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A numerically based investigation on the symmetry breaking and asymptotic behavior of the ground states to the \(p\)-Hénon equation. (English) Zbl 1220.58004

This paper studies some symmetry and asymptotic properties of solutions of the so-called \(p\)-Hénon equation. That is, a quasi-linear elliptic boundary value problem defined by the equation
\[ \Delta_p u+|x|^r u^{q-2}u=0, \quad x \in \Omega \subset \mathbb R^n,\;u\in W_0^{1,p}(\Omega), \tag \(*\) \]
where \(\Delta_p\) is the Laplacian operator given by \(\Delta_p u= \text{div}(|\nabla u|^{p-2}\nabla u)\) \((p>1)\), \(\Omega\) is a bounded open domain, \(|x|\) the Euclidean norm of \(x\), \(p<q<p^*\) (\(p^*\) is the Sobolev exponent), and \(r\geq0\). In the original Hénon equation \((p=2\), \(r=0)\), if \(\Omega\) is the unit ball in \(\mathbb R^n\) then the positive ground state of \((*)\) is radial. However, under different conditions on the parameters of \((*)\), symmetry breaking phenomena have been detected numerically and confirmed theoretically in the last years.
In the paper under consideration, the authors present the results of some numerical experiments for the \(p\)-Hénon equation not only over the standard unit ball domain but also over special non-radial domains, namely hypercubic domains \((-a,a)^n\), \(a>0\). A numerical computation of positive states is carried out taking into account that they are critical points of some energy functional \(J=J(u)\), \(u\in W_0^{1,p}(\Omega)\), that are obtained here by a minimax method especially adapted to the instabilities of the problem. A flow chart of the algorithm is given and the results of a number of numerical experiments on the unit disk and the square domain in \(\mathbb R^2\) are presented. On the basis of these results, some analytical properties on ground states are verified and others are conjectured in the final section.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E30 Variational principles in infinite-dimensional spaces
37M05 Simulation of dynamical systems
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
35J65 Nonlinear boundary value problems for linear elliptic equations
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