We consider the drifting p-Laplace operator

\begin{equation*}\Delta_{p,v}u=e^{-v} \text{div}\,(e^v |\nabla u|^{p-2}\nabla u)\end{equation*}

and discuss generalized weighted Hardy-type inequalities associated with the measure $d\mu=e^{v(x)}dx$. As an application, we obtain several Liouville-type results for positive solutions of the non-linear elliptic problem with singular lower order term

\begin{equation*}-\Delta_{p,v} u\ge c(x) u^{p-1}+B \frac{|\nabla u|^p}{u} \quad \text{in}\ \Omega,\end{equation*}

where Ω is a bounded or an unbounded exterior domain in ${\mathbb{R}}^N$, $N \gt p \gt 1$, $B+p-1 \gt 0$, as well as of the non-autonomous quasilinear elliptic problem

\begin{equation*}-\Delta_{p,v} u+b(x)|\nabla u|^{p-1} \ge c(x) u^{p-1}\quad \text{in}\ \Omega,\end{equation*}

with general weights $b\ge0$ and c > 0. Liouville-type results are also discussed for a class of higher order differential equations.

In this paper, we study the Cauchy problem for pseudo-parabolic equations with a logarithmic nonlinearity. After establishing the existence and uniqueness of weak solutions within a suitable functional framework, we investigate several qualitative properties, including the asymptotic behaviour and blow-up of solutions as $t\to +\infty$. Moreover, when the initial data are close to a Gaussian function, we prove that these weak solutions exhibit either super-exponential growth or super-exponential decay.

Here we consider the following fractional Hamiltonian system

\begin{equation*}\begin{cases}\begin{aligned}(-\Delta)^{s} u&=H_v(u,v) \;\;&&\text{in}~\Omega,\\(-\Delta)^{s} v&=H_u(u,v) &&\text{in}~\Omega,\\u &= v = 0 &&\text{in} ~ \mathbb{R}^N\setminus\Omega,\end{aligned}\end{cases}\end{equation*}

where $s\in (0,1)$, $N \gt 2s$, $H \in C^1(\mathbb{R}^2, \mathbb{R})$, and $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain. To apply the variational method for this problem, the key question is to find a suitable functional setting. Instead of usual fractional Sobolev spaces, we use the solution space of $(-\Delta)^{s}u=f\in L^r(\Omega)$ for $r\ge 1$, for which we show the (compact) embedding properties. When H has subcritical and superlinear growth, we construct two frameworks, respectively with the interpolation space method and the dual method, to show the existence of nontrivial solution. As byproduct, we revisit the fractional Lane–Emden system, i.e. $H(u, v)=\frac{1}{p+1}|u|^{p+1}+\frac{1}{q+1}|v|^{q+1}$, and consider the existence, uniqueness of (radial) positive solutions under subcritical assumption.

This paper is the latter part of a series of our studies on the concentration and oscillation analysis of semilinear elliptic equations with exponential growth $e^{u^p}$. In the first one [17], we completed the concentration analysis of blow-up positive solutions in the supercritical case p > 2 via a scaling approach. As a result, we detected infinite sequences of concentrating parts with precise quantification. In the present paper, we proceed to our second aim, the oscillation analysis. Especially, we deduce an infinite oscillation estimate directly from the previous infinite concentration ones. This allows us to investigate intersection properties between blow-up solutions and singular functions. Consequently, we show that the intersection number between blow-up and singular solutions diverges to infinity. This leads to a proof of infinite oscillations of bifurcation diagrams, which ensures the existence of infinitely many solutions. Finally, we also remark on infinite concentration and oscillation phenomena in the limit cases $p\to2^+$ and $p\to \infty$.

We focus on the existence of ground state solutions to the following class of elliptic equations

\begin{equation*}-\Delta_{\mathbb{B}^N} u+V(x)u=f(u) \quad \hbox{in} \quad \mathbb{B}^N,\end{equation*}

where $\mathbb{B}^N$ is the disc model of the Hyperbolic space and $\Delta_{\mathbb{B}^N}$ denotes the Laplace–Beltrami operator with $N \geq 2$, $V:\mathbb{B}^N \to \mathbb{R}$ and $f:\mathbb{R} \to \mathbb{R}$ are continuous functions that satisfy some technical conditions. With different types of the potential V, by introducing some new tricks handling the hurdle that the Hyperbolic space is not a compact manifold, we are able to obtain at least a positive ground state solution using variational methods.

As some applications for the methods adopted above, we derive the existence of normalized solutions to the elliptic problems

\begin{equation*}\left\{\begin{aligned}&-\Delta_{\mathbb{B}^N} u+V(x)u=\mu u+f(u) \quad \hbox{in} \quad \mathbb{B}^N,\\&\int_{\mathbb{B}^N}|u|^{2}\,dV_{{\mathbb{B}^N}}=a^{2},\end{aligned}\right.\end{equation*}

where a > 0, $\mu\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and f is a continuous function that fulfils the L2-subcritical or L2-supercritical growth. We do believe that it seems the first results to deal with normalized solutions for the Schrödinger equations in the Hyperbolic space.

We investigate radial and non-radial solutions to a class of (p, q)-Laplace equations involving weights. More precisely, we obtain existence and multiplicity results for nontrivial nonnegative radial and non-radial solutions, which extend results in the literature. Moreover, we study the non-radiality of minimizers in Hénon type (p, q)-Laplace problems and symmetry-breaking phenomena.

Dedicated to Professor Pavel Drábek on the occasion of his seventieth birthday

We establish the higher fractional differentiability for the minimizers of non-autonomous integral functionals of the form

\begin{align*}\mathcal{F}(u,\Omega):=\int_\Omega \left[ f(x,Du)- g \cdot u \right] dx ,\end{align*}

under (p, q)-growth conditions. Besides a suitable differentiability assumption on the partial map $x \mapsto D_\xi f(x,\xi)$, we do not need to assume any differentiability assumption on the function g.

In this paper, we deal with the following nonlinear time-harmonic Maxwell’s equations

\begin{equation*}\nonumber\begin{cases} \nabla \times(\nabla \times u_1)=\mu_1|u_1|^4u_1+\beta|u_1||u_2|^3u_1 & \text {in } \mathbb{R}^3, \\ \nabla \times(\nabla \times u_2)=\mu_2|u_2|^4u_2+\beta|u_1|^3|u_2|u_2 & \text {in } \mathbb{R}^3, \end{cases}\end{equation*}

$\nabla\times$

$\mathbb{R}^3$

$\mu_1,\mu_2 \gt 0$

$\beta\in\mathbb{R}\backslash\{0\}$

$\beta\in\mathbb{R}\backslash\{0\}$

We consider the two-dimensional nonlinear Schrödinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide proof by employing Kato’s method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance.

In this paper, we study the following high-order elliptic equation involving the GJMS operator:

\begin{align*}\alpha P_{\mathbb{S}^n}v_{\alpha}+2Q_{g_{\mathbb{S}^n}}=2Q_{g_{\mathbb{S}^n}}e^{nv_{\alpha}}.\end{align*}

We establish that if α > 1 and $n\geq3$ or if $\alpha\in (1-\epsilon_0, 1)$ with $n=2m\geq4$, then $v_{\alpha}\equiv0$. As an application, we present a new proof of the classical Beckner inequality.

In this article, we deal with non-existence results, i.e., Liouville type results, for positive radial solutions of quasilinear elliptic equations with weights both in the entire $\mathbb R^N$ and in a ball, in the latter case under Dirichlet boundary conditions. The presence of weights, possibly singular or degenerate, makes the study fairly delicate. The proofs use a Pohozaev type identity combined with an accurate qualitative analysis of solutions. In the last part of the article, a non-existence theorem is proved for a Dirichlet problem with a convection term.

We explore the regularity theory of optimal transport maps for costs satisfying a Ma–Trudinger–Wang condition, by viewing the graphs of the transport maps as maximal Lagrangian surfaces with respect to an appropriate pseudo-Riemannian metric on the product space. We recover the local regularity theory in two-dimensional manifolds.

This paper is focused on the existence and uniqueness of nonconstant steady states in a reaction–diffusion–ODE system, which models the predator–prey interaction with Holling-II functional response. Firstly, we aim to study the occurrence of regular stationary solutions through the application of bifurcation theory. Subsequently, by a generalized mountain pass lemma, we successfully demonstrate the existence of steady states with jump discontinuity. Furthermore, the structure of stationary solutions within a one-dimensional domain is investigated and a variety of steady-state solutions are built, which may exhibit monotonicity or symmetry. In the end, we create heterogeneous equilibrium states close to a constant equilibrium state using bifurcation theory and examine their stability.

In this article, we study the effect of the Hardy potential on existence, uniqueness, and optimal summability of solutions of the mixed local–nonlocal elliptic problem

\begin{equation*}-\Delta u + (-\Delta)^s u - \gamma \frac{u}{|x|^2}=f \,\text{in } \Omega, \ u=0 \,\text{in } {\mathbb R}^n \setminus \Omega,\end{equation*}

${\mathbb R}^n$

This article focuses on two kinds of generalized special Lagrangian type equations. We investigate the Dirichlet problem for these equations with supercritical phase and critical phase in $\mathbb {R}^n$, deriving the a priori estimates and establishing the existence under the assumption of a subsolution. Furthermore, we also consider the corresponding special Lagrangian curvature type equations with supercritical phase and critical phase.

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 $n$ 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 ($n\leq 10$) 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*}

coupled with a constraint condition:

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

where $\Omega\subset \mathbb{R}^N(N\ge3)$ denotes a smooth bounded domain, ν represents the unit outer normal vector to $\partial \Omega$, 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 $f(u)=|u|^{p-2}u$, $|u|^{q-2}u+ |u|^{p-2}u$, and $-|u|^{q-2}u+|u|^{p-2}u$, where $2 \lt q \lt 2+\frac{4}{N} \lt p \lt 2^*$. 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.

Let $n\ge 2$ and $\mathcal {L}=-\mathrm {div}(A\nabla \cdot )$ be an elliptic operator on $\mathbb {R}^n$. Given an exterior Lipschitz domain $\Omega $, let $\mathcal {L}_D$ be the elliptic operator $\mathcal {L}$ on $\Omega $ subject to the Dirichlet boundary condition. Previously, it was known that the Riesz transform $\nabla \mathcal {L}_D^{-1/2}$ is not bounded for $p>2$ and $p\ge n$, even if $\mathcal {L}=\Delta $ is the Laplace operator and $\Omega $ is a domain outside a ball. Suppose that A are CMO coefficients or VMO coefficients satisfying certain perturbation property, and $\partial \Omega $ is $C^1$. We prove that for $p>2$ and $p\in [n,\infty )$, it holds that

$$ \begin{align*}\inf_{\phi\in\mathcal{K}_p(\mathcal{L}_D^{1/2})}\left\|\nabla (f-\phi)\right\|_{L^p(\Omega)}\sim \left\|\mathcal{L}^{1/2}_D f\right\|_{L^p(\Omega)} \end{align*} $$

for $f\in \mathring {W}^{1,p}(\Omega )$. Here, $\mathcal {K}_p(\mathcal {L}_D^{1/2})$ is the kernel of $\mathcal {L}_D^{1/2}$ in $\mathring {W}^{1,p}(\Omega )$, which coincides with $\tilde {\mathcal {A}}^p_0(\Omega ):=\{f\in \mathring {W}^{1,p}(\Omega ):\ \mathcal {L}_Df=0\}$ and is a one-dimensional subspace. As an application, we provide a substitution of $L^p$-boundedness of $\sqrt {t}\nabla e^{-t\mathcal {L}_D}$ which is uniform in t for $p\ge n$ and $p>2$.

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) $ with Dirichlet boundary condition in a bounded smooth domain in $\mathbb R^n$ containing the origin, where $2^*(s)=\frac{2(n-s)}{n-2}$, $0\leq s \lt 2 \lt n$. We show that the almost necessary and sufficient condition obtained for the case s = 0 in [1] differs conceptually when $0 \lt s \lt 2$.

In this work, we consider a class of uniformly elliptic operators with a nonlocal term and mixed boundary conditions in bounded domains. We establish the existence of a principal eigenvalue and provide a result that offers both sufficient and necessary conditions for the validity of the maximum principle. As a consequence of these findings, we conduct a detailed study of an eigenvalue problem with an indefinite weight, as well as establish existence and uniqueness results for a logistic-type equation and prove some blow-up results.