The logarithmic convolution energy sup over E of the double integral of log(1/|x-y|) G(u(x))G(u(y)) is finite exactly when G is subcritical with exponent gamma >= 1, and maximizers exist in the strictly subcritical case.
A sharp threshold for Trudinger-Moser type inequalities with logarithmic kernels in dimension N
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In the paper we investigate Trudinger-Moser type inequalities in presence of logarithmic kernels in dimension N. A sharp threshold, depending on N, is detected for the existence of estremal functions or blow-up, where the domain is the ball or the entire space. We also show that the extremal functions satisfy suitable Euler-Lagrange equations. When the domain is the entire space, such equation can be derived by N-Laplacian Schrodinger equation strongly coupled with a higher order fractional Poisson's equation. The results extends [16] to any dimension N bigger than 2.
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Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials
The logarithmic convolution energy sup over E of the double integral of log(1/|x-y|) G(u(x))G(u(y)) is finite exactly when G is subcritical with exponent gamma >= 1, and maximizers exist in the strictly subcritical case.