We consider a class of quasilinear elliptic boundary problems,including the following Modified Nonlinear Schrödinger Equation as a special case:{Δu+1/2 uΔ(u^(2))−V(x)u+|u|^(q−2)u=0 in Ω,u=0 on∂Ω,whereΩis the...We consider a class of quasilinear elliptic boundary problems,including the following Modified Nonlinear Schrödinger Equation as a special case:{Δu+1/2 uΔ(u^(2))−V(x)u+|u|^(q−2)u=0 in Ω,u=0 on∂Ω,whereΩis the entire space R^(N) orΩ⊂R^(N) is a bounded domain with smooth boundary,q∈(2,22^(∗)]with 2^(∗)=2 N/(N−2)being the critical Sobolev exponent and 22^(∗)=4 N/(N−2).We review the general methods developed in the last twenty years or so for the studies of existence,multiplicity,nodal property of the solutions within this range of nonlinearity up to the new critical exponent 4 N/(N−2),which is a unique feature for this class of problems.We also discuss some related and more general problems.展开更多
In this paper,we consider the one dimensional third order p-Laplacian equation(Фp(u″))′+h(t)f(t,u(t))=0 with integral boundary conditions u(0)-αu′(0)=∫_(0)^(1) g1(s)u(s)ds,u(1)+βu′(1)=∫_(0)^(1)g2(s)u(s)ds,u″...In this paper,we consider the one dimensional third order p-Laplacian equation(Фp(u″))′+h(t)f(t,u(t))=0 with integral boundary conditions u(0)-αu′(0)=∫_(0)^(1) g1(s)u(s)ds,u(1)+βu′(1)=∫_(0)^(1)g2(s)u(s)ds,u″(0)=0.By using kernel functions and the Avery-Peterson fixed point theorem,we establish the existence of at least three positive solutions.展开更多
基金The work is supported by NSFC(Grant No.11831009).
文摘We consider a class of quasilinear elliptic boundary problems,including the following Modified Nonlinear Schrödinger Equation as a special case:{Δu+1/2 uΔ(u^(2))−V(x)u+|u|^(q−2)u=0 in Ω,u=0 on∂Ω,whereΩis the entire space R^(N) orΩ⊂R^(N) is a bounded domain with smooth boundary,q∈(2,22^(∗)]with 2^(∗)=2 N/(N−2)being the critical Sobolev exponent and 22^(∗)=4 N/(N−2).We review the general methods developed in the last twenty years or so for the studies of existence,multiplicity,nodal property of the solutions within this range of nonlinearity up to the new critical exponent 4 N/(N−2),which is a unique feature for this class of problems.We also discuss some related and more general problems.
基金supported by the National Natural Science Foundation of China(Grant No.10131050)the Program of 985 Innovation Engineering on Information in Xiamen University.
文摘The asymptotic behavior of the solutions for p-Laplacian equations as p →∞ is studied.
基金supported by the National Natural Science Foundation of China(No.12071491)。
文摘In this paper,we consider the one dimensional third order p-Laplacian equation(Фp(u″))′+h(t)f(t,u(t))=0 with integral boundary conditions u(0)-αu′(0)=∫_(0)^(1) g1(s)u(s)ds,u(1)+βu′(1)=∫_(0)^(1)g2(s)u(s)ds,u″(0)=0.By using kernel functions and the Avery-Peterson fixed point theorem,we establish the existence of at least three positive solutions.
