9 problems

Let $\Omega$ be a general bounded domain, and consider the Brezis–Nirenberg problem. A bubble-tower solution is a solution with multiple bubbles concentrating at the same point. Bu…

Hajaiej's mountain pass conjecture. Equation $$ possesses a positive mountain pass solution at level $c_M>0$.

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^3$, let $\lambda_1$ be the first eigenvalue of the Dirichlet Laplacian on $\Omega$, and for $\lambda\in(0,\lambda_1)$ let…

Sharp lower-bound conjecture. The problem has no solution when $\lambda\leq\lambda^*$.

Pucci–Serrin's conjecture. The necessary condition for the existence of a solution is that $\lambda$ be larger than some positive constant.

Let $u^m_{\lambda}$ be the $m$-nodal radial solution under consideration, and let $\bar{\lambda}$ denote the limiting parameter appearing as $\lambda$ approaches the endpoint of th…

Extremal-profile conjecture. The extremals for the embedding ${\mathcal {D}}^{s,p}({\mathbb {R}}^n)\subset L^{p^*_s}({\mathbb {R}}^n)$ are of the form $V_{\varepsilon,x_0}$.

Let $\Omega\subset\mathbb{R}^3$ be the domain under consideration, let $\lambda_1$ be the first Dirichlet eigenvalue of $-\Delta$ on $\Omega$, and let $g_\lambda$ be the Robin func…

Let $\Omega$ be a general bounded domain, and let $\bar\lambda_2=\bar\lambda_2(N,\Omega)$ be a limit value of the parameter. The preceding results concern nodal radial solutions wi…