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Register a new userCritical nonlocal systems with concave-convex powers
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By using the fibering method jointly with Nehari manifold techniques, we obtain the existence of multiple solutions to a fractional $p$-Laplacian system involving critical concave-convex nonlinearities provided that a suitable smallness condition on the parameters involved is assumed. The result is obtained despite there is no general classification for the optimizers of the critical fractional Sobolev embedding.
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By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.
We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.
It is established existence of ground and bound state solutions for Choquard equation considering concave-convex nonlinearities in the following form $$ begin{array}{rcl} -Delta u +V(x) u &=& (I_alpha* |u|^p)|u|^{p-2}u+ lambda |u|^{q-2}u, , u in H^1(mathbb{R}^{N}), end{array} $$ where $lambda > 0, N geq 3, alpha in (0, N)$. The potential $V$ is a continuous function and $I_alpha$ denotes the standard Riesz potential. Assume also that $1 < q < 2,~2_{alpha} < p < 2^*_alpha$ where $2_alpha=(N+alpha)/N$, $2_alpha=(N+alpha)/(N-2)$. Our main contribution is to consider a specific condition on the parameter $lambda > 0$ taking into account the nonlinear Rayleigh quotient. More precisely, there exists $lambda_n > 0$ such that our main problem admits at least two positive solutions for each $lambda in (0, lambda_n]$. In order to do that we combine Nehari method with a fine analysis on the nonlinear Rayleigh quotient. The parameter $lambda_n > 0$ is optimal in some sense which allow us to apply the Nehari method.
We consider a phase-field model where the internal energy depends on the order parameter in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for the order parameter. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, in the case of a potential defined on (-1,1) and singular at the endpoints, the existence of a finite-dimensional global attractor has been proven. Here we examine both the case of smooth potentials as well as the case of physically realistic (e.g., logarithmic) singular potentials. We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional attractors in the present cases as well.
We study a class of free boundary systems with nonlocal diffusion, which are natural extensions of the corresponding free boundary problems of reaction diffusion systems. As before the free boundary represents the spreading front of the species, but here the population dispersal is described by nonlocal diffusion instead of local diffusion. We prove that such a nonlocal diffusion problem with free boundary has a unique global solution, and for models with Lotka-Volterra type competition or predator-prey growth terms, we show that a spreading-vanishing dichotomy holds, and obtain criteria for spreading and vanishing; moreover, for the weak competition case and for the weak predation case, we can determine the long-time asymptotic limit of the solution when spreading happens. Compared with the single species free boundary model with nonlocal diffusion considered recently in cite{CDLL}, and the two species cases with local diffusion extensively studied in the literature, the situation considered in this paper involves several extra difficulties, which are overcome by the use of some new techniques.
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