In this paper,we investigate the minimization problem e_(s)(p)=_(u∈W_(V)^(1,N))(r^(N)),||u||_(N)^(N)=p>0 inf E(u),where E(u)=1/N∫_(R_(N))|▽_(u)|^(N)dx+1/N∫_(R_(N))V(x)|u|^(N)dx-1/s∫_(R_(N))|u|^(s)dx.Here s>...In this paper,we investigate the minimization problem e_(s)(p)=_(u∈W_(V)^(1,N))(r^(N)),||u||_(N)^(N)=p>0 inf E(u),where E(u)=1/N∫_(R_(N))|▽_(u)|^(N)dx+1/N∫_(R_(N))V(x)|u|^(N)dx-1/s∫_(R_(N))|u|^(s)dx.Here s>N,V is a spherically symmetric increasing function satisfying V(0)=0,|x|→∞lin V(x)=+∞We discuss the problem in three cases.First,for the case s>2N,e_(s)(ρ)=-∞for anyρ>0.Secondly,for the case N<s<2N,for anyρ>0,we prove that it admits a minimizer which is nonnegative,spherically symmetric and decreasing via the N-Laplacian GagliardoNirenberg inequality.When s=2N,the existence and nonexistence of minimizers of e_(s)(ρ)will also be given.During the arguments,we provide the detailed proof of the N-Laplacian Gagliardo-Nirenberg inequality and N-Laplacian Pohozaev identity.展开更多
Overweight and obesity has been a major public health problem globally.It was estimated that more than 2.1 billion adults were affected by overweight or obese in 2021 worldwide,about one fifth of whom lived in China^(...Overweight and obesity has been a major public health problem globally.It was estimated that more than 2.1 billion adults were affected by overweight or obese in 2021 worldwide,about one fifth of whom lived in China^([1]).By 2050,the country is forecast to remain the one with the largest population of overweight and obese globally^([1]),if no effective strategies were applied on overweight/obesity control.展开更多
基金supported by the Xingdian Talents Support Program of Yunnan Province of Youthsthe Yunnan Province Basic Research Project for General Program(202401AT070441)+5 种基金the Yunnan Key Laboratory of Modern Analytical Mathematics and Applications(202302AN360007)supported by the NNSF(12261031)supported by the NNSF(12401145)supported by the NNSF(12371120)the Yunnan Province Basic Research Project for Key Program(202401AS070024)the General Program(202301AT070141)。
文摘In this paper,we investigate the minimization problem e_(s)(p)=_(u∈W_(V)^(1,N))(r^(N)),||u||_(N)^(N)=p>0 inf E(u),where E(u)=1/N∫_(R_(N))|▽_(u)|^(N)dx+1/N∫_(R_(N))V(x)|u|^(N)dx-1/s∫_(R_(N))|u|^(s)dx.Here s>N,V is a spherically symmetric increasing function satisfying V(0)=0,|x|→∞lin V(x)=+∞We discuss the problem in three cases.First,for the case s>2N,e_(s)(ρ)=-∞for anyρ>0.Secondly,for the case N<s<2N,for anyρ>0,we prove that it admits a minimizer which is nonnegative,spherically symmetric and decreasing via the N-Laplacian GagliardoNirenberg inequality.When s=2N,the existence and nonexistence of minimizers of e_(s)(ρ)will also be given.During the arguments,we provide the detailed proof of the N-Laplacian Gagliardo-Nirenberg inequality and N-Laplacian Pohozaev identity.
文摘Overweight and obesity has been a major public health problem globally.It was estimated that more than 2.1 billion adults were affected by overweight or obese in 2021 worldwide,about one fifth of whom lived in China^([1]).By 2050,the country is forecast to remain the one with the largest population of overweight and obese globally^([1]),if no effective strategies were applied on overweight/obesity control.
