In this paper, we are interested in the long-time behavior of solutions to a nonlocal interaction equation in dimension $1$. We show that, up to an extraction, the solution converges to a steady-state. Then, we study the structure of stable steady-states.

We are concerned with a four-point boundary-value problem of the $p$-Laplacian dynamic equation on time scales where the nonlinear term contains the first-order derivatives of the dependent variable. By using Krasnosel'skii's fixed-point theorem, some new sufficient conditions are obtained for the existence of at least single or twin positive pseudo-symmetric solutions to this problem. We also establish the existence of at least triple or arbitrary odd positive pseudo-symmetric solutions to this problem by using the Avery-Peterson fixed-point theorem. As applications, two examples are given to illustrate and explain our main results.

We consider the equation \begin{equation} - y'(x)+q(x)y(x)=f(x),\quad x\in \mathbb R \tag*{(0.1)} \end{equation} where $f\in L_p(\mathbb R),$ $p\in[1,\infty]$ $(L_\infty(\mathbb R):=C(\mathbb R))$ and $0\le q\in L_1^{\rm loc}(\mathbb R).$ We assume that equation (0.1)} is correctly solvable in $L_p(\mathbb R).$ Let $y\in L_p(\mathbb R)$ be a solution of (0.1). In the present paper we find minimal requirements for the weight function $r\in L_p^{\rm loc}(\mathbb R)$ under which the following estimate holds: $$ \|ry\|_p\le c(p)\|f\|_p,\quad\forall f\in L_p(\mathbb R) $$ with an absolute constant $c(p)\in (0,\infty).$

The concept of resonance with the first eigenvalue ($\lambda = 0$) of the scalar $T$-periodic problem $$ x''+\lambda x =0, \ \ \ x(0)=x(T), \; x'(0)=x'(T) $$ is considered for first-order planar systems, by dealing with positively homogeneous Hamiltonians which can vanish at some points on $\mathbb{S}^1$. By means of degree methods, an existence result at double resonance for a planar system of the kind $$ Ju'=F(t, u), \quad J= \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \\ \end{array} \right), $$ is then proved, under the assumption that $F(t, u)$ is controlled from below by the gradient of one of such Hamiltonians described above, complementing the main theorem in [7] and including some classical results for the scalar case.

We study the following Brezis-Nirenberg type critical exponent problem: $$ \begin{cases}-\Delta u = \lambda u^q+ u^{2^{\ast}-1}\,\,\,\hbox{in} \,\,B_R,\\ u > 0\,\,\,\hbox{in}\,\, B_R,\quad u=0 \,\,\,\hbox{on} \,\, \partial B_R,\end{cases} $$ where $B_R$ is a ball with radius $R$ in $\mathbb R^N(N\ge3)$, ${\lambda} > 0$, $1\le q < 2^\ast-1 $, and $2^{\ast}$ is the critical Sobolev exponent. We prove the uniqueness results of the least-energy solution when $3\leq N\leq 5 $ and $1\le q < 2^\ast-1 $. We give extremely accurate energy estimates of the least-energy solutions as $R\to 0$ for $N\ge 4$ and $q=1$.

In this paper we present a very simple proof of the existence of at least one nontrivial solution for a Kirchhoff-type equation on ${{\mathbb{R}^N}}$, for $N\ge 3$. In particular, in the first part of the paper we are interested in studying the existence of a positive solution to the elliptic Kirchhoff equation under the effect of a nonlinearity satisfying the general Berestycki-Lions assumptions. In the second part we look for ground states using minimizing arguments on a suitable natural constraint.

We give existence and non-uniqueness results for simple planar curves with prescribed geodesic curvature.

This paper is concerned with the existence of a nontrivial solution for \begin{equation} -\Delta u=\lambda u+|x|^{\alpha}|u|^{2^*-2}u,\,\,\, {\rm in}\ \ \Omega, \quad u=0,\,\,\, {\rm on} \ \ \partial \Omega, \tag*{(0.1)} \end{equation} where $\lambda > 0$ and $\Omega \subset \mathbb{R}^n$ is a smooth bounded domain. Let $\lambda_k,$ $ k =1,2,\ldots$, be eigenvalues of the operator $-\Delta$; we show for $\lambda_k < \lambda < \lambda_{k+1}$ that problem (0.1) possesses at least a solution and each $\lambda_k$ is a bifurcation point.

In this paper, we consider minimization formulas which arise typically in optimal control and weak KAM theory for Hamilton-Jacobi equations. Given a minimization formula, we define a uniqueness set for the formula, which replaces the original region of minimization without changing its values. Our goal is to provide a necessary and sufficient condition that a given set be a uniqueness set. We also provide a characterization of the existence of a minimal uniqueness set with respect to set inclusion.

We study the stability for the viscosity solutions of the differential equation $$ \sum u_{x_i}u_{x_j}u_{x_i x_j}+ | {\nabla u} | ^2\ln( | {\nabla u} | )\langle\nabla u, \nabla \ln p \rangle=0 $$ under perturbations of the function $p(x).$ The differential operator is the so-called $\infty(x)$-Laplacian.