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

We consider the problem {equation*} (1) \{ {array}{ll}-Δ u = uᵖ + 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 $p>1$. Merle and Peletier showed that for p>N+2/N-2 there is a unique value λ =λ _* >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 \[ p < {N-2√N-1}{N-2√N-1-4} or N≤ 10 , \] then for λ close to λ _*, a large number of classical solutions of (1) exist. In particular infinitely many solutions are present if λ = λ _*. We establish a similar assertion for the problem \[ \{ {array}{ll}-Δ u = λ f(u+1) in B , u>0 in B , u=0 on ∂ B , {array} . \] where f(s) = sᵖ + sq, $1<q<p$, and p satisfies the same condition as above.