It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical.The potential V is bounded from below and above by positive constants.Because we are consi...It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical.The potential V is bounded from below and above by positive constants.Because we are considering a critical term which interacts with higher eigenvalues for the linear problem,we need to apply a linking theorem.Notice that the lack of compactness,which comes from critical problems and the fact that we are working in the whole space,are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems.The main feature in this article is to restore some compact results which are essential in variational methods.Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting.This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.展开更多
We study nonautonomonus second order periodic systems with a nonslnooth potential. Using the nonsmooth critical theory, we establish the existence of at least two nontrivial solutions. Our framework incorporates large...We study nonautonomonus second order periodic systems with a nonslnooth potential. Using the nonsmooth critical theory, we establish the existence of at least two nontrivial solutions. Our framework incorporates large classes of both subquadratic and superquadratic potentials at infinity.展开更多
In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|^(N-2) u + f(x, u), x ∈ ?,u ∈ W_0^(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R...In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|^(N-2) u + f(x, u), x ∈ ?,u ∈ W_0^(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R^N(N > 2), f(x, u) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ > λ_1, λ = λ_?(? = 2, 3, · · ·), and λ_? is the eigenvalues of the operator(-?_N, W_0^(1,N)(?)),which is defined by the Z_2-cohomological index.展开更多
基金partially supported by CNPq with(429955/2018-9)partially suported by CNPq(309026/2020-2)FAPDF with(16809.78.45403.25042017)。
文摘It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical.The potential V is bounded from below and above by positive constants.Because we are considering a critical term which interacts with higher eigenvalues for the linear problem,we need to apply a linking theorem.Notice that the lack of compactness,which comes from critical problems and the fact that we are working in the whole space,are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems.The main feature in this article is to restore some compact results which are essential in variational methods.Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting.This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.
基金supported by the State Committee for Scientific Research of Poland (KBN) under research grants nr 2 P03A 003 25 and nr 4T07A 027 26
文摘We study nonautonomonus second order periodic systems with a nonslnooth potential. Using the nonsmooth critical theory, we establish the existence of at least two nontrivial solutions. Our framework incorporates large classes of both subquadratic and superquadratic potentials at infinity.
基金Supported by Shanghai Natural Science Foundation(15ZR1429500)NNSF of China(11471215)
文摘In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|^(N-2) u + f(x, u), x ∈ ?,u ∈ W_0^(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R^N(N > 2), f(x, u) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ > λ_1, λ = λ_?(? = 2, 3, · · ·), and λ_? is the eigenvalues of the operator(-?_N, W_0^(1,N)(?)),which is defined by the Z_2-cohomological index.
