Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent

Abstract:

We consider the nonlinear eigenvalue problem \begin{equation*} (0.1) \qquad \qquad \qquad \qquad \qquad \left \{\begin {array}{l}-\Delta u=u^p+\lambda u \;\; \mbox {in $B$}, \\ u>0 \;\; \mbox {in $B$}, \;\;\; u=0 \;\; \mbox {on $\partial B$}, \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 $p>(N+2)/(N-2)$. According to classical bifurcation theory, the point $(\mu _1,0)$ is a bifurcation point from which emanates an unbounded branch $\mathscr {C}$ of solutions $(\lambda , u)$ of (0.1), where $\mu _1$ is the principal eigenvalue of $-\Delta$ in $B$ with Dirichlet boundary data. It is known that there is a unique value $\lambda =\lambda _* \in (0, \mu _1)$ such that (0.1) has a radial singular solution $u_* (|x|)$. Let $p_c>\frac {N+2}{N-2}$ be the Joseph-Lundgren exponent. We show that the structure of the branch $\mathscr {C}$ changes for $p \geq p_c$ and $(N+2)/(N-2)<p<p_c$. For $(N+2)/(N-2)<p<p_c$, $\mathscr {C}$ turns infinitely many times around $\lambda _*$, which implies that all the singular solutions have infinite Morse index. For $p \geq p_c$, we show that all solutions (regular or singular) have finite Morse index. For $N \geq 12$ and $p>p_c$ large, we show that all solutions (regular or singular) have exactly Morse index one. As a consequence, we prove that any regular solution intersects with the singular solution exactly once and any regular solution exists (and is unique) only when $\lambda \in (\lambda _{*}, \mu _1)$.