Nonlinear Differential Equations and Applications (NoDEA)最新文献

In this paper, we are interested in the following planar Schrödinger–Poisson system

where (p>2), (alpha _0>0) and (0<gamma le 2), the potential (a:{mathbb {R}}^2rightarrow {mathbb {R}}) is invariant under the action of a closed subgroup of the orthogonal transformation group O(2). As a consequence, we obtain infinitely many saddle type nodal solutions for equation (0.1) with their nodal domains meeting at the origin if (0<gamma <2) and (p>2). Furthermore, in the critical case (gamma =2) and (p>4), we prove that equation (0.1) possesses a positive solution which is invariant under the same group action.

We provide a representation of the weak solution of the continuity equation on the Heisenberg group ({mathbb {H}}^1) with periodic data (the periodicity is suitably adapted to the group law). This solution is the push forward of a measure concentrated on the flux associated with the drift of the continuity equation. Furthermore, we shall use this interpretation for proving that weak solutions to first order Mean Field Games on ({mathbb {H}}^1) are also mild solutions.

This paper investigates stationary mean-field games (MFGs) on the torus with Lipschitz non-homogeneous diffusion and logarithmic-like couplings. The primary objective is to understand the existence of (C^{1,alpha }) solutions to address the research gap between low-regularity results for bounded and measurable diffusions and the smooth results modeled by the Laplacian. We use the Hopf-Cole transformation to convert the MFG system into a scalar elliptic equation. Then, we apply Morrey space methods to establish the existence and regularity of solutions. The introduction of Morrey space methods offers a novel approach to address regularity issues in the context of MFGs.

This paper is concerned with the three trophic levels predator–prey system with alarm-taxis

under homogeneous Neumann boundary condition in smooth bounded domains (Omega subset {mathbb {R}}^n (nge 1)). We prove that the system possesses a unique global bounded classical solution for all sufficiently smooth initial data. Moreover, we show the large time behavior of the solution with convergence rates and perform some numerical simulations to verify the analytic results.

In this paper, we state the equivalence between weak and viscosity solutions for non-homogeneous problems involving the anisotropic ({{textbf {p}}}(cdot ))-Laplacian. The proof that viscosity solutions are weak solutions is performed by the inf-convolution technique. However, due to the anisotropic nature of the ({{textbf {p}}}(cdot ))-Laplacian we adapt the definition of inf-convolution to the non-homogeneity of this operator. For the converse, we develop comparison principles for weak solutions. Since the locally Lipschitz assumption is crucial to get the viscosity-weak implication, we prove that a class of bounded viscosity solutions are indeed locally Lipschitz.

In a smoothly bounded domain (Omega subset mathbb {R}^3), the chemotaxis-Stokes system

is considered along with the boundary conditions

where (c_star ge 0) is a given constant. It is shown that under a smallness condition on (c(cdot ,0)) and suitable assumptions on regularity of the initial data, global classical solutions exist which are uniformly bounded.

We prove new multiplicity results for some elliptic problems with critical exponential growth. More specifically, we show that the problems considered here have arbitrarily many solutions for all sufficiently large values of a certain parameter (mu > 0). In particular, the number of solutions goes to infinity as (mu rightarrow infty ). The proof is based on an abstract critical point theorem.

We consider the existence of solutions ((lambda _1,lambda _2, u, v)in mathbb {R}^2times (H^1(mathbb {R}^N))^2) to systems of coupled Schrödinger equations

satisfying the normalization

Here (mu _1,mu _2,beta >0) and the prescribed masses (a,b>0). We focus on the coupled purely mass super-critical case, i.e.,

with (2^*=frac{2N}{(N-2)_+}, 1le Nle 4) and give a partial affirmative answer to one open question in Bartsch et al. (J Math Pures Appl (9), 106(4):583–614, 2016). In particular, for (N=3,4) with (r_1,r_2in (1,2)), our result indicates the existence for all (a,b>0) and (beta >0).

We consider a Stackelberg control strategy applied to the Boussinesq system. More precisely, we act on this system with a hierarchy of two controls. The aim of the “leader” control is the null-controllability property whereas the objective of the “follower” control is to keep the state close to a given trajectory. By solving first the optimal control problem associated with the follower control, we are lead to show the null-controllability property of a system coupling a forward with a backward Boussinesq type systems. Our main result states that for an adequate weighted functional for the optimal control problem, this coupled system is locally null-controllable. To show this result, we first study the adjoint system of the linearized system and obtain a weighted observability estimate by combining several Carleman estimates and an adequate decomposition for the heat and the Stokes system.

We consider a general class of bulk-surface convective Cahn–Hilliard systems with dynamic boundary conditions. In contrast to classical Neumann boundary conditions, the dynamic boundary conditions of Cahn–Hilliard type allow for dynamic changes of the contact angle between the diffuse interface and the boundary, a convection-induced motion of the contact line as well as absorption of material by the boundary. The coupling conditions for bulk and surface quantities involve parameters (K,Lin [0,infty ]), whose choice declares whether these conditions are of Dirichlet, Robin or Neumann type. We first prove the existence of a weak solution to our model in the case (K,Lin (0,infty )) by means of a Faedo–Galerkin approach. For all other cases, the existence of a weak solution is then shown by means of the asymptotic limits, where K and L are sent to zero or to infinity, respectively. Eventually, we establish higher regularity for the phase-fields, and we prove the uniqueness of weak solutions given that the mobility functions are constant.