Infinitely many mountain pass solutions on a kind of fourth-order Neumann boundary value problem

In this paper, the existence of infinitely many mountain pass solutions are obtained for the fourth-order boundary value problem (BVP) u⁽⁴⁾(t)-2u^″(t)+u(t)=f(u(t)),0<t<1, u^′(0)=u^′(1)=u^?(0)=u^?(1)=0, where f:R→R is continuous. The study of the problem is based on the variational methods and critical point theory. We prove the conclusion by using sub-sup solution method, Mountain Pass Theorem in Order Intervals, Leray-Schauder degree theory and Morse theory.     

Existence of solutions to a class of nonlinear second order two-point boundary value problems

In this paper, the existence and multiplicity results of solutions are obtained for the second order two-point boundary value problem −u^″(t)=f(t,u(t)) for all t∈[0,1] subject to u(0)=u^′(1)=0, where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator and its properties play an important role.     

Existence of solutions for some discrete boundary value problems with a parameter

In this paper, the existence and multiplicity results of solutions are obtained for the discrete nonlinear two point boundary value problem (BVP) ; u(0)=0=Δu(T), where T is a positive integer, Z(1,T)={1,2,…,T}, Δ is the forward difference operator defined by Δu(k)=u(k+1)-u(k) and f:Z(1,T)×R→R is continuous, λ∈R⁺ is a parameter. By using the critical point theory and Morse theory, we obtain that the above (BVP) has solutions for λ being in some different intervals.     

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u_|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→⁰⁺g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary.     

Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u^″(t)+au^′(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,b∈R¹, f∈C¹([0,1]×R¹,R¹). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.     

Relative oscillation theory for Sturm-Liouville operators extended

We extend relative oscillation theory to the case of Sturm-Liouville operators Hu=r⁻¹(−^′(pu^′)+qu) with different p's. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.     

Social choice for bliss-point problems

This paper's concern is the axiomatic determination of social choice correspondences Ψ for a class of n-person problems that are characterized by some — generally — non-feasible bliss-point u(V)(?V∈τ . Meeting appropriate assumptions of ‘planner's rationality’ it is shown that Ψ is necessarily norm-induced, i.e. one can find some norm |·| in Rⁿ s.t. Ψ(V) = {u?V|6u-u(V)6=min{6v-u(V)6|v,V}}. The mathematical problem of recovering. from Ψ is one of integration which has its well-known parallel in the theory of revealed preference.     

Existence and multiple solutions for a nonlinear system with a parameter

In this paper, the critical point theory and the Morse theory are employed to discuss the system of nonlinear equations of the form Au=λf(u)$Au=\lambda f\left(u\right)$, where the matrix A$A$ is not necessarily positive definite. Some existence and multiple solutions are obtained.     

A second-order estimate for blow-up solutions of elliptic equations

We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, in a bounded smooth domain Ω⊂R^N. b(x) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). The main results show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.     

Second order estimates for large solutions of elliptic equations

This paper deals with the second term asymptotic behavior of large solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, x∈∂Ω, where Ω is a smooth bounded domain in R^N, and b(x) is a non-negative weight function. The absorption term f is regularly varying at infinite with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). Our analysis relies on the Karamata regular variation theory.