24 problems

Liouville nonexistence conjecture. The nonexistence result for (1.3) should hold whenever

Let $\Omega$ be an epigraph with aperture exceeding $\pi$, and consider the solutions of the semilinear elliptic problem constructed in the paper. A solution is strictly unstable i…

Lions's conjecture. Neither convexity of the domain $\Omega$ nor the technical condition

Conjecture A. The solution $u$ has a unique critical point in $\Omega$.

Consider a semilinear elliptic equation with a pure power nonlinearity of exponent $p>1$ on a domain $D$, and suppose solutions are studied through probabilistic representations in…

Exponential degeneration conjecture. For uniqueness, the condition

Dini condition conjecture. The Dini condition is also necessary for the uniqueness of the large solution.

Let $p\in(2,2^{*})$, where $2^{*}$ is the relevant critical Sobolev exponent, and let $g_z$ denote the function considered in the preceding analysis. Domain and exponent dependence…

Case VIII bifurcation conjecture. The branches of solutions to … .

S-shape conjecture. The interval $(\underline{\lambda},\overline{\lambda})$ should be expandable so that problem … should be removable.

Let $\overline{\Theta}$ be the angle from Corollary B(iv), and let $\Theta^*$ be the angle from Theorem C, for the Emden–Fowler problem on a spherical cap. Critical-exponent limit…

Let $N\ge 3$, let $p>p_{\rm S}$, and let $\Theta(\Gamma)$ denote the relevant solution-branch parameter for the Emden–Fowler equation … has no regular solution for…

Conjecture. If $n\leq 9$, then $u$ is bounded.

Let $n\geq 3$, let $\mathbb{R}^n_+=\{x=(x',x_n)\in\mathbb{R}^n:x_n>0\}$, and let $\mathbb{S}^{n-1}_+=\{\theta\in\mathbb{S}^{n-1}:\theta_n>0\}$. For a solution $u$ of the critical D…

Let $N\geqslant 3$, let $i\geqslant 2$, and assume $2+oubles$?

Let $f$ satisfy the three hypotheses stated in the source, let $u_0>0$ satisfy $f(u_0)=u_0$, and consider the Neumann problem on the ball $B$...

de Figueiredo–Lions–Nussbaum conjecture. The equation has a classical nontrivial positive solution. The conjecture asks whether the existence conclusion known under additional tech…

Closed-geodesic balanced-pattern conjecture. A possible balanced pattern may occur for spikes tightly aligned near a closed geodesic of $\partial\Omega$.

Alternate-sign peak alignment conjectures. (C1) There exists a solution with alternate-sign peaks aligned on an interior straight line intersecting $\partial\Omega$ orthogonally. (…

Atar–Athreya–Chen conjecture. The functions $H_D^\varphi$ and $H_D$ are not proportional when $\varphi(t)=t^p$ with $0\leq p<1$.

Let $L$ range from $L_{*}$ to $+infty$, and consider the least energy solutions discussed in the paper. Continuity conjecture. The least energy solutions form a continuous family a…

Lower-estimate conjecture. The function $u_{\infty\delta_y}$ always satisfies estimate.

Let $\Omega$ be the Lipschitz domain under consideration, let $y\in\partial\Omega$, and let $u_{k\delta_y}$ denote the positive solution associated with boundary measure…

Let $\Omega$ be the domain in the semilinear elliptic problem considered in the paper, let $\delta(x)$ denote the distance from $x$ to $\partial\Omega$, and let $U$ be an $XXL$-sol…