In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions： △2 u = K（x）u p , u 〉 0 in Ω , △u = u = 0 on Ω , where Ω is ...In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions： △2 u = K（x）u p , u 〉 0 in Ω , △u = u = 0 on Ω , where Ω is a smooth domain in R n , n 5, and p ＋ 1 = 2 n n 4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.展开更多
This paper introduces a notion of linear perturbed Palais-Smale condition for real-valued functions on Banach spaces. In terms of strongly exposed points, it presents a characterization which guarantees linear perturb...This paper introduces a notion of linear perturbed Palais-Smale condition for real-valued functions on Banach spaces. In terms of strongly exposed points, it presents a characterization which guarantees linear perturbed Palais-Smale condition holds for lower semicontinuous functions with bounded effective domains defined on a Banach space with the Radon-Nikody＇m property; and gives an example showing that linear perturbed P-S condition is strictly weaker than the P-S condition.展开更多
In this paper, we consider a p（x）-biharmonic problem with Navier boundary conditions. The existence of infinitely many solutions which tend to zero is investigated based on the symmetric Mountain Pass lemma. Our app...In this paper, we consider a p（x）-biharmonic problem with Navier boundary conditions. The existence of infinitely many solutions which tend to zero is investigated based on the symmetric Mountain Pass lemma. Our approach relies on the theory of variable exponent Sobolev space.展开更多
With the aids of variational method and concentration-compactness principle, infinitely many solutions are obtained for a class of fourth order elliptic equations with singular potentialΔ^2u=μ｜u｜^2＊＊（s）-2u/｜x...With the aids of variational method and concentration-compactness principle, infinitely many solutions are obtained for a class of fourth order elliptic equations with singular potentialΔ^2u=μ｜u｜^2＊＊（s）-2u/｜x｜^s＋λk（x）｜u｜^r-2 u, u∈H^2,2（R^N） （P）展开更多
We discussed a class of p-Laplacian boundary problems on a bounded smooth domain, the nonlinearity is odd symmetric and limit subcritical growing at infinite. A sequence of critical values of the variational functiona...We discussed a class of p-Laplacian boundary problems on a bounded smooth domain, the nonlinearity is odd symmetric and limit subcritical growing at infinite. A sequence of critical values of the variational functional was constructed after the general- ized Palais-Smale condition was verified. We obtain that the problem possesses infinitely many solutions and corresponding energy levels of the functional pass to positive infinite. The result is a generalization of a similar problem in the case of subcritical.展开更多
The purpose of this paper is to study a semilinear Schr<span style="white-space:nowrap;">ö</span>dinger equation with constraint in <em>H</em><sup>1</sup>(<str...The purpose of this paper is to study a semilinear Schr<span style="white-space:nowrap;">ö</span>dinger equation with constraint in <em>H</em><sup>1</sup>(<strong>R</strong><sup><em>N</em></sup>), and prove the existence of sign changing solution. Under suitable conditions, we obtain a negative solution, a positive solution and a sign changing solution by using variational methods.展开更多
Let Ω∈← 0 be an open bounded domain in RN （N 〉 3） and 2＊（s） = 2（N-8） =2（N-s）N-22. We consider the following elliptic system of two equations in where A, μ 〉 0 and α, β 〉 1 satisfy α ＋ β = 2＊（s...Let Ω∈← 0 be an open bounded domain in RN （N 〉 3） and 2＊（s） = 2（N-8） =2（N-s）N-22. We consider the following elliptic system of two equations in where A, μ 〉 0 and α, β 〉 1 satisfy α ＋ β = 2＊（s）. Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions.展开更多
文摘In this paper, we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier boundary conditions： △2 u = K（x）u p , u 〉 0 in Ω , △u = u = 0 on Ω , where Ω is a smooth domain in R n , n 5, and p ＋ 1 = 2 n n 4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers lower bounds for the number of solutions. As further applications of this Morse theoretical approach, we prove more existence results.
文摘This paper introduces a notion of linear perturbed Palais-Smale condition for real-valued functions on Banach spaces. In terms of strongly exposed points, it presents a characterization which guarantees linear perturbed Palais-Smale condition holds for lower semicontinuous functions with bounded effective domains defined on a Banach space with the Radon-Nikody＇m property; and gives an example showing that linear perturbed P-S condition is strictly weaker than the P-S condition.
基金supported in part by the NNSF of China(Grant No.11101145)Research Initiation Project for Highlevel Talents(201031)of North China University of Water Resources and Electric Power
文摘In this paper, we consider a p（x）-biharmonic problem with Navier boundary conditions. The existence of infinitely many solutions which tend to zero is investigated based on the symmetric Mountain Pass lemma. Our approach relies on the theory of variable exponent Sobolev space.
基金Supported by NSFC(No11001221)the Foundation of Shaanxi Province Education Department (No2010JK549)Zhejiang Provincial Natural Science Foundation of China(NoY6110118)
基金National Science Foundation of China (10471113)Natural Science Foundation of Zhejiang Province (Y606292)
文摘With the aids of variational method and concentration-compactness principle, infinitely many solutions are obtained for a class of fourth order elliptic equations with singular potentialΔ^2u=μ｜u｜^2＊＊（s）-2u/｜x｜^s＋λk（x）｜u｜^r-2 u, u∈H^2,2（R^N） （P）
基金Project supported by the National Natural Science Foundation of China(No.10371045)the Natural Science Foundation of Guangdong Province of China(No.5005930)
文摘We discussed a class of p-Laplacian boundary problems on a bounded smooth domain, the nonlinearity is odd symmetric and limit subcritical growing at infinite. A sequence of critical values of the variational functional was constructed after the general- ized Palais-Smale condition was verified. We obtain that the problem possesses infinitely many solutions and corresponding energy levels of the functional pass to positive infinite. The result is a generalization of a similar problem in the case of subcritical.
文摘The purpose of this paper is to study a semilinear Schr<span style="white-space:nowrap;">ö</span>dinger equation with constraint in <em>H</em><sup>1</sup>(<strong>R</strong><sup><em>N</em></sup>), and prove the existence of sign changing solution. Under suitable conditions, we obtain a negative solution, a positive solution and a sign changing solution by using variational methods.
基金supported by National Natural Science Foundation of China under grant Nos. 1110145011071239
文摘Let Ω∈← 0 be an open bounded domain in RN （N 〉 3） and 2＊（s） = 2（N-8） =2（N-s）N-22. We consider the following elliptic system of two equations in where A, μ 〉 0 and α, β 〉 1 satisfy α ＋ β = 2＊（s）. Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions.
