Abstract We investigate an unstirred chemostat model in which two species compete in a two-dimensional environment. The populations are assumed to disperse anisotropically, with distinct probabilities assigned to horizontal and vertical movements, which are interpreted as dispersal strategies. First, we analyze the dynamics of the single-species model and identify the conditions for the existence of positive steady states. Then, we classify the dynamical behavior of the two-species model into three scenarios based on the diffusion strategies: (i) extinction; (ii) competitive exclusion; and (iii) coexistence. Next, we provide sufficient conditions for the existence of coexistence steady states. Finally, our numerical simulations provide visual validation of our theoretical results and offer valuable insights for future researches.

Abstract Using the notion of a Bessel pair, we study the Hardy type inequalities in the setting of Dunkl operator. We also establish a general symmetrization principle for weighted Hardy type inequalities with Dunkl operator in the situation that the standard Schwarz symmetrization is not applicable.

Abstract In this paper, we first establish the Talenti comparison principle for anisotropic p -Laplacian equation with Robin boundary conditions. This achievement not only extends classical Talenti comparison result for Laplacian equation with Robin boundary conditions in (A. Alvino, C. Nitsch, and C. Trombetti, “A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions,” Commun. Pure Appl. Math. , vol. 76, no. 3, pp. 585–603, 2023, doi:10.1002/cpa.22090), but also for the anisotropic Laplacian cases in (R. Sannipoli, “Comparison results for solutions to the anisotropic Laplacian with Robin boundary conditions,” Nonlinear Anal. , vol. 214, no. 112615, 2022, doi:10.1016/j.na.2021.112615). Furthermore, we also obtain the rigidity for this kind of Talenti comparison principle. This naturally covers the rigidity for Talenti comparison principle involving anisotropic Laplacian operator which has not been proved before. As an application of rigidity result for anisotropic Talenti comparison principle, we furthermore establish the rigidity of anisotropic Faber–Krahn inequality with Robin boundary conditions and naturally solve the rigidity for classical Faber–Krahn inequality with Robin boundary condition established by Alvino-Nitsch-Trombetti in (A. Alvino, C. Nitsch, and C. Trombetti, “A Talenti comparison result for solutions to elliptic problems with Robin boundary conditions,” Commun. Pure Appl. Math. , vol. 76, no. 3, pp. 585–603, 2023, doi:10.1002/cpa.22090). Significantly, our results also answer some open problems contained in (A. L. Masiello and G. Paoli, “A rigidity result for the Robin torsion problem,” J. Geom. Anal. , vol. 33, no. 5, p. 149, 2023, doi:10.1007/s12220-023-01202-3) and (A. L. Masiello and G. Paoli, “Rigidity results for the p-Laplacian poisson problem with robin boundary conditions,” J. Optim. Theor. Appl. , vol. 202, pp. 628–648, 2024, doi:10.1007/s10957-024-02442-1).

Abstract A Schrödinger operator that is bounded below and has a unique positive ground state can be transformed into a Dirichlet form operator by the ground state transformation. If the resulting Dirichlet form operator is hypercontractive, Davies and Simon call the Schrödinger operator “intrinsically hypercontractive”. I will show that if one adds a suitable potential onto an intrinsically hypercontractive Schrödinger operator it remains intrinsically hypercontractive. The proof uses a fortuitous relation between the WKB equation and logarithmic Sobolev inequalities. All bounds are dimension independent. The main theorem will be applied to several examples.

Abstract In this paper, we obtain a general supercritical Sobolev inequality in W 0 , r a d 1 , p ( B ) ${W}_{0,rad}^{1, p}\left(B\right)$ , where B is the unit ball in R N ${\mathbb{R}}^{N}$ . In order to discuss the sharp growth, we consider p-Laplacian elliptic problem − Δ p u = u q ( r ) in B , u > 0 in B , u = 0 on ∂ B , $$\begin{cases}-{{\Delta}}_{p}u={u}^{q\left(r\right)}\quad \hfill & \quad \text{in}\quad B,\hfill \\ u{ >}0\quad \hfill & \quad \text{in}\quad B,\hfill \\ u=0\quad \hfill & \quad \text{on}\quad \partial B,\hfill \end{cases}$$ where Δ p u = div(|∇u| p−2∇u), 1 < p < N, p * = N p N − p ${p}^{{\ast}}=\frac{Np}{N-p}$ , r = |x| and the variable exponent q(r) ≥ p* − 1 is a differentiable function in [0,1]. We give a Pohozaev identity and prove nonexistence of positive solutions to supercritical p-Laplacian equation in the unit ball B, which is significant for studying the nonlinear problems with variable exponential growth. Moreover, we also give the result of existence for a general nonlinearity case. Finally, we also give some results for a class of concave-convex nonlinear problem.

Abstract The main result in this paper is that, for singular integral operators associated with standard kernels, local L 1-estimates imply global L p -estimates for every p ∈ (1, ∞). When combined with the result of Melnikov-Verdera, this yields a complete and self-contained proof of the L p -boundedness of the Cauchy operator on Lipschitz curves and chord-arc curves in the plane.

Abstract Moving from the seminal papers by Bobkov and Tanaka [“On positive solutions for (p, q)-Laplace equations with two parameters,” Calc. Var. Partial Differ. Equ., vol. 54, pp. 3277–3301, 2015, “Remarks on minimizers for (p, q)-Laplace equations with two parameters,” Commun. Pure Appl. Anal., vol. 17, pp. 1219–1253, 2018, “Multiplicity of positive solutions for (p, q)-Laplace equations with two parameters,” Commun. Contemp. Math., vol. 24, 2022, Art. no. 2150008] on the spectrum of the (p, q)-Laplacian, we analyze the case of the double-phase operator. We discuss the region of parameters in which existence and non-existence of positive solutions occur. The proofs are based on normalization procedures, the Nehari manifold, and truncation techniques, exploiting Picone-type inequalities and an ad-hoc strong maximum principle.

Abstract In this paper, we study a nonlinear system involving a generalized tempered fractional p-Laplacian in B 1(0): ∂ t u ( x , t ) + ( − Δ − λ f ) p s u ( x , t ) = g ( t , u ( x , t ) ) , ( x , t ) ∈ B 1 ( 0 ) × [ 0 , + ∞ ) , u ( x ) = 0 , ( x , t ) ∈ B 1 c ( 0 ) × [ 0 , + ∞ ) , $$\begin{cases}_{t}u\left(x,t\right)+{\left(-{\Delta}-{\lambda }_{f}\right)}_{p}^{s}u\left(x,t\right)=g\left(t,u\left(x,t\right)\right),\quad \hfill & \left(x,t\right)\in {B}_{1}\left(0\right){\times}\left[0,+\infty \right),\hfill \\ u\left(x\right)=0,\quad \hfill & \left(x,t\right)\in {B}_{1}^{c}\left(0\right){\times}\left[0,+\infty \right),\hfill \end{cases}$$ where 0 < s < 1, p > 2, n ≥ 2. We establish Hopf’s lemma for parabolic equations involving a generalized tempered fractional p-Laplacian. Hopf’s lemma will become powerful tools in obtaining qualitative properties of solutions for nonlocal parabolic equations.

Abstract In this paper, we extend our convexity theory for C 2 cost functions in optimal transportation to more general generating functions, which were originally introduced by the second author to extend the framework of optimal transportation to embrace near field geometric optics. In particular we provide an alternative geometric treatment to the previous analytic approach using differential inequalities, which also gives a different derivation of the invariance of the fundamental regularity conditions under duality. We also extend our local theory to cover the strict version of these conditions for C 2 cost and generating functions.