Nontrivial Solutions Research Articles

A Critical p(x)-Laplacian Steklov Type Problem with Weights

ABSTRACT This paper is concerned with the existence of nontrivial weak solutions for a class of p(x)-Laplacian problem under Steklov boundary condition. The variable exponent theory of generalized Lebesgue-Sobolev spaces and the concentration-compactness principle for weighted variable exponent spaces are used for this purpose.

Nonexistence of Nontrivial Weak Solutions of Some Nonlinear Inequalities with Gradient Nonlinearity

Nonexistence of Nontrivial Weak Solutions of Some Nonlinear Inequalities with Gradient Nonlinearity

Optimal existence, uniqueness and blow-up for a quasilinear diffusion equation with spatially inhomogeneous reaction

Well-posedness and a number of qualitative properties for solutions to the Cauchy problem for the following nonlinear diffusion equation with a spatially inhomogeneous source∂tu=Δum+|x|σup, posed for (x,t)∈RN×(0,T), with exponents 1<p<m and σ>0, are established. More precisely, we identify the optimal class of initial conditionsu0 for which (local in time) existence is ensured and prove non-existence of solutions for the complementary set of data. We establish then (local in time) uniqueness and a comparison principle for this class of data. We furthermore prove that any non-trivial solution to the Cauchy problem blows up in a finite timeT∈(0,∞) and finite speed of propagation holds true for t∈(0,T): if u0∈L∞(RN) is an initial condition with compact support and blow-up time T>0, then u(t) is compactly supported for t∈(0,T). We also establish in this work the absence of localization at the blow-up time T for solutions stemming from compactly supported data.

Five nontrivial solutions of superlinear elliptic problem

In this paper, we consider the following superlinear elliptic problem(P){−Δu=λ|u|p−2u+f(x,u),inΩ,u=0,on∂Ω, where λ>0 and 2<p<2+δ for some δ>0 small. The nonlinearity f satisfies the Ambrosetti-Rabinowitz condition and other appropriate hypotheses such that u=0 is a local minimizer of the associated energy functional of equation (P).Our main novelties are threefold. Firstly, using the properties of Gromoll-Meyer pairs in Morse theory, we prove that equation (P) has at least one nontrivial solution close to 0. Moreover, four nontrivial solutions are obtained with assumptions on f at infinity, and none of these solutions depends on the gaps of consecutive eigenvalues of operator −Δ. Therefore, our results differ significantly from those of the paper by Li and Li (2016) [16]. Secondly, under the assumptions of the paper above, we can obtain the existence of a fifth nontrivial solution of equation (P) for λ=1. Finally, by using minimax methods and Morse theory, we also obtain the existence of five nontrivial solutions of equation (P) based on the relationship between parameter λ and eigenvalues of operator −Δ.

Existence of Ground State Solutions for Choquard Equation with the Upper Critical Exponent

In this article, we investigate the existence of a nontrivial solution for the nonlinear Choquard equation with upper critical exponent see Equation (6). The Riesz potential in this case has never been studied. We establish the existence of the ground state solution within bounded domains Ω⊂RN. Variational methods are used for this purpose. This method proved to be instrumental in our research, enabling us to address the problem effectively. The study of the existence of ground state solutions for the Choquard equation with a critical exponent has applications and relevance in various fields, primarily in theoretical physics and mathematical analysis.

Model based component development and analysis with ComMA

The lack of explicit and precise specifications of software interfaces between components often leads to integration issues during development and maintenance. To address this, we have developed a framework named ComMA (Component Modeling and Analysis) that supports model-based engineering of high-tech systems by precisely defining components and their interfaces. The framework is a family of Domain Specific Languages (DSLs) for modeling component interfaces, protocol state machines, time and data constraints, and constraints on relations between events of multiple interfaces. From these models a number of artifacts can be generated automatically to support analysis and various engineering tasks. ComMA has been developed in close collaboration with the Philips IGT business unit that develops minimally-invasive X-ray systems. This paper presents the experience we gained in creating the ComMA framework and its application in industrial practice. We describe and reflect on the technical, organizational and process-related aspects of deploying a non-trivial MDE solution in an industrial setting.

Nondegeneracy implies the existence of parametrized families of free boundaries

In this paper, we introduce the notion of variational free boundary problem. Namely, we say that a free boundary problem is variational if its solutions can be characterized as the critical points of some shape functional. Moreover, we extend the notion of nondegeneracy of a critical point to this setting. As a result, we provide a unified functional-analytical framework that allows us to construct families of solutions to variational free boundary problems whenever the shape functional is nondegenerate at some given solution.As a clarifying example, we apply this machinery to construct families of nontrivial solutions to the two-phase Serrin's overdetermined problem in both the degenerate and nondegenerate case.

Analysis of a reaction–diffusion dengue model with vector bias on a growing domain

In this paper, we consider a reaction–diffusion dengue model on a varying domain that monotonically increases in time and gradually approaches saturation arising from environmental change. By the upper and lower solutions, comparison principle, asymptotic autonomous semiflows and the technique of Lyapunov function, we investigate the stabilities of equilibria in terms of the basic reproduction number . The results show that (i) if , the nontrivial solutions starting from the upper and lower solutions of the model approach to the set formulated by the maximal and minimal solutions of its related elliptic problem; (ii) the disease-free equilibrium is globally asymptotically stable when . Comparing our problem in different settings including growing domain, fixed domain and without spatial structure, our results demonstrate that the disease can spread in the growing domain, while vanish in the fixed domain; and the spatial model decreases the transmission risk compared with the system without spatial structure.

Ricci Vector Fields

We introduce a special vector field ω on a Riemannian manifold (Nm,g), such that the Lie derivative of the metric g with respect to ω is equal to ρRic, where Ric is the Ricci tensor of (Nm,g) and ρ is a smooth function on Nm. We call this vector field a ρ-Ricci vector field. We use the ρ-Ricci vector field on a Riemannian manifold (Nm,g) and find two characterizations of the m-sphere Smα. In the first result, we show that an m-dimensional compact and connected Riemannian manifold (Nm,g) with nonzero scalar curvature admits a ρ-Ricci vector field ω such that ρ is a nonconstant function and the integral of Ricω,ω has a suitable lower bound that is necessary and sufficient for (Nm,g) to be isometric to m-sphere Smα. In the second result, we show that an m-dimensional complete and simply connected Riemannian manifold (Nm,g) of positive scalar curvature admits a ρ-Ricci vector field ω such that ρ is a nontrivial solution of the Fischer–Marsden equation and the squared length of the covariant derivative of ω has an appropriate upper bound, if and only if (Nm,g) is isometric to m-sphere Smα.

On stable solutions of a weighted elliptic equation involving the fractional Laplacian

In this paper, we study the following fractional Choquard equation with weight where and is a positive weight function satisfying at infinity, for some . We establish, in this paper, a Liouville type theorem saying that if then the above equation has no nontrivial stable solution. Our result, in particular, extends the result in [Le, Phuong. Bull. Aust. Math. Soc. 102 (2020), no. 3, 471–478.] from the Laplace operator to the fractional Laplacian.

Two results on 𝑥^{𝑟}+𝑦^{𝑟}=𝑑𝑧^{𝑝}

This note proves two theorems regarding Fermat-type equation x r + y r = d z p x^r + y^r = dz^p where r ≥ 5 r \geq 5 is a prime. Our main result shows that, for infinitely many integers d d , the previous equation has no non-trivial primitive solutions such that 2 ∣ x + y 2 \mid x+y or r ∣ x + y r \mid x+y , for a set of exponents p p of positive density. We use the modular method with a symplectic argument to prove this result.

Large time behavior of signed fractional porous media equations on bounded domains

Following the methodology of Brasco (Adv Math 394:108029, 2022), we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane–Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.

Big-bang limit of 2 + 1 gravity and Thurston boundary of Teichmüller space

We study the asymptotic behavior of the solution curves of the dynamics of spacetimes of the topological type Σp×R, p &amp;gt; 1, where Σp is a closed Riemann surface of genus p, in the regime of 2 + 1 dimensional classical general relativity. The configuration space of the gauge fixed dynamics is identified with the Teichmüller space (TΣp≈R6p−6) of Σp. Utilizing the properties of the Dirichlet energy of certain harmonic maps, estimates derived from the associated elliptic equations in conjunction with a few standard results of the theory of the compact Riemann surfaces, we prove that every non-trivial solution curve runs off the edge of the Teichmüller space at the limit of the big bang singularity and approaches the space of projective measured laminations/foliations (PMLPMF), the Thurston boundary of the Teichmüller space.

Small perturbations of critical nonlocal equations with variable exponents

Abstract In this article, we are concerned with the following critical nonlocal equation with variable exponents: ( − Δ ) p ( x , y ) s u = λ f ( x , u ) + ∣ u ∣ q ( x ) − 2 u in Ω , u = 0 in R N \ Ω , \left\{\begin{array}{ll}{\left(-\Delta )}_{p\left(x,y)}^{s}u=\lambda f\left(x,u)+{| u| }^{q\left(x)-2}u&amp; \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0&amp; \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right. where Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with Lipschitz boundary, N ≥ 2 N\ge 2 , p ∈ C ( Ω × Ω ) p\in C(\Omega \times \Omega ) is symmetric, f : C ( Ω × R ) → R f:C\left(\Omega \times {\mathbb{R}})\to {\mathbb{R}} is a continuous function, and λ \lambda is a real positive parameter. We also assume that { x ∈ R N : q ( x ) = p s ∗ ( x ) } ≠ ∅ \left\{x\in {{\mathbb{R}}}^{N}:q\left(x)={p}_{s}^{\ast }\left(x)\right\}\ne \varnothing , and p s ∗ ( x ) = N p ˜ ( x ) ⁄ ( N − s p ˜ ( x ) ) {p}_{s}^{\ast }\left(x)=N\tilde{p}\left(x)/\left(N-s\tilde{p}\left(x)) is the critical Sobolev exponent for variable exponents. We prove the existence of non-trivial solutions in the case of low perturbations ( λ \lambda small enough) by using the mountain pass theorem, the concentration-compactness principles for fractional Sobolev spaces with variable exponents, and the Moser iteration method. The features of this article are the following: (1) the function f f does not satisfy the usual Ambrosetti-Rabinowitz condition and (2) this article contains the presence of critical terms, which can be viewed as a partial extension of the previous results concerning the the existence of solutions to this problem in the case of s = 1 s=1 and subcritical case.

Non-Trivial Periodic Solutions for a Class of Second Order Differential Equations with Large Delay

Non-Trivial Periodic Solutions for a Class of Second Order Differential Equations with Large Delay

Periodic solutions of seasonal epidemiological models with information‐dependent vaccination

In this paper, we study nontrivial periodic solutions for seasonal epidemiological models with information‐dependent vaccination. Using Leray–Schauder degree theory, we establish new results concerning the existence of at least one nontrivial periodic solution for the and models with seasonality, state‐dependent vaccination coverage and delayed information. Some numerical simulations are presented to illustrate the effect of such vaccination coverage.

Maxwell-scalar system and scalar field with impurity: Novel approach to find nontrivial solutions

In this work, we investigate a Maxwell-scalar model that couples the scalar and gauge fields through the electric permittivity and another model, in which the scalar field lives in the presence of impurity. By considering a single spatial dimension, we determine the conditions under which the model with impurity can be seen as an effective model for the Maxwell-scalar system, having similar solutions. This correspondence shows that the impurity can be used to describe the presence of a charge density, and we use it to verify that the impurity-free case, which supports minimum energy configurations, is related to the case of point charges. We also investigate a class of impurities which modifies the core of the scalar field, and find the corresponding nontrivial charge densities and electric fields. In particular, the asymptotic behavior in terms of the impurity is also studied, leading to solutions exhibiting long-range or quasi-compact profile.

On Abel’s Problem and Gauss Congruences

Abstract A classical problem due to Abel is to determine if a differential equation $y^{\prime}=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta $ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $\eta $, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $\eta $ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty \}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y^{\prime}=\eta y$ if and only if the coefficients of the Puiseux expansion of $x\eta (x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y^{\prime}=\eta y$ with an algebraic solution when $x\eta (x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present two other applications, namely to diagonals of rational fractions and to directed two-dimensional walks.

Concentration-compactness type principle for systems with critical terms in [formula omitted

In this paper, we obtain some important variants of the Lions and Chabrowski Concentration-compactness principle, for nonlinear systems in the context of fractional Sobolev spaces with variable exponents. As an application of the results, we show the existence and asymptotic behaviour of nontrivial solutions for elliptic systems involving a new class of general nonlocal integrodifferential operators with variable exponents and critical growth conditions in RN.

A class of elliptic problems driven by a mixed local-nonlocal operator

In this paper, we consider a class of elliptic problems driven by a mixed local-nonlocal operator. By estimating the critical groups and using Morse theory, the existence of nontrivial solutions is obtained in different cases. Our results extend some classical theorems for semilinear elliptic equations to the mixed local-nonlocal setting.