We consider the minimizers of $L^{2}$-critical inhomogeneous variational problems with a spatially decaying nonlinear term in an open bounded domain $\Omega$ of $\mathbb{R}^{N}$ which contains $0$. We prove that there is a threshold $a^{*}> 0$ such that minimizers exist for $0< a< a^{*}$ and the minimizer does not exist for any $a> a^{*}$. In contrast to the homogeneous case, we show that both the existence and nonexistence of minimizers may occur at the threshold $a^*$ depending on the value of $V(0)$, where $V(x)$ denotes the trapping potential. Moreover, under some suitable assumptions on $V(x)$, based on a detailed analysis on the concentration behavior of minimizers as $a\nearrow a^*$, we prove local uniqueness of minimizers when $a$ is close enough to $a^*$.

This paper studies the existence and multiplicity of normalized solutions to the lower critical Choquard equation with a $L^2$-subcritical local perturbation and kinds of bounded potentials\begin{equation*}\begin{cases}-\Delta u+V(x)u\\\qquad =\lambda u+\big(I_{\alpha}\ast|u|^{({N+\alpha})/{N}}\big)|u|^{({N+\alpha})/{N}-2}u+\mu|u|^{q-2}u& \text{in } \mathbb{R}^N, \\\displaystyle\int_{\mathbb{R}^N}|u|^2dx=a^2,\end{cases}\end{equation*}where $N\geq 1$, $\mu, a> 0$, $2< q< 2+{4}/{N}$, $\alpha\in (0,N)$, $I_{\alpha}$ is the Riesz potential, $V(x)$ is a bounded potential and $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier.

This study provides a general multifractal formalism that overcomes the limitations of the traditional one. The generic Hewitt-Stromberg measures are used to introduce and study a multifractal formalism. The generic Hewitt-Stromberg dimensions' upper and lower bounds are estimated, producing results even at places $q$ where the upper and lower multifractal Hewitt-Stromberg dimension functions diverge.

The aim of this paper is to investigate some multiplicity results for a Hamiltonian system with Neumann-type boundary conditions. A critical point theory is applied in order to show that the the problem has multiple solutions. The crucial part of this paper is that, in contrast to periodic problems where the Poincaré-Birkhoff Theorem has a significant role, no twist condition is required.

First we study the Dirichlet problem ${\rm{Re}}\,\omega(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ with continuous boundary data $\varphi \colon \partial D\to\mathbb R$ for semi-linear Beltrami equations $\omega_{\overline{z}}-\mu(z) \omega_z=\sigma (z)q({\rm Re}\,\omega(z))$. We assume here that $D$ is an arbitrary bounded domain of the complex plane $\mathbb C$, which is either simply connected or has no boundary components degenerated to a single point, and that the equations are locally uniform elliptic with possible singularities at the boundary. For $\sigma\in L_p(D)$, $p> 2$, with compact support, and continuous $q\colon \mathbb R\to\mathbb C$, $q(t)/t\to 0$ as $t\to\infty$, we establish a series of effective criteria for existence of solutions of the Dirichlet problem in terms of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz integral means. We also establish representation and regularity of these solutions. Then, we prove existence, representation and regularity of weak solutions of the Dirichlet problem $u(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ to semi-linear Poisson type equations ${\rm div} [A(z)\nabla u(z)] = g(z)Q(u(z))$ for $g\in L_p(D)$, $p> 1$, with compact support, and continuous $Q\colon \mathbb R\to\mathbb R$, $Q(t)/t\to 0$ as $t\to\infty$. We also assume here conditions on the matrix coefficients $A(z)$ guaranteing locally uniform ellipticity of these equations. Finally, we give examples of possible applications of the obtained results to various semi-linear equations of the mathematical physics in anisotropic and inhomogeneous media.

This paper deals with the following $(p,N)$-Laplacian equation with logarithmic and critical exponential nonlinearities. Precisely, we study the problem\begin{equation*}\begin{cases}-\Delta_p u -\Delta_N u = |u|^{q-2}u \ln|u|^2 + \lambda f(u)& \text{in }\Omega,\\u=0& \text{on }\partial \Omega,\end{cases}\end{equation*}where $\Omega \subset \mathbb{R}^N$ is a bounded domain, $N \geq 2$, $1< p< N< q$, $\lambda > 0$ is a positive real parameter. By applying variational methods, we obtain the existence of solutions.

Let $\Omega$ be a domain in $\mathbb{R}^n$ ($n\geq 2$) of the form $\Omega=\Omega_{\text{out}}\setminus \overline{\Omega_{\text{in}}}$. Set $\Omega_D$ to be either $\Omega_{\text{out}}$ or $\Omega_{\text{in}}$. For $p\in (1,\infty)$, and $q\in [1,p]$, let $\tau_{1,q}(\Omega)$ be the first eigenvalue of \begin{alignat*}2 -\Delta_p u &=\tau \bigg(\int_{\Omega}|u|^q dx \bigg)^{({p-q})/{q}} |u|^{q-2}u &\quad&\text{in }\Omega,\\ u &=0&\quad&\text{on } \partial\Omega_D,\\\frac{\partial u}{\partial \eta}&=0&\quad& \text{on } \partial \Omega\setminus \partial \Omega_D. \end{alignat*}Under the assumption that $\Omega_D$ is convex, we establish the following reverse Faber-Krahn inequality$$\tau_{1,q}(\Omega)\leq \tau_{1,q}({\Omega}^\star),$$where ${\Omega}^\star=B_R\setminus \overline{B_r}$ is a concentric annular region in $\mathbb{R}^n$ having the same Lebesgue measure as $\Omega$ and such that

(i) (when $\Omega_D=\Omega_{\text{out}}$) $W_1(\Omega_D)= \omega_n R^{n-1}$, and $(\Omega^\star)_D=B_R$,

Here $W_{i}(\Omega_D)$ is the $i^{\text{th}}$ quermassintegral of $\Omega_D$. We also establish Sz.-Nagy's type inequalities for parallel sets of a convex domain in $\mathbb{R}^n$ ($n\geq 3$) for our proof.

We consider nonlinear perturbations of the hyperbolic equation in the Hilbert space. Necessary and sufficient conditions for the existence of solutions of boundary-value problem for the corresponding equation and iterative procedures for their finding are obtained in the case when the operator in linear part of the problem hasn't inverse and can have nonclosed set of values. As an application we consider boundary-value problem for van der Pol equation in a separable Hilbert space.

We investigate some self-similar Cantor sets $C(l,r,p)$, which we call S-Cantor sets, generated by numbers $l,r,p \in \mathbb N$, $l+r< p$. We give a full characterization of the set $C(l_1,r_1,p)-C(l_2,r_2,p)$ which can take one of the form: the interval $[-1,1]$, a Cantor set, an L-Cantorval, an R-Cantorval or an M-Cantorval. As corollaries we give examples of Cantor sets and Cantorvals, which can be easily described using some positional numeral systems.

The purpose of this paper is to present the existence of mild solutions and approximate controllability for a class of non-autonomous measure driven evolution systems with nonlocal conditions in Banach spaces. Firstly, we obtain the existence of mild solutions for the concerned problem by the semigroup theory and Schauder's theorem. Secondly, some sufficient conditions of approximate controllability are proved. At the end, an example is also given to illustrate the feasibility of our theoretical results.