Non-Radial Ground States for the Hénon Equation

Abstract

We analyse symmetry breaking for ground states of the Hénon equation [7] in a ball. Asymptotic estimates of the transition are also given when p is close to either 2 or 2 * .

Existence and asymptotic behavior of solutions for Hénon type equations

“…So it can be expected that problem (1.2) possesses non-radial solutions. Such solutions were found in [13] for 2 < p < 2n n−2 and in [12] for p = 2n n−2 . Furthermore, it was proved in [4] that the maximum point of the ground state solution u p of (1.2) approaches a boundary point of ∂Ω as p → 2 * .…”

“…Such a problem has been extensively studied, see for instance [4,11,12] and [13] etc. Interesting phenomenon concerning problem (1.2) that was revealed recently includes, among other things, that the exponent α affects the critical exponent for the existence of solutions.…”

Non radial solutions for a non homogeneous Hénon equation

“…Borrowing some ideas and some results from [9], we also prove existence of non radial solutions for a range of growth of f including supercritical growth. We will find the solutions as minima on the Nehari manifold of the functional usually associated to (1). So the main points of the present paper can be summarized as follows: non homogeneous nonlinearity, supercritical growth, Nehari manifold.…”

“…In the case in which f (t) is a power, say f (t) = |t| p−2 t, the problem is known as Hénon's equation (see [10]). A seminal paper of Smets, Su and Willem [1] showed that, for large α's, there is a breaking of symmetry and a new, non radial, solution appears. After that, much work has been made to study these non radial solutions: multiplicity, shape, asymptotic behavior.…”

“…0,rad (Ω)/{0}, non-negative solution of (1), that realizes the minimum on the Nehari manifold N α,r that is:…”