We investigate the constrained fractional Choquard equation■where m>0,N>2s with s∈(0,1)being the order of the fractional Laplacian operator and I_(α)forα∈(0,N)denotes the Riesz potential.The parameterμ∈ℝa...We investigate the constrained fractional Choquard equation■where m>0,N>2s with s∈(0,1)being the order of the fractional Laplacian operator and I_(α)forα∈(0,N)denotes the Riesz potential.The parameterμ∈ℝappears as a Lagrange multiplier.By imposing general mass-supercritical conditions on F,we confirm the existence of normalized solutions that characterize the global minimizer on the Pohozaev manifold.Our proof does not depend on the assumption that all weak solutions satisfy the Pohozaev identity,a challenge that remains unsolved for this doubly nonlocal equation.展开更多
Using the domain variation estimate,we give a new proof of the local Pohozaev identities for weak solutions of elliptic equations involving p-Laplacian operators under only C^(1)-regularity of the solutions.As an appl...Using the domain variation estimate,we give a new proof of the local Pohozaev identities for weak solutions of elliptic equations involving p-Laplacian operators under only C^(1)-regularity of the solutions.As an application,we obtain the Pohozaev identities for C^(1) solution of p-Laplacian equation in ℝ^(N) with 1<p<N.展开更多
In this paper, we study the Pohozaev identity associated with a Henon-Lane-Emden sys- tem involving the fractional Laplacian：{（-△）^su=｜x｜^aυ^p,x ∈Ω,（-△）^sυ=｜x｜^av^p,x ∈Ω u=υ=0, x∈R^b／Ω,in a star-s...In this paper, we study the Pohozaev identity associated with a Henon-Lane-Emden sys- tem involving the fractional Laplacian：{（-△）^su=｜x｜^aυ^p,x ∈Ω,（-△）^sυ=｜x｜^av^p,x ∈Ω u=υ=0, x∈R^b／Ω,in a star-shaped and bounded domain Ω for s E （0, 1）. As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritieal cases.展开更多
This paper deals with the following prescribed boundary mean curvature problem in B^(N){−Δu=0,u>0,∂_(u)∂_(ν)+N−2/2 u=N−2/2 K˜(y)u^(2−1),y∈B^(N)y∈S^(N−1),where K˜(y)=K˜(|y|,y˜)is a bounded nonnegative function w...This paper deals with the following prescribed boundary mean curvature problem in B^(N){−Δu=0,u>0,∂_(u)∂_(ν)+N−2/2 u=N−2/2 K˜(y)u^(2−1),y∈B^(N)y∈S^(N−1),where K˜(y)=K˜(|y|,y˜)is a bounded nonnegative function with y=(y,y˜)∈R^(2)×R^(N−3),2=2(N−1)/N−2.Combining the finite-dimensional reduction method and local Pohozaev type of identities,we prove that if N≥5 and K˜(r,y˜)has a stable critical point(r_(0),y˜_(0))with r0>0 and K˜(r0,y˜0)>0,then the above problem has infinitely many solutions,whose energy can be made arbitrarily large.Here our result fill the gap that the above critical points may include the saddle points of K˜(r,y˜).展开更多
In this paper, we prove some sharp non-existence results for Dirichlet prob- lems of complex Hessian equations. In particular, we consider a complex Monge- Ampere equation which is a local version of the equation of K...In this paper, we prove some sharp non-existence results for Dirichlet prob- lems of complex Hessian equations. In particular, we consider a complex Monge- Ampere equation which is a local version of the equation of Kahler-Einstein metric. The non-existence results are proved using the Pohozaev method. We also prove existence results for radially symmetric solutions. The main difference of the complex case with the real case is that we don＇t know if a priori radially symmetric property holds in the complex case.展开更多
基金supported by the Guangdong Basic and Applied Basic Research Foundation(2022A1515012138)the NSFC(12271436,12371119)supported by the Natural Science Basic Research Program of Shaanxi(2022JC-04).
文摘We investigate the constrained fractional Choquard equation■where m>0,N>2s with s∈(0,1)being the order of the fractional Laplacian operator and I_(α)forα∈(0,N)denotes the Riesz potential.The parameterμ∈ℝappears as a Lagrange multiplier.By imposing general mass-supercritical conditions on F,we confirm the existence of normalized solutions that characterize the global minimizer on the Pohozaev manifold.Our proof does not depend on the assumption that all weak solutions satisfy the Pohozaev identity,a challenge that remains unsolved for this doubly nonlocal equation.
基金Supported by the National Natural Science Foundation of China(12261053)the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities Association(2019FH001-078,202101BA070001-132)Introduction of Talents Research Project of Kunming University(XJ20210020,YJL20019)。
基金supported by the NSFC(11931012,12471112,12571118).
文摘Using the domain variation estimate,we give a new proof of the local Pohozaev identities for weak solutions of elliptic equations involving p-Laplacian operators under only C^(1)-regularity of the solutions.As an application,we obtain the Pohozaev identities for C^(1) solution of p-Laplacian equation in ℝ^(N) with 1<p<N.
基金supported by NSFC(Grant No.11571176)the second author is supported by NSFC(Grant No.11571057)
文摘In this paper, we study the Pohozaev identity associated with a Henon-Lane-Emden sys- tem involving the fractional Laplacian：{（-△）^su=｜x｜^aυ^p,x ∈Ω,（-△）^sυ=｜x｜^av^p,x ∈Ω u=υ=0, x∈R^b／Ω,in a star-shaped and bounded domain Ω for s E （0, 1）. As an application of our identity, we deduce the nonexistence of positive solutions in the critical and supercritieal cases.
基金Supported by NSFC(Grant Nos.12226324,11961043,11801226)。
文摘This paper deals with the following prescribed boundary mean curvature problem in B^(N){−Δu=0,u>0,∂_(u)∂_(ν)+N−2/2 u=N−2/2 K˜(y)u^(2−1),y∈B^(N)y∈S^(N−1),where K˜(y)=K˜(|y|,y˜)is a bounded nonnegative function with y=(y,y˜)∈R^(2)×R^(N−3),2=2(N−1)/N−2.Combining the finite-dimensional reduction method and local Pohozaev type of identities,we prove that if N≥5 and K˜(r,y˜)has a stable critical point(r_(0),y˜_(0))with r0>0 and K˜(r0,y˜0)>0,then the above problem has infinitely many solutions,whose energy can be made arbitrarily large.Here our result fill the gap that the above critical points may include the saddle points of K˜(r,y˜).
文摘In this paper, we prove some sharp non-existence results for Dirichlet prob- lems of complex Hessian equations. In particular, we consider a complex Monge- Ampere equation which is a local version of the equation of Kahler-Einstein metric. The non-existence results are proved using the Pohozaev method. We also prove existence results for radially symmetric solutions. The main difference of the complex case with the real case is that we don＇t know if a priori radially symmetric property holds in the complex case.
