非线性数量场方程基态能量的整体性态  

Global behavior of the ground state energy of the nonlinear scalar field equation

作　　者：吴元泽 王志强 Yuanze Wu;Zhi-Qiang Wang

机构地区：[1]中国矿业大学数学学院,徐州221116 [2]Department of Mathematics and Statistics,Utah State University,Logan,UT 84322,USA

出　　处：《中国科学：数学》2023年第1期25-40,共16页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11701554,11771319,11971339,11831009)资助项目。

摘　　要：本文研究p-Laplace型的非线性数量场方程并给出其基态能量整体性态的完整刻画.具体而言,令ε_(λ,q)(u)=1/p(||▽u||_(p)^(p)+λ||u||_(p)^(p))−1/q||u||_(q)^(q)为W^(1,p)(R^(N))中的能量泛函,其中,λ>0,1<p<N,p<q<p^(∗):=pN/N−p,p^(∗)是Sobolev临界指数.众所周知,ε_(λ,q)(u)有唯一的径向对称的山路解Uλ,q.Uλ,q也是ε_(λ,q)(u)对应方程的基态解,其能量m(λ,q)被称为基态能量.本文证明,存在一个定义在[p,p^(∗)]上的严格单调递减函数λ_(0)(q)满足λ_(0)(p)=1且λ_(0)(p^(∗))=0,使得当λ∈(0,λ_(0)(q))固定时,m(λ,r)作为r的函数在(p,q)内严格单调递增;当λ∈(λ_(0)(q),1)固定时,m(λ,r)作为r的函数在(q,p^(∗))内严格单调递增;当λ∈[1,+∞)固定时,m(λ,r)作为r的函数在(p,p^(∗))内严格单调递减.通过进一步建立幂次型数量场方程与对数型数量场方程的联系,本文给出λ_(0)(q)在q→p和q→p^(∗)时的精确渐近行为.We study the global behavior of the ground state energy for the nonlinear scalar field equation(including the p-Laplacian version),and give a complete description of this energy in terms of nonlinearity power.More precisely,letε_(λ,q)(u)=1/p(‖▽u‖_(p)^(p)+λ‖u‖_(p)^(p))-1/q‖u‖_(q)^(q)be the energy functional in W1,p(RN),whereλ>0,1N/(N-p),and p^(∗)is the critical Sobolev exponent.It is known thatε_(λ,q)(u)has a unique radially symmetric mountain-pass critical point Uλ,q,which is called the ground state solution to the corresponding nonlinear scalar field equation and whose energyε_(λ,q)(Uλ,q)is called the ground state energy m(λ,q).We show that there is a decreasing functionλ_(0)(q)defined in q∈[p,p^(∗)]withλ_(0)(p)=1 andλ_(0)(p^(∗))=0 such that m(λ,r)is strictly increasing for r∈(p,q)withλ∈(0,λ_(0)(q))being fixed,m(λ,r)is strictly decreasing for r∈(q,p^(∗))withλ∈(λ_(0)(q),1)being fixed,and m(λ,r)is strictly decreasing for all r∈(p,p^(∗))withλ∈[1,+∞)being fixed.We also deduce the precise asymptotic behavior ofλ_(0)(q)as q→p and q→p^(∗).This is done by establishing a relation between power-law scalar field equations and logarithmic-law scalar field equations,which is of the independent interest.

关 键 词：数量场方程 基态能量 整体性态 

分 类 号：O175[理学—数学]