In this paper, we treat the existence of solutions for a class of general elliptic problems whose prototype is the following: \begin{equation} \begin{cases}-\Delta _{p}u+h(x)|u|^{q-1}u=\beta |\nabla u|^{p}+\lambda f(x) &\mbox {in } \Omega , \\ u=0 &\mbox {on } \partial \Omega ,\end{cases} \end{equation} where $\Omega $ is a bounded open subset of $\mathbb {R}^{N}$ with $N>1$, $1\lt p\lt N$, $q\geq 1$, $\lambda \in \mathbb {R}$, $\beta \in \mathbb {R}$, $h\in L^{1}(\Omega )$ with $h\geq 0$ and $f\in L^{1}(\Omega )$. Assuming that the source term $f$ satisfies $$\lambda _{1}(f)=\inf \bigg \{ \frac {\int _{\Omega }\vert \nabla w\vert ^{p}dx}{\int _{\Omega }|f|\vert w\vert ^{p}dx}:w\in W_{0}^{1,p} (\Omega )\setminus \{ 0\} \bigg \}>0,$$ we obtain the existence of a solution $u\in W_{0}^{1,p}(\Omega )$ when $|\lambda |$ is sufficiently small.

In this paper, we deal with the following fractional nonlocal $p$-Laplacian problem: \begin{equation} \begin{cases}(-\Delta )^{s}_{p}u= \lambda \beta (x) u^q + f(u) &\mbox {in } \Omega ,\\ u\geq 0,\ u\not \equiv 0 &\mbox {in } \Omega ,\\ u=0 &\mbox {in } \mathbb{R} ^{N}\setminus \Omega , \end{cases} \end{equation} where $\Omega \subset \mathbb{R} ^{N}$ is a bounded domain with a smooth boun\-dary of $\mathbb{R} ^N$, $s\in (0,1)$, $p\in (1, \infty )$, $N> s p$, $\lambda $ is a real parameter, $\beta \in L^\infty (\Omega )$ is allowed to be indefinite in sign, $q>0$ and $f:[0,+\infty )\to \mathbb{R} $ is a continuous function oscillating near the origin or at infinity. By using variational and topological methods, we obtain the existence of infinitely many solutions for the problem under consideration. The main results obtained here represent some new interesting phenomena in the nonlocal setting.

In this paper, we derive a new class of solutions for the Kadomtsev-Petviashvili (KP) equation, and we discuss their possible relevance to rogue waves. The nonlinear interaction of these solutions is considered.

The geometry of closed surfaces of genus $g\geq 2$ equipped with a Riemannian metric of variable bounded curvature with finitely many conical points is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.

In this paper, we examine differential subordination related to the harmonic mean in the case where a dominant is a linear function. A result for the first order Euler differential subordination of the nonlinear type is also discussed.

The paper presents a detailed description of the K-theory and K-homology of C*-algebras generated by $q$-normal operators, including generators and index pairing. The C*-algebras generated by $q$-normal operators can be viewed as a $q$-deformation of the quantum complex plane. In this sense, we find deformations of the classical Bott projections describing complex line bundles over the 2-sphere, but there are also simpler generators for the $K_0$-groups, for instance, one dimensional Powers-Rieffel type projections and elementary projections belonging to the C*-algebra. The index pairing between these projections and generators of the even K-homology group is computed, and the result is used to express the $K_0$-classes of quantized line bundles of any winding number in terms of the other projections.

A characterization is given for directed graphs that yield graph $C^*$-algebras with continuous trace. This is established for row-finite graphs with no sources first using a groupoid approach, and extended to the general case via the Drinen-Tomforde desingularization. A characterization of continuous-trace AF $C^*$-algebras is obtained. Partial results are given to characterize higher-rank graphs that yield $C^*$-algebras with continuous trace.

We improve the known expressions of the $\mathcal {C}^r$ differential systems in the plane having a given $\mathcal {C}^{r+1}$ invariant curve, or a given $\mathcal {C}^{r+1}$ first integral. Their application to polynomial differential systems having either an invariant algebraic curve, or a first integral, also improves the known results on such systems.

Given an arbitrary spectral space $X$, we consider the set $\mathcal{X} (X)$ of all nonempty subsets of $X$ that are closed with respect to the inverse topology. We introduce a Zariski-like topology on $\mathcal{X} (X)$ and, after observing that it coincides the upper Vietoris topology, we prove that $\mathcal{X} (X)$ is itself a spectral space, that this construction is functorial, and that $\mathcal{X} (X)$ provides an extension of $X$ in a more ``complete'' spectral space. Among the applications, we show that, starting from an integral domain $D$, $\mathcal{X} (Spec (D))$ is homeomorphic to the (spectral) space of all the stable semistar operations of finite type on $D$.

In this paper, we define the local $S$-class group of an integral domain $D$. A nonzero fractional ideal $I$ of $D$ is said to be $S$-invertible if there exist an $s\in S$ and a fractional ideal $J$ of $D$ such that $sD \subseteq I, J \subseteq D$. The local $S$-class group of $D$, denoted G$ (D)$, is the group of fractional $t$-invertible $t$-ideals of $D$ under $t$-multiplication modulo its subgroup of $S$-invertible $t$-invertible $t$-ideals of $D$. We study the case {G }$(D)=0$, and we generalize some known results developed for the classic contexts of Krull and P$\upsilon $MD domains. Moreover, we investigate the case of isomorphism {G }$(D) \simeq$ {G }$(D[[X]])$. In particular, we give with an additional condition an answer to the question of Bouvier, that is, when is G$ (D)$ isomorphic to G$ (D[[X]])?$

Consider the Caputo fractional $h$-difference equation \[ _a\Delta ^\nu _{h,*}x(t)=c(t)x(t+\nu ), \quad 0\lt \nu \lt 1,\ t\in (h\mathbb{N} )_{a+(1-\nu )h}, \] where $_a\Delta ^\nu _{h,*}x(t)$ denotes the Caputo-like delta fractional $h$-difference of $x(t)$ on sets $(h\mathbb{N} )_{a+(1-\nu )h}$. Our main results are found in Theorems A and B in Section 1. In Section 3, we show that the proof of a recent result in Baleanu, Wu, Bai and Chen is incorrect. Finally, four numerical examples are given to illustrate the main results.

In this paper, we introduce some new concepts of almost factoriality of integral domains. More precisely, we investigate nearly Krull domains, nearly Prufer $v$-multiplication domains and some related integral domains.

We are concerned with the nonlinear fractional Schrödinger system \begin{equation} \begin{cases}(-\Delta )^{s} u+V_{1}(x)u=f(x,u)+\Gamma (x)|u|^{q-2}u|v|^{q} &\mbox {in } \mathbb {R}^{N},\\ (-\Delta )^{s} v+V_{2}(x)v=g(x,v)+\Gamma (x)|v|^{q-2}v|u|^{q} &\mbox {in } \mathbb {R}^{N},\\ u,v\in H^{s}(\mathbb {R}^{N}), \end{cases} \end{equation} where $(-\Delta )^{s}$ is the fractional Laplacian operator, $s\in (0,1)$, $N>2s$, $4\leq 2q\lt p\lt 2^{\ast }$, $2^{\ast }={2N}/({N-2s})$. $V_{i}(x)=V^{i}_{per }(x)+V^{i}_{loc }(x)$ is closed-to-periodic for $i=1,2$, $f$ and $g$ have subcritical growths and $\Gamma (x)\geq 0$ vanishes at infinity. Using the Nehari manifold minimization technique, we first obtain a bounded minimizing sequence, and then we adopt the approach of Jeanjean-Tanaka (2005) to obtain a decomposition of the bounded Palais-Smale sequence. Finally, we prove the existence of ground state solutions for the nonlinear fractional Schrödinger system.

In this work, global stability of a free boundary problem modeling solid avascular tumor growth is studied. The model is considered with time delays during the proliferation process. We prove that the unique positive constant steady state is globally asymptotically stable under some assumptions. The proof uses the comparison principle and the iteration method.

This paper investigates the global existence and the decay rate in time of a solution to the Cauchy problem for an incompressible Oldroyd model with a deformation tensor damping term. There are three major results. The first is the global existence of the solution for small initial data. Second, we derive the sharp time decay of the solution in $L^{2}$-norm. Finally, the sharp time decay of the solution of higher order Sobolev norms is obtained.

In this paper, by a vanishing viscosity approach, we investigate Riemannian solutions containing delta shock waves with Dirac delta functions in both state variables for a class of non-strictly hyperbolic systems of conservation laws. The existence and uniqueness of solutions for the viscous problem are shown.