Multiple Boundary Concentrating Solutions to Dirichlet Problem of Hénon Equation

Abstract

Let Ω be the unit ball centered at the origin in $\mathbb{R}^{N} {\left( {N \geqslant 4} \right)},2^{ * } = \frac{{2N}} {{N - 2}},\tau > 0,\varepsilon > 0$. We study the following problem

$$ \left\{ {\begin{array}{*{20}l} {{ - \Delta u = {\left| x \right|}^{\tau } u^{{2^{ * } - 1 - \varepsilon }} } \hfill} & {{x \in \Omega ,} \hfill} \\ {{u > 0} \hfill} & {{x \in \Omega ,} \hfill} \\ {{u = 0} \hfill} & {{x \in \partial \Omega .} \hfill} \\ \end{array} } \right. $$

By a constructive argument, we prove that for any k = 1, 2, • • •, if ε is small enough, then the above problem has positive a solution u_ε concentrating at k distinct points which tending to the boundary of Ω as ε goes to 0⁺.