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Funktsional. Anal. i Prilozhen., 2015, Volume 49, Issue 2, Pages 88–92 (Mi faa3193)  

Brief communications

Multiplicity of 1D-concentrated positive solutions to the Dirichlet problem for an equation with $p$-Laplacian

S. B. Kolonitskii

St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: We consider the Dirichlet problem for the equation $-\Delta_p = u^{q-1}$ with $p$-Laplacian in a thin spherical annulus in $\mathbb R^n$ with $1 < p < q < p^*_{n-1}$, where $p^*_{n-1}$ is the critical Sobolev exponent for embedding in $\mathbb R^{n-1}$ and either $n=4$ or $n \ge 6$. We prove that this problem has a countable set of solutions concentrated in neighborhoods of certain curves. Any two such solutions are nonequivalent if the annulus is thin enough. As a corollary, we prove that the considered problem has as many solutions as required, provided that the annulus is thin enough.

Keywords: $p$-Laplacian, multiplicity of solutions

Funding Agency Grant Number
Saint Petersburg State University 6.38.670.2013
Supported by SPbSU grant no. 6.38.670.2013.


DOI: https://doi.org/10.4213/faa3193

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English version:
Functional Analysis and Its Applications, 2015, 49:2, 151–154

Bibliographic databases:

UDC: 517.956.25
Received: 21.01.2014

Citation: S. B. Kolonitskii, “Multiplicity of 1D-concentrated positive solutions to the Dirichlet problem for an equation with $p$-Laplacian”, Funktsional. Anal. i Prilozhen., 49:2 (2015), 88–92; Funct. Anal. Appl., 49:2 (2015), 151–154

Citation in format AMSBIB
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