This work investigates the online machine learning problem of prediction with expert advice in an adversarial setting through numerical analysis of, and experiments with, a related partial differential equation. The problem is a repeated two-person game involving decision-making at each step informed by n$n$ experts in an adversarial environment. The continuum limit of this game over a large number of steps is a degenerate elliptic equation whose solution encodes the optimal strategies for both players. We develop numerical methods for approximating the solution of this equation in relatively high dimensions (n\leq 10$n\leq 10$) by exploiting symmetries in the equation and the solution to drastically reduce the size of the computational domain. Based on our numerical results we make a number of conjectures about the optimality of various adversarial strategies, in particular about the non-optimality of the COMB strategy.

In this article, we investigate the following non-linear Schrödinger (NLS) equation with Neumann boundary conditions:

\begin{equation*}\begin{cases}-\Delta u+ \lambda u= f(u) & {\mathrm{in}} \,~ \Omega,\\\displaystyle\frac{\partial u}{\partial \nu}=0 \, &{\mathrm{on}}\,~\partial \Omega\end{cases}\end{equation*} $$\begin{equation*}\begin{cases}-\Delta u+ \lambda u= f(u) & {\mathrm{in}} \,~ \Omega,\\\displaystyle\frac{\partial u}{\partial \nu}=0 \, &{\mathrm{on}}\,~\partial \Omega\end{cases}\end{equation*}$$

coupled with a constraint condition:

\begin{equation*}\int_{\Omega}|u|^2 dx=c,\end{equation*} $$\begin{equation*}\int_{\Omega}|u|^2 dx=c,\end{equation*}$$

where \Omega\subset \mathbb{R}^N(N\ge3)$\Omega\subset \mathbb{R}^N(N\ge3)$ denotes a smooth bounded domain, ν represents the unit outer normal vector to \partial \Omega$\partial \Omega$, c is a positive constant, and λ acts as a Lagrange multiplier. When the non-linearity f exhibits a general mass supercritical growth at infinity, we establish the existence of normalized solutions, which are not necessarily positive solutions and can be characterized as mountain pass type critical points of the associated constraint functional. Our approach provides a uniform treatment of various non-linearities, including cases such as f(u)=|u|^{p-2}u$f(u)=|u|^{p-2}u$, |u|^{q-2}u+ |u|^{p-2}u$|u|^{q-2}u+ |u|^{p-2}u$, and -|u|^{q-2}u+|u|^{p-2}u$-|u|^{q-2}u+|u|^{p-2}u$, where 2 \lt q \lt 2+\frac{4}{N} \lt p \lt 2^*$2 \lt q \lt 2+\frac{4}{N} \lt p \lt 2^*$. The result is obtained through a combination of a minimax principle with Morse index information for constrained functionals and a novel blow-up analysis for the NLS equation under Neumann boundary conditions.

In this article, we investigate necessary and sufficient conditions on the perturbation ρ for the existence of positive least energy solutions of the critical singular semilinear elliptic equation -\Delta u = \frac{|u|^{2^{*}(s)-2}}{|x|^s}u + \rho(u) $-\Delta u = \frac{|u|^{2^{*}(s)-2}}{|x|^s}u + \rho(u)$ with Dirichlet boundary condition in a bounded smooth domain in \mathbb R^n$\mathbb R^n$ containing the origin, where 2^*(s)=\frac{2(n-s)}{n-2}$2^*(s)=\frac{2(n-s)}{n-2}$, 0\leq s \lt 2 \lt n$0\leq s \lt 2 \lt n$. We show that the almost necessary and sufficient condition obtained for the case s = 0 in [1] differs conceptually when 0 \lt s \lt 2$0 \lt s \lt 2$.

In the article, we investigate Trudinger–Moser type inequalities in presence of logarithmic kernels in dimension N. A sharp threshold, depending on N, is detected for the existence of extremal functions or blow-up, where the domain is the ball or the entire space \mathbb{R}^N$\mathbb{R}^N$. We also show that the extremal functions satisfy suitable Euler–Lagrange equations. When the domain is the entire space, such equations can be derived by a N-Laplacian Schrödinger equation strongly coupled with a higher order fractional Poisson’s equation. The results extends [16] to any dimension N \geq 2$N \geq 2$.

We study the asymptotic behaviour of the least energy solutions to the following class of nonlocal Neumann problems:

\begin{align*} \begin{cases} { d(-\Delta)^{s}u+ u= \vert u\vert^{p-1}u } & \text{in } \Omega, \\ {u>0} & \text{in } \Omega, \\ { \mathcal{N}_{s}u=0 } & \text{in } \mathbb{R}^{n}\setminus \overline{\Omega}, \end{cases} \end{align*} $$\begin{align*} \begin{cases} { d(-\Delta)^{s}u+ u= \vert u\vert^{p-1}u } & \text{in } \Omega, \\ {u>0} & \text{in } \Omega, \\ { \mathcal{N}_{s}u=0 } & \text{in } \mathbb{R}^{n}\setminus \overline{\Omega}, \end{cases} \end{align*}$$

where \Omega \subset \mathbb {R}^{n}$\Omega \subset \mathbb {R}^{n}$ is a bounded domain of class C^{1,1}$C^{1,1}$, 1<p<({n+s})/({n-s}),\,n>\max \{1, 2s \}, 0<s<1, d>0$1<p<({n+s})/({n-s}),\,n>\max \{1, 2s \}, 0<s<1, d>0$ and \mathcal {N}_{s}u$\mathcal {N}_{s}u$ is the nonlocal Neumann derivative. We show that for small d,$d,$ the least energy solutions u_d$u_d$ of the above problem achieve an L^{\infty }$L^{\infty }$-bound independent of d.$d.$ Using this together with suitable L^{r}$L^{r}$-estimates on u_d,$u_d,$ we show that the least energy solution u_d$u_d$ achieves a maximum on the boundary of \Omega $\Omega$ for d sufficiently small.

We consider the radially symmetric positive solutions to quasilinear problem

\begin{equation*}-\triangle u-u\triangle u^{2}+\lambda u=f(u),\quad{\rm in} \ \mathbb{R}^{N},\end{equation*} $$\begin{equation*}-\triangle u-u\triangle u^{2}+\lambda u=f(u),\quad{\rm in} \ \mathbb{R}^{N},\end{equation*}$$

having prescribed mass \int_{\mathbb{R}^{N}}|u|^2 =a^2,$\int_{\mathbb{R}^{N}}|u|^2 =a^2,$ where a > 0 is a constant, λ appears as a Lagrange multiplier. We focus on the pure L2-supercritical case and combination case of L2-subcritical and L2-supercritical nonlinearities

\begin{equation*}f(u)=\tau |u|^{q-2}u+|u|^{p-2}u,\quad \tau \gt 0,\qquad{\rm where}\ \ 2 \lt q \lt 2+\frac{4}{N} \ {\rm and} \quad \ p \gt \bar{p},\end{equation*} $$\begin{equation*}f(u)=\tau |u|^{q-2}u+|u|^{p-2}u,\quad \tau \gt 0,\qquad{\rm where}\ \ 2 \lt q \lt 2+\frac{4}{N} \ {\rm and} \quad \ p \gt \bar{p},\end{equation*}$$

where \bar{p}:=4+\frac{4}{N}$\bar{p}:=4+\frac{4}{N}$ is the L2-critical exponent. Our work extends and develops some recent results in the literature.

In this paper, we study the singular boundary value problem

\begin{align*} \begin{cases} \Delta_\infty^h u=\lambda f(x,u,Du) \quad &\mathrm{in}\; \Omega, \\ u>0\quad &\mathrm{in}\; \Omega,\\ u=0 \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$\begin{align*} \begin{cases} \Delta_\infty^h u=\lambda f(x,u,Du) \quad &\mathrm{in}\; \Omega, \\ u>0\quad &\mathrm{in}\; \Omega,\\ u=0 \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*}$$

where \lambda>0$\lambda>0$ is a parameter, h>1$h>1$ and \Delta _\infty ^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $\Delta _\infty ^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle$ is the highly degenerate and h-homogeneous operator related to the infinity Laplacian. The nonlinear term f(x,t,p):\Omega \times (0,\infty )\times \mathbb {R}^{n}\rightarrow \mathbb {R}$f(x,t,p):\Omega \times (0,\infty )\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ is a continuous function and may exhibit singularity at t\rightarrow 0^{+}$t\rightarrow 0^{+}$. We establish the comparison principle by the double variables method for the general equation \Delta _\infty ^h u=F(x,u,Du)$\Delta _\infty ^h u=F(x,u,Du)$ under some conditions on the term F(x,t,p)$F(x,t,p)$. Then, we establish the existence of viscosity solutions to the singular boundary value problem in a bounded domain based on Perron’s method and the comparison principle. Finally, we obtain the existence result in the entire Euclidean space by the approximation procedure. In this procedure, we also establish the local Lipschitz continuity of the viscosity solution.

In this paper, we consider the closed spacelike solution to a class of Hessian quotient equations in de Sitter space. Under mild assumptions, we obtain an existence result using standard degree theory based on a priori estimates.

The paper is concerned with positive solutions to problems of the type

-\Delta_{\mathbb{B}^{N}} u - \lambda u = a(x) |u|^{p-1}\;u + f \text{ in }\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})}, $$-\Delta_{\mathbb{B}^{N}} u - \lambda u = a(x) |u|^{p-1}\;u + f \text{ in }\mathbb{B}^{N}, \quad u \in H^{1}{(\mathbb{B}^{N})},$$

\mathbb {B}^N$\mathbb {B}^N$

1< p<2^*-1:=\frac {N+2}{N-2}$1< p<2^*-1:=\frac {N+2}{N-2}$

\;\lambda < \frac {(N-1)^2}{4}$\;\lambda < \frac {(N-1)^2}{4}$

f \in H^{-1}(\mathbb {B}^{N})$f \in H^{-1}(\mathbb {B}^{N})$

f \not \equiv 0$f \not \equiv 0$

a\in L^\infty (\mathbb {B}^N)$a\in L^\infty (\mathbb {B}^N)$

\lim _{d(x, 0) \rightarrow \infty } a(x) \rightarrow 1,$\lim _{d(x, 0) \rightarrow \infty } a(x) \rightarrow 1,$

d(x,\, 0)$d(x,\, 0)$

a(x) \leq 1$a(x) \leq 1$

a(x) \geq 1$a(x) \geq 1$

\mu ( \{ x : a(x) \neq 1\}) > 0.$\mu ( \{ x : a(x) \neq 1\}) > 0.$

a(x) \equiv 1$a(x) \equiv 1$

In this paper, we consider the following non-linear system involving the fractional Laplacian0.1

\begin{equation} \left\{\begin{array}{@{}ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. \end{equation} $$\begin{equation} \left\{\begin{array}{@{}ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. \end{equation}$$

0< s<1$0< s<1$

x_n$x_n$

In the present paper we deal with a quasi-linear elliptic equation depending on a sublinear nonlinearity involving the gradient. We prove the existence of a nontrivial nodal solution employing the theory of invariant sets of descending flow together with sub-supersolution techniques, gradient regularity arguments, strong comparison principle for the p$p$-Laplace operator. The same conclusion is obtained for an eigenvalue problem under a different set of assumptions.

We consider the Dirichlet problem for p(x)$p(x)$-Laplacian equations of the form

\begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*} $$\begin{align*} -\Delta_{p(x)}u+b(x)\vert u\vert ^{p(x)-2}u=f(x,u),\quad u\in W_{0}^{1,p(x)}(\Omega). \end{align*}$$

The odd nonlinearity f(x,u)$f(x,u)$ is p(x)$p(x)$-sublinear at u=0$u=0$ but the related limit need not be uniform for x\in \Omega $x\in \Omega$. Except being subcritical, no additional assumption is imposed on f(x,u)$f(x,u)$ for |u|$|u|$ large. By applying Clark’s theorem and a truncation method, we obtain a sequence of solutions with negative energy and approaching the zero function u=0$u=0$.

For s_1,\,s_2\in (0,\,1)$s_1,\,s_2\in (0,\,1)$ and p,\,q \in (1,\, \infty )$p,\,q \in (1,\, \infty )$, we study the following nonlinear Dirichlet eigenvalue problem with parameters \alpha,\, \beta \in \mathbb {R}$\alpha,\, \beta \in \mathbb {R}$ driven by the sum of two nonlocal operators:

(-\Delta)^{s_1}_p u+(-\Delta)^{s_2}_q u=\alpha|u|^{p-2}u+\beta|u|^{q-2}u\ \text{in }\Omega, \quad u=0\ \text{in } \mathbb{R}^d \setminus \Omega, \quad \mathrm{(P)} $$(-\Delta)^{s_1}_p u+(-\Delta)^{s_2}_q u=\alpha|u|^{p-2}u+\beta|u|^{q-2}u\ \text{in }\Omega, \quad u=0\ \text{in } \mathbb{R}^d \setminus \Omega, \quad \mathrm{(P)}$$

\Omega \subset \mathbb {R}^d$\Omega \subset \mathbb {R}^d$

\alpha,\,\beta$\alpha,\,\beta$

(\alpha,\, \beta )$(\alpha,\, \beta )$

p$p$

q$q$

Let \Omega \subset \mathbb {R}^N$\Omega \subset \mathbb {R}^N$ (N\geq 3$N\geq 3$) be a C^2$C^2$ bounded domain and \Sigma \subset \partial \Omega$\Sigma \subset \partial \Omega$ be a C^2$C^2$ compact submanifold without boundary, of dimension k$k$, 0\leq k \leq N-1$0\leq k \leq N-1$. We assume that \Sigma = \{0\}$\Sigma = \{0\}$ if k = 0$k = 0$ and \Sigma =\partial \Omega$\Sigma =\partial \Omega$ if k=N-1$k=N-1$. Let d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ and L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$, where \mu \in {\mathbb {R}}$\mu \in {\mathbb {R}}$. We study boundary value problems (P_\pm$P_\pm$) -{L_\mu} u \pm |u|^{p-1}u = 0$-{L_\mu} u \pm |u|^{p-1}u = 0$ in \Omega$\Omega$ and \mathrm {tr}_{\mu,\Sigma}(u)=\nu$\mathrm {tr}_{\mu,\Sigma}(u)=\nu$ on \partial \Omega$\partial \Omega$, where p>1$p>1$, \nu$\nu$ is a given measure on \partial \Omega$\partial \Omega$ and \mathrm {tr}_{\mu,\Sigma}(u)$\mathrm {tr}_{\mu,\Sigma}(u)$ denotes the boundary trace of u$u$ associated to L_\mu$L_\mu$. Different critical exponents for the existence of a solution to (P_\pm$P_\pm$) appear according to concentration of \nu$\nu$. The solvability for problem (P_+$P_+$) was proved in [3, 29] in subcritical ranges for p$p$, namely for p$p$ smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of -L_\mu$-L_\mu$, we provide conditions on \nu$\nu$ expressed in terms of capacities for the existence of a (unique) solution to (P_+$P_+$) in supercritical ranges for p$p$, i.e. for p$p$ equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to (P_-$P_-$) under a smallness assumption on \nu$\nu$.

We study the possible singularities of an m-subharmonic function \varphi $\varphi$ along a complex submanifold V of a compact Kähler manifold, finding a maximal rate of growth for \varphi $\varphi$ which depends only on m and k, the codimension of V. When k < m$k < m$, we show that \varphi $\varphi$ has at worst log poles along V, and that the strength of these poles is moreover constant along V. This can be thought of as an analogue of Siu’s theorem.

We will present the proof of existence and uniqueness of renormalized solutions to a broad family of strongly non-linear elliptic equations with lower order terms and data of low integrability. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak–Orlicz spaces. The setting considered in this paper generalized known results in the variable exponents, anisotropic polynomial, double phase and classical Orlicz setting.

In this paper, we are concerned with the non-existence of positive solutions of a Hartree–Poisson system:

\begin{equation*}\left\{\begin{aligned}&-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\&-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n},\end{aligned}\right.\end{equation*} $$\begin{equation*}\left\{\begin{aligned}&-\Delta u=\left(\frac{1}{|x|^{n-2}}\ast v^p\right)v^{p-1},\quad u \gt 0\ \text{in} \ \mathbb{R}^{n},\\&-\Delta v=\left(\frac{1}{|x|^{n-2}}\ast u^q\right)u^{q-1},\quad v \gt 0\ \text{in} \ \mathbb{R}^{n},\end{aligned}\right.\end{equation*}$$

n \geq3$n \geq3$

\min\{p,q\} \gt 1$\min\{p,q\} \gt 1$

The paper deals with the existence of positive solutions with prescribed L^2$L^2$ norm for the Schrödinger equation

-\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\quad u\in H^1_0(\Omega),\quad\int_\Omega u^2{\rm d}\,x=\rho^2,\quad\lambda\in\mathbb{R}, $$-\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\quad u\in H^1_0(\Omega),\quad\int_\Omega u^2{\rm d}\,x=\rho^2,\quad\lambda\in\mathbb{R},$$

\Omega =\mathbb {R}^N$\Omega =\mathbb {R}^N$

\mathbb {R}^N\setminus \Omega$\mathbb {R}^N\setminus \Omega$

\rho >0$\rho >0$

V\ge 0$V\ge 0$

V\equiv 0$V\equiv 0$

p\in (2,2+\frac 4 N)$p\in (2,2+\frac 4 N)$

\bar u$\bar u$

V$V$

\|V\|_{L^q}$\|V\|_{L^q}$

q\ge \frac N2$q\ge \frac N2$

q>1$q>1$

N=2$N=2$

V$V$

\mathbb {R}^N\setminus \Omega$\mathbb {R}^N\setminus \Omega$

\bar u$\bar u$

V\equiv 0$V\equiv 0$

\Omega =\mathbb {R}^N$\Omega =\mathbb {R}^N$

We consider the non-linear Schrödinger equation(Pμ)

\begin{equation*}\begin{array}{lc}-\Delta u + V(x) u = \mu f(u) + |u|^{2^*-2}u, &\end{array}\end{equation*} $$\begin{equation*}\begin{array}{lc}-\Delta u + V(x) u = \mu f(u) + |u|^{2^*-2}u, &\end{array}\end{equation*}$$

\mathbb{R}^N$\mathbb{R}^N$

N\geq3$N\geq3$

f(s)/s$f(s)/s$

We establish a priori bounds, existence and qualitative behaviour of positive radial solutions in annuli for a class of nonlinear systems driven by Pucci extremal operators and Lane-Emden coupling in the superlinear regime. Our approach is purely nonvariational. It is based on the shooting method, energy functionals, spectral properties, and on a suitable criteria for locating critical points in annular domains through the moving planes method that we also prove.