Lorentz Spaces and Nonlinear Elliptic Systems

Abstract

In this paper we study the following system of semilinear elliptic equations:

$$ \left\{ {\begin{array}{*{20}c} \begin{gathered} - \Delta u \hfill \\ - \Delta v \hfill \\ u = 0 \hfill \\ \end{gathered} & {\begin{array}{*{20}c} \begin{gathered} = \hfill \\ = \hfill \\ and \hfill \\ \end{gathered} & \begin{gathered} g\left( v \right), \hfill \\ f\left( u \right), \hfill \\ v = 0, \hfill \\ \end{gathered} & \begin{gathered} in \hfill \\ in \hfill \\ in \hfill \\ \end{gathered} & \begin{gathered} \Omega , \hfill \\ \Omega , \hfill \\ \partial \Omega , \hfill \\ \end{gathered} \\ \end{array} } \\ \end{array} } \right. $$

where Ω is a bounded domain in $\mathbb{R}$ ^N, and f, g ∈ C($\mathbb{R}$) are superlinear nonlinearities. The natural framework for such systems are Sobolev spaces, which give in most cases an adequate answer concerning the “maximal growth” on f and g such that the problem can be treated variationally. However, in some limiting cases the Sobolev imbeddings are not sufficiently fine to capture the true maximal growth. We consider two cases, in which working in Lorentz spaces gives better results.

a) N ≥ 3: we assume that g(s) = s ^p, with $p + 1 = \frac{N} {{N - 2}}$ , which means that p lies on the asymptote of the so-called “critical hyperbola”, see below. In the Sobolev space setting there exist several different variational formulations, which (surprisingly) yield different maximal growths for f. We show that this is due to the non-optimality of the Sobolev embeddings theorems; indeed, by using instead a Lorentz space setting (which gives optimal embeddings), the different maximal growths disappear: we then infer that the critical growth for f is $f\left( u \right) \sim e^{\left| u \right|^{N/\left( {N - 2} \right)} }$.

b) N = 2: in two dimensions the maximal growth is of exponential type, given by Trudinger-Moser type inequalities. Using the Lorentz space setting, we show that for $f \sim e^{\left| s \right|^p }$ and $g \sim e^{\left| s \right|^q }$ we have maximal (critical) growth for

$$ \frac{1} {p} + \frac{1} {q} = 1, $$

which is an analogue of the critical hyperbola in dimensions N ≥ 3.