In this paper, we consider the following fractional logarithmic Schrödinger equation\begin{equation*}\varepsilon^{2s}(-\Delta)^s u + V(x)u=u\log u^2\quad \text{in } \mathbb R^N,\end{equation*}where $\varepsilon> 0$, $N\ge 1$ and $V(x)\in C\big(\mathbb R^N,\mathbb R\big)$ is a potential which can be unbounded below at infinity. By considering a new penalization, we show that the problem has a nontrivial solution $u_{\varepsilon}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$.

In this paper, we study the existence and uniqueness of $PC$-mild solutions for fractional impulsive evolution systems with Caputo derivative. First, we give a compact result of the solution operators of linear fractional evolution system while the cosine family is compact. Second, the representation of $PC$-mild solutions of linear fractional impulsive systems with initial value conditions and nonlocal initial value conditions are given by using the Laplace transform method. Third, several sufficient conditions for judging the existence and uniqueness of solutions of our problem are established via some fixed point theorems and operator semigroup theory. Finally, an example is given to illustrate the results of our paper.

For every nonorientable closed surface $\mathbb U$, we present a strong surjection $f\colon X\to\mathbb U$, where $X$ is a finite two-dimensional cw-complex with trivial second integer cohomology group. This provides an answer, for all nonorientable closed surfaces, to a problem in topological root theory for which we have hitherto known solutions only for the sphere, the torus, the projective plane and the Klein bottle.

Let $\Delta$ be a $g_2$-minimal normal 3-pseudomanifold. A vertex in $\Delta$ whose link is not a sphere is called a singular vertex. When $\Delta$ contains at most two singular vertices, its combinatorial characterization is known [9]. In this article, we present a combinatorial characterization of such a $\Delta$ when it has three singular vertices, including one $\mathbb{RP}^2$-singularity, or four singular vertices, including two $\mathbb{RP}^2$-singularities. In both cases, we prove that $\Delta$ is obtained from a one-vertex suspension of a surface, and some boundary complexes of $4$-simplices by applying the combinatorial operations of types connected sums, vertex foldings, and edge foldings.

In this paper we study a double phase problem which involves the double phase operator, and the nonlinear term has an oscillatory behavior. By using variational methods and the theory of the Musielak-Orlicz-Sobolev space, we establish the existence of infinitely many solutions whose $W_0^{1,H}(\Omega)$-norms tend to zero (to infinity, respectively) whenever the nonlinearity oscillates at zero (at infinity, respectively).

In this article, we study the fractional Brézis-Nirenberg type problem on whole domain $\mathbb R^N$ associated with the fractional $p$-Laplace operator. To be precise, we want to study the following problem:\begin{equation} \label{Pr1} (-\Delta_{p})^{s}u - \lambda w |u|^{p-2}u= |u|^{p_{s}^{*}-2}u \quad \text{in } \mathcal{D}^{s,p}\big(\mathbb{R}^{N}\big) , \tag{P}\end{equation}where $s\in (0,1), p\in (1, {N}/{s})$, $p_{s}^{*}={Np}/({N-sp})$ and the operator $(-\Delta_{p})^{s}$ is the fractional $p$-Laplace operator. The space $\mathcal{D}^{s,p}\big(\mathbb{R}^{N}\big)$ is the completion of $C_c^\infty\big(\mathbb R^N\big)$ with respect to the Gagliardo semi-norm. In this article, we prove the existence of a positive solution to problem (P) by allowing the Hardy weight function $w$ to change its sign.

A formal framework for the analysis of Hopf bifurcations for a kind of delayed equation with $\varphi$-Laplacian and with a discrete time delay is presented, thus generalizing known results for the sunflower equation given by Somolinos in 1978. Also, under appropriate assumptions we prove the gradient-like behavior of the equation which, in turn, implies the non-existence of nonconstant periodic solutions. Our conditions improve previous results known in the literature for the standard case $\varphi(x)=x$.

In this article, we investigate $p(x)$-biharmonic equations involving two kinds of different Hardy potentials, in which the $r(x)$-Hardy potentials are seldom mentioned in previous papers. New criteria for the existence of generalized solutions are reestablished when the nonlinear terms satisfying appropriate assumptions. The results are based on variational methods and the theory of variable exponent Sobolev spaces.

Consider the quasilinear elliptic equation$$-\text{div}(\mathcal{A}(u)\nabla u)+\frac{1}{2}\mathcal{A}'(u)|\nabla u|^2+V(x)u=(I_{\alpha}\ast |u|^p) |u|^{p-2}u\quad \text{in } \mathbb R^N,$$where $\mathcal{A}\in C^1(\mathbb R,\mathbb R)$ is a positive bounded function, $V$ is a given potential and $I_\alpha$ denotes the Riesz potential with $0< \alpha< N$. While most existing works in the literature are concerned with the case where $\mathcal{A}$ is unbounded, little is known about the case where $\mathcal{A}$ is bounded. Under some general conditions on $\mathcal{A}$ and $V$, we establish the existence of a positive solution for the above equation by variational approach.

In this work, our interest lies in proving the existence of critical values of the following Rayleigh-type quotients$$\mathcal{Q}_{\mathbf{p}}(u) = \frac{\|\nabla u\|_{\mathbf{p}}}{\|u\|_{\mathbf{p}}}\quad\text{and}\quad\mathcal{Q}_{\mathbf{s},\mathbf{p}}(u) = \frac{[u]_{\mathbf{s},\mathbf{p}}}{\|u\|_{\mathbf{p}}},$$ where $\mathbf{p} = (p_1,\dots,p_n)$, $\mathbf{s}=(s_1,\dots,s_n)$ and$$\|\nabla u\|_{\mathbf{p}} = \sum_{i=1}^n \|u_{x_i}\|_{p_i}$$ is an anisotropic Sobolev norm, $[u]_{\mathbf{s},\mathbf{p}}$ is a fractional version of the same anisotropic norm, and$$\|u\|_{\mathbf{p}} =\bigg(\int_{\mathbb R}\bigg(\dots \bigg(\int_{\mathbb R}|u|^{p_1}dx_1\bigg)^{{p_2}/{p_1}}dx_2\dots \bigg)^{p_n/p_{n-1}}dx_n\bigg)^{1/p_n}$$ is an anisotropic Lebesgue norm. Using the Lusternik-Schnirelmann theory, we prove the existence of a sequence of critical values and we also find an associated Euler-Lagrange equation for critical points. Additionally, we analyze the connection between the fractional critical values and its local counterparts.

This work tackles a class of nonlinear parabolic equations in divergence form having fractional time order derivatives. We consider parabolic equations involving second-order spacial operators with Riemann-Liouville time-fractional derivatives. We establish the existence and uniqueness of weak solutions to the studied models. Our theoretical approach relies essentially on the use of Leray-Schauder topological degree and involves some new technical estimates.

In this paper, we consider and study the concept of the Banach sphere. The usual spherical distance of the unit sphere of a Hilbert space is defined by its inner product and the inverse of the cosine function. Therefore, we cannot apply this notion to Banach spheres in general. We first introduce a two-variable function like a spherical distance and a notion of convex combination on a Banach sphere. After that, we define a projection onto a subset of a Banach sphere, and prove a fixed point theorem and fixed point approximation for a mapping having a kind of nonspreadingness. Our work is a challenge to construct optimisation theory on spherical planes without geodesics.

Under mild assumptions on the kernel $K\ge0$, the non-local $K$-perimeter $P_K$ satisfies the monotonicity property on nested convex bodies; i.e. if $A\subset B\subset\mathbb{R}^n$ are two convex bodies, then $P_K(A)\le P_K(B)$. In this note, we prove quantitative lower bounds on the difference of the $K$-perimeters of $A$ and $B$ in terms of their Hausdorff distance, provided that $K$ satisfies suitable symmetry properties.

In this work we study the following fractional critical problem\begin{align*}\begin{cases}(-\Delta)^{s}u=\lambda|u|^{q-2}u+|u|^{2_{s}^{\ast}-2}u& \mbox{in } \Omega, \\u=0 & \text{in }\mathbb{R}^{N}\setminus \Omega,\end{cases}\end{align*}where $\Omega$ is an open unbounded strip-like domain in $\mathbb{R}^{N}$, $N> 2s$ with $s\in(0,1)$, $\lambda> 0$, $q\in[2,2_{s}^{\ast})$ and $2_{s}^{\ast}=2N/(N-2s)$. By variational methods, we prove the existence of positive ground state solutions to the problem. Further, we study the regularity of these solutions. Precisely, using a Brézis-Kato type estimate for unbounded domains, we establish the $L^{\infty}$-bound on nonnegative solutions of the equation for certain range of $q$. The present work extends the existence and regularity results for fractional Laplace equations to unbounded domains.

We study the existence of positive ground or bound state solutions for a class of nonlinear Schrödinger equations:\begin{equation*}\label{P_a}-\Delta u+\lambda u=a(x)f(u), \quad u \in H^1\big(\mathbb{R}^N\big), \ N \geq 3,\end{equation*}where the function a is positive, symmetric under some group action $G$,and $\lambda$ is a positive constant. The inhomogeneous nonlinearity $f$, under very mild assumptions, is asymptotically linear or superlinear and subcritical at infinity, with $f(s)/s$, $s> 0$, not satisfying any monotonicity condition.

This corrigendum makes a correction to K. Salame, Weakly almost periodic functions invariant means and fixed point properties in locally convex topological vector spaces, Topol. Methods Nonlinear Anal. 60 (2022), 135-152, and brings some new contributions to this open problem.

This paper is devoted to the study of normalized solutions to $p$-Laplacian equation involving Sobolev critical exponent. Under the focussing condition, we obtain existence of ground states in mass subcritical case, mass critical case and mass supercritical case, respectively. Furthermore, we show some asymptotic behaviors of ground states. Meanwhile, under the defocussing condition, we prove some non-existence results.