In this paper, we study the following nonlinear time-harmonic Maxwell equations\begin{equation}\tag*{(0.1)}\nabla\times(\nabla \times E)-\omega^2\varepsilon(x)E=P(x)|E|^{p-2}E+Q(x)|E|^{q-2}E,\end{equation}where $\varepsilon(x)$ is the permittivity of the material, $x\in\mathbb{R}^{3}$, $1< q< {p}/({p-1})< 2< p< 6$, $P(x),Q(x)\in C(\mathbb{R}^{3},\mathbb{R})$. Under some special cylindrical symmetric conditions for $\varepsilon(x)$, $P(x)$ and $Q(x)$,we obtain infinite many cylindrically symmetric solutions of (0.1) by using variational method and fountain theorems without $\tau$-upper semi-continuity.

Let ${\mathbb X} \subset \mathbb R^n$ be a bounded Lipschitz domain and consider the $\sigma_{2,p}$-energy functional\begin{equation*}{{\mathbb F}_{\sigma_{2,p}}}[u; {\mathbb X}] := \int_{\mathbb X} \big|{\wedge}^2 \nabla u\big|^p dx,\end{equation*}with $p\in ]1, \infty]$ over the space of measure preserving maps\begin{equation*}{\mathcal A}_p(\mathbb X) =\big\{u \in W^{1,2p}\big(\mathbb X, \mathbb R^n\big) : u|_{\partial \mathbb X} = {x},\ \det \nabla u =1\mbox{ for ${\mathcal L}^n$-a.e. in $\mathbb X$} \big\}.\end{equation*}In this article we address the question of multiplicity versus uniqueness for extremals and strong local minimizers of the $\sigma_{2,p}$-energy funcional $\mathbb F_{\sigma_{2,p}}[\cdot; {\mathbb X}]$ in ${\mathcal A}_p({\mathbb X})$. We use a topological class of maps referred to as generalised twists and examine them in connection with the Euler-Lagrange equations associated with $\sigma_{2,p}$-energy functional over ${\mathcal A}_p({\mathbb X})$. Most notably, we prove the existence of a countably infinite of topologically distinct twisting solutions to the later system in all even dimensions by linking the system to a set of nonlinear isotropic ODEs on the Lie group ${\rm SO}(n)$. In sharp contrast in odd dimensions the only solution is the map $u\equiv x$. The result relies on a careful analysis of the full versus the restricted Euler-Lagrange equations. Indeed, an analysis of curl-free vector fields generated by symmetric matrix fields plays a pivotal role.

This paper is concerned with the reconstruction of the support of the potential term for a time-fractional diffusion equation from the final measured data. The aim of this paper is to propose an accurate approach based on the topological derivative method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing a given cost function. We derive a topological asymptotic expansion for the fractional model. The unknown support is reconstructed using the level-set curve of the topological gradient. We finally make some numerical examples proving the efficiency and accuracy of the proposed algorithm.

In this paper we consider the following supercritical biharmonic problem:$$\begin{cases}\Delta^2 u= K(x)u^{p+\epsilon}&\text{in } \Omega,\\u> 0 &\text{in }\Omega,\\u=\Delta u=0&\text{on }\partial\Omega,\end{cases}$$where $K(x)\in C^3(\overline{\Omega})$ is a nonnegative function, $p=({N+4})/({N-4})$, $\epsilon> 0$, $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq6$. We show that, for $\epsilon$ small enough, there exists a family of concentrating solutions under certain assumptions on the critical points of the function $K(x)$.

We consider the Cahn-Hilliard-Oono equation in the whole of $\mathbb{R}^N$, $N\leq 3$. We prove the existence of an exponential attractor in $H^1\big(\mathbb{R}^N\big)$, which contains a global attractor. We also estimate from above fractal dimension of the attractors.

In this paper we are concerned with normalized solutions to the Schrödinger-Poisson system with doubly critical growth \[\begin{cases}-\Delta u-\phi |u|^3u=\lambda u+\mu|u|^{q-2}u+|u|^4u, &x \in \mathbb R^{3},\\ -\Delta \phi=|u|^5, &x \in \mathbb R^{3},\end{cases} \]and prescribed mass\[ \int_{\mathbb R^3}|u|^2dx=a^2,\]where $a> 0$ is a constant, $\mu> 0$ is a parameter and $2< q< 6$. In the $L^2$-subcritical case, we study the multiplicity of normalized solutions by applying the truncation technique, and the genus theory; and in the $L^2$-supercritical case, we obtain a couple of normalized solutions by developing a fiber map. Under both cases, to recover the loss of compactness of the energy functional caused by the critical growth, we need to adopt the concentration-compactness principle. Our results complement and improve some related studies for the Schrödinger-Poisson system with nonlocal critical term in the literature.

In this paper, we establish several new fixed point results in the framework of topological spaces endowed with a partial order. Special attention is paid to the case that the topology is induced by a metric. Our conclusions generalize many well-known results. Several examples and illustrative applications are provided to support the exposed results.

We consider capacity (fuzzy measure, non-additive probability) on a compactum as a monotone cooperative normed game. Then it is natural to consider probability measures as elements of core of such game. We prove a topological version of the Bondareva-Shapley theorem that non-emptiness of the core is equivalent to balancedness of the capacity. We investigate categorical properties of balanced capacities and give characterizations of some fuzzy integrals of balanced capacities.

In this paper, based on regulated functions and fixed point theorem, a class of nonlocal problem of multi-term time-fractional measure differential equations involving nonlocal conditions in Banach spaces. Firstly, we introduce the concept of $S$-asymptotically $\omega$-periodic mild solution, on the premise of by utilizing $(\beta,\gamma_k)$-resolvent family and measure functional (Henstock-Lebesgue-Stieltjes integral), the existence of $S$-asymptotically $\omega$ periodic mild solutions for the mentioned system are obtained. Finally, as the application of abstract results, the existence $S$-asymptotically $\omega$-periodic mild solution for a classes of measure driven differential equation are discussed.

In this paper, we are concerned with the parabolic Lichnerowicz equation involving the $\Delta_\lambda$-Laplacian $$ v_t-\Delta_\lambda v=v^{-p-2}-v^p,\quad v> 0, \quad \mbox{ in }\mathbb R^N\times\mathbb R, $$ where $p> 0$ and $\Delta_\lambda$ is a sub-elliptic operator of the form $$\Delta_\lambda=\sum_{i=1}^N\partial_{x_i}\big(\lambda_i^2\partial_{x_i}\big).$$Under some general assumptions of $\lambda_i$ introduced by A.E. Kogojand E. Lanconelli in Nonlinear Anal. 75 (2012), no. 12, 4637-4649, we shall prove a uniform lower bound of positive solutions of the equation provided that $p> 0$. Moreover, in the case $p> 1$, we shall show that the equation has only the trivial solution $v=1$. As a consequence, when $v$ is independent of the time variable, we obtain the similar results for the elliptic Lichnerowicz equation involving the $\Delta_\lambda$-Laplacian $$ -\Delta_\lambda u=u^{-p-2}-u^p,\quad u> 0,\quad \mbox{in }\mathbb R^N. $$

In this paper we study, in a separable metric space, a class of Hausdorff measures ${\mathcal H}_\mu^{q, \xi}$ defined using a measure $\mu$ and a premeasure $\xi$. We discuss a Hausdorff structure of product sets. Weighted Hausdorff measures ${\mathcal W}_\mu^{q, \xi}$ appear as an important tool when studying the product sets. When $\mu$ and $\xi$ satisfy the doubling condition, we prove that ${\mathcal H}_\mu^{q, \xi} = {\mathcal W}_\mu^{q, \xi}$. As an application, the case where $\xi$ is defined as the Hausdorff function is considered.

This paper is devoted to the study of the existence of periodic solutions for a class of semilinear wave equations with variable coefficients. The forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media is governed by this mathematical model. When the frequency is a sufficiently large irrational number with bounded partial quotients, the existence of weak solutions is established. Then, under some suitable conditions, we improve the regularity of weak solutions. Our results can also be applied to the corresponding constant coefficients wave equation.

We provide a direct proof of existence and uniqueness of weak solutions to a broad family of strongly nonlinear elliptic equations with lower-order terms. The leading part of the operator satisfies general growth conditions settling the problem in the framework of fully anisotropic and inhomogeneous Musielak-Orlicz spaces generated by an $N$-function $M\colon \Omega\times\mathbb R^d\to [0,\infty)$. Neither $\nabla_2$ nor $\Delta_2$ conditions are imposed on $M$. Our results cover among others problems with anisotropic polynomial, Orlicz, variable exponent, and double phase growth.

In this paper we study the following nonlinear Schrödinger equations in $\mathbb{R}^N$: $$-\Delta u + V(x)u= g(u),\quad u \in H^1(\mathbb{R}^N), $$where $N \ge 2$, $V \in C^1(\mathbb{R}^N,\mathbb{R})$ and $g \in C(\mathbb{R},\mathbb{R})$. For a wide class of nonlinearities, which satisfy the Berestycki-Lions type condition, we show the existence and multiplicity of radially symmetric solutions. We use a new deformation argument under a new version of the Palais-Smale condition.

In the present paper we consider the problem:\begin{equation} \label{0}\tag{N$_\varepsilon$}\begin{cases}-\varepsilon^{2}\Delta u+u=u^{3}+\lambda v& \text{in } \Omega, \\-\varepsilon^{2}\Delta v+v=v^{3}+\lambda u& \text{in } \Omega,\\u> 0,\ v> 0& \text{in } \Omega,\\\dfrac{\partial u}{\partial n}=\dfrac{\partial v}{\partial n}=0& \text{on }\partial\Omega,\end{cases}\end{equation}where $\varepsilon> 0$, $0< \lambda< 1$, $\Omega\subset\mathbb{R}^{3}$ is smooth and bounded, and $n$ denotes the outer normal vector defined on $\partial\Omega$, the boundary of $\Omega$. By the Lyapunov-Schmidt reduction method and the maximum principle of elliptic equations, we construct synchronized solutions of (\ref{0}) with mixed interior and boundary peaks for any $0< \varepsilon< \varepsilon_0$ and $\lambda\in(0,1)\backslash\{\lambda_0\}$, where $\lambda_0\in(0,1)$ is given and $\varepsilon_0> 0$ is sufficiently small. As $\varepsilon$ approaches $0$, the interior peaks concentrate at sphere packing points in $\Omega$ and the boundary peaks concentrate at the critical points of the mean curvature function of the boundary.

We prove new results on the existence of multiple solutions for elliptic inclusions with nonlinear boundary conditions of Neumann type. Our approach is topological and relies on the fixed point index for multivalued map.