Search Results (74)

In this article, we study a double-phase variable-exponent Kirchhoff problem and show the existence of at least three solutions. The proposed model, as a generalization of the Kirchhoff equation, is interesting since it is driven by a double-phase operator that governs anisotropic and heterogeneous diffusion associated with the energy functional, as well as encapsulating two different types of elliptic behavior within the same framework. To tackle the problem, we obtain regularity results for the corresponding energy functional, which makes the problem suitable for the application of a well-known critical point result by Bonanno and Marano. We introduce an n-dimensional vector inequality, not covered in the literature, which provides a key auxiliary tool for establishing essential regularity properties of the energy functional such as $C^1$-smoothness, the $\left(S_{+}\right)$-condition, and sequential weak lower semicontinuity. Full article

In this paper, we investigate the existence of multiple solutions for a Kirchhoff-type equation with Dirichlet boundary conditions defined on locally finite graphs. Our study extends some previous results on nonlinear Laplacian equations to the more complex Kirchhoff equation which incorporates a nonlocal term. By employing an abstract three critical points theorem that is based on Morse theory, we provide sufficient conditions that guarantee the existence of at least three distinct solutions, including two strictly positive solutions. We also present an example to verify our results. Full article

In this paper, we delve into the following nonlinear fractional Kirchhoff-type problem ${(a+b||(-\Delta )}^{{\displaystyle \frac{s}{2}}}{u\left|\right|}_2^2{)(-\Delta )}^{s}u+\lambda u=g\left(u\right)+{\left|u\right|}^{2_s^{*}-2}u$ in ${\mathbb{R}}^3$ with prescribed mass ${\int }_{{\mathbb{R}}^3}{\left|u\right|}^{2}dx=\rho >0,$ where $s\in ({\displaystyle \frac{3}{4}},1),\lambda \in \mathbb{R},2_s^{*}={\displaystyle \frac{6}{3-2s}}$. Under some general growth assumptions imposed on g, we employ minimization of the energy functional on the linear combination of Nehari and Poho$\breve{z}$aev constraints intersected with the closed ball in the $L^2\left({\mathbb{R}}^3\right)$ of radius $\rho$ to prove the existence of normalized ground state solutions to the equation. Moreover, we provide a detailed description for the asymptotic behavior of the ground state energy map. Full article

This paper mainly studies the existence of sign-changing solutions for the following $p\left(x\right)$-biharmonic Kirchhoff-type equations: $-\left(a+b{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{1}{p\left(x\right)}}{|\Delta u|}^{p\left(x\right)}\mathrm{d}x\right){\Delta }_{p\left(x\right)}^{2}u+V\left(x\right){|u|}^{p\left(x\right)-2}u$ = $K\left(x\right)f\left(u\right),x\in {\mathbb{R}}^N$, where ${\Delta }_{p\left(x\right)}^{2}u=\Delta \left({|\Delta u|}^{p\left(x\right)-2}\Delta u\right)$ is the $p\left(x\right)$ biharmonic operator, $a,b>0$ are constants, $N\ge 2$, $V\left(x\right),K\left(x\right)$ are positive continuous functions which vanish at infinity, and the nonlinearity f has subcritical growth. Using the Nehari manifold method, deformation lemma, and other techniques of analysis, it is demonstrated that there are precisely two nodal domains in the problem’s least energy sign-changing solution $u_b$. In addition, the convergence property of $u_b$ as $b\to 0$ is also established. Full article

This study derives the uniqueness of positive solutions to Brézis–Oswald-type problems involving the fractional $(r,q)$-Laplacian operator and discontinuous Kirchhoff-type coefficients. The Brézis–Oswald-type result and Ricceri’s abstract global minimum principle are critical tools in identifying this uniqueness. We consider an eigenvalue problem associated with fractional $(r,q)$-Laplacian problems to confirm the existence of a positive solution for our problem without the Kirchhoff coefficient. Moreover, we establish the uniqueness result of the Brézis–Oswald type by exploiting a generalization of the discrete Picone inequality. Full article

The aim of this paper is to investigate the existence of positive solutions for a discrete Robin problem of the Kirchhoff type involving the p-Laplacian by the means of critical point theory. Our results demonstrate that the problem admits at least three solutions, or at least two solutions under different conditions on the nonlinear term f. We establish a strong maximum principle for the problem and obtain the existence and multiplicity of positive solutions. Finally, we give three examples to verify our results. Full article

In this paper, we have undertaken the challenging and novel task of establishing the existence of weak solutions for four types of hyperbolic Kirchhoff-type problems: the classical hyperbolic Kirchhoff problem, the problem with a free boundary, the problem with a volume constraint, and the problem combining both a volume constraint and a free boundary. These problems are characterized by the presence of non-local terms arising from the Kirchhoff term, the free boundary, and the volume constraint, which introduces significant analytical complexities. To address these challenges, we utilize the discrete Morse flow (DMF) approach, reformulating the original continuous problem into a sequence of discrete minimization problems. This method guarantees the existence of a minimizer for the discretized functional, which subsequently serves as a weak solution to the primary problem. Full article

Nano-structures play a crucial role in advancing technology due to their unique properties and applications in various fields. This study examines the forced vibration behavior of an orthotropic nano-system consisting of an elastically connected nanoplate and a doubly curved shallow nano-shell. Both nano-elements are simply supported and embedded in a Winkler-type elastic medium. Utilizing the Eringen constitutive elastic relation, Kirchhoff–Love plate theory, and Novozhilov’s linear shallow shell theory, we derive a system of four coupled nonhomogeneous partial differential equations (PDEs) describing the forced transverse vibrations of the system. We perform forced vibration analysis using modal analysis. The developed model is a novel approach that has not been extensively researched by other authors. Therefore, we provide insights into the nano-system of an elastically connected nanoplate and a doubly curved shallow nano-shell, offering a detailed analytical and numerical analysis of the PDEs describing transverse oscillations. This includes a clear insight into natural frequency analysis and the effects of the nonlocal parameter. Additionally, damping proportional coefficients and external excitation significantly influence the transverse displacements of both the nanoplate and nano-shell. The proposed mathematical model of the ECSNPS aids in developing new nano-sensors that respond to transverse vibrations based on the geometry of the nano-shell element. These sensors are often used to adapt to curved surfaces in medical practice and gas sensing. Full article

The enumeration of spanning trees in various graph forms has been made easier by the study of electrically equivalent transformations, which was motivated by Kirchhoff’s work on electrical networks. In this work, using knowledge of difference equations, the electrically equivalent transformations and rules of weighted generating function are used to calculate the explicit formulas of the number of spanning trees of novel pyramid graph types based on some nonahedral graphs. Lastly, we compare our graphs’ entropy with that of other average-degree graphs that have been researched. Full article

This paper is devoted to establishing multiplicity results of nontrivial weak solutions to the fractional p-Laplacian equations of the Kirchhoff–Schrödinger type with Hardy potentials. The main features of the paper are the appearance of the Hardy potential and nonlocal Kirchhoff coefficients, and the absence of the compactness condition of the Palais–Smale type. To demonstrate the multiplicity results, we exploit the fountain theorem and the dual fountain theorem as the main tools, respectively. Full article

The aim of this paper is to establish the existence and uniqueness of solutions to non-local problems involving a discontinuous Kirchhoff-type function via a global minimum principle of Ricceri. More precisely, we first obtain the uniqueness result of weak solutions to nonlinear fractional Laplacian problems of Brézis–Oswald type. We then demonstrate the existence of a unique positive solution to Kirchhoff-type problems driven by the non-local fractional Laplacian as its application. The main features of the present paper are the lack of the continuity of the Kirchhoff function in $[0,\infty )$ and the localization of a positive solution. Full article

In the present work, we study a fractional elliptic Kirchhoff-type problem that has a singular term. More precisely, we start by proving some properties related to the energy functional associated with the studied problem. Then, we use the variational method combined with the min–max method to prove that the energy functional reaches its global minimum. Finally, since the energy functional has a singularity, we use the implicit function theorem to show that the point where the minimum is reached is a weak solution for the main problem. To illustrate our main result, we give an example at the end of this paper. Full article

This paper is concerned with the following $L^2$-subcritical Kirchhoff-type equation $-\left(a+b{\left({\int }_{{\mathbb{R}}^2}{|\nabla u|}^{2}dx\right)}^s\right)\Delta u+V\left(x\right)u=\mu u+\beta {\left|u\right|}^{2}u,x\in {\mathbb{R}}^2$, with ${\int }_{{\mathbb{R}}^2}{\left|u\right|}^{2}dx=1$. We give a detailed analysis of the limit property of the $L^2$-normalized solution when exponent s tends toward 0 from the right (i.e., $s\searrow 0$). Our research extends previous works, in which the authors have displayed the limit behavior of $L^2$-normalized solutions when $s=1$ as $a\searrow 0$ or $b\searrow 0$. Full article

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth: $\left\{\begin{array}{c}-M\left({\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p}dx}\right){\Delta }_{p}u={\left|u\right|}^{p^{\ast }-2}u+\lambda {\left|u\right|}^{p-2}u-{\left|u\right|}^{p-2}u\ln {\left|u\right|}^2\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}x\in \mathsf{\Omega },\\ u=0 x\in \mathsf{\partial }\mathsf{\Omega },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$ where $\mathsf{\Omega }\subset {ℝ}^N$ is a bounded domain with a smooth boundary, $2<p<p^{\ast }<N$, and both $p$ and $N$ are positive integers. By using the Nehari manifold and the Mountain Pass Theorem without the Palais-Smale compactness condition, it was proved that the equation had at least one nontrivial solution under appropriate conditions. It addresses the challenges posed by the critical term, the Kirchhoff nonlocal term and the logarithmic nonlinear term. Additionally, it extends partial results of the Brézis–Nirenberg problem with logarithmic perturbation from p = 2 to more general p-Kirchhoff type problems. Full article