t)/μ^(n-α),g(μ^(n-α)s,μ^(n-β)t)/μ^(n-β)are nonincreasing in μ>0 for all s,t≥0.The main technique we use is the method of moving spheres in integral forms.Since our assumptions are more general than those in the previous literature,some new ideas are introduced to overcome this difficulty.
基金Supported by University of Economics and Law,VNU-HCM。
文摘We prove Liouville type theorems for stable and finite Morse index H_(loc)^(1)∩L_(loc)^(∞)solutions of the nonlinear Schrodinger equation -Δu+λ|x|^(a)u=|x|b|^(u)|^(p-1)u in R^(N),where N≥2,λ>0,a,b>-2 and p>1,Our analysis reveals that all stable solutions of the equation must be zero for all p>1,Furthermore,finite Morse index solutions must be zero if N≥3 an p≥(N+2+2b)/(N-2).The main tools we use are integral estimates,a Pohozaev type identity and a monotonicity formula.
