Existence and asymptotic behavior of solutions for Hénon type equations

Abstract

Abstract. This paper is concerned with ground state solutions for the Hénon type equationWe study the existence of cylindrically symmetric and non-cylindrically symmetric ground state solutions for the problem. We also investigate asymptotic behavior of the ground state solution when p tends to the critical exponent 2 * = 2n n−2 if n ≥ 3.

Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space

“…Using (24) we infer that I is bounded on K 0 . Consider {u k } ⊂ K 0 such that I (u k ) → c. By the compactness of K 0 we may assume, passing to a subsequence if necessary, that there exists u ∈ K 0 such that u k → u in C. It follows that G(u k ) → G(u) and Ψ (u) lim inf k→∞ Ψ (u k ).…”

“…Problems of this type have been widely studied (see for instance [23,24,27,29] and the references therein). It is worth to point out that, as shown in [27], problem (34) has a positive radial solution provided that q ∈ 1, N + 2m + 2 N − 2 (N 3, m > 0).…”