In this paper, we study the existence and stability of solitary wave solutions for the generalized Benjamin equation in both the $L^2$-critical and $L^2$-supercritical cases by applying the variational methods and the non-homogeneous Gagliardo–Nirenberg inequality. Our main results generalize and complement the existing results in the literature.

In this paper, we are concerned with the following Dirichlet problems for nonlinear equations involving the fractional $p$-Laplacian:

\begin{equation*}\begin{cases}(-\Delta)_p^\alpha u=f(x,u,\nabla u),\ \ u \gt 0,\ \ \text{in}\ \ E,\\\ \ \ \ \ \ u\equiv0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ \ \mathbb{R}^{n}\setminus E,\end{cases}\end{equation*}

where $E \subseteq \mathbb{R}^{n}$ is a coercive epigraph, i.e., there exists a continuous function $\phi: \, \mathbb{R}^{n-1} \rightarrow \mathbb{R}$ satisfying

\begin{equation*}\lim_{|x'|\rightarrow+\infty}\phi(x')=+\infty,\end{equation*}

such that $E:=\{x=(x',x_{n}) \in \mathbb{R}^{n}|\,x_{n} \gt \phi(x')\}$, where $x':= (x_{1},...,x_{n-1}) \in \mathbb{R}^{n-1}$. Under some mild assumptions on the nonlinearity $f(x,u,\nabla u)$, we prove strict monotonicity of positive solutions to the above Dirichlet problems involving fractional $p$-Laplacian in coercive epigraph $E$.

Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of $H^s$ energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-frequency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size $n\leq 6$ a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions $7$ and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov’s Lemma for ODE that is to our knowledge new.

Given $p\in[1,\infty)$ and a bounded open set $\Omega\subset\mathbb{R}^d$ with Lipschitz boundary, we study the $\Gamma$-convergence of the weighted fractional seminorm

\begin{equation*}[u]_{s,p,f}^p=\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|\tilde u(x)- \tilde u(y)|^p}{\|x-y\|^{d+sp}}\,f(x)\,f(y)\,\mathrm{d} x\,\mathrm{d} y,\end{equation*}

as $s\to1^-$ for $u\in L^p(\Omega)$, where $\tilde u=u$ on $\Omega$ and $\tilde u=0$ on $\mathbb{R}^d\setminus\Omega$. Assuming that $(f_s)_{s\in(0,1)}\subset L^\infty(\mathbb{R}^d;[0,\infty))$ and $f\in\mathrm{Lip}_b(\mathbb{R}^d;(0,\infty))$ are such that $f_s\to f$ in $L^\infty(\mathbb{R}^d)$ as $s\to1^-$, we show that $(1-s)[u]_{s,p,f_s}^p$ $\Gamma$-converges to the Dirichlet $p$-energy weighted by $f^2$. In the case $p=2$, we also prove the convergence of the corresponding gradient flows.

In recent years, there has been significant interest in the effect of different types of adversarial perturbations in data classification problems. Many of these models incorporate the adversarial power, which is an important parameter with an associated trade-off between accuracy and robustness. This work considers a general framework for adversarially perturbed classification problems, in a large data or population-level limit. In such a regime, we demonstrate that as adversarial strength goes to zero that optimal classifiers converge to the Bayes classifier in the Hausdorff distance. This significantly strengthens previous results, which generally focus on $L^1$-type convergence. The main argument relies upon direct geometric comparisons and is inspired by techniques from geometric measure theory.

This paper investigates sharp stability estimates for the fractional Hardy–Sobolev inequality:

\begin{align*}\ \ \ \ \ \ \ \ \ \mu_{s,t}\left(\mathbb{R}^N\right) \left(\int_{\mathbb{R}^N} \frac{|u|^{2^*_s(t)}}{|x|^t} \,\mathrm{d}x \right)^{\frac{2}{2^*_s(t)}} \leq \int_{\mathbb{R}^N} \left|(-\Delta)^{\frac{s}{2}} u \right|^2 \,\mathrm{d}x, \quad \text{for all } u \in \dot{H}^s\left(\mathbb{R}^N\right)\end{align*}

where $s \in (0,1)$, $0 \lt t \lt 2s$, $N \gt 2s$ is an integer, and $2^*_s(t) = \frac{2(N-t)}{N-2s}$. Here, $\mu_{s,t}\left(\mathbb{R}^N\right)$ represents the best constant in the inequality. The primary focus is on the quantitative stability results of the above inequality and the corresponding Euler–Lagrange equation near a positive ground-state solution. Additionally, a qualitative stability result is established for the Euler–Lagrange equation, offering a thorough characterization of the Palais–Smale sequences for the associated energy functional. These results generalize the sharp quantitative stability results for the classical Sobolev inequality in $\mathbb{R}^N$, originally obtained by Bianchi and Egnell [J. Funct. Anal. 1991] as well as the corresponding critical exponent problem in $\mathbb{R}^N$, explored by Ciraolo, Figalli, and Maggi [Int. Math. Res. Not. 2017] in the framework of fractional calculus.

We consider the existence, nonexistence and multiplicity of normalized solutions to the $p$-Laplacian equation on a bounded domain(0.1)

\begin{equation}\left\{\begin{array}{ll}-\Delta_p u+\lambda |u|^{p-2}u=g(u),&\text{in }\Omega,\\\int_\Omega |u|^p=\rho, u=0&\text{on }\partial \Omega,\end{array}\right.\end{equation}

where $\Omega$ is a bounded domain, $p\geq 2$. Firstly, under suitable assumptions on $\rho$, if $g$ is at most mass-critical at infinity, we prove the existence of infinitely many solutions. Secondly, for $\rho$ large, if $g$ is mass-supercritical, we perform a blow-up analysis to show the nonexistence of finite Morse index solutions. At last, for $\rho$ suitably small, combining with the monotonicity argument, we obtain a multiplicity result. In particular, when $p=2$, we obtain the existence of infinitely many normalized solutions.

We study the existence and multiplicity of positive bounded solutions for a class of nonlocal, non-variational elliptic problems governed by a nonhomogeneous operator with unbalanced growth, specifically the double phase operator. To tackle these challenges, we employ a combination of analytical techniques, including the sub-super solution method, variational and truncation approaches, and set-valued analysis. Furthermore, we examine a one-dimensional fixed-point problem.To the best of our knowledge, this is the first workaddressing nonlocal double phase problems using these methods.

This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schrödinger equations. We establish the following key property of these solutions: they are all in $L^2(\mathbb{R})$ with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this paper, we define general rogue waves of infinite order, establish their basic exact and asymptotic properties, and provide computational tools for calculating them accurately.

The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.

In this article, we are concerned with the static Schrödinger equation with the van der Waals potential. Such an equation can be transformed into a system of integral equations. We present the nonexistence of two classes of positive solutions. One is a class of locally bounded positive super-solutions, and the other is a class of positive solutions with some integrability. To prove the nonexistence of positive integrable solutions, we investigate their qualitative properties, involving the radial symmetry, the better integrability, the differentiability, and the decay estimates at infinity. Here, the regularity lifting lemma, the method of moving planes in integral forms, and the Pohozaev identity in integral forms play important roles.

This paper is concerned with the following problem involving almost critical growth

\begin{equation*}\begin{cases}(-\Delta) ^su=|u|^{2_{s}^{*}-2-\varepsilon}u,& \text{in } \varOmega ,\\u=0,& \text{on } \Sigma ,\\\end{cases}\end{equation*}

where $2_{s}^{*}=\frac{2N}{N-2s}$, $s\in(\frac{1}{2},1)$, $N \gt 2s$, Ω is a bounded domain in $\mathbb{R}^N$, ɛ is a small parameter, and the boundary Σ is given in different ways according to the different definitions of the fractional Laplacian operator $(-\Delta)^{s}$. The operator $(-\Delta)^{s}$ is defined in two types: the spectral fractional Laplacian and the restricted fractional Laplacian. For the spectral case, Σ stands for $\partial \Omega$; for the restricted case, Σ is $\mathbb{R}^{N}\setminus \Omega$. Firstly, we provide a positive confirmation of the fractional Brezis–Peletier conjecture, that is, the above almost critical problem has a single bubbling solution concentrating around the non-degenerate critical point of the Robin function. Furthermore, the non-degeneracy andlocal uniqueness of this bubbling solution are established.

In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation

\begin{align*}\left\{\begin{array}{ll}-(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u=\mu u+\beta^*|u|^{\frac{8}{3}}u &\text{in}\ {\Omega}, \\[0.1cm]u=0&\text{on}\ {\partial\Omega}, \\[0.1cm]\int_{\Omega}|u|^2\mathrm{d}x=1, \\[0.1cm]\end{array}\right.\end{align*}

$a\geq 0$

$\Omega\subset\mathbb R^3$

$\beta^*:=\frac{b}{2}|Q|_2^{\frac{8}{3}}$

$-2\Delta u+\frac{1}{3}u-|u|^{\frac{8}{3}}u=0$

$\mathbb R^3.$

$a\searrow 0$

We study the long time dynamic properties of the nonlocal Kuramoto–Sivashinsky (KS) equation with multiplicative white noise. First, we consider the dynamic properties of the stochastic nonlocal KS equation via a transformation into the associated conjugated random differential equation. Next, we prove the existence and uniqueness of solution for the conjugated random differential equation in the theory of random dynamical systems. We also establish the existence and uniqueness of a random attractor for the stochastic nonlocal equation.

In this paper, we study the existence of solutions to the following Hartree equation

\begin{align*}\begin{cases}-\Delta u+\lambda V(x) u+\mu u=\left(\int_{\mathbb{R}^N}\frac{|u|^p}{|x-y|^{N-\alpha}}\right)|u|^{p-2}u,\ \text{in}\ \mathbb{R}^N,\\\int_{\mathbb{R}^N}|u|^2=\omega,\end{cases}\end{align*}

Where $N\geq 3$, $\omega,\lambda \gt 0$, $p\in \left(\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}\right)\setminus\left\{\frac{N+\alpha+2}{N}\right\}$ and µ will appear as a Lagrange multiplier. We assume that $0\leq V\in L^{\infty}_{loc}(\mathbb{R}^N)$ has a bottom $int V^{-1}(0)$ composed of $\ell_0$ $(\ell_{0}\geq1)$ connected components $\{\Omega_i\}_{i=1}^{\ell_0}$, where $int V^{-1}(0)$ is the interior of the zero set $V^{-1}(0)=\{x\in\mathbb{R}^N| V(x)=0\}$ of V. It is worth pointing out that the penalization technique is no longer applicable to the local sublinear case $p\in \left(\frac{N+\alpha}{N},2\right)$. Therefore, we develop a new variational method in which the two deformation flows are established that reflect the properties of the potential. Moreover, we find a critical point without introducing a penalization term and give the existence result for $p\in \left(\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}\right)\setminus\left\{\frac{N+\alpha+2}{N}\right\}$. When ω is fixed and satisfies $\omega^{\frac{-(p-1)}{-Np+N+\alpha+2}}$ sufficiently small, we construct a $\ell$-bump $(1\leq\ell\leq \ell_{0})$ positive normalization solution, which concentrates at $\ell$ prescribed components $\{\Omega_i\}^{\ell}_{i=1}$ for large λ. We also consider the asymptotic profile of the solutions as $\lambda\rightarrow\infty$ and $\omega^{\frac{-(p-1)}{-Np+N+\alpha+2}}\rightarrow 0$.

We study existence and uniqueness of spherically symmetric solutions of

\begin{equation*}S_k(D^2v)+\beta \xi\cdot\nabla v+\alpha v+\left\vert v\right\vert^{q-1}v=0\;\; \mbox{in}\;\; \mathbb{R}^n,\end{equation*}

where $\alpha,\beta$ are real parameters, $n \gt 2,\, q \gt k\geqslant 1$ and $S_k(D^2v)$ stands for the k-Hessian operator of v. Our results are based mainly on the analysis of an associated dynamical system and energy methods. We derive some properties of the solutions of the above equation for different ranges of the parameters α and β. In particular, we describe with precision its asymptotic behaviour at infinity. Further, according to the position of q with respect to the first critical exponent $\frac{(n+2)k}{n}$ and the Tso critical exponent $\frac{(n+2)k}{n-2k}$ we study the existence of three classes of solutions: crossing, slow decay or fast decay solutions. In particular, if k > 1 all the fast decay solutions have a compact support in $\mathbb{R}^n$. The results also apply to construct self-similar solutions of type I to a related nonlinear evolution equation. These are self-similar functions of the form $u(t,x)=t^{-\alpha}v(xt^{-\beta})$ with suitable α and β.

We consider the propagation dynamics of a single species with a birth pulse and living in a shifting environment driven by climate change. We describe how birth pulse and environment shift jointly impact the propagation properties. We show that a moderate environment shifting speed promotes the spatial–temporal propagation represented by a stable forced KPP wave, and that the birth pulse shrinks the survival region.

We consider the asymptotics of long-time behavior of a solution u of the semilinear parabolic problem $\partial _tu=\Delta u-u+u|u|^{p-2}$ in ${\mathbb {R}^N}\times (0,\infty )$, $u(0)=u_0\in H^1({\mathbb {R}^N})\cap L^\infty ({\mathbb {R}^N})$. Since the spatial domain on which the problem is posed is noncompact, we cannot expect the relative compactness of the solution orbit, e.g., in $H^1({\mathbb {R}^N})$ in general. In this article, we prove that the compactness of the orbit holds up to the ground state energy level, namely, if $\lim _{t\to \infty }I(u(t))\leq d_\infty $, where I is the energy functional associated with (P) and $d_\infty $ its ground state energy, then the orbit of $u(t)$ is compact in $H^1({\mathbb {R}^N})$. Our result includes the previous results in [4, 5].

In this article, we consider the global-in-time existence and singularity formation of smooth solutions for the radially symmetric relativistic Euler equations of polytropic gases. We introduce the rarefaction/compression character variables for the supersonic expanding wave with relativity and derive their Riccati type equations to establish a series of priori estimates of solutions by the characteristic method and the invariant domain idea. It is verified that, for rarefactive initial data with vacuum at the origin, smooth solutions will exist globally. On the other hand, the smooth solution develops a singularity in finite time when the initial data are compressed and include strong compression somewhere.

We consider radially symmetric solutions of the degenerate Keller–Segel system

\begin{align*}\begin{cases}\partial_t u=\nabla\cdot (u^{m-1}\nabla u - u\nabla v),\\0=\Delta v -\mu +u,\quad\mu =\frac{1}{|\Omega|}\int_\Omega u,\end{cases}\end{align*}

in balls $\Omega\subset\mathbb R^n$, $n\ge 1$, where m > 1 is arbitrary. Our main result states that the initial evolution of the positivity set of u is essentially determined by the shape of the (nonnegative, radially symmetric, Hölder continuous) initial data u0 near the boundary of its support $\overline{B_{r_1}(0)}\subsetneq\Omega$: It shrinks for sufficiently flat and expands for sufficiently steep u0. More precisely, there exists an explicit constant $A_{\mathrm{crit}} \in (0, \infty)$ (depending only on $m, n, R, r_1$ and $\int_\Omega u_0$) such that if $u_0(x)\le A(r_1-|x|)^\frac{1}{m-1}$ for all $|x|\in(r_0, r_1)$ and some $r_0\in(0,r_1)$ and $A \lt A_{\mathrm{crit}}$ then there are T > 0 and ζ > 0 such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\le r_1 -\zeta t$ for all $t\in(0, T)$, while if $u_0(x)\ge A(r_1-|x|)^\frac{1}{m-1}$ for all $|x|\in(r_0, r_1)$ and some $r_0 \in (0, r_1)$ and $A \gt A_{\mathrm{crit}}$ then we can find T > 0 and ζ > 0 such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\ge r_1 +\zeta t$ for all $t\in(0, T)$.