In this paper, we consider the following noncooperative elliptic systems where Ω is a bounded domain in R<sup>N</sup> with smooth boundary ∂Ω, λ,δ,γ are real parameters, and . We assume that F is subq...In this paper, we consider the following noncooperative elliptic systems where Ω is a bounded domain in R<sup>N</sup> with smooth boundary ∂Ω, λ,δ,γ are real parameters, and . We assume that F is subquadratic at zero with respect to the variables u,v. By using a variant Clark’s theorem, we obtain infinitely many nontrivial solutions (u<sub>k</sub><sub></sub>,v<sub>k</sub>) with as k → ∞. Compared with the existing literature, we do not need to assume the behavior of the nonlinearity ∇F at infinity.展开更多
This paper deals with the Neumann problem for a class of semilinear elliptic equations -△u + u =|u|2*-2u+ μ|u|q-2u in Ω, au/ar= |u|(?)*-2u on aΩ, where 2 = 2N/N-2, s=2(N-1)/N-2, 1 <q<2,N(?)3,μ>γ denotes...This paper deals with the Neumann problem for a class of semilinear elliptic equations -△u + u =|u|2*-2u+ μ|u|q-2u in Ω, au/ar= |u|(?)*-2u on aΩ, where 2 = 2N/N-2, s=2(N-1)/N-2, 1 <q<2,N(?)3,μ>γ denotes the unit outward normal to boundary aΩ. By vaxiational method and dual fountain theorem, the existence of infinitely many solutions with negative energy is proved.展开更多
By using variational method, the multiplicity of solutions for nonlinear biharmonic equation involving critical parameter and critical exponent are established.
This paper considers the following quasilinear elliptic problem [GRAPHICS] where Omega is a bounded regular domain in R-N (N greater than or equal to 3), N > p > 1. When g(u) satisfies suitable conditions and g(...This paper considers the following quasilinear elliptic problem [GRAPHICS] where Omega is a bounded regular domain in R-N (N greater than or equal to 3), N > p > 1. When g(u) satisfies suitable conditions and g(u)u - beta integral (u)(0) g(s)ds is unbounded, a(x) is a Holder continuous function which changes sign on Omega and integral (Omega-) \a(x)\ dx is suitably small. The authors prove the existence of a nonnegative nontrivial solution for N > p > 1. in particular, the existence of a positive solution to the problem for N > p greater than or equal to 2. Our main theorem generalizes a recent result of Samia Khanfir and Leila Lassoued (see [1]) concerning the case where p = 2. They prove also that if g(u) = \u \ (q-2)u with p < q < p* and Omega (+) = {x is an element ofQ \a(x) > 0} is a nonempty open set, then the above problem possesses infinitely many solutions.展开更多
This paper is concerned with the existence of positive solutions of the followingDirichlet problem for p-mean curvature operator with critical exponent: -div((1 +|↓△u|^2 )p-2/2 ↓△u) = λup-1+μ u=q-1,u 〉...This paper is concerned with the existence of positive solutions of the followingDirichlet problem for p-mean curvature operator with critical exponent: -div((1 +|↓△u|^2 )p-2/2 ↓△u) = λup-1+μ u=q-1,u 〉 0,x x∈Ω,u=0,x∈ δΩ,where u ∈ W01,P is a bounded domain in R^N(N 〉 p 〉 1) with smooth boundary δΩ, 2≤p ≤q〈p,p=Np/N-p,λ,μ〉0. It reaches the conclusions that this problem has at least one positive solution in the different cases. It is discussed the existences of positivesolutions of the Dirichlet problem for the p-mean curvature operator with critical exponentby using Nehari-type duality property firstly. As p = 2, q = p, the result is correspond tothat of Laplace operator.展开更多
The authors study the p(x)-Laplacian equations with nonlinear boundary condi- tion. By using the variational method, under appropriate assumptions on the perturbation terms f1(x,u), f2(x,u) and h1(x), h2(x), such that...The authors study the p(x)-Laplacian equations with nonlinear boundary condi- tion. By using the variational method, under appropriate assumptions on the perturbation terms f1(x,u), f2(x,u) and h1(x), h2(x), such that the associated functional satisfies the "mountain pass lemma" and "fountain theorem" respectively, the existence and multiplicity of solutions are obtained. The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.展开更多
The existence and multiplicity results are obtained for periodic solutions of second order systems at resonance with unbounded nonlinearity. The proofs rely on the minimax methods and an interesting integral inequality.
This paper is concerned with the existences of positive solutions of the following Dirichlet problem for p-mean curvature operator with supercritical potential:{-div((1+| u|^2)p-2/2 u)=λu^r-1+μ(u^1q-1,|x...This paper is concerned with the existences of positive solutions of the following Dirichlet problem for p-mean curvature operator with supercritical potential:{-div((1+| u|^2)p-2/2 u)=λu^r-1+μ(u^1q-1,|x|^s),u〉0 x∈Ω,u=0 x∈ЭΩ where u∈ W0^1,P(Ω),Ω is a bounded domain in R^N(N 〉 p 〉 1) with smooth boundary ЭΩ and 0∈Ω,0 〈 q 〈p, 0≤s〈 N/p(p-q)+q, p≤r〈p*, p* = Np/N-p,μ〉0. It reaches the conclusion where this problem has two positive solutions in the different cases. It discusses the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with supercritical potential firstly. Meanwhile, it extends some results of the p-Laplace operator to that of p-mean curvature operator for p ≥2.展开更多
文摘In this paper, we consider the following noncooperative elliptic systems where Ω is a bounded domain in R<sup>N</sup> with smooth boundary ∂Ω, λ,δ,γ are real parameters, and . We assume that F is subquadratic at zero with respect to the variables u,v. By using a variant Clark’s theorem, we obtain infinitely many nontrivial solutions (u<sub>k</sub><sub></sub>,v<sub>k</sub>) with as k → ∞. Compared with the existing literature, we do not need to assume the behavior of the nonlinearity ∇F at infinity.
文摘This paper deals with the Neumann problem for a class of semilinear elliptic equations -△u + u =|u|2*-2u+ μ|u|q-2u in Ω, au/ar= |u|(?)*-2u on aΩ, where 2 = 2N/N-2, s=2(N-1)/N-2, 1 <q<2,N(?)3,μ>γ denotes the unit outward normal to boundary aΩ. By vaxiational method and dual fountain theorem, the existence of infinitely many solutions with negative energy is proved.
文摘By using variational method, the multiplicity of solutions for nonlinear biharmonic equation involving critical parameter and critical exponent are established.
文摘This paper considers the following quasilinear elliptic problem [GRAPHICS] where Omega is a bounded regular domain in R-N (N greater than or equal to 3), N > p > 1. When g(u) satisfies suitable conditions and g(u)u - beta integral (u)(0) g(s)ds is unbounded, a(x) is a Holder continuous function which changes sign on Omega and integral (Omega-) \a(x)\ dx is suitably small. The authors prove the existence of a nonnegative nontrivial solution for N > p > 1. in particular, the existence of a positive solution to the problem for N > p greater than or equal to 2. Our main theorem generalizes a recent result of Samia Khanfir and Leila Lassoued (see [1]) concerning the case where p = 2. They prove also that if g(u) = \u \ (q-2)u with p < q < p* and Omega (+) = {x is an element ofQ \a(x) > 0} is a nonempty open set, then the above problem possesses infinitely many solutions.
基金Supported by the National Natural Science Foundation of China(10171032) Supported by the Guangdong Provincial Natural Science Foundation of China(011606)
文摘This paper is concerned with the existence of positive solutions of the followingDirichlet problem for p-mean curvature operator with critical exponent: -div((1 +|↓△u|^2 )p-2/2 ↓△u) = λup-1+μ u=q-1,u 〉 0,x x∈Ω,u=0,x∈ δΩ,where u ∈ W01,P is a bounded domain in R^N(N 〉 p 〉 1) with smooth boundary δΩ, 2≤p ≤q〈p,p=Np/N-p,λ,μ〉0. It reaches the conclusions that this problem has at least one positive solution in the different cases. It is discussed the existences of positivesolutions of the Dirichlet problem for the p-mean curvature operator with critical exponentby using Nehari-type duality property firstly. As p = 2, q = p, the result is correspond tothat of Laplace operator.
基金supported by the National Natural Science Foundation of China (No. 10771141)the ZhejiangProvincial Natural Science Foundation of China (No. Y7080008).
文摘The authors study the p(x)-Laplacian equations with nonlinear boundary condi- tion. By using the variational method, under appropriate assumptions on the perturbation terms f1(x,u), f2(x,u) and h1(x), h2(x), such that the associated functional satisfies the "mountain pass lemma" and "fountain theorem" respectively, the existence and multiplicity of solutions are obtained. The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.
文摘The existence and multiplicity results are obtained for periodic solutions of second order systems at resonance with unbounded nonlinearity. The proofs rely on the minimax methods and an interesting integral inequality.
基金National Natural Science Foundation of China(10171032) and the Guangdong Provincial Natural Science Foundation of China(011606)
文摘This paper is concerned with the existences of positive solutions of the following Dirichlet problem for p-mean curvature operator with supercritical potential:{-div((1+| u|^2)p-2/2 u)=λu^r-1+μ(u^1q-1,|x|^s),u〉0 x∈Ω,u=0 x∈ЭΩ where u∈ W0^1,P(Ω),Ω is a bounded domain in R^N(N 〉 p 〉 1) with smooth boundary ЭΩ and 0∈Ω,0 〈 q 〈p, 0≤s〈 N/p(p-q)+q, p≤r〈p*, p* = Np/N-p,μ〉0. It reaches the conclusion where this problem has two positive solutions in the different cases. It discusses the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with supercritical potential firstly. Meanwhile, it extends some results of the p-Laplace operator to that of p-mean curvature operator for p ≥2.