Existence of Many Nonradial Positive Solutions of the Hénon Equation in R³

N. Shioji

Journal of convex analysis, Tome 22 (2015) no. 1, pp. 61-8

Résumé

Let $B_1$ be the open unit ball in $\mathbf{R}^3$ and let $2$. We show that for each $m\in \mathbf{N}$, there exists $\alpha_0>0$ such that for each $\alpha\geq \alpha_0$, there exist at least $m$ nonradial positive solutions of $$ -\Delta u = |x|^\alpha |u(x)|^{p-2}u(x) \quad\text{in $B_1$,}\qquad u = 0 \quad\text{on $\partial B_1$,} $$ which are mutually nonequivalent if $m\geq 2$.

Détail

Classification : 35J20, 35J61
Mots-clés : Henon equation, multiplicity of positive solutions, concentration compactness principle, Poincare's inequalities