Global Solution Branch and Morse Index Estimates of a Semilinear Elliptic Equation with Super-critical Exponent
نویسندگان
چکیده
We consider the nonlinear eigenvalue problem (0.1) { −Δu = up + λu in B, u > 0 in B, u = 0 on ∂B, 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 (μ1, 0) is a bifurcation point from which emanates an unbounded branch C of solutions (λ, u) of (0.1), where μ1 is the principal eigenvalue of −Δ in B with Dirichlet boundary data. It is known that there is a unique value λ = λ∗ ∈ (0, μ1) 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∗, μ1).
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