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

S. B. Kolonitskii

Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 2, pp. 88-92

Résumé

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.

Détail

Keywords: $p$-Laplacian, multiplicity of solutions.