We consider a critical problem in a bounded domain involving the p-Grushin operator. After a truncation argument, we obtain infinitely many solutions to our problem via Krasnoselskii's genus, extending a previous result of Garcia Azorero and Peral Alonso in [13] to the p-Grushin operator. A central part of our analysis is the verification of the Palais-Smale condition of the associated functional under a certain level. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/59/abstr.html
. Classical infectious disease models often neglect the interplay of spatial diffusion, infection-age structure, and time-dependent controls. We address this gap by analyzing a spatiotemp oral SIQRV reaction-diffusion model structured by infection age, incorporating vaccination, quarantine, and waning immunity. The simultaneous inclusion of nonlo cal infection-age structure, spatial diffusion, and time-dependent controls introduces major analytical challenges, including coupled transport-diffusion dynamics, nonlinear nonlo cal incidence, and the derivation of optimality conditions in infinite-dimensional spaces. Using semigroup theory and spectral methods, we establish well-posedness, derive the basic reproduction number R0, and analyze equilibrium stability. An optimal control framework is introduced with vaccination, social distancing, and quarantine as time-varying interventions. Necessary conditions are derived via an adjoint system and solved using a forward-backward sweep method. Simulations with measleslike parameters show that spatially targeted, age-structured interventions effectively suppress outbreaks and improve resource allocation. This study underscores the importance of integrating spatial and temporal heterogeneities in epidemic control strategies.
In this work we study the traveling wave solutions for a generalized nonlocal dispersal Leslie-Gower predator-prey system. Employing the truncation method and Schauder fixed-point theorem, we establish the existence of traveling waves connecting the semi-trivial equilibrium to the coexistence equilibrium for all wave speeds $c \geq c^*$, where $c^*$ denotes the minimal wave speed. Particularly, the right-hand tail limit of the wave profile is obtained by using the idea of a contracting rectangle. Meanwhile, by analyzing the characteristic equation obtained from linearizing the system at the semi-trivial equilibrium, we showed the nonexistence of traveling waves for $0
This article concerns the singularly of perturbed Gross-Pitaevskii equations in trapped dipolar quantum gases, $$\displaylines{ - \varepsilon^2 \Delta u + V(x)u +{\lambda _1}{| u |^2}u + {\lambda _2}(K * {| u |^2})u= 0 \quad \text{in } \mathbb{R}^3, \cr u > 0,\quad u \in {H^1}({\mathbb{R}^3}), }$$ where $\varepsilon $ is a small positive parameter, $\lambda_1, \lambda_2\in \mathbb{R}$, $*$ denotes the convolution, $ K(x) = \frac{{1 - 3{{\cos }^2}\theta }}{{{{| x |}^3}}}$ and $\theta = \theta(x)$ is the angle between the dipole axis determined by $(0,0,1)$ and the vector $x$. Moreover, the potential $V$ satisfies $\liminf_{|x| \to \infty } V(x) > \inf_{\mathbb{R}^3} V(x) = 0$. Under certain assumptions on $(\lambda_1, \lambda_2)\in \mathbb{R}^2$, we construct a family of positive solutions ${u_\varepsilon } \in {H^1}({\mathbb{R}^3})$ whose ${L^\infty }$ norm approaches $0$ as $\varepsilon \to 0$. Our main results extend the results in Byeon and Wang [6] which dealt with singularly perturbed Schrodinger equations with a local nonlinearity, to the nonlocal Gross-Pitaevskii type equation.
This article concerns the nonlinear Schrodinger equation (NLS) on $\mathbb{R}^3$ with subcubic and cubic nonlinearities. For the cubic NLS, with initial data in $H^m(\mathbb{R}^3)$, we establish exponential growth of high-order Sobolev norms by using modified energy functionals and lossless Strichartz estimates. For the subcubic NLS, with data in $H^2(\mathbb{R}^3)$, we establish polynomial growth.
This article concerns the stability and long-time behavior of 3D anisotropic micropolar systems with fractional dissipation in R3. By using delicate energy estimates, we prove the stability of solutions to this system when the Hm (m >= 2) norm of the initial data is suitably small, and obtain the decay rate for the horizontal derivatives of this solution.
In this article, we study elliptic systems consisting of $p$-Laplacian equations and $q$-Laplacian equations (for short $(p,q)$-Laplacian elliptic system). By employing topological methods, we discuss the existence, nonexistence, and multiplicity of positive solutions depending on multiple parameters. We also study the asymptotic behavior of the solutions.
We study the existence of nontrivial weak solutions for anisotropic $p$-Laplacian systems. A sequence of eigenvalues is characterized through the $\mathbb{Z}_2$-cohomological index on a suitable constraint manifold of Finsler type. By combining cohomological local splitting with Morse theoretic tools, we prove the existence of nontrivial solutions to the anisotropic $p$-Laplacian system under local assumptions on the nonlinear terms, both in the nonresonant and resonant cases.
We study a low-dimensional system of ordinary differential equations modeling the interactions between nature and society. The variables are renewable and non-renewable resources (nature), and population, wealth and pollution (society). We find equilibria for the system and study their stability. Also we obtain several results on the stability of trajectories, possible collapse trajectories, and societal "safe-harbors."
We prove the nonlinear ill-posedness of Kelvin-Helmholtz problems for compressible Euler fluids with a radiation field. For this we assume that the magnitude of the Mach number is between some small enough constant $\epsilon_0$ and $\sqrt{2(1+\ell)}$ in which $\ell$ is the radiation parameter.
We study the following two-species doubly degenerate nutrient-taxis system in a smoothly bounded domain \(\Omega \subset \mathbb{R}^2\) under homogeneous Neumann boundary conditions: $$\displaylines{ (u_1)_t = \nabla \cdot(u_1 v \nabla u_{1}) -\chi_1 \nabla \cdot(u_1^2 v \nabla v) + \ell_1 u_1v, \quad x \in \Omega, \; t > 0, \cr (u_2)_t = \nabla \cdot(u_2 v \nabla u_2) -\chi_2 \nabla \cdot(u_2^2 v \nabla v) + \ell_2 u_2 v, \quad x \in \Omega,\; t > 0, \cr v_t = \Delta v - (u_1 + u_2) v, \quad x \in \Omega, \; t > 0, }$$ with \(\chi_i,\ell_i > 0\) for \(i = 1, 2\). This system describes the spatio-temporal dynamics of two interacting populations which consume nutrients. We show that for all reasonably regular initial data, the system possesses a global bounded weak solution. We also identify a criterion regarding the initial smallness of the third component such that the solution stabilizes to a nonconstant steady state.
In this article, we investigate the growth of solutions of higher-order complex linear differential equations in the unit disc, with analytic coefficients of finite $(\alpha,\beta,\gamma)$-order. By using the concepts of $(\alpha ,\beta ,\gamma )$-order and $(\alpha,\beta,\gamma)$-type, we establish new results concerning the growth of such solutions. Our results extend and generalize earlier works of Heittokangas et al., Hamouda, Semochko, Tu and Huang, as well as those of the second author with Biswas.
We study nonlo cal elliptic problems driven by the fractional (p1(x, y), p2(x, y))-Laplacian operator under Dirichlet boundary conditions, where p1(& centerdot;, & centerdot;) and p2(& centerdot;, & centerdot;) are continuous functions defined on a bounded domain ohm subset of R-N (N >= 2). The model includes indefinite weight functions, which may change the sign within the domain. By applying variational methods, we establish the existence of at least one nontrivial weak solution. Our results extend recent contributions in the literature on nonlo cal problems with variable exponent operators, and provide new insights into the interaction between fractional order, and sign-changing weights.
This article studies the Cauchy problem of the 3D generalized incompressible magnetohydrodynamic equations in critical Fourier-Triebel-Lizorkin-Morrey spaces. The introduction of the Fourier-Triebel-Lizorkin-Morrey spaces facilitates the estimation of nonlinear terms in the system via Fourier transforms. Moreover, the Fourier-Triebel-Lizorkin-Morrey spaces are strictly larger than the Fourier-Triebel-Lizorkin spaces. When the initial data are sufficiently small, the global well-posedness of solutions to the Cauchy problem for the 3D generalized incompressible magnetohydrodynamic equations is established using the Littlewood-Paley theory and the Banach-Picard contraction principle. Furthermore, we derive Gevrey-class regularity of the solutions in the Fourier-Triebel-Lizorkin-Morrey spaces.
In this article, we study the degenerated quasilinear elliptic equations -div(B(x, v(1), Dv(1))) + g(x, v(1)) = f (x) in Omega(1), - div(B(x, v(2), Dv(2))) + g(x, v(2)) = f (x) in Omega(2), v(1) = 0 on partial derivative Omega, B(x, v(1), Dv(1))center dot nu(1) = B(x, v(2), Dv(2)) center dot nu 1 on Gamma, B(x, v(1), Dv(1))center dot nu(1) = -h(x)|v(1)-v(2)|p-2(v(1)-v(2)) on Gamma, where Omega is a connected bounded open set of R-N (N >= 2), and can be decomposed as Omega = Omega(1) boolean OR Omega(2) boolean OR Gamma, where Omega(2) is an open subset included in Omega, and Omega(1) = Omega\(Omega(2)) over bar ( )and Gamma = partial derivative Omega(2), with 2 + lambda - 1/N< p < N, and also we assume that g(center dot, s) satisfying some growth condition. We will show the existence of entropy solutions for this class of equation and we conclude some regularity results.
. The real Jacobian conjecture in j2 claims that if F = (f, g) : j2 -> j2 is a polynomial map such that det DF(x, y) =/ 0 for all (x, y) is an element of j2, then F is globally injective. However, it is known that there exists a counterexample on this conjecture. Since then, various sufficient conditions have been proposed to ensure that the real Jacobian conjecture holds. In this article, we generalize a result in [26] to the quasi-homogeneous case. Moreover, we provide several examples to illustrate the relationships between some existing sufficient conditions regarding this conjecture.
. This article studies approximations to the set of trajectories, attainable sets and integral funnel of a control system described by an ordinary differential equation. It is assumed that the equation is nonlinear with respect to the phase state vector and affine with respect to the control vector. The system includes control functions, some of which satisfy the Lp (p is an element of (1, infinity)) norm constraint, while the others satisfy the L infinity norm constraint. Step by step, the set of admissible control functions is replaced by a set consisting of a finite number of piecewiseconstant control functions that generate a finite number of trajectories. Error evaluations are provided for the Hausdorff distances between the set of trajectories, attainable sets, integral funnel, and their approximations, which depend on discretization parameters.
This article concerns the well posedness of solutions for model of a three-dimensional non-homogeneous incompressible nematic liquid crystal flows with density-dependent viscosity. We establish the existence of global strong solutions when the initial data satisfies (rho 0, u0, del d0) is an element of L infinity(R3) & times; H(center dot)1/2(R3) & times; H(center dot)1/2(R3), and uniqueness when (rho 0, u0, del d0) is an element of B(center dot)3/qq,1(R3) & times; B(center dot)1/2 2,1 (R3) & times; B(center dot)1/2 2,1 (R3). These results refines the corresponding results obtained by Ye and Zhang [23].
. This article explores the existence of weak solutions to a parabolic problem governed by the tau-Laplacian-like operator -triangle & ell;tau S and a nonlinear source term rl div cb(y, t, S). Under suitable growth conditions on the nonlinear function cb, we ensure that the weak formulation of the problem is well- posed, leading to the existence result. This result is obtained through the application of Galerkin's approximation technique to build approximate solutions, as well as the Young measures theory, which provides a framework for handling the complexities introduced by the nonlinearity.
In this article, we study the long-time behavior of solutions of the non-autonomous stochastic reaction-diffusion equations with dynamical boundary conditions. When the diffusion terms and drift terms are general nonlinear functions, we prove the existence of the weak pullback mean random attractors for deterministic equations with random initial data and stochastic equations, respectively.
