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Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted nonlinearity

Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted... We consider the boundary value problem $\Delta u + \left| x \right|^{2\alpha } \left| u \right|^{p - 1} u = 0, - 1 < \alpha \ne 0$ , in the unit ball B with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, we prove that for any positive integer m, there exists a multi-peak nodal solution u p whose maxima and minima are located alternately near the origin and the other m points $\widetilde{q_l } = (\lambda \cos \frac{{2\pi (l - 1)}} {m},\lambda \sin \frac{{2\pi (l - 1)}} {m}),l = 2, \cdots ,m + 1 $ , such that as p goes to +∞, $p\left| x \right|^{2\alpha } \left| {u_p } \right|^{p - 1} u_p \rightharpoonup 8\pi e(1 + \alpha )\delta _0 + \sum\limits_{l = 2}^{m + 1} {8\pi e( - 1)^{l - 1} \delta _{\widetilde{q_l }} } $ , where λ ∈ (0, 1), m is an odd number with (1+α)(m+2)−1 > 0, or m is an even number. The same techniques lead also to a more general result on general domains. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Multi-peak nodal solutions for a two-dimensional elliptic problem with large exponent in weighted nonlinearity

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References (36)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-015-0465-5
Publisher site
See Article on Publisher Site

Abstract

We consider the boundary value problem $\Delta u + \left| x \right|^{2\alpha } \left| u \right|^{p - 1} u = 0, - 1 < \alpha \ne 0$ , in the unit ball B with the homogeneous Dirichlet boundary condition, when p is a large exponent. By a constructive way, we prove that for any positive integer m, there exists a multi-peak nodal solution u p whose maxima and minima are located alternately near the origin and the other m points $\widetilde{q_l } = (\lambda \cos \frac{{2\pi (l - 1)}} {m},\lambda \sin \frac{{2\pi (l - 1)}} {m}),l = 2, \cdots ,m + 1 $ , such that as p goes to +∞, $p\left| x \right|^{2\alpha } \left| {u_p } \right|^{p - 1} u_p \rightharpoonup 8\pi e(1 + \alpha )\delta _0 + \sum\limits_{l = 2}^{m + 1} {8\pi e( - 1)^{l - 1} \delta _{\widetilde{q_l }} } $ , where λ ∈ (0, 1), m is an odd number with (1+α)(m+2)−1 > 0, or m is an even number. The same techniques lead also to a more general result on general domains.

Journal

Acta Mathematicae Applicatae Sinica – Springer Journals

Published: Apr 12, 2015

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