29 problems

Let $a\in\mathcal{C}([0,L])$ satisfy the sign condition that it is positive on $n$ separated intervals $I_i^+=(\sigma_i,\tau_i)$ and negative on the intervening intervals, together…

Let $\Omega\subset\mathbb R^3$ be a bounded set, let $q\in(2,4)$, and consider the Dirichlet problem … with parameter $\lambda>0$. Brezis–Nirenberg conjecture. There exists a const…

For $\varepsilon\in(0,1)$, consider problem … possesses at least 5 positive solutions whenever $\lambda<\lambda_t$, and … The conjecture is based on bifurcation diagrams and comput…

Let $a(x)=a_{\kappa,\varepsilon}(x)$ with $\kappa\geq1$ and $\varepsilon<1$ arbitrarily close to $1$. Arbitrarily large multiplicity conjecture. By taking $\kappa$ sufficiently lar…

Let problem … possesses $2^{\kappa+1}-1$ positive solutions. The conjecture is motivated by the almost-everywhere convergence of positive solutions to zero on the zero set of the w…

Kouchnirenko's conjecture. The bound $s_1\cdots s_n$ is sharp for the number of positive solutions of $C\cdot x^A=0$ when $C$ has real coefficients.

The paper considers the more general problems referred to as … , in the one-dimensional case $N=1$. Conjecture. A similar strategy to the one used for the ball case with $N=1$ and…

Consider the Cauchy problem studied in the paper, with parameters $m$, $p$, and $\sigma$, and let $L$ be the quantity defined by the equation referenced as $(\mathrm{const})$. A so…

The polyharmonic heat equation is obtained by replacing the biharmonic operator with a higher-order polyharmonic operator, and its linear counterpart is the corresponding homogeneo…

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…

Let $K_4$ be the constant appearing in Theorem 3, and let $σ$ be the parameter used there. Threshold constant conjecture. In Theorem 3, $K_4$ is larger than … The quantity on the r…

Let $\lambda$ be the parameter in the indefinite parameter-dependent Neumann problem, and let $\mu^{*}_0(\lambda)$, $\mu^{*}_1(\lambda)$, $\mu^{**}_0(\lambda)$, and…

Let $a\in L^{\infty}(0,T)$ satisfy the assumptions in (hp-a), and let $\gamma\in\mathbb{R}\setminus\{-1,0,1\}$. Consider the Neumann and periodic boundary value problems associated…

Takáč's conjecture. The assumption that $\partial\Omega$ is connected can be omitted for sufficiently regular domains.

Generic asymmetric bifurcation conjecture. Generically, the set of positive solutions consists of $C^+$ together with $n$ supercritical folds $D_j$, $j\in\{1,\ldots,n\}$, and, as…

Gómez-Reñasco–López-Gómez conjecture. There exists $lambda_c<pi^2$ such that, for every $lambda<lambda_c$, the problem admits at least $2^{n+1}-1$ positive solutions.

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

All-time positivity conjecture. For every feedback gain satisfying

Let $N\geq 2$, let $\Omega\subset\mathbb{R}^N$ be a convex domain, and consider the Lane–Emden Dirichlet problem … Here $p$ is such that a solution exists. Gidas–Ni–Nirenberg uniqu…

Consider a superlinear indefinite problem with homogeneous Dirichlet boundary conditions, parameter $\lambda$ sufficiently negative, and a weight function $a(t)$ having $n e 0$ com…

Bihan's conjecture. There is an affine relation basis for $\mathcal A$ whose coefficients all have absolute value at most $2$.

Let $\varphi_3$ be the function used in the paper, let $u$ be the independent variable, and let $g$ be the function associated with a pair $(\varphi,f)$. A pair is of Type IV-…

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

Consider problem … left{left((1+2nu)^{-1/2}zmu^1,(1+2nu)^{-1/2}zmu^1right):mu0right} … . The theorem preceding this conjecture proves the analogous assertion for positive ground st…

For a power-type nonlinear elliptic equation, consider positive solutions in a convex domain. Uniqueness conjecture. It has been conjectured that, for some power-type nonlinearitie…