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

Abstract:

We consider the problem \begin{equation*} (1)\qquad \qquad \qquad \left \{\begin {array}{ll}-\Delta u = u^{p} + \lambda u \quad \mathrm {in}\quad B , u>0 \quad \mathrm {in}\quad B ,\quad u=0 \quad \mathrm {on}\quad \partial B, \end{array}\right . \qquad \qquad \qquad \qquad \end{equation*} 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 \[ p <\frac {N-2\sqrt {N-1}}{N-2\sqrt {N-1}-4} \quad \mathrm {or} \quad N\leq 10 , \] 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 \[ \left \{\begin {array}{ll}-\Delta u = \lambda f(u+1) \quad \mathrm {in}\quad B , u>0 \quad \mathrm {in}\quad B ,\quad u=0 \quad \mathrm {on}\quad \partial B ,\end {array} \right . \] where $f(s) = s^p + s^q$, $1<q<p$, and $p$ satisfies the same condition as above.