In this paper,we study the uniqueness of positive solutions to the following semilinear equations{-Δu=λ|x|^(α)ue^(u^(2)),in B_(1),u=0,onδB_(1)ueu2;in B_(1);u=0;on@B_(1);whereλ>0,α>-2;B_(1)denotes the unit ...In this paper,we study the uniqueness of positive solutions to the following semilinear equations{-Δu=λ|x|^(α)ue^(u^(2)),in B_(1),u=0,onδB_(1)ueu2;in B_(1);u=0;on@B_(1);whereλ>0,α>-2;B_(1)denotes the unit disk in R^(2):By delicate and relatively complicated computation of radial solutions to the above equation and the asymptotic expansion of solutions near the boundary of B_(1),the uniqueness of positive solutions is obtained.The results of this paper extend the uniqueness result for the semilinear equation with critical exponential growth in CHEN et al.(2022)to the case that includes a Henon term.展开更多
This paper investigates the following mixed local and nonlocal elliptic problem fea-turing concave-convex nonlinearities and a discontinuous right-hand side:{L(u)=H(u−μ)|u|^(p−2)u+λ|u|^(q−2)u,x∈Ω,u≥0,x∈Ω,u=0,x...This paper investigates the following mixed local and nonlocal elliptic problem fea-turing concave-convex nonlinearities and a discontinuous right-hand side:{L(u)=H(u−μ)|u|^(p−2)u+λ|u|^(q−2)u,x∈Ω,u≥0,x∈Ω,u=0,x∈R^(N)\Ω,where Ω ⊂R^(N)(N>2)is a bounded domain,μ≥0 and λ>0 are real parameters,H denotes the Heaviside function(H(t)=0 for t<0,H(t)=1 for t>0),and the mixed local and nolocal operator is defined as L(u)=−Δu+(−Δ)^(s)u with(−Δ)^(s) being the restricted fractional Laplace(0<s<1).The exponents satisfy 1<q<2<p.By employing a novel non-smooth variational principle,we establish the existence of an M-solution for this problem and identify a range for the exponent p.展开更多
In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to...In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball.展开更多
Let(M,g)be a compact Riemann surface with unit area,h a smooth function on M.The Kazdan-Warner problem is that under what kind of conditions on h the equationΔu=8π-8πhe^(u) has a solution.In this survey article,we ...Let(M,g)be a compact Riemann surface with unit area,h a smooth function on M.The Kazdan-Warner problem is that under what kind of conditions on h the equationΔu=8π-8πhe^(u) has a solution.In this survey article,we shall review the development of this problem along the variational method.展开更多
We consider the multiplicity of solutions to a p(x)-Laplacian problem involving supercritical Sobolev growth via Ricceri’s principle.By means of truncation combined with De Giorgi iteration,we can extend the results ...We consider the multiplicity of solutions to a p(x)-Laplacian problem involving supercritical Sobolev growth via Ricceri’s principle.By means of truncation combined with De Giorgi iteration,we can extend the results of subcritical and critical growth to supercritical growth and obtain at least three solutions to the p(x)-Laplacian problem.展开更多
基金Supported by the Natural Science Foundation of China(12571122,12061010)。
文摘In this paper,we study the uniqueness of positive solutions to the following semilinear equations{-Δu=λ|x|^(α)ue^(u^(2)),in B_(1),u=0,onδB_(1)ueu2;in B_(1);u=0;on@B_(1);whereλ>0,α>-2;B_(1)denotes the unit disk in R^(2):By delicate and relatively complicated computation of radial solutions to the above equation and the asymptotic expansion of solutions near the boundary of B_(1),the uniqueness of positive solutions is obtained.The results of this paper extend the uniqueness result for the semilinear equation with critical exponential growth in CHEN et al.(2022)to the case that includes a Henon term.
基金Supported by the National Natural Science Foundation of China(Grant No.12361026)the Discipline Construction Fund Project of Northwest Minzu University.
文摘This paper investigates the following mixed local and nonlocal elliptic problem fea-turing concave-convex nonlinearities and a discontinuous right-hand side:{L(u)=H(u−μ)|u|^(p−2)u+λ|u|^(q−2)u,x∈Ω,u≥0,x∈Ω,u=0,x∈R^(N)\Ω,where Ω ⊂R^(N)(N>2)is a bounded domain,μ≥0 and λ>0 are real parameters,H denotes the Heaviside function(H(t)=0 for t<0,H(t)=1 for t>0),and the mixed local and nolocal operator is defined as L(u)=−Δu+(−Δ)^(s)u with(−Δ)^(s) being the restricted fractional Laplace(0<s<1).The exponents satisfy 1<q<2<p.By employing a novel non-smooth variational principle,we establish the existence of an M-solution for this problem and identify a range for the exponent p.
文摘In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball.
文摘Let(M,g)be a compact Riemann surface with unit area,h a smooth function on M.The Kazdan-Warner problem is that under what kind of conditions on h the equationΔu=8π-8πhe^(u) has a solution.In this survey article,we shall review the development of this problem along the variational method.
基金supported by the Fundamental Research Funds for the Central Universities(2024KYJD2006).
文摘We consider the multiplicity of solutions to a p(x)-Laplacian problem involving supercritical Sobolev growth via Ricceri’s principle.By means of truncation combined with De Giorgi iteration,we can extend the results of subcritical and critical growth to supercritical growth and obtain at least three solutions to the p(x)-Laplacian problem.
