Standing waves for a class of fractional p‐Laplacian equations with a general critical nonlinearity

In this paper, we concern with the following fractional p‐Laplacian equation with critical Sobolev exponent $$\left\{\begin{array}{l}{\epsilon }^{ps}{\left\{?\mathrm{\Delta }\right.}_p^{s}u+V(x){\left|u\right|}^{p?2}u=\lambda f(x){\left|u\right|}^{q?2}u+{\left|u\right|}^{p_s^{?}?2}u\text{in}{?}^N,\\ u\in W^{s,p}\left\{{?}^N\right.,u>0,\end{array}\right.$$ where ε > 0 is a small parameter,  λ > 0 , N is a positive integer, and N > ps with s ∈ (0, 1) fixed, $1<q\le p,p_s^{?}:=Np/\left(N?ps\right)$. Since the nonlinearity $h(x,u):=\lambda f(x){\left|u\right|}^{q?2}u+{\left|u\right|}^{p_s^{?}?2}u$ does not satisfy the following Ambrosetti‐Rabinowitz condition: $$0<\mu H(x,u):=\mu {\int }_0^{u}h(x,t)dt\le h(x,u)u,x\in {?}^N,0\ne u\in ?,$$ with μ > p , it is difficult to obtain the boundedness of Palais‐Smale sequence, which is important to prove the existence of positive solutions. In order to overcome the above difficulty, we introduce a penalization method of fractional p‐Laplacian type.     

Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions of 2N‐point type

This paper is devoted to the study of a nonlinear wave equation with initial conditions and nonlocal boundary conditions of 2N‐point type, which connect the values of an unknown function u(x,t) at x = 1, x = 0, x = η_i(t) , and x = θ_i(t), where $0<{\eta }_1(t)<{\eta }_2(t)<\dots <{\eta }_{{}_{N?1}}(t)<1,$ $0<{\theta }_1(t)<{\theta }_2(t)<\dots <{\theta }_{{}_{N?1}}(t)<1,$ for all t ≥ 0. First, we prove local existence of a unique weak solution by using density arguments and applying the Banach's contraction principle. Next, under the suitable conditions, we show that the problem considered has a unique global solution u(t) with energy decaying exponentially as t → +∞ . Finally, we present numerical results.     

Order plus size of τ‐critical graphs

Let $G=\left(V,E\right)$ be a $\tau$‐critical graph with $\tau \left(G\right)=t$. Erd?s and Gallai proved that $\mid V\mid \le 2t$ and the bound $\mid E\mid \le \left(\genfrac{}{}{0ex}{}{t+1}{2}\right)$ was obtained by Erd?s, Hajnal, and Moon. We give here the sharp combined bound $\mid E\mid +\mid V\mid \le \left(\genfrac{}{}{0ex}{}{t+2}{2}\right)$ and find all graphs with equality.     

Dynamic stability of a class of fractional-order nonlinear systems via fixed point theory

In this paper, the problem of the uniform stability for a class of fuzzy fractional-order genetic regulatory networks with random discrete delays, distributed delays, and parameter uncertainties is studied. Although there is a portion of literature on using fixed point theorems to study the stability of fractional neural networks, most of them required the fractional order to be in $\left(\frac{1}{2},1\right)$. However, the case of the fractional-order belonging to $(0,\frac{1}{2})$ has not been discussed. To solve it, this work proposes a novel idea of using fixed point theory to study the stability of fuzzy (0,1) order neural networks, the problem of the uniqueness of the solution of the considered genetic regulatory networks is resolved, and a novel sufficient condition to guarantee the uniform stability of above genetic regulatory networks is also derived. Eventually, an example is given to demonstrate that the obtained result is effective.     

Large ∣k∣ Behavior of Complex Geometric Optics Solutions to d-bar Problems

Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations are studied for large values of the spectral parameter k. For potentials $q\in {⟨\cdot ⟩}^{-2}{H}^s\left(ℂ\right)$ for some $s\in \left]1,2\right]$, it is shown that the solution converges as the geometric series in $1/{\left|k\right|}^{s-1}$. For potentials q being the characteristic function of a strictly convex open set with smooth boundary, this still holds with s = 3/2, i.e., with $1/\sqrt{\mid k\mid }$ instead of $1/{\left|k\right|}^{s-1}$. The leading-order contributions are computed explicitly. Numerical simulations show the applicability of the asymptotic formulae for the example of the characteristic function of the disk. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.     

A critical Kirchhoff‐type problem driven by a p (·)‐fractional Laplace operator with variable s (·) ‐order

The paper deals with the following Kirchhoff‐type problem $$\left\{\begin{array}{lll}& M\left({?}_{{?}^{2N}}\frac{1}{p(x,y)}\frac{|v(x)?v(y){|}^{p(x,y)}}{|x?y{|}^{N+p(x,y)s(x,y)}}dxdy\right){(?\mathrm{\Delta })}_{p(·)}^{s(·)}v(x)=\mu g(x,v)+|v{|}^{r(x)?2}v& \text{in}\mathrm{\Omega },\\ & v=0& \text{in}{?}^N\backslash \mathrm{\Omega },\end{array}\right.$$ where M models a Kirchhoff coefficient, ${(?\mathrm{\Delta })}_{p(·)}^{s(·)}$ is a variable s(·) ‐order p(·) ‐fractional Laplace operator, with $s(·):{?}^{2N}\to (0,1)$ and $p(·):{?}^{2N}\to (1,\infty )$. Here, $\mathrm{\Omega }?{?}^N$ is a bounded smooth domain with N > p(x, y)s(x, y) for any $(x,y)\in \overline{\mathrm{\Omega }}\times \overline{\mathrm{\Omega }}$, μ is a positive parameter, g is a continuous and subcritical function, while variable exponent r(x) could be close to the critical exponent $p_s^{?}(x)=N\overline{p}(x)/(N?\overline{s}(x)\overline{p}(x))$, given with $\overline{p}(x)=p(x,x)$ and $\overline{s}(x)=s(x,x)$ for $x\in \overline{\mathrm{\Omega }}$. We prove the existence and asymptotic behavior of at least one non‐trivial solution. For this, we exploit a suitable tricky step analysis of the critical mountain pass level, combined with a Brézis and Lieb‐type lemma for fractional Sobolev spaces with variable order and variable exponent.     

Blow‐up for wave equation with the scale‐invariant damping and combined nonlinearities

In this article, we study the blow‐up of the damped wave equation in the scale‐invariant case and in the presence of two nonlinearities. More precisely, we consider the following equation: $$u_{tt}?\mathrm{\Delta }u+\frac{\mu }{1+t}{u}_t=|u_t{|}^p+|u{|}^q,\text{in}{?}^N\times [0,\infty ),$$ with small initial data. For $\mu <\frac{N(q?1)}{2}$ and μ ∈ (0, μ_?) , where μ_? > 0 is depending on the nonlinearties' powers and the space dimension (μ_? satisfies $(q?1)\left((N+2{\mu }_{?}?1)p?2\right)=4$), we prove that the wave equation, in this case, behaves like the one without dissipation (μ = 0 ). Our result completes the previous studies in the case where the dissipation is given by $\frac{\mu }{{(1+t)}^{\beta }}{u}_t;\beta >1$, where, contrary to what we obtain in the present work, the effect of the damping is not significant in the dynamics. Interestingly, in our case, the influence of the damping term $\frac{\mu }{1+t}{u}_t$ is important.     

Multibump solutions for nonlinear Schrödinger-Poisson systems

In this paper, we study the following Schrödinger-Poisson equations: $$\left\{\begin{array}{ll}-{\epsilon }^2\mathrm{\Delta }u+V(x)u+K(x)\varphi u=|u{|}^{p-2}u,& x\in {\mathbb{R}}^3,\\ -{\epsilon }^2\mathrm{\Delta }\varphi =K(x)u^2,& x\in {\mathbb{R}}^3,\end{array}\right.$$ where $p\in (4,6)$, $\epsilon >0$ is a parameter and $V$ and $K$ satisfy the critical frequency conditions. By using variational methods and penalization arguments, we show the existence of multibump solutions for the above system. Furthermore, the heights of these bumps are different order.     

Poincaré's inequality and Sobolev spaces with monomial weights

In this paper, we use a weighted version of Poincaré's inequality to study density and extension properties of weighted Sobolev spaces over some open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$. Additionally, we study the specific case of monomial weights $w(x_1,\text{…},x_N)={\prod }_{i=1}^N{\left|x_i\right|}^{a_i},a_i\ge 0$, showing the validity of a weighted Poincaré inequality together with some embedding properties of the associated weighed Sobolev spaces.     

Two solutions for a class of singular Kirchhoff‐type problems with Hardy–Sobolev critical exponent II

In this article, we devote ourselves to investigate the following singular Kirchhoff‐type equation: $$\left\{\begin{array}{l}?\left(a+b{\int }_{\mathrm{\Omega }}|?u{|}^{2}dx\right)\mathrm{\Delta }u=\frac{u^{5?2s}}{|x{|}^s}+\frac{\lambda }{|x{|}^{\beta }{u}^{\gamma }},x\in \mathrm{\Omega },\\ u>0,x\in \mathrm{\Omega },\\ u=0,x\in ?\mathrm{\Omega },\end{array}\right.$$ where $\mathrm{\Omega }?{?}^3$ is a bounded domain with smooth boundary ?Ω,0∈Ω,a≥0,b,λ>0,0<γ,s<1, and $0\le \beta <\frac{5+\gamma }{2}.$ By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.     

Norm of a multilinear integral operator from product of weighted-type spaces to weighted-type space

We investigate the following multilinear integral operator $$T_K^m(\overrightarrow{f})(x)={\int }_0^{\infty }\dots {\int }_0^{\infty }K(x,t_1,\dots ,t_m){\displaystyle \prod _{j=1}^m}{f}_j(t_j)d{t}_1\dots d{t}_m,$$ where $m\in ?$ and $K:{?}_{+}^{m+1}\to {?}_{+}$ is a continuous kernel function satisfying the condition $$K(x,g_1(x)s_1,\dots ,g_m(x)s_m)=h(x)K(1,s_1,\dots ,s_m),$$ for some functions $g_j,j=\stackrel{￣}{1,m}$, which are continuous, increasing, $g_j({?}_{+})={?}_{+},j=\stackrel{￣}{1,m}$, and a function $h:{?}_{+}\to {?}_{+}$, from a product of weighted-type spaces to weighted-type spaces of real functions. We calculate the norm of the operator, extending and complementing some results in the literature. We also give an explanation for a relation between integrals of an L^p integrable function and its radialization on ${?}^n$.     

Ground state solutions for a modified fractional Schrödinger equation with critical exponent

In this paper, we study the existence of ground state solutions for the modified fractional Schrödinger equations $${(-\mathrm{\Delta })}^{\alpha }u+\mu u+\kappa [{(-\mathrm{\Delta })}^{\alpha }{u}^2]u=\sigma |u{|}^{p-1}u+|u{|}^{q-2}u,x\in R^N,$$ where $N\ge 2$, $\alpha \in (0,1)$, $\mu$, $\sigma$ and $\kappa$ are positive parameters, $2<p+1<q\le 2_{\alpha }^{\ast }:=\frac{2N}{N-2\alpha }$, ${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian of order $\alpha$. For the case $2<p+1<q<2_{\alpha }^{\ast }$ and the case $q=2_{\alpha }^{\ast }$, the existence results of ground state solutions are given, respectively.     

Regularity and energy conservation for compressible isentropic magnetohydrodynamic equations

In this paper, we mainly study the local energy equation of the weak solutions of the compressible isentropic MHD equation defined on ${\mathrm{??}}^3$. We prove that the regularity of the solution is sufficient to guarantee the balance of the total energy in the $B_3^{\alpha ,\infty }((0,T)\times {\mathrm{??}}^3)$ space. We adopt a variant of the method of Feireisl et al.     

Unique weak solutions of the d‐dimensional micropolar equation with fractional dissipation

This article examines the existence and uniqueness of weak solutions to the d‐dimensional micropolar equations (d=2 or d=3) with general fractional dissipation (?Δ)^αu and (?Δ)^βw. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when $\alpha \ge \frac{1}{2}$ and $\beta \ge \frac{1}{2}$, any initial data (u₀,w₀) in the critical Besov space $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{1+\frac{d}{2}?2\beta }({?}^d)$ yields a unique weak solution. For α ≥ 1 and β=0, any initial data $u_0\in B_{2,1}^{1+\frac{d}{2}?2\alpha }({?}^d)$ and $w_0\in B_{2,1}^{\frac{d}{2}}({?}^d)$ also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely, α=β=1, have a unique weak solution for $(u_0,w_0)\in B_{2,1}^0$. The proof involves the construction of successive approximation sequences and extensive a priori estimates in Besov space settings.     

Positive ground state solutions for Schrödinger–Poisson system with critical nonlocal term in ℝ3

This paper is dedicated to studying the following Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\mathrm{\Delta }u+V(x)u-K(x)\varphi |u{|}^{3}u=a(x)f(u),& x\in {ℝ}^3,\\ -\mathrm{\Delta }\varphi =K(x)|u{|}^5,& x\in {ℝ}^3.\end{array}\right.$$ Under some different assumptions on functions V(x), K(x), a(x) and f(u), by using the variational approach, we establish the existence of positive ground state solutions.