This article studies the modified fourth-order elliptic equation $$\displaylines{ \Delta^2 u - \Delta u + u - \frac{1}{2}u\Delta(u^2) = f(u), \quad \text{in } \mathbb{R}^N, \cr u \in H^2(\mathbb{R}^N), }$$ where \(N=5\) or \(6\), and the nonlinearity \(f:\mathbb{R}\to \mathbb{R}\) satisfies Berestycki-Lions type conditions. By using Jeanjean's augmented functional technique and constructing a Pohozaev-Palais-Smale sequence, we prove that the problem admits a ground state solution. Moreover, if \(f\) is odd, we prove, via a symmetric mountain-pass lemma, that the problem admits infinitely many radially symmetric solutions For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/87/abstr.html
In this work we study the traveling wave solutions for a generalized nonlo cal 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 >= c(& lowast;), where c(& lowast;) 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 < c < c & lowast;. The main technical challenges arise from the combined effects of nonlo cal dispersal and the lack of monotonicity in the system.
This article concerns the nonlinear Schro & uml;dinger equation (NLS) on R3 with sub-cubic and cubic nonlinearities. For the cubic NLS, with initial data in Hm(R3), we establish exponential growth of high-order Sobolev norms by using modified energy functionals and loss-less Strichartz estimates. For the sub cubic NLS, with data in H2(R3), we establish polynomial growth.
This article studies two classes of \(\psi\)-Caputo fractional equations with impulsive conditions and operator pairs in Banach spaces: one is the impulsive Sobolev-type integro-differential equations with operator pairs on bounded intervals, the other is the impulsive differential equations with operator pairs and finite delays on unbounded intervals. By combining the theory of propagating theory and the \(\psi\)-Laplace transform as well as fractional calculus, we firstly derive a proper definition of mild solutions for these impulsive equations. Depending on the definition, we establish existence and uniqueness theorems and a continuous dependence theorem for the impulsive integro-differential equations on bounded intervals. Next, we investigate the piecewise pseudo S-asymptotically \((\omega,c)\)-periodic solutions for the impulsive delay equations on unbounded intervals. Finally, some examples are given to illustrate the main results. Here we do not assume the operator \(B\) has bounded or compact inverse \(B^{-1}\). Actually, our results are even new in the case when \(B=I\) and \(\psi(t)=t\). For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/86/abstr.html
We study a fourth-order evolution equation describing the transverse vibrations of a deep-sea mining pipeline with a nonlocal stiffness coefficient. The bending rigidity depends on the total deformation of the pipeline through the integral $$ \int_0^L |v_s(\sigma,t)|^2d\sigma. $$ The equation also contains axial loading, gyroscopic effects, submerged weight and linear velocity damping. We prove local-in-time existence of weak solutions by the Faedo-Galerkin method, using energy estimates, compactness arguments and the positivity of the nonlocal stiffness coefficient. We also derive a continuation criterion and establish a small-energy continuation result, which yields existence on arbitrary prescribed finite time intervals under a suitable smallness condition on the initial energy. Under additional regularity and compatibility assumptions, we prove the existence of local strong solutions. Finally, we establish uniqueness in the strong class and prove continuous dependence on the initial data in a regular energy class. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/85/abstr.html
