SEMILINEAR ELLIPTIC EQUATIONS HARDY TERM

. We study the elliptic equation ∆ u + µ/ | x | 2 + K ( | x | ) u p = 0 in R n \{ 0 } , where n ≥ 1 and p > 1. In particular, when K ( | x | ) = | x | l , a classiﬁcation of radially symmetric solutions is presented in terms of µ and l . Moreover, we explain the separation structure for the equation, and study the stability of positive radial solutions as steady states.


SOOHYUN BAE
The second transformation for σ < n−2 2 , leads to the equation (1.6) Under condition (1.6), we see that (1.5) possesses a local radial solution w such that w(r) = O(r −σ ) at 0. This paper is organized as follows. Classification of radial solutions for Lane-Emden equation is reviewed in Section 2. In Section 3 we study the existence of regular and singular solutions of (1.1). Moreover, we explain the separation structure for (1.1) and study the stability. In Section 4 we provide the graphs of the regions where separation phenomena occur.
By the Kelvin transform, we observe that there exist infinitely many singular solutions for n+l n−2 < p < n+2+2l n−2 . When p > n+2+2l n−2 , r −m L is a unique singular solution, which is defined for p > n+l n−2 . When p = n+2+2l n−2 , there are singular solutions with different asymptotic behavior. More precisely, for each 0 < d 1 < L, there exists a singular solution u s (r) such that 0 < d 1 := min r m u s (r) < L < d 2 := max r m u s (r) < (n+l)(n−2) )/(p + 1), and r m u s (r) is periodic in t = log r.
where µ < ( n−2 2 ) 2 and p > 1. Set If |x| l is replaced byK(|x|), then this condition corresponds to (1.6) which is necessary to have local radial solutions satisfying u(r) = O(r −ν− ) at 0 where r = |x|. We regard this type as regular solution. Let We consider the nonexistence of positive solutions. Proof. Letū be the spherical average of u and V (t) = r mū , t = log r. Then,ū and V satisfy that ∆ū + µ |x| 2ū + |x| lūp ≤ 0 by Jensen's inequality and where a = n − 2 − 2m.
for T large, we see that V ≥ 0 near +∞. Multiplying (3.2) by e at and integrating over [T, t] for T large, we have ), a contradiction for t large.
. Then W satisfies that W − aW + W p ≤ 0 which has no positive solution near +∞ as observed in Case 1. Hence, (3.2) has no positive solution near −∞.
In particular, we have the nonexistence if µ ≥ ( n−2 2 ) 2 =:μ, the Hardy constant. In fact, L p−1 =μ is the maximum at p = n+2+2l n−2 . For any p > 1, the nonexistence holds when • n = 1 and µ ≥ 1 4 ; • n = 2 and µ ≥ 0. Now, we mention the following two well-known nonexistence results. By making use of the second transformation from Lane-Emden equation, we have the nonexistence of regular radial solutions.
where a = n − 2 − 2m and L := L p−1 − µ 1 p−1 . By making use of the inverse Fowler transform, we observe that (3.3) for n = 1, 2 also possesses one-parameter family of solutions under the assumptions. In particular, we have the critical cases.
• For n = 1, assume m 2 + m + µ < 0 and 1 < p ≤ −2l − 3 < −2ν − (p − 1) + 4. The critical problem is We regard the solutions as even functions in R\{0}. • For n = 2, assume m 2 + µ < 0 and l ≤ −2. The critical problem is • For n ≥ 3, assume µ < L p−1 and p ≥ n+2+2l n−2 for supercritical case. The asymptotic behavior of solutions has two types: the first is fast decay for the critical case; the second is slow decay for the supercritical case.
(i) If p(n − 2) = n + 2 + 2l, then there are two types: the first has the selfsimilar singularity, r −m L; the second is of periodic type. Precisely, for each 4 ] − 1 2(2+l) for n = 1, and r m u s (r) is periodic in t = log r. (ii) If p(n − 2) > n + 2 + 2l, then r −m L is the unique singular radial solution.
Theorem 3.7. Let a = n − 2 − 2m and p(n − 2) ≥ n + 2 + 2l. Assume L p−1 > µ ≥ L p−1 − a 2 4(p−1) . Then, any two positive solutions of (3.3) do not intersect. We omit the proof of Theorem 3.7 since the arguments are the same as in the proofs of Theorem 3.2 and Theorem 1.2 in [4]. Another way is to make use of the second transformation in the introduction. Importantly, (2.1) and (3.1) are closely related via (3.4). In order to clarify the relationship among parameters, we analyze each case and explain the conditions. For p(n − 2) ≥ n + 2 + 2l, there exist µ − (n, p, l) < µ + (n, p, l) <μ such that the separation occurs for 0 < µ − ≤ µ < µ + .
There is an upper bound of p for this case. Observe that µ − ≤ µ < µ + ≤μ = 1 4 and lim For given −∞ < µ < 1 4 , l+2 ν− + 1 = p − < p ≤ p + . lim In the last section, we describe the above conditions as the regions surrounded by the graphs in the coordinate (µ, p)-plane.