In this paper, using the fixed point index method, we present a result on the existence of sign-changing solutions for a class of nonlocal elliptic problem in an annulus.

We obtain an explicit form of supports of strongly mixing evolution system of measures naturally connected with non-homogenous Markov process induced by time-dependent SPDEs. We show that considered supports one can get as a net of limit sets determined by a two-parameters semigroup of set-valued maps induced by transition probabilities.

This paper is devoted to the study of existence of positive periodic solutions of a tumor-immune competition model with vaccine interaction. By using the continuation theorem of coincidence degree theory developed by Gains and Mahwin, we establish the sufficient conditions for the existence of periodic solutions.

Bifurcation theory is used to prove the existence of connected components of positive solutions for some classes of singular superlinear semi-positone problems with nonlinear boundary conditions.

The paper is concerned with the inverse source problem for an ADMB-KdV equation, which describes the nonlinear waves generated by a long-wave instability in a viscous film flowing down an inclined rigid surface. The inverse problem aims to determine a spatially varying source function from internal observation data on a suitable subdomain and the whole spatial observation data at a time. We first prove a Carleman in- equality for ADMB-KdV equation, and then apply this Carleman inequality to derive Lipschitz stability for this inverse source problem.

We prove the existence of at least one globally attractive mild solution to the equation $$ \partial_t (b*[x-h(\cdot,x(\cdot))])(t) + A(x(t) - h(t,x(t))) = f(t,x(t)), \quad t\geq 0, $$ under the assumption, among other hypothesis, that $A$ is an almost sectorial operator on a Banach space $X$ and the kernel $b$ belongs to a large class, which covers many relevant cases from physics applications, in particular the important case of time-fractional evolution equations of neutral type.

In this paper, we consider the asymptotic dynamics of non-autonomous fractional reaction-diffusion equations of the form \[u_{t}+(-\Delta)^{s}u+f(u)=g(t)\] complemented with the Dirichlet boundary condition on a bounded domain. First, we obtain some higher-attraction results for pullback attractors, that is, without any additional $t$-differentiability assumption on the forcing term $g$, for any space dimension $N$ and any growth power $p\geq2$ of $f$, the known $(L^{2}(\Omega), L^{2}(\Omega))$ pullback attractor can indeed attract every $L^{2}(\Omega)$-bounded set in the $L^{2+\delta}(\Omega)$-norm for every $\delta\in [0,\infty)$ as well as in the $W_{0}^{s,2}(\Omega)$-norm. Then, we construct a family of Borel probability measures $\{\mu_{t}\}_{t\in\mathbb{R}}$, whose supports satisfy the higher-attraction results. Finally, we investigate the relationship between such the Borel probability measures and time-dependent statistical solutions for this fractional Laplacian equation.

The paper is devoted to the investigation of a reaction-diffusion system of equations describing the process of blood coagulation. Existence of pulse solutions, that is, positive stationary solutions with zero limit at infinity is studied. It is shown that such solutions exist if and only if the speed of the travelling wave described by the same system is positive. The proof is based on the Leray-Schauder method using topological degree for elliptic problems in unbounded domains and a priori estimates of solutions in some appropriate weighted spaces.

In the Euclidean space $\mathbb R^k$, we consider the perturbed eigenvalue problem $Lx + \varepsilon N(x) = \lambda x$, $\|x\| = 1$, where $\varepsilon,\lambda$ are real parameters, $L$ is a linear endomorphism of $\mathbb R^k$, and $N\colon S^{k-1} \to \mathbb R^k$ is a continuous map defined on the unit sphere of $\mathbb R^k$. We prove a global continuation result for the solutions $(x,\varepsilon,\lambda)$ of this problem. Namely, under the assumption that $x_* \in S^{k-1}$ is one of the two unit eigenvectors of $L$ corresponding to a~simple eigenvalue $\lambda_* \in \mathbb R$, we show that, in the set of all the solutions, the connected component containing $(x_*,0,\lambda_*)$ is either unbounded or meets a~solution $(x^*,0,\lambda^*)$ having $x^* \not= x_*$. Our result is inspired by a paper of R. Chiappinelli concerning the local persistence property of eigenvalues and eigenvectors of a perturbed self-adjoint operator in a real Hilbert space.

The aim of this paper is to investigate the existence and asymptotic behavior of the solutions for the critical pseudo-relativistic Hartree equation $$ \sqrt{-\Delta+m^{2}}u+(\beta V(x)-\lambda)u =\bigg(\int_{\mathbb{R}^{N}}\frac{|u(z)|^{2_{\mu}^{\ast}}}{|x-z|^{\mu}}dz\bigg)|u| ^{2_{\mu}^{\ast}-2}u $$ for $\mathbb{R}^{N}$, where $m, \lambda, \beta\in\mathbb{R}^+$, $0< \mu< N$, $N\geq3$, $2_{\mu}^{\ast}=({2N-\mu})/({N-1})$ plays the role of critical exponent due to the Hardy-Littlewood-Sobolev inequality. By transforming the nonlocal problem into a local one via the Dirichlet-to-Neumann map, we are able to obtain the existence of the solutions by variational methods. Suppose that $0< \lambda< \lambda_{1}(\Omega)$ with $\lambda_{1}(\Omega)$ the first eigenvalue and the parameter $\beta$ is large enough, we can prove the existence of ground state solutions. Furthermore, for any sequences $\beta_{n}\rightarrow\infty$, we can show that the ground state solutions $\{u_{n}\}$ converges to a solution of $$ \sqrt{-\Delta+m^{2}}u-\lambda u= \bigg(\int_{\Omega}\frac{|u(z)|^{2_{\mu}^{\ast}}}{|x-z|^{\mu}}dz\bigg) |u|^{2_{\mu}^{\ast}-2}u \quad \mbox{in } \Omega, $$ where $\Omega :=\mbox{int} V^{-1}(0)$ is a nonempty bounded set with smooth boundary. By the way we also establish the existence and nonexistence results for the ground state solutions of the problems set on bounded domain.

In this paper we prove the $C^{0,\alpha}$ regularity of bounded and uniformly continuous viscosity solutions of some degenerate fully nonlinear equations in the first Heisenberg group.

We prove the existence of a positive solution to a semipositone $N$-Laplacian problem with a critical Trudinger-Moser nonlinearity. The proof is based on obtaining uniform $C^{1,\alpha}$ a priori estimates via a compactness argument. Our result is new even in the semilinear case $N = 2$, and our arguments can easily be adapted to obtain positive solutions of more general semipositone problems with critical Trudinger-Moser nonlinearities.

In this paper, we study the following singular nonlinear elliptic problem \begin{equation} \begin{cases} \displaystyle (-\Delta)^{ \alpha/ 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}} &\text{in }\Omega, \\ u=0 &\text{on } \partial\Omega, \end{cases} \tag{P} \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb R^N(N\geq 2)$ with $0\in \Omega$, $\lambda,\mu> 0$, $0< s\leq\alpha$, $(-\Delta)^{\alpha/ 2}$ is the spectral fractional Laplacian operator with $0< \alpha< 2$. We establish existence results and nonexistence results of problem (P) for subcritical, Sobolev critical and Hardy-Sobolev critical cases.

The well-posedness of a generalized Coleman-Gurtin equation equipped with dynamic boundary conditions with memory was recently established by C.G. Gal and the author. Additionally, it was established by the author that the problem admits a finite dimensional global attractor and a robust family of exponential attractors in the case where singularly perturbed memory kernels defined on the interior of the domain and on the boundary of the domain coincide. In the present article we report advances concerning the asymptotic behavior of this heat transfer model when the memory kernels do not coincide. In this setting we obtain a weak exponential attractor whose basin of attraction is the entire phase space, that is, a finite dimensional exponentially attracting compact set in the weak topology of the phase space. This result completes an analysis of the finite dimensional attractors for the generalized Coleman-Gurtin equation equipped with dynamic boundary conditions with memory.

We investigate the following Kirchhoff-Hénon type equation involving the critical Sobolev exponent with Dirichlet boundary condition: \[ - \bigg( a + b \bigg( \int_\Omega \lvert Du \rvert^2 dx \bigg)^{(p-2)/2} \bigg) \Delta u = \Psi u^{q-1} + \lvert x \rvert^\alpha u^{2^* - 1} \] in $\Omega$ included in a unit ball under several conditions. Here, $a, b \geq 0$, $a + b > 0$, $2 < p < q < 2^*$ and $\Psi \in L^\infty(\Omega) \setminus \{ 0 \}$ is a given non-negative function with several conditions. We show that, if either $N = 3$ with $4 < q < 2^* = 6$ or $N \geq 4$, there exists a positive solution for small $\alpha \geq 0$. Our methods includes the mountain pass theorem and the Talenti function.

Using bifurcation analysis we investigate the structure of the set of positive solutions for the coupled nonlinear Schrödinger system \begin{equation*} \begin{cases} -\Delta u_1+ u_1= u_1^3+\beta u_1u_2^2 & \text{in } \mathbb{R}^N,\\ -\Delta u_2+\lambda u_2=\mu u_2^3+\beta u_2u_1^2 &\text{in } \mathbb{R}^N,\\ u_1(x),u_2(x)\rightarrow 0 &\text{as } \vert x\vert\rightarrow+\infty, \end{cases} \end{equation*} where $N=1,2,3$, $\mu$ is a positive constant, $\lambda$ and $\beta$ are positive real parameters. We prove the existence of two two-dimensional continua $\mathcal{S}_1$ and $\mathcal{S}_2$ emanating from the two sets of semi-positive solutions which cover some regions in term of $(\beta,\lambda)\in \mathbb{R}_+^2$. To do this, we establish a multi-parameter unilateral global bifurcation theorem.

The objective of this work is to study the existence theory for a class of variational inequalities of generalized Marguerr-von Kármán's type, which model unilateral problem for the buckling of generalized Marguerre-von Kármán shallow shells. More specifically, we reduce this problem to a variational inequality with cubic operator. Then, we prove the existence of solutions to this problem by using the Leray-Schauder degree.