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### Open Access Journals

PROC

We prove existence of a counterpart of the Talenti solution in the
critical semilinear problem −$\Delatu = f(u)$ in $\mathbb{R}^N$, $N$ > 3, where the nonlinearity $f$ oscillates about the critical "stem" $f(s) = s^((N+2)/(N − 2))$ : specifically, $f(2^((N − 2)/2j)s) = 2^(( N + 2)/ 2 j)f(s)$ for all $j \in \mathbb{Z}$, $s \in \mathb{R}.

PROC

While the critical nonlinearity $\int |u|^2^$* for the Sobolev space $H^1$ in
dimension N > 2 lacks weak continuity at any point, Trudinger-Moser nonlinearity
$\int e^(4\piu^2)$ in dimension

*N*= 2 is weakly continuous at any point except zero. In the former case the lack of weak continuity can be attributed to invariance with respect to actions of translations and dilations. The Sobolev space $H^1_0$f the unit disk $\mathbb{D}\subset\mathbb{R}^2$ possesses transformations analogous to translations (Möbius transformations) and nonlinear dilations $r\to r^s$. We present improvements of the Trudinger-Moser inequality with nonlinearities sharper than $\int e^(4\piu^2)$ that lack weak continuity at any point and possess (separately), translation and dilation invariance. We show, however, that no nonlinearity of the form $\int F(|x|, u(x))dx$ is both dilation- and Möbius shift-invariant. The paper also gives a new, very short proof of the conformal-invariant Trudinger-Moser inequality obtained recently by Mancini and Sandeep [10] and of a sharper version of Onofri-type inequality of Beckner [4].
PROC

The mountain pass statement for semilinear elliptic equations −$\Deltau = f(u)$ in $\mathbb{R}^N, N > 2$, with a critical exponent nonlinearity, namely $C_1 <= f(s)s/|s|^2^* <= C_2$, satisfies the (PS)$_c$ condition provided that the critical sequences are bounded and that the nonlinearity either has log-periodic oscillations or dominates its asymptotic values (relative to $|s|^2^*$ ) at zero and at infinity.

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