Schrödinger–Newton equations in dimension two via a Pohozaev–Trudinger log-weighted inequality

We study the following Choquard type equation in the whole plane $$\begin{aligned} (C)\quad -\Delta u+V(x)u=(I_2*F(x,u))f(x,u),\quad x\in \mathbb {R}^2 \end{aligned}$$ ( C ) - Δ u + V ( x ) u = ( I 2 ∗ F ( x , u ) ) f ( x , u ) , x ∈ R 2 where $$I_2$$ I 2 is the Newton logarithmic kernel, V is a bounded Schrödinger potential and the nonlinearity f(x, u), whose primitive in u vanishing at zero is F(x, u), exhibits the highest possible growth which is of exponential type. The competition between the logarithmic kernel and the exponential nonlinearity demands for new tools. A proper function space setting is provided by a new weighted version of the Pohozaev–Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to (C).

Local Muckenhoupt class for variable exponents

This work extends the theory of Rychkov, who developed the theory of $A_{p}^{\mathrm{loc}}$ A p loc weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class $A_{p(\cdot )}^{\mathrm{loc}}$ A p ( ⋅ ) loc is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained.

A weighted inequality for potential type operators

We establish a weighted inequality for fractional maximal and convolution type operators, between weak Lebesgue spaces and Wiener amalgam type spaces on {\mathbb{R}} endowed with a measure which needs not to be doubling.