Geometry of phase space and solutions of semilinear elliptic equations in a ball

We consider the problem

$(1)\qquad\qquad\qquad \left\{\begin{array}{ll}-\Delta u = u^{p} ... ...=0 \quad\hbox{on}\quad \partial B, \end{array}\right. \qquad\qquad\qquad\qquad$

where $B$ denotes the unit ball in $\mathbb{R}^N$, $N\geq 3$, $\lambda > 0$ and $p>1$. Merle and Peletier showed that for $p>\tfrac{N+2}{N-2}$ there is a unique value $\lambda =\lambda_* >0$ such that a radial singular solution exists. This value is the only one at which an unbounded sequence of classical solutions of (1) may accumulate. Here we prove that if additionally

$p <\frac{N-2\sqrt{N-1}}{N-2\sqrt{N-1}-4} \quad\hbox{or} \quad N\leq 10\,,$

then for $\lambda$ close to $\lambda_*$, a large number of classical solutions of (1) exist. In particular infinitely many solutions are present if $\lambda = \lambda_*$. We establish a similar assertion for the problem

$\left\{\begin{array}{ll}-\Delta u = \lambda\, f(u+1) \quad\hbox{i... ...ox{in}\quad B\,,\quad u=0 \quad\hbox{on}\quad \partial B\,,\end{array} \right.$

where $f(s) = s^p + s^q$, $1<q<p$, and $p$ satisfies the same condition as above.