In this work, we consider the following nonlocal quasilinear elliptic problem $$ -M(\mathcal{A}(u))\operatorname{div}\left(a(|\nabla u|^{p(b(u))})|\nabla u|^{p(b(u))-2}\nabla u\right) = f(x,u),$$ with a homogeneous Dirichlet boundary condition. Under suitable conditions on the functions $M,a,p,b$ and $f$, employing an approximation method combined with fixed point arguments, we obtain the existence of a weak solution for the above problem.
Quasilinear boundary value problems with periodic conditions have been extensively investigated because of their importance in nonlinear analysis, mechanics, and dynamical systems. In this paper, we examine a class of second-order quasilinear differential equations with derivative-dependent principal part and periodic boundary conditions. Under a framework of structural and growth assumptions, we prove the existence of infinitely many weak solutions. Our approach is variational and based on Ricceri’s infinite critical points theorem, which ensures the existence of an unbounded sequence of critical points of the associated energy functional. The obtained results generalize and complement recent contributions in the literature concerning one, two, and three-solution problems. Several corollaries and examples are also presented to illustrate the applicability of the main theorem.
In this paper, the bifurcation structure and stability of the nonconstant positive steady states in a ratio-dependent predator-prey model with prey-taxis are described. First, the stability results for the positive constant equilibrium will extend to the derivative of prey's functional response with prey is positive. It is found that attractive prey-taxis can stabilize equilibrium even self-diffusion-driven instability has occurred, and repulsive prey-taxis can destabilize equilibrium. Then, several numerical simulations are performed to visualize the complex dynamic behavior: The model exhibits taxis-controlled spatial patterns such as spots pattern, spots-stripe pattern and stripe pattern. Finally, the stationary problem in one-dimensional domain is investigated. Choosing prey-taxis coefficient as bifurcation parameter, we derive the existence and stability of nonconstant positive steady states by applying Crandall–Rabinowitz bifurcation theory, and obtain the stable bifurcating solutions near the bifurcation point under suitable conditions.
Fractional differential equations with impulses have been studied over the past decades. A methodology often used for the Caputo fractional derivative case is to change the problem into an equivalent integral equation in a space of piecewise continuous functions. One approach is to change the fractional derivative at each impulse point, thereby studying a sequence of problems, one on each subinterval. A second approach is to have one fixed fractional derivative on the whole interval. These approaches have often been confused. In the second case an integral equation was proposed by Fečkan, Zhou and Wang (Commun. Nonlinear Sci. Numer. Simul., 2012). This paper and some successors have been cited a large number of times. However, it is shown here, with simple counter-examples, that the proposed piecewise continuous solutions of the integral equation do not give solutions of the fractional differential equation with an impulse. Moreover, it is proved that no solution of this type is actually possible. A similar approach for the non-instantaneous impulse case is also shown to be not correct. An impulse problem involving the Riemann–Liouville fractional derivative is also discussed.
We study classes of quasilinear singular problems, involving the p -Laplacian operator and a potential, with homogeneous Dirichlet boundary condition. First, we prove the existence and regularity of positive solutions to a purely singular problem. Then, we investigate the existence of positive solutions with respect to a parameter depending on the behavior of the nonlinearities at infinity and at the origin. Finally, we extend some of these results to coupled system of equations. We use sub-super solution techniques to establish our results.
In this work, we consider a one-dimensional linear thermoelastic shear beam model coupled with a heat equation, where the heat conduction is given by the generalized dual-phase-lag model. We establish the well-posedness of the system by using the semigroup theory. Then, the exponential stability is obtained by applying the Gearhart–Herbst–Prüss–Huang theorem under the hypothesis $2\tau _{\theta }>\tau _{q}$. In addition, we conduct several numerical tests to illustrate our theoretical results.
In this paper, a nonlocal-in-time semilinear heat equation is studied. The goal of this paper is prove the existence of insensitizing controls for a function that depends on the L 2 -norm of solutions of the nonlocal-in-time semilinear heat equation. This problem is equivalent to the null controllability result for a nonlinear cascade system of the heat equations with terms that are nonlocal in time. To prove the null controllability of this nonlinear cascade system, we first obtained a Carleman inequality for an adjoint system, and after, we use Liusternik's theorem.
Hand-foot-mouth disease (HFMD) is an infectious disease caused by enteroviruses common in everyone and reinfection can occur. In order to investigate the impact of covert infection and temporary immunity on the transmission of HFMD, we establish an age-structured model, which is based on a system of partial differential equations, describing the interactions among susceptible individuals, latently infected individuals, and temporarily immune individuals. At first, a careful analysis of the existence and local stability of equilibrium states shows the disease-free equilibrium is globally asymptotically stable for R 0 < 1 , the endemic equilibrium is unstable for R 0 > 1 , by means of reformulating the model as an abstract Cauchy problem. Then, an application of the Lyapunov–Schmidt approach reveals the conditions for the occurrence of backward bifurcation, and the numerical simulations of obtained conclusions are examined.
In this paper, we study ordinary differential equations of the Kirchhoff type. These equations exhibit a nonlinearity that is monotonically increasing and depends on one positive parameter. Using a fixed-point theorem and the sub-supersolution method, we prove the existence of positive solutions. In some cases, we also prove the non-existence of positive solutions.
This article concerns the following Klein–Gordon–Maxwell systems with the term of Choquard \begin{equation*} \begin{cases} -\Delta u+ V(x)u-(2\omega+\phi)\phi u=f(x,u)+(I_\alpha \ast |u|^p)|u|^{p-2}u,\quad & x\in \mathbb{R}^{3},\\ \Delta \phi=(\omega+\phi)u^2, \quad & x\in \mathbb{R}^{3}, \end{cases} \end{equation*} and \begin{equation*} \begin{cases} -\Delta u+ V(x)u-(2\omega+\phi)\phi u=\lambda|u|^{q-2}u+(I_\alpha \ast |u|^p)|u|^{p-2}u, \quad & x\in \mathbb{R}^{3},\\ \Delta \phi=(\omega+\phi)u^2, \quad & x\in \mathbb{R}^{3}, \end{cases} \end{equation*} where $\omega> 0$ is a constant, $\lambda>0$ is a parameter, $q\geq 6$, $I_\alpha$ is a Riesz potential whose order is $\alpha$. Under some suitable assumptions on $\alpha$, $p$, $V$ and $f$, we establish the existence and multiplicity of solutions for the systems by variational methods and Moser iteration. Our results extend several recent findings on solutions to these systems.
This paper investigates an elliptic boundary value problem involving the Laplace equation on the unit disk in two-dimensional space, subject to a nonlinear exponential Robin boundary condition on a portion of the boundary. This model is physically motivated by the study of corrosion localized on metallic structures, where the flux law exhibits an exponential dependence on the electrochemical potential. We establish the existence and uniqueness of the solution in an appropriate Hilbert space under suitable smallness assumptions on the boundary data. Our mathematical approach is constructive: we define an iterative sequence of linearized problems and prove its strong convergence toward the solution of the original nonlinear system. A key technical contribution of this work is the derivation of explicit Sobolev embedding constants for the periodic setting on the unit disk, providing refined estimates necessary to control the exponential nonlinearity.
In this paper, we investigate the existence of ground state sign-changing solutions for Kirchhoff-type equation with critical exponent \begin{equation*} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2 dx\right)\Delta u+\lambda V(x)u=|u|^{4}u+\mu f(u),\quad x\in \mathbb{R}^3, \end{equation*} where $a,b,\mu,\lambda>0$ are positive parameters. Under suitable conditions on $f$ and $V$, by working on a sign-changing Nehari manifold and combining minimax arguments with deformation techniques, we establish the existence of least energy sign-changing solutions. At the same time, we also study the asymptotic behavior of sign-changing solutions as $\lambda \to \infty$.
This paper deals with the following logarithmic Schrödinger–Bopp–Podolsky system $$ \begin{cases} - \Delta u+\lambda V(x) u+ q(x)\phi u=u \log u^2, & \text {in } \mathbb{R}^3, \\ - \Delta \phi+ \Delta^2 \phi=4 \pi q(x) u^2, & \text {in } \mathbb{R}^3, \end{cases} $$ where $\lambda>0$, $V(x)\in C(\mathbb{R}^3, \mathbb{R})$ and $q(x) \geq 0$. Under suitable conditions on potentials $V(x)$ and $q(x)$, we demonstrate the existence of a positive solution $u_\lambda \in H^1(\mathbb{R}^3)$ of the above problem for $\lambda>0$ large enough via the variational method developed by Szulkin forhe functional that is the sum of a smooth and a convex lower semicontinuous term, and we also study the concentration behavior of positive solutions $\left\lbrace u_\lambda\right\rbrace $ as $\lambda\rightarrow +\infty$.
In this article, we consider the following quasilinear elliptic equation $$\begin{cases} -\operatorname{div}\dfrac{\nabla u}{(1+u)^{2\theta}} - \dfrac{\gamma\theta|\nabla u|^2}{(1+u)^{1+2\theta}} = u^p,\ x \in \mathbb{R}^N,\\[8pt] u > 0, \ \lim\limits_{|x| \to +\infty} u(x) = 0, \end{cases}$$ where \(\theta > 0\), \(\gamma \in \mathbb{R}\), and \(p > \frac{N+2}{N-2}\). We first perform a change of variables, which is more general because it generalizes the study of the cases \(\gamma = 1\) and \(\gamma = 0\) to all real \(\gamma\). Then, by using the Pohozaev identity in the entire space, we show that the equation does not admit solutions in \(H^1\left(\mathbb{R}^N\right)\) in some cases, which leads us to find solutions in a larger space. Lastly, we introduce a parameter and apply a perturbation method to get a continuum of solutions with decay \(O\big(|x|^{-\frac{2}{p-1}}\big)\) at infinity.
In this study, we develop a delayed predator–prey model that incorporates two stage structures for both predator and prey populations, along with two constant maturation delays and fear effect. The model is formulated as a delay differential equation system, and its well-posedness is rigorously demonstrated. We conduct a comprehensive analysis of the system, including stability evaluations and the identification of Hopf bifurcations caused by the delay terms. Numerical simulations are used to validate and extend the theoretical findings, leading to several key insights. Notably, variations in the delay parameters significantly affect the stability of the system. With a single delay, the system undergoes a transition from stability to oscillation and back to stability as the delay increases, showing a characteristic stability-oscillation-stability pattern. Furthermore, when both delays are taken into account, extrema-based bifurcation diagrams and time series reveal multi-branch structures, periodic windows, and complex irregular oscillatory segments. Delay-embedding phase portraits uncover typical patterns such as period-doubling cascades and the reappearance of periodic windows. Additionally, two types of numerical diagnostics provide further evidence: significant sensitivity to initial histories is observed in the complex segment, while coexisting attractors are detected in the branch-splitting segment, suggesting bistability in the system. These results indicate that stage structure coupled with dual maturation delays can significantly enrich the bifurcation structure and long-term dynamical behavior of the system.
This paper is concerned with normalized solutions for a class of fractional Kirchhoff equations with an inhomogeneous perturbation in $\mathbb{R}^2$. We study the constrained minimization problem associated with the corresponding nonlocal energy functional under a prescribed $L^2$-mass. The interaction between the Kirchhoff term, the fractional Laplacian and the spatially dependent coefficient $f(x)$ leads to several compactness and asymptotic difficulties. For the zero-potential case, we establish existence and nonexistence results for constrained minimizers. When a trapping potential is present, we prove the existence of minimizers in the subcritical case and characterize the threshold in the critical case $p=4s$. In the mass-critical case $p=2s$, we analyze the concentration behavior of non-negative minimizers as the Kirchhoff parameter tends to zero. We prove that the minimizers concentrate at global minimum points of the potential and, after rescaling, converge to the unique positive radial solution of the limiting fractional scalar field equation. A sharper description of the concentration location and the energy asymptotics is obtained when the potential is almost homogeneous near its minimum points.
This study investigates both the homogeneous and non-homogeneous forms of the k -confluent hypergeometric differential equation. In the neighborhood of the regular singular point, series solutions are constructed using the Frobenius method. For the irregular singular point at infinity, an asymptotic analysis is carried out to describe the behavior of the solutions in that region. These analyses lead to the definition and characterization of k -Kummer and k -Tricomi-type hypergeometric functions, which are shown to constitute fundamental solutions of the differential equation. Moreover, an explicit series representation is obtained for the non-homogeneous k -confluent hypergeometric differential equation with a polynomial right-hand side. The results provide analytical tools for obtaining explicit and asymptotic solutions of differential equations of confluent hypergeometric type, including models arising in wave propagation, heat conduction, and quantum mechanics.
In this paper, we prove the existence of nontrivial solutions for perturbed Schrödinger systems with pinching and mixed nonlinearities, where the nonlinearities are composed of sublinear terms, asymptotically linear terms (pinching terms) and superlinear terms. When the system is time-independent and perturbed, we obtain the existence of nontrivial homoclinic solutions. When the system is time-dependent and non-perturbed, we obtain the existence of nontrivial standing waves and homoclinic solutions. In addition, we also give some function examples that are suitable for our models.
This paper establishes new oscillation criteria for odd-order half-linear differential equations with retarded arguments in the non-canonical case. By employing a linearization technique, the problem is reduced to studying an associated linear equation, allowing us to analyze oscillatory behavior via comparison with first-order delay differential equations whose properties are well known. The obtained results generalize, improve, and consolidate several existing criteria in the literature. Examples are included to illustrate the practical utility and effectiveness of the proposed method.
We study the normalized ground state solutions for the following mass critical Kirchhoff equations with inhomogeneous interactions \begin{equation*} \begin{cases} - \Big(a+b\int_{{\mathbb{R}}^N}|\nabla u|^2\mathrm{d}x \Big)\Delta u+V(x)u =\lambda u +\beta m(x)u^{p+1}&\text{in }\mathbb{R}^N, \\ \int_{{\mathbb{R}}^N}|u|^2\mathrm{d}x=1,& \end{cases} \end{equation*} where $a\geq0$, $b>0$, $\beta>0$, $p=\frac{8}{N}$, $00$ such that the constraint minimizers exist if the interaction strength $\beta$ satisfies $\beta\leq\beta^*$ when $a>0$ and if $\beta<\beta^*$ when $a=0$. Further, we investigate the limit behavior of minimizers as $a\to0$ for $\beta=\beta^*$. We also prove that all the mass of the minimizers concentrate at a global minimum point $x_0$ of $V(x)$ as $a\to0$, where $x_0$ is also a global maximum point of $m(x)$. Specially, we find that the limit behavior is related to the behavior of the functions $V(x)$ and $m(x)$ at their critical points.
