The study of the boundary value problem with arbitrary measurable data originated in the dissertation of Luzin where he investigated the Dirichlet problem for harmonic functions in the unit disk. Recently, in [26], we studied the Hilbert, Poincaré and Neumann boundary value problems with arbitrary measurable data for generalized analytic and generalized harmonic functions and provided applications to relevant problems in mathematical physics. The present paper is devoted to the study of the boundary value problem with arbitrary measurable boundary data in a domain with rectifiable boundary corresponding to semi-linear equation with suitable nonlinear source. We construct a completely continuous operator and generate nonclassical solutions to the Hilbert and Poincaré boundary value problems with arbitrary measurable data for Vekua type and Poisson equations, respectively. Based on that, we prove the existence of solutions of the Hilbert boundary value problem for the nonlinear Vekua type equation with arbitrary measurable data in a domain with rectifiable boundary. It is necessary to point out that our approach differs from the classical variational approach in PDE as it is based on the geometric interpretation of boundary values as angular (along non-tangential paths) limits. The latter makes it possible to also obtain a theorem on the boundary value problem for directional derivatives, and, in particular, of the Neumann problem with arbitrary measurable data for the Poisson equation with nonlinear sources in any Jordan domain with rectifiable boundary. As a result we arrive at applications to some problems of mathematical physics.

In this paper, a class of degenerative quasilinear Schrödinger equations with vanishing potentials and critical Sobolev exponents is considered. The main operator involved in these equations is not strictly elliptic. Under suitable conditions, the existence of nontrivial solutions to the equations is obtained by employing variational methods and the decay rate of the solutions is established.

In this paper, we are concerned with the following fractional Schrödinger-Poisson systems with concave-convex nonlinearities:$$\begin{cases}(-\Delta )^{s}u+u+\mu l(x)\phi u=f(x)|u|^{p-2}u+g(x)|u|^{q-2}u& \text{in }\mathbb{R}^{3}, \\(-\Delta )^{t}\phi =l(x)u^{2} & \text{in }\mathbb{R}^{3},\end{cases}$$where ${1}/{2}< t\leq s< 1$, $1< q< 2< p< \min \{4,2_{s}^{\ast }\}$, $2_{s}^{\ast }={6}/({3-2s})$, and $\mu > 0$ is a parameter,$f\in C\big(\mathbb{R}^{3}\big)$ is sign-changing in $\mathbb{R}^{3}$ and $g\in L^{p/(p-q)}\big(\mathbb{R}^{3}\big)$. Undersome suitable assumptions on $l(x)$, $f(x)$ and $g(x)$, we explore that the energy functional corresponding to the system is coercive and bounded below on $H^{\alpha }\big(\mathbb{R}^{3}\big)$ which gets a positive solution. Furthermore, we constructed some new estimation techniques, and obtained other two positive solutions. Recent results from the literature are generally improved and extended.

In this work, we consider an activator-inhibitor system, known as the Gierer-Meinhardt model. Using the linear stability analysis at the unique positive equilibrium, we derive the conditions of the Hopf bifurcation. We compute the normal form of this bifurcation up to the third degree and obtain the direction of the Hopf bifurcation. Finally, we provide numerical simulations to illustrate the theoretical results of this paper. In this study, we will use the technique of normal form and center manifold theorem.

In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural conditions. Those conditions, roughly speaking, state that the norm can be approximated (on the equinormed sets even uniformly) by the elements of this family. As we are given some freedom of choice of the underlying semi-normed structure that is used to define equinormed sets, our approach opens a new perspective for building compactness criteria in specific normed spaces. As an example we show that natural selections of families of semi-norms in spaces $C(X,\mathbb R)$ and $l^p$ for $p\in[1,+\infty)$ lead to the well-known compactness criteria (including the Arzela-Ascoli theorem). In the second part of the paper, applying the abstract theorems, we construct a simple compactness criterion in the space of functions of bounded Schramm variation.

In this paper we study the existence of solutions to the following generalized nonlinear two-parameter problem \begin{equation*}a(u, v) = \lambda b(u, v) + \mu m(u, v) + \varepsilon F(u, v),\end{equation*}for a triple $(a, b, m)$ of continuous, symmetric bilinear forms on a real separable Hilbert space $V$ and nonlinear form $F$. This problem is a natural abstraction of nonlinear problems that occur for a large class of differential operators, various elliptic pde's with nonlinearities in either the differential equation and/or the boundary conditions being a special subclass. First, a Fredholm alternative for the associated linear two-parameter eigenvalue problem is developed, and then this is used to construct a nonlinear version of the Fredholm alternative. Lastly, the Steklov-Robin Fredholm equation is used to exemplify the abstract results.

In this paper we look for ground state solutions of the elliptic system $$\begin{cases} -\Delta u+V(x)u+\gamma\phi K(x)u = Q(x)f(u), &x\in\mathbb{R}^{2}, \\ \Delta \phi =K(x) u^{2}, &x\in\mathbb{R}^{2}, \end{cases}$$where $\gamma> 0$ and the continuous potentials $V$, $K$, $Q$ satisfy some mild growth conditions and the nonlinearity $f$ has exponential critical growth. The key point of our approach is a new version of the Trudinger-Moser inequality for weighted Sobolev space.

This paper is concerned with the following supercritical Hénon equation with variable exponent$$\begin{cases}-\Delta u=|x|^{\alpha}|u|^{2^*_\alpha-2+|x|^\beta}u&\text{in } B,\\u=0 &\text{on } \partial B,\end{cases}$$where $B\subset\mathbb{R}^N$ $(N\geq 3)$ is the unit ball, $\alpha> 0$, $ 0< \beta< \min\{(N+\alpha)/2,N-2\}$ and $2^*_\alpha=({2N+2\alpha})/({N-2})$. We obtain the existence of positive ground state solution by applying the mountain pass theorem, concentration-compactness principle and approximation techniques.

This paper is concerned with the existence of traveling wave solutions of a higher dimensional lattice delayed cooperation system with nonlocal diffusion. For sufficiently small intraspecific cooperative delays, we construct upper and lower solutions under two different parameters conditions. And then, by using the monotone iterative and Schauder's fixed point theorem, we obtain the existence of traveling wave solutions. The lower bound of the wave speed is in accordance with the properties of linear determined.

In this paper, we study the existence of periodic solutions for a Liénard type $\phi$-Laplacian equation$$(\phi(x'))'+f(x)x'+g(x)=p(t).$$We prove a continuation lemma and use it to prove the existence of periodic solutions for above equation when $g$ or $G$ (the primitive of $g$) satisfies some one-sided or bilateral growth conditions and $F$ (the primitive of $f$) satisfies sublinear condition.

In 2010, M. Sakai and the first author showed that the topological complexity of a space $X$ coincides with the fibrewise unpointed L-S category of a pointed fibrewise space ${\rm pr}_{1} \colon X \times X \to X$ with the diagonal map $\Delta \colon X \to X \times X$ as its section. In this paper, we describe our algorithm how to determine the fibrewise L-S category or the Topological Complexity of a topological spherical space form. Especially, for $S^3/Q_8$ where $Q_8$ is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.

This paper is a continuation of the investigation done in dimension two, this time for the Gutierrez-Sotomayor vector fields on singular $3$-manifolds. The singularities of Gutierrez-Sotomayor flows (GS flows, for short) in this setting are the 3-dimensional counterparts of cones, cross-caps, double and triple crossing points. First, we prove the existence of a Lyapunov function in a neighborhood of a given singularity of a GS flow, i.e. a GS singularity. In these neighbourhoods, index pairs are defined and allow a direct computation of the Conley indices for the different types of GS singularities. The Conley indices are used to prove local necessary conditions on the number of connected boundary components of an isolating block for a GS singularity as well as their Euler characteristic. Lyapunov semi-graphs are introduced as a tool to record this topological and dynamical information. Lastly, we construct isolating blocks so as to prove the sufficiency of the connectivity bounds on the boundaries of isolating blocks given by the Lyapunov semi-graphs.

In this paper, we study the Kirchhoff elliptic equations of the form $$ -M(\|\nabla_\lambda u\|^2)\Delta_\lambda u=w(x)f(u)\quad \mbox{in }\mathbb R^{N}, $$where $M$ is a smooth monotone function, $w$ is a weight function and $f(u)$ is of the form $u^p, e^u$ or $-u^{-p}$. The operator $\Delta_\lambda$ is strongly degenerate and given by $$\Delta_\lambda=\sum_{j=1}^N \frac{\partial}{\partial x_j}\bigg(\lambda_j^2(x)\frac{\partial }{\partial x_j}\bigg).$$We shall prove some classifications of stable solutions to the equation above under general assumptions on $M$ and $\lambda_j$, $j=1,\ldots,N$.

Our main objective of this article is to investigate a class of singular $k_i$-Hessian systems. Among others, we obtain new theorems on the existence and multiplicity of positive radial solutions. Several nonexistence theorems are also derived.

Fessler and Gutierrez [10], [13] proved that if a non-singular planar map has Jacobian matrix without eigenvalues in $(0,+\infty)$, then it is injective. We prove that the same holds replacing $(0,+\infty)$ with any unbounded curve disconnecting the upper (lower) complex half-plane. Additionally we prove that a Jacobian map $(P,Q)$ is injective if $\partial P/\partial x + \partial Q/\partial y$ is not a surjective function.

We prove the existence of at least one sign-changing solution for a third-order nonlocal boundary value problem by applying Leray-Schauder Continuation Principle. To illustrate the applicability of the obtained results, we consider an example.