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Full-Text Articles in Physical Sciences and Mathematics
An Elliptic Equation With No Monotonicity Condition On The Nonlinearity, Gregory S. Spradlin
An Elliptic Equation With No Monotonicity Condition On The Nonlinearity, Gregory S. Spradlin
Greg S. Spradlin Ph.D.
An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree theory argument must be used.
Interfering Solutions Of A Nonhomogeneous Hamiltonian System, Gregory S. Spradlin
Interfering Solutions Of A Nonhomogeneous Hamiltonian System, Gregory S. Spradlin
Greg S. Spradlin Ph.D.
A Hamiltonian system is studied which has a term approaching a constant at an exponential rate at infinity. A minimax argument is used to show that the equation has a positive homoclinic solution. The proof employs the interaction between translated solutions of the corresponding homogeneous equation. What distinguishes this result from its few predecessors is that the equation has a nonhomogeneous nonlinearity.