This article is cited in 2 papers

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.

UDC: 517.956.25

Received: 21.01.2014

DOI: 10.4213/faa3193

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

Bibliographic databases:

• 
• 
• 
• 
• 
•