9 problems

Li–Shafrir quantization conjecture. Each concentration weight satisfies

Let $C_{\text{\tiny{BE}}}$ denote the best Bianchi–Egnell constant for the one-dimensional fractional Sobolev inequality, and let $C_{\text{\tiny{BE}}}^{\text{\tiny{loc}}}$ denote…

Strictness characterization.

Lane–Emden concentration conjecture. As $\beta\to\infty$,

Let $d=1$ and $s\in(0,\frac{1}{2})$. For $f\in\dot{H}^s(\mathbb{R})\setminus\mathcal B$, let $\mathcal Q(f)$ denote the Bianchi–Egnell quotient, let $c_{BE}(s)$ be its infimum, let…

Let $\lambda_*$ and $\beta_*$ be limiting parameters for the semilinear elliptic problem, with $\lambda_*>\Lambda_1$ and $\beta_*\leq1$. The alternatives (iii)–(vi) refer to the co…

Let $\Lambda^*<\lambda_*<\Lambda_1$ and set $\beta_*=1$. Let $S_{1,\lambda,\beta}$ denote the class of radial sign-changing solutions with the index specified in the paper, and let…

Consider sign-changing radial solutions of the semilinear elliptic problem with perturbation parameter $\beta$, in the regime $\beta\geq3/2$. Concentrating solutions are solution s…

The problem concerns sign-changing radial solutions of the semilinear elliptic equation studied in the paper, with perturbation $h(t)=t e^{|t|^{\beta}}$ and $\beta\in(0,2)$. In the…