In this paper, we are concerned with the compactness of the Sobolev type embeddings with variable exponents both in bounded domains and in the whole space. We give a sufficient condition for the compactness on the growth rate of the variable exponent that touches the critical exponent at a point and involves the analysis of the noncompactness caused by the concentration. The results in ℝ^N need an additional consideration on another mode of the compactness called “vanishing” which is related with the noncompactness of W^1,p_rad(ℝ^N)↪ L^p(ℝ^N) . We give a sharp condition on the relationship between p(· ) and q(· ) for the compactness of the embedding W^1,p(· )_rad(ℝ^N)↪ L^q(· )(ℝ^N) under suitable smoothness assumptions on p(· ) , when q(· ) touches either of the two critical exponents p(· ) and p^*(· )(:=Np(· )/(N-p(· ))) . Applications to elliptic equations are also discussed.
As an extension of the study on boundedness in the parabolic-elliptic chemotaxis system with nonlocal logistic absorption, this paper advances to investigate the more challenging quasilinear fully parabolic model: u_t=∇· (D(u)∇ u-uS(u)∇ v)+μ u^α(1-σ∫ _Ωu^β) and v_t=Δ v-v+f(u) in bounded domains, subject to no-flux boundary conditions, where α ,β >1 , μ≥ 0 and σ >0 , the self-diffusion rate D and the sensitivity S generalize the prototypes D(s)=(s+1)^λ and S(s)=(s+1)^δ -1 for all s≥ 0 with λ ,δ∈ℝ , and the nonlinear production function f satisfies 0≤ f(s)≤ K_fs^γ for all s≥ 0 with γ >0 . By expanding upon the strategy developed in [J. Differential Equations 268 (2020) 7047–7091, Z. Angew. Math. Mech. 105 (2025) e70064], we establish the global boundedness of solutions under various parameter assumptions, depending on whether the self-diffusion alone suppresses both the cross-diffusion and the local source term μ u^α , or the self-diffusion and the nonlocal absorption -μσ u^α∫ _Ωu^β synergistically control the cross-diffusion and the local source.
We review Dirichlet-type problems of the form -Δ u=λu/|x|^2+ g(· ,∇ u) in Ω , u=0 on ∂Ω , where N⩾ 3 , Ω⊂ℝ^N is a bounded regular domain containing the origin, and 0<λ < Λ _N=( N-2/2) ^2 , the optimal Hardy constant. We first consider the Hardy–Poisson problem, i.e., the case g(· ,z)=f with f in a weighted Lebesgue space, and we provide sharp weighted regularity estimates for the SOLA (solution obtained as the limit of approximations) of the linear problem. These results allow us to extend and refine earlier existence results for a KPZ-type problem. In particular, for nonlinearities of the form g(· ,z)=|z|^q+ρ f , with f belonging to a suitable weighted L^m-space , we establish the existence of a weak solution under suitable conditions on m and q, for ρ >0 small.
Consider the system { Δ u ≥ p(x)g(v) in ℝ^n, Δ v ≥ q(x)f(|∇ u|) in ℝ^n, u>0, v>0 in ℝ^n, . where n≥ 2 , f,g∈ C[0,∞ ) and p,q∈ C(ℝ^n) . If f, g are non-decreasing and convex, using a comparison argument we prove a sufficient Keller-Osserman type condition for the nonexistence of entire solutions. This result is sharp for a wide class of systems. Similar results are also shown to hold for systems without gradient terms.
In this work, we investigate the existence of solutions for quasicritical elliptic equations involving the 1-Laplacian operator with Robin and Neumann boundary conditions. Assuming that the nonlinearity satisfies suitable hypotheses, we establish the existence of solutions using approximation techniques. To overcome the challenges associated with the lack of compactness, we employ the Vitali Convergence Theorem.
A system of integro-differential equations arising in a price formation model in mean field games introduced by Gomes and Saúde [21] is studied. The system consists of a Hamilton–Jacobi equation and a Fokker–Planck equation coupled through a time-dependent price variable determined by a market clearing condition requiring that the aggregate demand coincides with a prescribed supply. The main result establishes the existence and uniqueness of classical solutions in a multidimensional setting. The main difficulty lies in the global integral constraint determining the price variable, which couples the Hamilton–Jacobi and Fokker–Planck equations. The analysis relies on suitable a priori estimates for the Hamilton–Jacobi and Fokker–Planck equations with a fixed price parameter. In particular, the regularity estimates depend only on the L^∞ norm of the price parameter, which allows the construction of a fixed point argument for the full system. This provides a direct PDE approach for treating the multidimensional case and Hamiltonians depending on space and time.
We examine three singular Dirichlet problems driven by the double phase operator. One of the problems is nonparametric and the other two are parametric. In all problems, the perturbation is “superlinear”, but does not satisfy the Ambrosetti-Rabinowitz condition. We prove existence and multiplicity results for the problems. For the parametric problems, the results are global in the parameter λ >0 . Our approach uses variational tools from the critical point theory, truncations and comparisons and critical groups.
In this article, we investigate the existence and nonexistence of weak solutions to higher-order doubly critical elliptic problems with weights, driven by a polyharmonic double phase operator. More precisely, we deal with the following problem {[ ℒ^m_p,q(u) = f(x,u) in Ω ,; u=∇ u=⋯∇ ^m-1 u=0 on ∂Ω, ]. where Ω⊂ℝ^N with N ≥ 2 is a smooth bounded domain with Lipschitz boundary ∂Ω , m ∈ℕ , 1< p< q < N/m with (N-1)q≤ Np , the nonlinear term f:Ω×ℝ→ℝ is a Carathéodory function, which has doubly critical growth, and ℒ^m_p,q represents a polyharmonic double phase operator. By establishing new compactness results within a suitable Musielak–Orlicz–Sobolev framework and applying variational methods, we prove the existence of nontrivial weak solutions. In addition, we derive nonexistence results under appropriate assumptions by establishing a Pohozaev-type identity for higher-order derivatives. Our approach extends classical techniques to capture the intricate features of the double-phase operator for higher-order derivatives, and addresses the difficulties arising from critical nonlinearities, in particular extending the results of [F. Colasuonno, K. Perera, J. Differ. Equ., 422 (2025), 426–488] in a polyharmonic double phase setup overcoming the non-closedness of truncations in higher-order Sobolev spaces.
Troesch’s problem is well known for its extreme sensitivity to large parameters, which poses challenges for both numerical computation and asymptotic analysis. In this paper, we study a broader class of highly sensitive boundary-value problems generalizing the classical Troesch’s problem. We develop a refined asymptotic framework that captures the solution profiles near the boundary and establishes the emergence of boundary concentration as the parameter tends to infinity. While numerical studies have been carried out extensively, to the best of our knowledge, this appears to be the first rigorous asymptotic analysis characterizing the boundary behavior of solutions to Troesch-type problems in the large-parameter regime.
We consider a quasilinear Choquard problem { -div(β (|∇ u|^2)∇ u)+(λ +V(x))u=(I_θ *|u|^p)|u|^p-2u in ℝ^N≥ 3, u(x)→ 0, |x|→ +∞. with the prescribed mass ∫ _ℝ^N|u|^2dx=a^2 for a fixed a>0 , where λ∈ℝ is a Lagrange multiplier and I_θ is the Riesz potential of θ -th order with θ∈ (0,N) . We first prove the existence of normalized ground state solutions to this problem with the mass subcritical case and V(x)≡ 0 by imposing some suitable conditions on β (· ) . We next attain the existence of radially normalized ground state solutions to this problem with the mass supercritical case and V(x)≤ 0 by means of mountain pass theorem. Finally, we show the nonexistence of its normalized solutions with the mass critical case.
We study a nonlinear Schrödinger system with three-wave interaction: { - Δ u_1 = f_1(u_1) + α u_2u_3 in ℝ^N, - Δ u_2 = f_2(u_2) + α u_3u_1 in ℝ^N, - Δ u_3 = f_3(u_3) + α u_1u_2 in ℝ^N, u⃗=(u_1,u_2,u_3)∈ (H_rad^1(ℝ^N))^3, . where 3≤ N≤ 5 , α∈ℝ and each nonlinearity f_i(ξ ) satisfies the Berestycki-Lions conditions. Let S_i denote the set of all least energy solutions of the scalar equation -Δ u = f_i(u) in H_rad^1(ℝ^N) . A solution of the systems is called vector if all its components are nontrivial. We establish the existence of two distinct families of vector solutions {u⃗_α} with different asymptotic behaviors as α→ 0 . One family satisfies dist(u⃗_α,S_1× S_2× S_3) → 0 , while another satisfies dist(u⃗_α,S_1× S_2×{0}) → 0 . By contrast, we prove that no family of vector solutions satisfies dist(u⃗_α,S_1×{0}×{0}) → 0 . Together, these results give a complete description of the asymptotic structure of vector solutions when the three-wave interaction is weak.
Lie symmetries, conservation laws, reductions, and new exact solutions of the extended form of the (2+1) -dimensional generalized Zakharov–Kuznetsov (ZK) equation are studied. This generalized ZK model includes three arbitrary exponents m, n and k in the nonlinear terms. The Lie algebra for the generalized ZK model is four dimensional. Conservation laws are derived using the multiplier method and five cases arise under different restrictions on the parameters m, n and k of the model. Lie symmetries associated with the conserved vectors are determined for each case. Furthermore, several reductions and new exact solutions of ZK equation are established using the generalized double reduction theory. These solutions advance the analysis of nonlinear dynamics in physical systems including plasmas, shallow water flows, and other dispersive media.
In this paper, we study the 3D dissipative fluid–dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the Navier–Stokes/Poisson–Nernst–Planck system. We prove that if the partial derivatives of two velocity components (∂ _1 u_1, ∂ _2 u_2) satisfy ∫ _0^T‖ (∂ _1 u_1, ∂ _2 u_2)(· , t) ‖ ^8/5-4α_Ḃ_∞ ,∞^-α/1 + ln (e + ‖ (∂ _1 u_1, ∂ _2 u_2)(· , t) ‖ _Ḃ_∞ , ∞^-α ) dt< ∞ for 0< α < 1, then the local solution can be smoothly extended past the time t = T . Particularly, a regularity criterion is further established for the critical case α =0 . These results represent further improvements of previous studies by Zhao et al. (2016; 2019; 2025) and Wu (2019). Moreover, we extend the results by Zhang (2008) and Dong et al. (2010; 2011) for the incompressible Navier–Stokes equations.
We investigate the phenomenon of Landau damping within the context of the Vlasov-Riesz system. Employing a nonlinear analysis technique developed by Ionescu, Pausader, Wang, and Widmayer, our primary objective is to extend the understanding of Landau damping to kinetic models beyond the traditional Vlasov-Poisson system. Our analysis is grounded in the use of the Gevrey- γ norm, and we demonstrate that the critical Gevrey index γ =-4/3α +5/3 is dependent on the parameter α from the Riesz interaction, defined as U=( -Δ) ^-α(ρ -1) . This study not only broadens the applicative scope of Landau damping but also deepens the theoretical understanding of its mechanisms in more complex kinetic frameworks.
This paper investigates a class of stochastic nonlinear fractional partial differential equations of complex Ginzburg-Landau (CGL) type posed on the multidimensional positive orthant. The model incorporates both interior multiplicative noise and Robin-type stochastic boundary forcing, acting independently along each coordinate hyperplane. The governing equation includes a vectorial Caputo-type fractional Laplacian of order β∈( 3/2,2) , a nonlinear term of the cubic type and non-Gaussian noise introduced via the modulus of Brownian motions convolved with deterministic kernels. We focus on the formulation and analysis of mild solutions under minimal smoothness assumptions. The main analytical challenges stem from the nonlocal nature of fractional diffusion, the loss of martingale structure due to non-centered boundary noise, and the intricate coupling induced by Robin-type conditions in multiple spatial dimensions. We develop a novel framework combining infinite-dimensional Itô calculus, Laplace-transform methods, and weighted fractional Sobolev estimates. Our contributions include the construction of a well-posed mild solution framework, the derivation of probabilistic a priori bounds, and second moment estimates. We also characterize the long-time behavior of solutions, identifying decay rates and stochastic regularization phenomena driven by the fractional Laplacian. The novelty of this work lies in the synthesis of fractional diffusion, nonlinear complex dynamics, and non-Gaussian boundary noise—a setting that remains largely unexplored. Unlike previous studies restricted to Gaussian interior noise, our approach captures realistic dynamics involving delayed boundary responses and spatially distributed stochastic inputs. These results offer new insights into the behavior of boundary-driven fractional SPDEs and provide a foundation for future work in stochastic modeling of anomalous transport and interface phenomena.
In this work, we provide an analytical proof of the existence of attracting invariant tori supporting quasiperiodic dynamics in a family of three-dimensional dissipative systems modeling self-oscillatory electronic devices. This family was originally proposed by Kuznetsov, Kuznetsov, and Stankevich, who observed numerically such quasiperiodic dynamics and evidences for the existence of invariant tori. Our results establish the existence of normally hyperbolic attracting tori using recent advances in averaging theory. We also show that the dynamics on these tori are always either periodic or quasiperiodic, and that quasiperiodic motion occurs for a set of parameter values with positive Lebesgue measure. These findings not only confirm and extend the prior numerical observations, but also demonstrate that quasiperiodicity arises with positive probability when parameters are randomly chosen. Moreover, our study highlights the effectiveness of averaging theory as a powerful analytical tool for detecting attractors in applied dynamical systems.
In this paper, we deal with the following Neumann-initial boundary value problem for a quasilinear chemotaxis model describing tumor angiogenesis: {[ u_t = ∇· (D(u)∇ u) - χ∇· (u∇ v) + ξ _1∇· (u^m∇ w) , x∈Ω , t> 0,; v_t = Δ v + ξ _2∇· (v∇ w) - v + u, x∈Ω , t> 0,; 0 = Δ w - w + u, x∈Ω , t> 0,; ∂ u/∂ν = ∂ v/∂ν = ∂ w/∂ν = 0, x∈∂Ω , t > 0,; u(x,0) = u_0(x), v(x,0) = v_0(x), x∈Ω , ]. in a bounded smooth domain Ω⊂ℝ^n (n≤ 3) , where the parameter χ , ξ _1,ξ _2 > 0 , D(u) is supposed to satisfy the following property D(u) ≥ (u + 1)^α with α > 0. Assume that ξ _1≥λ _1^*χ ^2 , where the parameter λ _1^* = λ _1^*(u_0, v_0, Ω ) > 0 ; then the system admits a global classical solution (u, v, w) via subtle energy estimates. Moreover, it is asserted that the corresponding solution exponentially converges to the constant stationary solution (u̅_0, u̅_0, u̅_0) provided the initial data u_0 is sufficiently small, where u̅_0 = ∫ _Ω u_0/|Ω | .
We consider the nonlinear Schrödinger equations with nonlinear Neumann boundary conditions on the right half-line ℝ_+= ( 0,∞) . Our purpose is to study the maximal existence time of strong solutions to the nonlinear Schrödinger equations under certain compatibility boundary conditions.
