This paper addresses the quasilinear Schrödinger equation with critical growth in the zero-mass case:−Δu−uΔ(u2)=λg(x)|u|p−2u+|u|2·2*−2u,x∈RN,where N ≥ 3, λ > 0, p ∈ (4, 2 · 2*) or p ∈ (1, 2) with 2*=2NN−2 the critical Sobolev exponent, and g∈Lp0(RN), wherep0=2·2*2·2*−pifp∈(4,2·2*),p0=2*2*−pifp∈(1,2).To address the inherent lack of compactness in the zero-mass case, we develop a perturbative variational framework inspired by established techniques. For the case p ∈ (4, 2 · 2*), we establish the existence of a positive-energy ground state solution when λ is sufficiently large. For the case p ∈ (1, 2), we prove that for small λ > 0, the equation admits a ground state solution with negative energy. Additionally, for p ∈ (1, 2), we show the multiplicity of solutions to the above equation for small λ.
We study a free boundary problem on the evolution of a radial tumor with triple-layered structure and time-varying angiogenesis. The model is based on the diffusion of the internal nutrient σ and the changes of the tumor radius R(t) resulting from the growth and decay of cells in different layers. In Robin boundary condition, a time-dependent function β(t), which depicts the strength of tumor vascular system, is taken into account to illustrate the effects of the tumor aggressiveness on the dynamics of the tumor. We exhibit the asymptotic behavior of the tumor under different assumptions on β(t): (i) β(t) → β* > 0; (ii) β(t) vanishes; (iii) β(t) blows up, where cases (ii) and (iii) connect the tumor models in Neumann and Dirichlet boundary conditions.
Understanding the impact of temporal and spatial heterogeneity on population dynamics has significantly advanced mathematical theories in reaction-diffusion systems. This manuscript proposes and investigates the global dynamics of a reaction-diffusion system with time delay, where all model parameters are spatially and temporally dependent. The main contributions are threefold: (i) formulating a stage-structured model that incorporates diffusion and temporally and spatially inhomogeneous development delay; (ii) proposing a dynamical systems framework that employs a quotient phase space to establish strong monotonicity of the periodic semiflow; (iii) establishing a threshold-type result under minimal assumptions, for both increasing and unimodal birth functions. This synthesized approach is expected to motivate further studies when strong monotonicity of the solution semiflow fails in conventional phase spaces.
In this paper we prove a comparison result by Steiner symmetrizzation for solutions to Cauchy problem for parabolic equations whose prototype is ut−Δp,xu−uyy+cu=f. Our approach is based on an analogous mass comparison result for Dirichlet boundary value problem for the quasilinear elliptic equation −Δp,xu−uyy+cu=f obtained by a finite-differences discretization in y, and a discretization technique applied to the parabolic operator. We extend our results for −Δp,x to general operators of the form −div(a(|∇xu|)∇xu) where a is non-decreasing and behaves like |·|p−2 at infinity.
We collect several results related to the Symmetrized Fractional Variation model for signal and image denoising (shortly denoted SFV): a variational approach based on L1 fitting data term together with regularizing terms exploiting a distributional version of Riemann-Liouville fractional derivatives. We enhance the analysis of the one-dimensional case through the study of the space BV*s of admissible signals on a bounded interval, say the functions with bounded variation of both sides fractional derivatives for a prescribed real positive order s. We show that the embedding in BV*s of the Sobolev space of the same fractional order is strict. We exhibit some nontrivial borderline examples of admissible or non admissible functions in the space BV*s. We prove several relationships between related fractional calculus and the integral transforms. The SFV model is discretized based on a second-order consistent Grunwald Letnikov scheme and coupled with an automatic selection procedure of all model parameters relying on the whiteness principle: some numerical simulations are presented to show the efficacy of the proposed approach in denoising one-dimensional signals corrupted by impulsive noise modelled by the Laplace distribution.
We establish Schauder-type estimates for fully nonlinear elliptic equations with oblique boundary conditions under assumptions more general than Hölder continuity. Specifically, our analysis requires only Dini continuity of the source term, controlled oscillation of the operator’s coefficients, and suitable regularity of the boundary data. Our approach is based on a robust iterative argument, coupled with perturbative arguments and compactness techniques, which yields classical regularity up to the boundary. As a consequence, we characterize the nodal sets of uniformly elliptic equations subject to oblique boundary conditions. Finally, the obtained estimates remain remarkable even for linear models with Dini-continuous data.
In this manuscript, we prove the existence of strictly positive solutions for a class of p&q systems whose nonlinearities involved may exhibit semipositone behavior, which arise naturally in several applications that includes models of competing species and the dynamics of particle movement. Our approach combines variational methods, comparison principles and a novel regularity result for systems. This estimate, which may be of independent interest, is new even in the classical Laplacian case.
We study the Dirichlet problem for nonlinear elliptic systems of convection-diffusion type with singular coefficients in the convective term. Our operators are monotone, with principal part of p-Laplacian type, satisfy appropriate ellipticity conditions and have critical growth. This may produce lack of coercivity and compactness. We extend to systems the existence results contained in the paper [1], where the scalar case is studied. To treat the vectorial case, we assume a suitable asymptotic structural condition.Dedicated to Gioconda, on the occasion of her 70th birthday.
This paper considers a class of degenerate mixed Yamabe type equations on smooth compact Riemannian manifolds with totally geodesic boundary. Under some sufficient conditions, we establish the a priori estimates and obtain an existence and uniqueness theorem for the Neumann problem of these equations.
We establish new sharp regularity results for convex supersolutions of fully nonlinear elliptic equations with Hamiltonian terms, extending the classical Hopf-Oleĭnik boundary point lemma to a broader class of operators. Specifically, we analyze fully nonlinear equations of the formPλ,Λ,γ,μ,m(D2u,Du,x)=f(x),where Pλ,Λ,γ,μ,m represents a class of extremal Pucci-type operators with gradient dependence and unbounded coefficients. By refining barrier techniques introduced in [1] and employing viscosity solution methods, we derive sharp interior regularity estimates for semiconvex supersolutions of Pλ,Λ,γ,μ,m. Our results not only generalize classical boundary type estimates for fully nonlinear PDEs but also provide a unified framework to handle gradient-dependent structures with different growth regimes. In particular, we establish a non-homogeneous Hopf-Oleĭnik lemma that remains valid in the presence of unbounded nonlinearities, significantly strengthening previous results in this direction. These findings contribute to the broader understanding of nonlinear PDEs, with implications in geometric analysis and optimal control.
In this wok, the Cauchy problem on the line of the modified Korteweg-de Vries equation with higher-order dispersion is studied, and for data in Sobolev and analytic Gevrey spaces its optimal well-posedness is established. The proofs are based on the derivation of sharp trilinear estimates in Bourgain spaces suggested by the corresponding linear problem. Furthermore, for analytic initial data improved lower bounds for the radius of spatial analyticity of the solution are derived.
In this paper, which is a continuation of our recent paper [1], we give two new definitions of a weak solution of the one-dimensional, elliptic, nonlinear singular problem which is formally written as{−ddx(a(x)dudx)=−dϕ(u)dx−dg(x)dxin(0,L),u(0)=u(L)=0,where ϕ:R→R∪{+∞} is a singular function whose model is given by ϕ(s)=ϕγ(s)=1|s|γ with γ > 0. In these two definitions the solutions belong respectively to the space u∈W01,1(0,L) with ϕ(u) ∈ L1(0, L), and to the space u∈W01,m(0,L) with ϕ(u) ∈ Lm(0, L) for m > 1. In these frameworks we state and prove results of non-existence, existence, and non-isolation of the solutions.
This paper is concerned with the existence and multiplicity of symmetric periodic solutions for a class of non-autonomous differential system with multiple distributed delays in Rn, using index theory as the main tool. When the nonlinear term is the gradient of a function F which is asymptotically quadratic at infinity, some existence results for symmetric periodic solutions are obtained. When the function F is also asymptotically quadratic at the origin, some existence and multiplicity results for nontrivial symmetric periodic solutions are obtained.
We analyze fractional phase-transition energies with periodic heterogeneities at the critical H1/2 scaling. We prove that the Γ-limit is a sharp-interface functional whose surface energy density combines homogenization and averaging effects. The resulting coefficient is a weighted combination of the minimum and the mean of the oscillatory parameter, reflecting the coexistence of multiple interaction scales. This behavior is specific to the critical regime and differs from the case s > 1/2.
We introduce a game–theoretical framework for the doubly nonlinear parabolic equation|∂tu|p−2∂tu−Δpu=0.where Δpu=∇·(|∇u|p−2∇u) with p > 2 is the standard p−Laplacian. A key feature to our approach is a new asymptotic mean value formula (AMVF) for the p−Laplacian that is robust even when the gradient vanishes and is independent of the sign of the p−Laplacian. This new AMVF leads naturally to a dynamic programming principle (DPP) whose solutions converge to the viscosity solution of the boundary value problem for the differential equation. In addition, solutions to the DPP coincide with value functions for a stochastic, two-players, zero-sum game that we introduce and analyze here.
In this paper we study an overdetermined problem which is directly related to the well known torsion problem studied by J. Serrin. A perturbed version of the latter is tackled by using asymptotic series as well as tools borrowed from the celebrated Nekhoroshev Theorem. In a similar fashion to this class of results, we establish the existence of infinitely many approximants for the perturbed problem’s solution, whose approximation error is so small which can be regarded as negligible for practical applications. The approach is fully constructive and this feature is demonstrated via an example in a final section.
This paper studies a one-dimensional Mean-Field Planning (MFP) system with a non-local, rank-based coupling. Using a potential formulation, we rewrite the system as an associated scalar partial differential equation. We prove an equivalence between classical solutions to the ranking MFP system with positive density and classical solutions to the associated potential problem, and we derive explicit reconstruction formulas. We then identify a monotonicity structure in the operator associated with the potential formulation, which, under strict convexity assumptions, yields uniqueness of classical solutions to the associated problem and, hence, uniqueness of the ranking MFP system up to an additive constant in the value function. Finally, under superlinear growth assumptions, we exploit monotonicity to address existence for the potential formulation in a low-regularity setting. By formulating a variational inequality for a q-Laplacian regularized operator, we apply Minty’s method to establish the existence of weak solutions in the space of functions of bounded variation for a relaxed potential formulation.
We study existence of solutions in the variational sense for a class of stochastic phase-field models describing moving boundary problems. The models consist of stochastic reaction-diffusion equations with singular diffusion forced by a phase-field. We investigate both the case of an independently evolving phase-field and of coupled phase-field evolution driven by a viscous Hamilton-Jacobi equation. Such systems are used in the modelling of single-cell chemotaxis [1], where the contour of the cell shape corresponds to a level set of the phase-field. The technical challenge lies in the singularities at zero level sets of the phase-field. For large classes of initial data, we establish global existence of probabilistically weak solutions in L2-spaces with weights which compensate for the singularities.
In this paper, we study infinite dimensional holomorphic vector fields on sequence spaces, having a fixed point at 0. Under suitable hypotheses we prove the existence of germs of analytic invariant submanifolds passing through the fixed point. The restricted dynamics is analytically conjugate to the linear one under some Diophantine-like condition.
The symmetric decreasing rearrangement of functions on Rn features in several seminal inequalities, such as the Pólya-Szegő inequality. The latter was shown by the authors [6] to hold for all smoothing rearrangements, a class that includes the more general (k, n)-Steiner rearrangement, as well as others introduced by Brock and by Solynin. The theory of rearrangements and their associated set maps is developed, with an emphasis on approximation, particularly by polarizations. The Pólya-Szegő inequality holds with equality for polarizations, so is proved relatively easily for rearrangements that can be suitably approximated by them. One goal here is to show that the Brock rearrangements cannot be approximated in such a way. It turns out that under mild conditions, each set map associated with a rearrangement has in turn an associated contraction map from Rn to Rn. With this new analytical tool, several general results on the approximation of rearrangements are also proved.
