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Abstract In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical non‐linearities on an unbounded domain, and show that such an equation with positive or sign‐changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik–Schnirelmann category, we investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets related to the weight potential.

This paper is devoted to studying the existence and nonexistence of traveling wave solution for a nonlocal dispersal delayed predator-prey system with the Beddington-DeAngelis functional response and harvesting. By constructing the suitable upper-lower solutions and applying Schauder's fixed point theorem, we show that there exists a positive constant $ c^* $ such that the system possesses a traveling wave solution for any given $ c > c^* $. Moreover, the asymptotic behavior of traveling wave solution at infinity is obtained by the contracting rectangles method. The existence of traveling wave solution for $ c = c^* $ is established by means of Corduneanu's theorem. The nonexistence of traveling wave solution in the case of $ c < c^* $ is also discussed.

In this article we study the Schrodinger-Poisson system $$\displaylines{ -\Delta u +V(|x|)u+\lambda\phi u = f(u), \quad x\in\mathbb{R}^3, \cr -\Delta \phi =u^2, \quad x\in\mathbb{R}^3, }$$ where V is a singular potential with the parameter $\alpha$ and the nonlinearity f satisfies critical growth. By applying a generalized version of Lions-type theorem and the Nehari manifold theory, we establish the existence of the nonnegative ground state solution when $\lambda=0$. By the perturbation method, we obtain a nontrivial solution to above system when $\lambda\neq 0$.

This article concerns the existence and multiplicity of positive solutions to the fractional Kirchhoff equation with critical indefinite nonlinearities by applying the Nehari manifold approach and fibering maps.

In this article, we consider the equation $$ u_t= \sum_{i=1}^N \big(a_i(x)|u_{x_i}|^{p_i(x)-2}u_{x_i}\big)_{x_i}, $$ with $a_i(x), p_i(x)\in C^1(\overline{\Omega})$ and $p_i(x)>1$. Where $a_i(x)=0$ if $x\in\partial \Omega$, and $a_i(x)>0$ if $x\in \Omega$, without any boundary conditions. We propose an analytical method for studying the stability of weak solutions. We also study the uniqueness of a weak solution, and establish its stability under certain conditions.

We prove a global Calderon-Zygmund type estimate in the framework of Lorentz spaces for the variable power of the gradient of weak solution pair (u,P) to the generalized steady Stokes system over a bounded non-smooth domain. It is assumed that the leading coefficients satisfy the small BMO condition, the boundary of domain belongs to Reifenberg flatness, and the variable exponent p(x) is log-Holder continuous.

In this paper, we study a ratio-dependent predator-prey system with diffusion and cross-diffusion under the homogeneous Neumann boundary condition. By applying the maximum principle and Harnack's inequality, we present a priori estimates of the positive steady state of the system. The existence and non-existence of non-constant positive steady states are established. Our findings show that under certain hypotheses, non-constant positive steady states can exist due to the emergence of cross-diffusion, which reveals that cross-diffusion can induce stationary patterns but the random diffusion fails.

This article studies the existence and multiplicity of component-wise positive solutions for systems of nonlinear Hammerstein integral equations. In this system one nonlinear term is uniformly superlinear or uniformly sublinear, and the other is locally uniformly superlinear or locally uniformly sublinear. Discussions are undertaken by means of the fixed point index theory in cones. As applications, we show the existence and multiplicity of component-wise positive solutions for systems of second-order ordinary differential equations with the Dirichlet boundary value conditions and mixed boundary value conditions, respectively.

In this article, we study an Euler-Bernoulli viscoelastic control system which is dissipative due to the presence of the viscoelastic term. The main feature which distinguishes this paper from other related works lies in the fact that we no longer impose traditional conditions such as complete monotonicity and decay property on the kernel function g. Without loss of generality, we study the system in the case of $g\equiv 1$. By means of the duality principle and the Hahn-Banach theorem, we show that the system with g=1 is approximately controllable in the appropriate function space.

We study a non-Newtonian polytropic filtration equation with a convection term. We introduce new type of weak solutions and show the existence of weak solutions. We show that when $\int_{\Omega} [a(x)]^{-1(p-1)}dx

In this article, we explore the existence of multiple solutions for a p-Kirchhoff equation with the nonlinearity containing both singular and critical terms. By means of the concentration compactness principle and Ekeland's variational principle, we obtain two positive weak solutions.

In this article we define a q-analogue of Kummer's equation. It has two singular points. Near the singular point at zero, using the Frobenius method, we obtain two linearly independent series solutions in any one of three cases according to the difference of roots of the characteristic equation. Near the singular point at infinity, given that the only formal series solution is divergent, we find two integral solutions which are convergent under some condition. Finally, using the q-analogue of Kummer's equation, we deduce six contiguous relations about the q-hypergeometric series ${}_1\Phi_1$.

In this article, we study a density-dependent predator-prey system with the Beddington-DeAngelis functional response for stability and Hopf bifurcation under certain parametric conditions. We start with the condition of the existence of the unique positive equilibrium, and provide two sufficient conditions for its local stability by the Lyapunov function method and the Routh-Hurwitz criterion, respectively. Then, we establish sufficient conditions for the global stability of the positive equilibrium by proving the non-existence of closed orbits in the first quadrant $\mathbb{R}^2_{+}$. Afterwards, we analyze the Hopf bifurcation geometrically by exploring the monotonic property of the trace of the Jacobean matrix with respect to $r$ and analytically verifying that there is a unique $r^*$ such that the trace is equal to 0. We also introduce an auxiliary map by restricting all the five parameters to a special one-dimensional geometrical structure and analyze the Hopf bifurcation with respect to all these five parameters. Finally, some numerical simulations are illustrated which are in agreement with our analytical results.

In this article, we study a class of elliptic systems with Hardy potentials. We analyze the possible behavior of radial solutions to the system when p,t>1, q,s>0$ and \lambda,\mu > (N-2)^2/4, and obtain the existence of positive solutions to the system with the Dirichlet boundary condition under certain conditions. When $\lambda,\mu\leq 0$, p,t>1 and q,s>0, we show that any radial positive solution is decreasing in r.

This article concerns the bifurcation of limit cycles from a cubic integrable and non-Hamiltonian system. By using the averaging theory of the first and second orders, we show that under any small cubic homogeneous perturbation, at most two limit cycles bifurcate from the period annulus of the unperturbed system, and this upper bound is sharp. By using the averaging theory of the third order, we show that two is also the maximal number of limit cycles emerging from the period annulus of the unperturbed system.

This article concerns the bifurcation of limit cycles for a quartic system with an isochronous center. By using the averaging theory, it shows that under any small quartic homogeneous perturbations, at most two limit cycles bifurcate from the period annulus of the considered system, and this upper bound can be reached. In addition, we study a family of perturbed isochronous systems and prove that there are at most three limit cycles bifurcating from the period annulus of the unperturbed one, and the upper bound is sharp.

In this article, we are concerned with the existence of positive solutions for nonlinear fractional differential equation whose nonlinearity contains the first-order derivative, $$displaylines{ D_{0^+}^{alpha}u(t)+f(t,u(t),u'(t))=0,quad tin (0,1),; n-14 $ $(ninmathbb{N})$, $D_{0^+}^{alpha}$ is the standard Riemann-Liouville fractional derivative of order $alpha$ and $f(t,u,u'):[0,1] imes [0,infty)imes(-infty,+infty) o [0,infty)$ satisfies the Caratheodory type condition. Sufficient conditions are obtained for the existence of at least one or two positive solutions by using the nonlinear alternative of the Leray-Schauder type and Krasnosel'skii's fixed point theorem. In addition, several other sufficient conditions are established for the existence of at least triple, n or 2n-1 positive solutions. Two examples are given to illustrate our theoretical results.

In this article, under certain parametric conditions, we study the first integrals of the Duffing-van der Pol-type oscillator equations which include the van der Pol and the Duffing oscillator systems, as particular cases. After making a series of variable transformations and applying the Preller-Singer method, we find the first integrals of the simplified equations without complicated calculations. Through the inverse transformations we obtain the first integrals of the original equations. Some statements in the literature are indicated and clarified.