A Concentration Phenomenon for Semilinear Elliptic Equations |
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Authors: | Nils Ackermann Andrzej Szulkin |
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Institution: | 1. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510, Mexico, DF, Mexico 2. Department of Mathematics, Stockholm University, 10691, Stockholm, Sweden
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Abstract: | For a domain ${\Omega \subset \mathbb{R}^{N}}$ we consider the equation $$-\Delta{u} + V(x)u = Q_n(x)|{u}|^{p-2}u$$ with zero Dirichlet boundary conditions and ${p\in(2, 2^*)}$ . Here ${V \geqq 0}$ and Q n are bounded functions that are positive in a region contained in ${\Omega}$ and negative outside, and such that the sets {Q n > 0} shrink to a point ${x_0 \in \Omega}$ as ${n \to \infty}$ . We show that if u n is a nontrivial solution corresponding to Q n , then the sequence (u n ) concentrates at x 0 with respect to the H 1 and certain L q -norms. We also show that if the sets {Q n > 0} shrink to two points and u n are ground state solutions, then they concentrate at one of these points. |
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