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

We consider the nonlinear eigenvalue problem {equation*} (0.1) \{ {array}{l}-Δ u=u^p+λ u \;\; in B, \\ u>0 \;\; in B, \;\;\; u=0 \;\; on ∂ B, {array} . {equation*} where B denotes the unit ball in RN, N ≥ 3, λ >0 and $p>(N+2)/(N-2)$. According to classical bifurcation theory, the point (μ ₁,0) is a bifurcation point from which emanates an unbounded branch C of solutions (λ , u) of (0.1), where μ ₁ is the principal eigenvalue of -Δ in B with Dirichlet boundary data. It is known that there is a unique value λ =λ _* ∈ (0, μ ₁) such that (0.1) has a radial singular solution u_* (|x|). Let pc>N+2/N-2 be the Joseph-Lundgren exponent. We show that the structure of the branch C changes for p ≥ pc and (N+2)/(N-2)<p<pc. For (N+2)/(N-2)<p<pc, C turns infinitely many times around λ _*, which implies that all the singular solutions have infinite Morse index. For p ≥ pc, we show that all solutions (regular or singular) have finite Morse index. For N ≥ 12 and p>pc 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 λ ∈ (λ *, μ ₁).