Search

In this study, the Fučik spectrum is used to investigate Kirchhoff-type problems with jumping nonlinearities at infinity, and a new result on the existence of nontrivial solutions is obtained. [ABSTRACT FROM AUTHOR]

Abstract In this study, we focus on the Fučik spectrum for the Kirchhoff-type problem, which is defined as a set Σ comprising those (α , β) ∈ R 2 such that (0.1) − ∫ Ω | ∇ u | 2 Δ u = α (u +) 3 + β (u −) 3 , in Ω , u = 0 , on ∂ Ω has a nontrivial solution, where Ω is an open ball in R N for N = 1 , 2 , 3 ; or Ω ⊂ R 2 is symmetric in x and y , and convex in the x and y directions, u + = max { u , 0 } , u − = min { u , 0 } , and u = u + + u −. First, we prove that the curves { μ 1 } × R , R × { μ 1 } , and C ≔ { (s + c (s) , c (s)) : s ∈ R } belong to Σ , where c (s) = min { β : (s + β , β) ∈ Σ 0 } and Σ 0 comprises those (α , β) ∈ R 2 such that (0.1) has a sign changing solution. We refer to { μ 1 } × R and R × { μ 1 } as trivial curves in Σ in the sense that any solution of (0.1) with (α , β) ∈ { μ 1 } × R or R × { μ 1 } is signed. We denote C as the first nontrivial curve in Σ in the sense that any solution of (0.1) with (α , β) ∈ C is sign changing and for each s ∈ R , we consider the line that passes through (s , 0) with a slope of 1 in the α O β plane R 2 , then the first point on this line that intersects with Σ 0 is simply (s + c (s) , c (s)) ∈ C. Second, we investigate some properties of the function c and the curve C. In particular, c is Lipschitz continuous, decreasing on R and c (s) → μ 1 as s → ∞ , and C is asymptotic to the broken line ℒ 2 ≔ { μ 1 } × [ μ 1 , ∞) ∪ [ μ 1 , ∞) × { μ 1 } . Furthermore, we show that the point (α , β) corresponding to the signed solution of (0.1) is from ℒ ≔ ({ μ 1 } × R) ∪ (R × { μ 1 }) , the point (α , β) corresponding to the sign changing solution of (0.1) is on the upper right of ℒ 2 , and no nontrivial solution of (0.1) exists when (α , β) is between ℒ 2 and C. Finally, as an application, we establish the multiplicity of solutions to the following Kirchhoff-type problem: − 1 + ∫ Ω | ∇ u | 2 Δ u = f (x , u) , in Ω , u = 0 , on ∂ Ω , where the nonlinearity f is asymptotically linear at zero and asymptotically 3-linear at infinity. To the best of our knowledge, this is the first study to consider that the nonlinearity has an extension property at both the zero and infinity points. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an initial boundary value problem to a class of pseudo-parabolic partial differential equations with Newtonian nonlocal term. First, the local existence and uniqueness of a weak solution is established. In virtue of the energy functional and the related Nehari manifold, we also describe the exponent decay behavior and the blow up phenomenon of weak solutions with different kinds of initial data. Our second conclusion states that some solutions starting in a potential well exist globally, whereas solutions with suitable initial data outside the potential well must blow up. Furthermore, the instability of a ground state equilibrium solution is studied. [ABSTRACT FROM AUTHOR]

• Insensitive TN removal to longitudinal heterogeneity at non-excessive CH 4 supply. • Longitudinal gradient in liquid phase affected biomass stratification/CH 4 utilization. • Recommended plug flow operation for MBfR with co-current flow of wastewater and CH 4. Integrating anammox with denitrifying anaerobic methane oxidation (DAMO) in the membrane biofilm reactor (MBfR) is a promising technology capable of achieving complete nitrogen removal from wastewater. However, it remains unknown whether reactor configurations featuring longitudinal gradients parallel to the membrane surface would affect the performance of the CH 4 -driven MBfR. To this end, this work aims to study the impacts of longitudinal heterogeneity potentially present in the gas and liquid phases on a representative CH 4 -driven MBfR performing anammox/DAMO by applying the reported modified compartmental modeling approach. Through comparing the modeling results of different reactor configurations, this work not only offered important guidance for better design, operation and monitoring of the CH 4 -driven MBfR, but also revealed important implications for prospective related modeling research. The total nitrogen removal efficiency of the MBfR at non-excessive CH 4 supply (e.g., surface loading of ≤0.064 g-COD m−2 d−1 in this work) was found to be insensitive to both longitudinal gradients in the liquid and gas phases. Comparatively, the longitudinal gradient in the liquid phase led to distinct longitudinal biomass stratification and therefore played an influential role in the effective CH 4 utilization efficiency, which was also related to the extent of reactor compartmentation considered in modeling. When supplied with non-excessive CH 4 , the MBfR is recommended to be designed/operated with both the biofilm reactor and the membrane lumen as plug flow reactors (PFRs) with co-current flow of wastewater and CH 4 , which could mitigate dissolved CH 4 discharge in the effluent. For the reactor configurations with the biofilm reactor designed/operated as a PFR, multi-spot sampling in the longitudinal direction is needed to obtain a correct representation of the microbial composition of the MBfR. Graphical abstract [Display omitted]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of multiple solutions to the following nonhomogeneous generalized Schrödinger–Poisson system - Δ u + Ku + q ϕ f ( u ) = g ( u ) + h ( x ) , in R 3 , - Δ ϕ = 2 qF ( u ) , in R 3 , where q ⩾ 0 is a parameter, 0 ≠ h ( x ) = h ( | x | ) ∈ L 2 ( R 3 ) , and g is asymptotically linear or superliner at infinity. We show that there exists q 0 > 0 such that the system has at least two positive radial solutions for q ∈ [ 0 , q 0 ) . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the Sturm–Liouville boundary value problem subject to . By constructing a new Sobolev space , we discuss the existence of multiple solutions, especially the existence of multiple sign-changing solutions to this problem when the nonlinear f is resonant both at 0 and . By combining the methods of the Morse theory, the topological degree and the fixed point index, we establish a multiple solutions theorem which guarantees that the problem has at least six nontrivial solutions. If this problem has only finitely many solutions then, of these solutions, there are two positive solutions, two negative solutions and two sign-changing solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we use the topological degree theory and the critical groups to investigate the elliptic equation in and subject to on , and establish a multiple solutions theorem which guarantees that this problem has at least six nontrivial solutions under some resonant conditions. If this problem has only finitely many solutions then, of these solutions, there are two positive solutions, two negative solutions and two sign-changing solutions. [Copyright &y& Elsevier]

Abstract: In this paper, by using the fixed point index theory and Leray–Schauder degree theory, we consider the existence and multiplicity of sign-changing solutions to nonlinear second-order integral boundary value problem for all subject to and , where . We obtain some new existence results concerning sign-changing solutions by computing hardly eigenvalues and the algebraic multiplicities of the associated linear problem. If and satisfy certain conditions, then this problem has at least six different nontrivial solutions: two positive solutions, two negative solutions and two sign-changing solutions. Moreover, if and are also odd, then the problem has at least eight different nontrivial solutions, which are two positive, two negative and four sign-changing solutions. [Copyright &y& Elsevier]

Abstract: For some fourth-order boundary value problems, several new existence theorems on multiple positive, negative and sign-changing solutions are obtained. The critical point theory and the supersolution and subsolution method are employed to discuss this problem. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation where is an -periodic continuous function with , is continuous and is also an -periodic function for each . Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation. [Copyright &y& Elsevier]

Abstract: In this paper, the existence and multiplicity results of the solutions are obtained for the fourth-order boundary value problem for all subject to , where f is continuous. The monotone operator theory and critical point theory are employed to discuss this problem, respectively. [Copyright &y& Elsevier]

The existence of multiple nonnegative solutions of the equations -χ″ = f(χ, χ′) subject to χ(0) = χ(1) = 0 is studied. The result is obtained that there are at least three symmetric nonnegative solutions if certain conditions are imposed on f. [Copyright &y& Elsevier]

We study the reaction–diffusion Lotka–Volterra predator–prey model with Dirichlet boundary condition. In the case of equal diffusion rates and equal growth rates, the synchronized steady state solution is proved to be locally asymptotically stable. [ABSTRACT FROM AUTHOR]

[Display omitted] • Impacts of granule properties on reactors performing partial nitritation/anammox. • Big anammox-enriched granules inoculated for high TN removal & low N 2 O production. • Optimum boundary layer thickness increased with decreasing diffusivity in granules. This work studied the impacts of key granule properties on the granular reactor performing partial nitritation/anammox from the modeling perspective. The results could guide not only future reliable modeling but also practical startup/operation of the reactor. To achieve high total nitrogen (TN) removal whilst avoiding significant N 2 O production, inoculated granules should be big and anammox-enriched. The optimum boundary layer thickness for maximum TN removal increased with the decreasing diffusivity of soluble components in the granule structure. Even though a thick boundary layer could protect anammox bacteria from elevated dissolved oxygen (DO) (e.g., 0.5 g-O 2 /m3) and obtain high TN removal (>90.0%) and low N 2 O production (<1.8%), even complete removal of the boundary layer would fail to provide sufficient substrate for anammox and therefore couldn't increase TN removal to 90.0% and decrease N 2 O production to <2.4% at insufficient DO (e.g., 0.3 g-O 2 /m3 in the presence of lifted influent NH 4 + concentration). [ABSTRACT FROM AUTHOR]

Even though modeling has been frequently used to understand the autotrophic deammonification-based membrane-aerated biofilm reactor (MABR), the relationships between system-specific biofilm property settings and model predicted N 2 O production have yet to be clarified. To this end, this study investigated the impacts of 4 key biofilm property settings (i.e., biofilm thickness/compactness, boundary layer thickness, diffusivity of soluble components in the biofilm structure, and biofilm discretization) on one-dimensional modeling of the MABR, with the focus on its N 2 O production. The results showed that biofilm thickness/compactness (200–1000 μm), diffusivity of soluble components in the biofilm structure (reduction factor of diffusivity: 0.2–0.9), and biofilm discretization (12–28 grid points) significantly influenced the simulated N 2 O production, while boundary layer thickness (0–300 μm) only played a marginal role. In the studied ranges of biofilm property settings, distinct upper and lower bounds of N 2 O production factor (i.e., the percentage ratio of N 2 O formed to NH 4 + removed, 5.5% versus 2.3%) could be predicted. In addition to the microbial community structure, the N 2 O production pathway contribution differentiation was also subject to changes in biofilm property settings. Therefore, biofilm properties need to be quantified experimentally or set properly to model N 2 O production from the MABR correctly. As a good practice for one-dimensional modeling of N 2 O production from biofilm-based reactors, especially the MABR performing autotrophic deammonification, the essential information about those influential biofilm property settings identified in this study should be disclosed and clearly documented, thus ensuring both the reproducibility of modeling results and the reliable applications of N 2 O models. [Display omitted] • Impacts of 4 key biofilm properties on modeling N 2 O production from MABR. • Distinct upper and lower bounds of modeled N 2 O production factor (5.5% vs 2.3%). • Disclosure of quantified/set biofilm properties as a good modeling practice. [ABSTRACT FROM AUTHOR]