A dynamical system approach to the construction of singular solutions of some degenerate elliptic equations

Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity

“…If the assumption that u is non-negative is dropped, but suppose that (N + 1)/(N − 1) q < C N , then the same results hold except that the limit in the case (i 1 ) should now be q,N or − q,N ; see [29,Theorem 2]. Further results on sign-changing singular solutions can be found in [7,16,28].…”

Boundary singularities of solutions of N-harmonic equations with absorption

“…If we look for separable solutions of − div |Du| p−2 Du + |u| q−1 u = 0, (2.52) in R N , where q > p − 1 > 0, p not necessarily equal to N or to 2, under the form u(r, σ ) = r β ω(σ ), then β = β p,q = −p/(q + 1 − p) and ω is a solution of − div σ β 2 p,q ω 2 + |∇ σ ω| 2 (p−2)/2 ∇ σ ω − Λ(p, q) β 2 p,q ω 2 + |∇ σ ω| 2 (p−2)/2 ω + |ω| q−1 ω = 0 [8,21]. The existence of non-trivial solution of (2.53) is insured as soon Λ(p, q) > 0, or equivalently q < N(p − 1)/ (N − p) if p < N, and no condition if p N .…”

“…The existence of non-trivial solutions of the same equation in S N −1 + vanishing on ∂S N −1 + is much more complicated. However, it is proved in [8,21] that there exists a critical exponent q c > p − 1 such that, if q q c no non-trivial solution exists while if p − 1 < q < q c there exists a unique positive solution in S N −1 + vanishing on ∂S N −1 + . The uniqueness proof in the previous proposition is valid.…”

Separable solutions of some quasilinear equations with source reaction

“…Similarly, the study of the boundary behaviour of solutions of quasilinear equations has a natural starting point in the description of their isolated singularities on the boundary. Besides the historical results of Fatou, Herglotz and Doob on the boundary trace of positive harmonic and super harmonic functions, equations of types (1.2), (1.3) and (1.4) have already been considered [4,6,11,12,26,27]. In the present article we consider equations of type (1.5).…”

“…Apart the case p = 2, the existence counterpart of this theorem is not known in arbitrary dimension, except if q = q c in which case (1.5) is the Euler-Lagrange equation of the functional 12) and applications of the already mentioned variational methods lead to an existence result. However, when N = 2 the problem of finding solutions of (1.5) under the form (1.6) can be completely solved using dynamical systems methods.…”