Positive solutions of elliptic equations with a critical oscillatory nonlinearity
Kyril Tintarev
Conference Publications 2007, 2007(Special): 974-981 doi: 10.3934/proc.2007.2007.974
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}.
keywords: Talenti solution concentration compactness. critical exponent Semilinear elliptic equations
Is the Trudinger-Moser nonlinearity a true critical nonlinearity?
Kyril Tintarev
Conference Publications 2011, 2011(Special): 1378-1384 doi: 10.3934/proc.2011.2011.1378
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].
keywords: Poincare disk Palais-Smale sequences Hardy inequalities. weak convergence Trudinger-Moser inequality elliptic problems in critical dimension concentration compactness hyperbolic space
Mountain pass solutions to semilinear problems with critical nonlinearity
Ian Schindler Kyril Tintarev
Conference Publications 2007, 2007(Special): 912-919 doi: 10.3934/proc.2007.2007.912
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.
keywords: critical exponent Talenti solution Semilinear elliptic equations concentration compactness.

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