An Elliptic Equation with Spike Solutions Concentrating at Local Minima of the Laplacian of the Potential
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We consider the equation -∊2 ∆u + V(z)u = ƒ(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 ƒ satisfies super-linear and sub-critical growth conditions, we show that for small ∊ 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 variational mountain-pass and concentration compactness arguments. A penalization technique developed by Felmer and del Pino is used to handle the lack of compactness and the absence of the Palais-Smale condition in the variational framework.
CitationSpradlin, G. S. (2000). An elliptic equation with spike solutions concentrating at local minima of the Laplacian of the potential. Electronic Journal of Differential Equations, 2000(32), pp. 1-14.
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