In this paper, we show the existence of infinitely many solutions for Kirchhoff-type variable-order fractional Laplacian problems involving variable exponents. More precisely, consider M([u]s(⋅)...

In this paper, we study the existence of multi-bump solutions for following Schrödinger–Bopp–Podolsky system with steep potential well: <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="block"> <a:mrow> <a:mo>{</a:mo> <a:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <a:mtr> <a:mtd> <a:mo>−</a:mo> <a:mi mathvariant="normal">Δ</a:mi> <a:mi>u</a:mi> <a:mo>+</a:mo> <a:mo stretchy="false">(</a:mo> <a:mi>λ</a:mi> <a:mi>V</a:mi>...

&lt;abstract&gt;&lt;p&gt;In the present paper, we study following Kirchhoff-Schrödinger-Poisson system with logarithmic and critical nonlinearity:&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{align} \begin{array}{ll} \left \{ - \Bigr(a+b\int_\Omega|\nabla u|^2{\mathrm{d}}x \Bigr)\Delta u+V(x)u-\frac{1}{2}u\Delta (u^2)+\phi u = \lambda |u|^{q-2}u\ln|u|^2+|u|^4u, &amp;amp;x\in \Omega, \\ -\Delta \phi u^2,&amp;amp; x\in 0,&amp;amp;...

In this paper, we study the Kirchhoff type equation −a+b∫R3|∇u|2dxΔu+V(x)u=f(x,u),x∈R3,u∈H1(R3), where a, b>0 are constants and V(x) is a given positive potential. The nonlinearity covers pure power f(x,u)=Q(x)|u|p−2u with 2<p<4, case in which few existence results of sign-changing solutions known. By using method invariant sets descending flow, obtain solution for problem. Furthermore, if f odd respect to u, prove that problem admits infinitely many solutions.

In the present paper, we deal with following fractional Kirchhoff–Schrödinger–Poisson system logarithmic and critical nonlinearity: (a+b[u]s2)(−Δ)su+V(x)u+ϕu=λ|u|q−2uln⁡|u|2+|u|2s∗−2u,x∈Ω,(−Δ)tϕ=u2,x∈Ω,u=0,x∈R3∖Ω, where s∈34,1,t∈(0,1),λ,a,b>0,4<q<2s∗, [u]s2=∫R3∫R3|u(x)−u(y)|2|x−y|3+2sdxdy, Ω is a bounded domain in R3 Lipschitz boundary. Combining constraint variational methods, topological degree theory quantitative deformation arguments, prove that above problem has least energy...

Abstract We describe the group of all reflection-preserving automorphisms an imprimitive complex reflection group. also study some properties this automorphism

In this paper, we consider the following Schrödinger‐Poisson system: urn:x-wiley:mma:media:mma5694:mma5694-math-0001 where parameters α , β ∈(0,3), λ &gt;0, and are Hardy‐Littlewood‐Sobolev critical exponents. For &lt; prove existence of nonnegative groundstate solution to above system. Moreover, applying Moser iteration scheme Kelvin transformation, show behavior at infinity. &gt;0 small, apply a perturbation method study solution. is particular value, infinitely many solutions

{We study the existence and asymptotic behavior of positive solutions for following fractional magnetic Kirchhoff equation with steep potential well \begin{align*} \renewcommand{\arraystretch}{1.25} \begin{array}{ll} \ds \left \{ \left(a+b\int_{\R^{3}}|(-\Delta)_A^\frac{s}{2}u|^2\, dx\right)(-\Delta)_A^s u+V_{\lambda}u=|u|^{p-2}u~~~in~\mathbb{R}^3, \\ u\in H^s(\R^3), \end{array} \right . \end{align*} where $a,~b&gt;0$ are constants, $2

\begin{abstract} {In this paper we consider the following fractional Kirchhoff equation with steep potential well \begin{align*} \renewcommand{\arraystretch}{1.25} \begin{array}{ll} \ds \left \{ \left(a+b\int_{\R^{3}}|(-\Delta)^\frac{s}{2}u|^2\, dx\right)(-\Delta)^s u+\lambda V(x)u=|u|^{p-2}u,\,\,x\in\mathbb{R}^3, \\ u\in H^s(\R^3), \end{array} \right . \end{align*} where $a&gt;0$ is a constant, $b$ and $\lambda$ are positive parameters. $2