In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε≤...In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε≤ε; for any m ∈ N, it has m pairs solutions if ε≤εm, where ε,εm are sufficiently small positive numbers. Moreover, these solutions axe closed to zero in W1,p（RN） as ε→0.展开更多
In this paper,we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schr?dinger equation:-ε2Δv+Γ(I(x)+v^(2))/(1+I(x)+v^(2))v=λv fo...In this paper,we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schr?dinger equation:-ε2Δv+Γ(I(x)+v^(2))/(1+I(x)+v^(2))v=λv for x∈R2.For a negatively large coupling constantΓ,we show that there exists a family of normalized positive solutions(i.e.,with the L2-constraint)whenεis small,which concentrate around local maxima of the intensity function I(x)asε→0.We also consider the case where I(x)may tend to-1 at infinity and the existence of multiple solutions.The proof of our results is variational and the novelty of the work lies in the development of a new truncation-type method for the construction of the desired solutions.展开更多
基金Supported by the National Natural Science Foundation of China(No.11501186,11326145,11526088)
文摘In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε≤ε; for any m ∈ N, it has m pairs solutions if ε≤εm, where ε,εm are sufficiently small positive numbers. Moreover, these solutions axe closed to zero in W1,p（RN） as ε→0.
基金supported by National Natural Science Foundation of China(Grant No.11861053)supported by National Natural Science Foundation of China(Grant No.11831009)supported by National Natural Science Foundation of China(Grant No.11901582)。
文摘In this paper,we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schr?dinger equation:-ε2Δv+Γ(I(x)+v^(2))/(1+I(x)+v^(2))v=λv for x∈R2.For a negatively large coupling constantΓ,we show that there exists a family of normalized positive solutions(i.e.,with the L2-constraint)whenεis small,which concentrate around local maxima of the intensity function I(x)asε→0.We also consider the case where I(x)may tend to-1 at infinity and the existence of multiple solutions.The proof of our results is variational and the novelty of the work lies in the development of a new truncation-type method for the construction of the desired solutions.
