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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
Mathematics - Daytona Beach
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. © EDP Sciences, SMAI 2006.
An Elliptic Equation With Spike Solutions Concentrating At Local Minima Of The Laplacian Of The Potential, Gregory S. Spradlin
An Elliptic Equation With Spike Solutions Concentrating At Local Minima Of The Laplacian Of The Potential, Gregory S. Spradlin
Publications
We consider the equation −ԑ2∆u + V (z)u = f(u) which arises in the study of nonlinear Schrödinger equations. We seek solutions that are positive on RN and that vanish at infinity. Under the assumption that f satisfies super-linear and sub-critical growth conditions, we show that for small ԑ there exist solutions that concentrate near local minima of V. The local minima may occur in unbounded components, as long as the Laplacian of V achieves a strict local minimum along such a component. Our proofs employ vibrational mountain-pass and concentration compactness arguments. A penalization technique developed by Felmer and del …