On a Hénon-type problem for general domains

We consider the problem
\begin{equation}\tag{1}
\left\{
\begin{aligned}
-\mathrm{div}(a(x)\nabla u )&=   b(x)|x|^{\alpha}u^{p } && \text{in }\Omega,\\
u&=0 && \text{on }\partial\Omega,
\end{aligned}
\right.
\end{equation}
where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^{N}\), \(N\geq3,\) the functions \(a\in C^{1}(\overline{\Omega})\), \(b\in C^{1}(\overline{\Omega})\) are strictly positive on \(\overline{\Omega}\), \(p>1\) and \(\alpha\) is a positive real number.

When \(a\equiv b\equiv 1\) and \(\Omega\) is the unit ball in \(\mathbb{R}^{N}\), (1) is the well known Hénon problem.  In a classical paper [1], Ni prove that if \(p\in(1,\frac{N+2+2\alpha}{N-2})\) then the Hénon problem possesses a positive radial solution. If \( \Omega\) has no spherical symmetries the existence of a solution for \(p\in(1,\frac{N+2+2\alpha}{N-2})\) remains an open problem.

In this talk we will discuss some classical results to the problem (1) in general domains and present some concentration results for particular cases of the problem (1) when the exponent \(p\) is close, to both, the critical exponent \(\frac{n+2}{n-2}\) and the \(\alpha\)- critical exponent \(\frac{N+2+2\alpha}{N-2}\).

This is joint work with professors Angela Pistoia, Massimo Grossi, Juan Dávila and Fethi Mahmoudi

[1] W.M. Ni, A Nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. Jour. 31 (1982), 801-807.