Abstract
We study the following Choquard type equation in the whole plane (C)-Δu+V(x)u=(I2∗F(x,u))f(x,u),x∈R2\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} (C)\\quad -\\Delta u+V(x)u=(I_2*F(x,u))f(x,u),\\quad x\\in \\mathbb {R}^2 \\end{aligned}$$\\end{document}where 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).
Highlights
Consider the following class of nonlocal equations− u + V (x)u = (Iα ∗ F(u)) f (u), x ∈ RN (1.1)Communicated by A
Since we look for positive solutions, we may assume f (s) = 0 for s ≤ 0
Our approach here is different, from one side we look for a proper function space setting in which such a natural constraint turns out to be automatically satisfied and on the other side, we wonder if this can be done by allowing the nonlinearity to exhibit exponential growth which is what we expect in dimension two, since the seminal work of Pohozaev [31] and Trudinger [36]
Summary
We prove that the Sobolev space H 1 with a logarithmic weight on the L2 mass term of the norm gives the proper function space setting in which the energy in well defined up to the natural exponential growth in the nonlinearity. Our argument throws light on the fact that, roughly speaking, as concentration phenomena in the Moser functional are controlled by the L2 norm of the gradient whereas vanishing phenomena are controlled by the L2 norm, here we prove that a suitable logarithmic weight in the L2 component of the H 1 norm is enough to obtain a functional inequality which at the end yields a natural function space framework where to set up the problem We think this result is of independent interest and that could be useful elsewhere.
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