In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p＆q-Laplacian type involving the critical Sobolev exponent：{-△pu-△qu=│u│^p＊-2u＋μ│u│^r-2u in Ω u...In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p＆q-Laplacian type involving the critical Sobolev exponent：{-△pu-△qu=│u│^p＊-2u＋μ│u│^r-2u in Ω u│δΩ=0,where Ω belong to R^N is a bounded domain,N〉p,p^＊=Np/N-p is the critical Sobolev exponent and μ 〉0. We prove that if 1 〈 r 〈 q 〈 p 〈 N, then there is a μ0 〉 0, such that for any μ∈ （0, μ0）, the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.展开更多
In this paper,we consider the nonlinear equations involving the fractional p&qLaplace operator with a sign-changing potential.This model is inspired by the De Giorgi Conjecture.There are two main results in this p...In this paper,we consider the nonlinear equations involving the fractional p&qLaplace operator with a sign-changing potential.This model is inspired by the De Giorgi Conjecture.There are two main results in this paper.First,in the bounded domain,we use the moving plane method to show that the solution is radially symmetric.Second,for the unbounded domain,in view of the idea of the sliding method,we find the existence of the maximizing sequence of the bounded solution,then obtain that the solution is strictly monotone increasing in some direction.展开更多
In this article, we study the existence of infinitely many solutions to the degenerate quasilinear elliptic system -div（h1（x）｜△u｜p-2△u）=d（x）｜u｜r-2u＋Gu（x,u,v） in Ω -div（h2（x）｜△u｜q-2△v）=f（x）｜v...In this article, we study the existence of infinitely many solutions to the degenerate quasilinear elliptic system -div（h1（x）｜△u｜p-2△u）=d（x）｜u｜r-2u＋Gu（x,u,v） in Ω -div（h2（x）｜△u｜q-2△v）=f（x）｜v｜s-2v＋Gv（x,u,v） in Ω, u=v=0 on δΩ where Ω is a bounded domain in RN with smooth boundary δΩ, N ≥ 2, 1 〈 r 〈 p ∞, 1〈 s 〈 q 〈 ∞; h1（x） and h2（x） are allowed to have ＂essential＂ zeroes at some points in Ω; d（x）｜u｜r-2u and f（x）｜v｜s-2v are small sources with Gu（x,u,v）, Gv（x,u,v） being their high-order perturbations with respect to （u, v） near the origin, respectively.展开更多
In this paper, we consider the existence for periodic solutions of nonau- tonomons second-order differential systems with （q,p）-Laplacian by using the least action principle and the minimax methods.
基金Supported by NSFC (10571069 and 10631030) the Lap of Mathematical Sciences, CCNU, Hubei Province, China
文摘In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p＆q-Laplacian type involving the critical Sobolev exponent：{-△pu-△qu=│u│^p＊-2u＋μ│u│^r-2u in Ω u│δΩ=0,where Ω belong to R^N is a bounded domain,N〉p,p^＊=Np/N-p is the critical Sobolev exponent and μ 〉0. We prove that if 1 〈 r 〈 q 〈 p 〈 N, then there is a μ0 〉 0, such that for any μ∈ （0, μ0）, the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem.
基金partially supported by the NSFC(12271269)the Fundamental Research Funds for the Central Universitiespartially supported by the Fundamental Research Funds for the Central Universities(2021YJSB006)。
文摘In this paper,we consider the nonlinear equations involving the fractional p&qLaplace operator with a sign-changing potential.This model is inspired by the De Giorgi Conjecture.There are two main results in this paper.First,in the bounded domain,we use the moving plane method to show that the solution is radially symmetric.Second,for the unbounded domain,in view of the idea of the sliding method,we find the existence of the maximizing sequence of the bounded solution,then obtain that the solution is strictly monotone increasing in some direction.
基金Supported by Anhui Provincial Natural Science Foundation(1408085MA02,1508085QA01,1608085MA12)the Key Foundation of Anhui Education Bureau(KJ2012A019,KJ2013A028,KJ2014A010)+1 种基金211 Project of Anhui University(KJJQ1101,02303303-33030011,02303902-39020011,J18520207,XJYJXKC04)the National Natural Science Foundation of China(11271371,11301004,51479215,11471015)
基金supported by Anhui Provincial Nature Science Foundation(1208085MA13)the Research Fund for the Doctoral Program of Higher Education(20103401120002,20113401120001)+1 种基金211 Project of Anhui University(02303129,KJTD002B,02303303-33030011,02303902-39020011)the Key Foundation of Anhui Education Bureau(KJ2012A019)
基金Supported by the National Natural Science Foundation of China(11426122,11371153,and 11361029)the Specialized Research Fund for the Doctoral Program of Higher Education of Chinathe Natural Science Foundation of Jiangxi Province of China(20151BAB211003)
文摘In this article, we study the existence of infinitely many solutions to the degenerate quasilinear elliptic system -div（h1（x）｜△u｜p-2△u）=d（x）｜u｜r-2u＋Gu（x,u,v） in Ω -div（h2（x）｜△u｜q-2△v）=f（x）｜v｜s-2v＋Gv（x,u,v） in Ω, u=v=0 on δΩ where Ω is a bounded domain in RN with smooth boundary δΩ, N ≥ 2, 1 〈 r 〈 p ∞, 1〈 s 〈 q 〈 ∞; h1（x） and h2（x） are allowed to have ＂essential＂ zeroes at some points in Ω; d（x）｜u｜r-2u and f（x）｜v｜s-2v are small sources with Gu（x,u,v）, Gv（x,u,v） being their high-order perturbations with respect to （u, v） near the origin, respectively.
基金The NSF (10871059 and 10671028) of Chinathe Fundamental Research Founds (B09020181) for the Central Universities
文摘In this paper, we consider the existence for periodic solutions of nonau- tonomons second-order differential systems with （q,p）-Laplacian by using the least action principle and the minimax methods.
