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In this work, we develop a study involving some nonlinear partial differential equations on spheres and hemispheres, with zero Neumann boundary condition, which are so-called Brezis–Nirenberg type problems, and we provide conditions under which, these equations have only constant solutions. We also extend these results for some nonlinear partial differential systems. [ABSTRACT FROM AUTHOR]

In the present paper, we are concerned with the existence of nonnegative solutions for two Kirchhoff-type problems driven by a fractional p-Laplacian operator, (− Δ) p s , in a bounded smooth domain Ω of R N with N > p s . by employing approximation methods, we establish the existence of nonnegative solutions for each problems under some hypotheses on f , g : Ω × R → R and weak conditions on the diffusion coefficients M , N : (0 , + ∞) → R that we will present. The regularity of finite-energy solutions of both problems is also studied by imposing a few extra hypotheses on Ω, f, and g. [ABSTRACT FROM AUTHOR]

In this paper, we study the following asymptotically critical Choquard equation 0.1 - Δ u = ∫ Ω u 2 α ∗ - ε (y) | x - y | α d y u 2 α ∗ - 1 - ε , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω , where Ω is a smooth bounded domain in R N , N ≥ 3 , α ∈ (0 , N) , 2 α ∗ = 2 N - α N - 2 is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, and ε > 0 is a small parameter. We prove that any family of solutions to (0.1), under a suitable constrain on their gradients, blows up exactly at a critical point of the Robin function. Conversely, by a reduction argument, we construct a family of solutions to (0.1) concentrating around a critical point of the Robin function as ε → 0 . [ABSTRACT FROM AUTHOR]

The object of this work is to study the trichotomy dynamics of fractional heat equation with critical exponent in R n ∂ t u + (- Δ) s u = | u | 4 s n - 2 s u , in R n × (t 0 , + ∞) , u (x , t 0) = u 0 (x) , in R n , where 4 s < n < 6 s , 0 < s < 1. For t 0 sufficiently large, we construct the positive solution, which is smooth and globally defined in time, provided that the initial value satisfies u 0 (x) ∼ | x | - γ with γ > n - 2 s 2. The global solution has the approximate form: (i) for n - 2 s 2 < γ < 2 s , the solution exhibits a algebraic decay; (ii) for γ = 2 s , the solution exhibits a slow logarithmic decay; (iii) for 2 s < γ < n - 2 s , the solution converges to a constant. Our strategy of main proof is based on the inner–outer gluing method in the fractional parabolic equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Schrödinger system involving the tempered fractional p-Laplacian (- Δ - λ) p s u (x) + a u p - 1 (x) = f (u (x) , v (x)) , (- Δ - λ) p t v (x) + b v p - 1 (x) = g (u (x) , v (x)) , where n ≥ 2 , a , b > 0 , 2 < p < ∞ , 0 < s , t < 1 and λ is a sufficiently small positive constant. To effectively utilize the direct method of moving planes, we first establish the narrow region principle and the decay at infinity. Then we prove the radial symmetry and monotonicity of positive solutions for the system in the unit ball and the whole space. Our results are an extension of some content in Ma and Zhang (Appl Math J Chin Univ 37: 52–72, 2022). [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of normalized solutions to the following nonlinear Schrödinger equations with critical growth and potential - Δ u + V (ϵ x) u = λ u + μ | u | q - 2 u + f (u) in R N , ∫ R N | u | 2 d x = a , where a > 0 , μ > 0 , q ∈ (2 , 2 + 4 / N) , ϵ > 0 is a small parameter, λ ∈ R will arise as Lagrange multiplier, the potential V : R N → 0 , + ∞ is a continuous function that satisfies some suitable conditions and f ∈ C (R , R) enjoys critical exponential growth of Trudinger–Moser type when N = 2 , and f (u) = | u | 2 ∗ - 2 u when N ≥ 3 and 2 ∗ = 2 N N - 2 . When ϵ > 0 is sufficiently small, we prove that the existence of normalized solutions by applying truncation techniques, and the relationship between the number of positive solutions and the topology of the set where V attains its minimum is obtained by using Ljusternik–Schnirelmann theory. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Kirchhoff-type problem with critical nonlinearity - a + b ∫ Ω | ∇ u | 2 d x Δ u = λ f (x) | u | p - 2 u + | u | 4 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R 3 , a > 0 is a constant, b , λ are positive parameters and 2 < p < 4 . Under different assumptions on the nonlinearity, the equation has been extensively considered in the case 4 < p < 6 . By contrast, there is no existence result of solutions for the case 2 < p < 4 since the appearance of the nonlocal term. By using some innovative analytical skills, we obtain the existence results about the sign-changing solutions of this problem. Furthermore, we also present asymptotic behaviors of the sign-changing solutions as b ↘ 0 or λ ↘ 0 . [ABSTRACT FROM AUTHOR]

On a smooth, closed Einstein manifold (M, g) of dimension n ≥ 3 with positive scalar curvature and not conformally diffeomorphic to the standard sphere, we prove that the only conformal metrics to g with constant Q-curvature of order 4 are the metrics λ g with λ > 0 constant. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of multiple normalized solutions to the following class of elliptic problems - ϵ 2 Δ u + V (x) u = λ u + f (u) , in R N , ∫ R N | u | 2 d x = a 2 ϵ N , where a , ϵ > 0 , λ ∈ R is an unknown parameter that appears as a Lagrange multiplier, V : R N → [ 0 , ∞) is a continuous function, and f is a continuous function with L 2 -subcritical growth. It is proved that the number of normalized solutions is related to the topological richness of the set where the potential V attains its minimum value. In the proof of our main result, we apply minimization techniques, Lusternik-Schnirelmann category and the penalization method due to del Pino and Felmer (Calc. Var. Partial Differential Equations 4, 121–137 1996). [ABSTRACT FROM AUTHOR]

We investigate a class of semipositone nonlinear elliptic boundary value problems in unbounded domains D of R n , n ≥ 3 . Under some mild-assumptions, we obtain the existence of a unique positive continuous solution and provide information about its global behavior. [ABSTRACT FROM AUTHOR]

In this paper, we prove that under weak assumptions on the reaction terms and diffusion coefficients, a positive solution exists for a one-dimensional case and a positive radial solution to a multidimensional case of a nonlocal elliptic problem. Additionally, we establish the uniqueness of the solution, with the fixed point theorem being the primary tool employed. Our results are new and generalize several existing results. [ABSTRACT FROM AUTHOR]

In this paper, with the help of potential function, we extend the classical Brezis–Lieb lemma on Euclidean space to graphs, which can be applied to the following Kirchhoff equation - 1 + b ∫ V | ∇ u | 2 d μ Δ u + λ V (x) + 1 u = | u | p - 2 u in V , u ∈ W 1 , 2 (V) , on a connected locally finite graph G = (V , E) , where b , λ > 0 , p > 2 and V(x) is a potential function defined on V . The purpose of this paper is four-fold. First of all, using the idea of the filtration Nehari manifold technique and a compactness result based on generalized Brezis–Lieb lemma on graphs, we prove that there admits a positive solution u λ , b ∈ E λ with positive energy for b ∈ (0 , b ∗) when 2 < p < 4 . In the sequel, when p ⩾ 4 , a positive ground state solution w λ , b ∈ E λ is also obtained by using standard variational methods. What’s more, we explore various asymptotic behaviors of u λ , b , w λ , b ∈ E λ by separately controlling the parameters λ → ∞ and b → 0 + , as well as jointly controlling both parameters. Finally, we utilize iteration to obtain the L ∞ -norm estimates of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following quasilinear Schrödinger equation involving Stein–Weiss type nonlinearity: - Δ u + V (x) u - Δ (u 2) u = 1 | x | α ∫ R N G (u (y)) | y - x | μ | y | α d y g (u (x)) , in R N , where N ≥ 3 , 0 < μ < N , α ≥ 0 and 2 α + μ < min { N + 2 2 , 4 } and G is the primitive of function g. The potential V : R N → R may decay to zero at infinity. By using variational methods, penalization technique and L ∞ -estimates, we obtain the existence of a positive solution for the above quasilinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonhomogeneous quasilinear Schrödinger–Poisson system with exponential and logarithmic nonlinearities - Δ u + ϕ u = | u | p - 2 u log | u | 2 + λ f (u) + h (x) , in Ω , - Δ ϕ - ε 4 Δ 4 ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , where 4 < p < + ∞ , ε , λ > 0 are parameters, Δ 4 ϕ = div (| ∇ ϕ | 2 ∇ ϕ) , Ω ⊂ R 2 is a bounded domain, and f has exponential critical growth. First, using reduction argument, truncation technique, Ekeland’s variational principle, and the Mountain Pass theorem, we obtain that the above system admits at least two solutions with different energy for λ large enough and ε fixed. Finally, we research the asymptotic behavior of solutions with respect to the parameters ε . [ABSTRACT FROM AUTHOR]

In this paper, the existence of solutions, (λ , u) , of the problem - Δ u = λ u - a (x) | u | p - 1 u in Ω , u = 0 on ∂ Ω , is explored for 0 < p < 1 . When p > 1 , it is known that there is an unbounded component of such solutions bifurcating from (σ 1 , 0) , where σ 1 is the smallest eigenvalue of - Δ in Ω under Dirichlet boundary conditions on ∂ Ω . These solutions have u ∈ P , the interior of the positive cone. The continuation argument used when p > 1 to keep u ∈ P fails if 0 < p < 1 . Nevertheless when 0 < p < 1 , we are still able to show that there is a component of solutions bifurcating from (σ 1 , ∞) , unbounded outside of a neighborhood of (σ 1 , ∞) , and having u ⪈ 0 . This non-negativity for u cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described. [ABSTRACT FROM AUTHOR]

In this paper, we prove Hopf's lemma for the Logarithmic Laplacian. At first, we introduce the strong minimum principle. Then Hopf's lemma for the Logarithmic Laplacian in the ball is proved. On this basis, Hopf's lemma of the Logarithmic Laplacian is extended to any open set with the property of the interior ball. Finally, an example is given to explain Hopf's lemma can be applied to the study of the symmetry of the positive solution of the nonlinear Logarithmic Laplacian problem by the moving plane method. [ABSTRACT FROM AUTHOR]

This paper studies the following coupled k-Hessian system with different order fractional Laplacian operators: S k (D 2 w (x)) - A (x) (- Δ) α / 2 w (x) = f (z (x)) , S k (D 2 z (x)) - B (x) (- Δ) β / 2 z (x) = g (w (x)). Firstly, we discuss decay at infinity principle and narrow region principle for the k-Hessian system involving fractional order Laplacian operators. Then, by exploiting the direct method of moving planes, the radial symmetry and monotonicity of the nonnegative solutions to the coupled k-Hessian system are proved in a unit ball and the whole space, respectively. We believe that the present work will lead to a deep understanding of the coupled k-Hessian system involving different order fractional Laplacian operators. [ABSTRACT FROM AUTHOR]

This article considers the biharmonic equation Δ 2 u = K (x) f (u) in R N. Under suitable assumptions, the existence of positive solutions is obtained. The methods used here contain the integral operator and the Schauder fixed point theory. Since the form of fundamental solution of Δ 2 u = 0 in R N depends on N, we divide our discussions into three cases as (a) N = 2 ; (b) N = 4 ; (c) N > 2 but N ≠ 4 . The fundamental solution of Δ 2 plays an essential role in our results. [ABSTRACT FROM AUTHOR]

In this paper, the reaction-diffusion equation in space 1D with negative power nonlinearity is presented as an example the stationary solutions of which are characterized by the bifurcation at infinity. The all dynamics, including to infinity, of the ordinary differential equation concerning with the stationary problem of the reaction-diffusion equation are obtained by the Poincaré-type compactification. The results of this paper give relationship between the bifurcation at infinity of the stationary solutions for the reaction-diffusion equation and the connecting orbits of the ordinary differential equations, especially, that connect the equilibria at infinity. Then, for the bifurcation at infinity, the question of what kind of bifurcation structures are present in this ODEs not only before the solutions go to infinity but also after they go to infinity is answered by a precise analysis. Moreover, this result also answers the question of what kind of solutions appear (or disappear) associated with the bifurcation. In addition, from the bifurcation theory view point, to clarify the bifurcation structures of the equilibria in the ordinary differential equation, the dynamics of it through the one-point compactification called the Bendixson compactification is also shown. [ABSTRACT FROM AUTHOR]

We provide a classification result for positive solutions to - Δ u = 1 u γ in the half space, under zero Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

We study the Schrödinger–Poisson–Slater equation where p ∈ (2 , 3) , λ > 0 , V (x) ∈ C (R 3 , R +) and I 2 (x) : = (4 π | x |) - 1 is the Newtonian potential. We prove the nonexistence of nontrivial solutions, existence of positive nonradial (and radial) ground-state solutions, and mountain-pass-type solutions to (SPS), depending on the values of parameters p and λ . To our knowledge, this is the first study of the existence of ground-state solutions at positive energy levels when p ∈ (2 , 3) . Furthermore, we show that a symmetry breaking occurs for the ground-state solutions, which is a purely nonlocal phenomenon that cannot be observed in the local prototype case. [ABSTRACT FROM AUTHOR]

A general time-delay reaction–diffusion–advection system with the Dirichlet boundary condition and spatial heterogeneity is investigated in this paper. By using the implicit function theorem, we obtain the existence and asymptotic expression of the spatially non-homogeneous positive steady-state solution. This is the steady-state bifurcation from zero equilibrium. Via analyzing the corresponding characteristic equation, the stability of the spatially non-homogeneous positive steady-state solution and the occurrence of Hopf bifurcation at the positive steady-state solution are obtained, and the spatially non-homogeneous periodic solution is derived from Hopf bifurcation, this is the secondary bifurcation behavior of the system. Utilizing the normal form method and center manifold theory, we prove that the direction of Hopf bifurcation is supercritical and the bifurcating spatially non-homogeneous periodic solution is stable. Furthermore, We show that there exist two sequences Hopf bifurcation values and the orders of two sequences Hopf bifurcation values are given. Moreover, theoretical and numerical results are applied to competition and cooperation systems, respectively. Finally, the effect of the advection rate and spatial heterogeneity are discussed. [ABSTRACT FROM AUTHOR]

We revisit the well-known Brezis-Nirenberg problem - Δ u = u N + 2 N - 2 + ε u , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω , where ε > 0 and Ω ⊂ R N are a smooth bounded domain with N ≥ 3 . The existence of multi-bump solutions to above problem for small parameter ε > 0 was obtained by Musso and Pistoia (Indiana Univ Math J 51:541–579, 2002). However, to our knowledge, whether the multi-bump solutions are non-degenerate that is open. Here, we give some straightforward answer on this question under some suitable assumptions for the Green's function of - Δ in Ω , which enriches the qualitative analysis on the solutions of Brezis-Nirenberg problem and can be viewed as a generalization of Grossi (Nonlinear Differ Equ Appl 12:227–241, 2005) where the non-degeneracy of a single-bump solution has been proved. And the main idea is the blow-up analysis based on the local Pohozaev identities. [ABSTRACT FROM AUTHOR]

Existence of nodal (i.e., sign changing) solutions and constant-sign solutions for quasilinear elliptic equations involving convection–absorption terms are presented. A location principle for nodal solutions is obtained by means of constant-sign solutions whose existence is also derived. The proof is chiefly based on sub-supersolutions technique together with monotone operator theory. [ABSTRACT FROM AUTHOR]

This paper is dedicated to studying the steady state problem of a population-toxicant model with negative toxicant-taxis, subject to homogeneous Neumann boundary conditions. The model captures the phenomenon in which the population migrates away from regions with high toxicant density towards areas with lower toxicant concentration. This paper establishes sufficient conditions for the non-existence and existence of non-constant positive steady state solutions. The results indicate that in the case of a small toxicant input rate, a strong toxicant-taxis mechanism promotes population persistence and engenders spatially heterogeneous coexistence (see, Theorem 2.3). Moreover, when the toxicant input rate is relatively high, the results unequivocally demonstrate that the combination of a strong toxicant-taxis mechanism and a high natural growth rate of the population fosters population persistence, which is also characterized by spatial heterogeneity (see, Theorem 2.4). [ABSTRACT FROM AUTHOR]

We study the following nonlinear mass supercritical Kirchhoff equation: − (a + b ∫ R N | ∇ u | 2) △ u + μ u = f (u) in R N , where a , b , m > 0 are prescribed, and the normalized constrain ∫ R N | u | 2 d x = m is satisfied in the case 1 ≤ N ≤ 3 . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when 1 ≤ N ≤ 3 and obtain infinitely many radial solutions when 2 ≤ N ≤ 3 by constructing a particular bounded Palais–Smale sequence. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlocal elliptic problem involving a mixed local and nonlocal operator, (P) ∫ Ω f (x , u) d x β L p , s (u) = f α (x , u) in Ω , u > 0 in Ω , u = 0 in R N \ Ω , where Ω ⊂ R N is a bounded regular domain, L p , s ≡ - Δ p + (- Δ) p s , 0 < s < 1 < p < N , α , β ∈ R and f : Ω × R → R be a nonnegative function which is defined almost everywhere with respect to the variable x. Using Schauder and Tychonoff fixed point theorems, we get two existence theorems of weak positive solutions under some hypothesis on α , β and f. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following problem (- Δ) s u = g (x) u 2 s ∗ - 1 + λ u - γ , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω , where Ω ⊂ R N (N > 2 s) is a smooth bounded domain, s ∈ (0 , 1) , λ is a positive constant, 0 < γ < 1 , 2 s ∗ = 2 N N - 2 s and (- Δ) s is the spectral fractional Laplacian. Based upon the Nehari manifold and using variational method we relate the number of positive solutions to the global maximum of the coefficient of the critical nonlinearity g. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear systems involving fully nonlinear nonlocal operators { F α (u (x)) = v p (x) + k 1 (x) u r (x) , x ∈ ℝ N , G β (v (x)) = u q (x) + k 2 (x) v s (x) , x ∈ ℝ N and { F α (u (x)) = v p (x) | x | a + u r (x) | x | b , x ∈ ℝ N \ { 0 } , G β (v (x)) = u q (x) | x | c + v s (x) | x | d , x ∈ ℝ N \ { 0 } , where ki(x) ≥ 0, i = 1, 2, 0 < α, β < 2, p, q, r, s > 1, a, b, c, d > 0. By proving a narrow region principle and other key ingredients for the above systems and extending the direct method of moving planes for the fractional p-Laplacian, we derive the radial symmetry of positive solutions about the origin. During these processes, we estimate the local lower bound of the solutions by constructing sub-solutions. [ABSTRACT FROM AUTHOR]

Here, considering - ∞ < a < N - p p , a ≤ e ≤ a + 1 , d = 1 + a - e and p ∗ : = p ∗ (a , e) = Np N - d p , the existence of positive solution of a weighted p-Laplace equation involving vanishing potentials - Δ ap u + V (x) | x | - e p ∗ | u | p - 2 u = | x | - e p ∗ f (u) in R N is proved, where the potential V can vanish at infinity with exponential decay and f is a function with subcritical growth of class C 1 . We use Del Pino & Felmer’s arguments to overcome the lack of compactness and the Moser iteration method with Caffarelli–Kohn–Nirenberg inequality to obtain estimates of the solution in L ∞ (R N). [ABSTRACT FROM AUTHOR]

We are concerned with the parametrized family of problems P L u = λ a (x) (f (u) - l) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a bounded domain of R N (N ≥ 3) with regular boundary ∂ Ω , L is a general second-order uniformly elliptic operator, λ , l > 0 , a : Ω ¯ → R is a continuous function which may change sign, f : R + → R is subcritical and superlinear at infinity. Under some suitable conditions, we obtain there exists λ 0 > 0 such that (P) has positive solutions for all 0 < λ ≤ λ 0 by topological degree argument and a priori estimates. In doing so, we require f to be of regular variation at infinity. [ABSTRACT FROM AUTHOR]

We consider positive solutions of the Dirichlet problem for the fractional Laplace equation with singular nonlinearity (- Δ) s u (x) = K (x) u - α (x) + μ u p - 1 (x) in Ω , u > 0 in Ω , u = 0 in Ω c : = R N \ Ω , where s ∈ (0 , 1) , α > 0 and Ω ⊂ R N is a bounded domain with smooth boundary ∂ Ω and N > 2 s. Under some appropriate assumptions of α , p , μ and K, we obtain the existence of multiple weak solutions, and among them, including the minimal solution and a ground state solution. Radial symmetry of C loc 1 , 1 ∩ L ∞ solutions are also established for subcritical exponent p when the domain is a ball. Nonexistence of C 1 , 1 ∩ L ∞ solutions are obtained for star-shaped domain under a condition of K. [ABSTRACT FROM AUTHOR]

This paper aims to consider normalized solutions of Schrödinger equations in bounded domains, that is, find (λ , u) ∈ R × H 0 1 (Ω) such that - Δ u + V (x) u + λ u = | u | p - 2 u + | u | q - 2 u in Ω , ∫ Ω u 2 d x = α , where Ω ⊂ R N is a bounded smooth domain, V(x) is the potential, α > 0 is prescribed, 2 < p < 2 + 4 N < q < 2 ∗ . In view of the inherent characteristics of the problem with prescribed mass, there are only few approaches and results on the study of normalized solutions in bounded domains, and the multiplicity of normalized solutions to the problem is completely open so far. In the present paper, for any integer k ∈ N + , we construct k normalized solutions by establishing some special links and using the deformation method on the mass constraint manifold. We further consider the asymptotic behavior of these solutions as mass α tending to 0. Our methods can also be applied to settle another open problem, that is, the multiplicity of normalized solutions of Schrödinger equations with combined mass supercritical nonlinearities ( ∑ i = 1 m | u | q i - 2 u with m ∈ N + and 2 + 4 N < q i < 2 ∗ for any 1 ≤ i ≤ m ) in bounded domains. Our results are new even for the autonomous case V (x) ≡ 0 . [ABSTRACT FROM AUTHOR]

In the present paper, we study the existence of solutions for the following classes of elliptic problems where Ω ⊂ R 2 is a smooth bounded domain and where V ∈ C 0 (R 2) is periodic in Z 2 with 0 ∉ σ (- Δ + V) . In the both problems above, f is a continuous function of the form f (t) = h (t) e α 0 | t | τ , t ∈ R for some α 0 > 0 and τ ≥ 2 and h satisfying some technical conditions. By using variational methods, we show that problems (P) and (P V) have a nontrivial solution for different types of α 0 > 0 and τ ≥ 2 . [ABSTRACT FROM AUTHOR]

We prove existence of weak solutions of a fractional thin film type equation with linear mobility in any space dimension and for any order of the equation. The proof is based on a gradient flow technique in the space of Borel probability measures endowed with the Wasserstein distance. [ABSTRACT FROM AUTHOR]

We prove continuity and Harnack's inequality for bounded solutions to the equation div (∣ ∇ u ∣ p (x) - 2 ∇ u) = 0 , p (x) = p ¯ + L log log 1 ∣ x - x 0 ∣ log 1 ∣ x - x 0 ∣ , p ¯ > 1 , L > 0 , under the precise non-logarithmic condition on the function p(x). [ABSTRACT FROM AUTHOR]

We investigate positive solutions of some classes of stationary Kirchhoff equations. The nonexistence, existence and multiplicity of solutions are expressed by means of precise relationships between the parameters of the equations. The solutions, if they exist, are explicitly determined. [ABSTRACT FROM AUTHOR]

This manuscript is devoted to constructing complete metrics with constant higher fractional curvature on punctured spheres with finitely many isolated singularities. Analytically, this problem is reduced to constructing singular solutions for a conformally invariant integro-differential equation that generalizes the critical GJMS problem. Our proof follows the earlier construction in Ao et al. (Math Ann 369:109–151, 2017), based on a gluing method, which we briefly describe. Our main contribution is to provide a unified approach for fractional and higher order cases. This method relies on proving Fredholm properties for the linearized operator around a suitably chosen approximate solution. The main challenge in our approach is that the solutions to the related blow-up limit problem near isolated singularities need to be fully classified; hence we are not allowed to use a simplified ODE method. To overcome this issue, we approximate solutions near each isolated singularity by a family of half-bubble tower solutions. Then, we reduce our problem to solving an (infinite-dimensional) Toda-type system arising from the interaction between the bubble towers at each isolated singularity. Finally, we prove that this system’s solvability is equivalent to the existence of a balanced configuration. [ABSTRACT FROM AUTHOR]

Let Ω be a bounded smooth domain in ℝN (N ≥ 3). Assuming that 0 < s < 1, 1 < p , q ≤ N + 2 s N − 2 s with (p , q) ≠ (N + 2 s N − 2 s , N + 2 s N − 2 s) , and a, b > 0 are constants, we consider the existence results for positive solutions of a class of fractional elliptic system below, { (a + b [ u ] s 2) (− Δ) s u = v p + h 1 (x , u , v , ∇ u , ∇ v) , x ∈ Ω , (− Δ) s v = u q + h 2 (x , u , v , ∇ u , ∇ v) , x ∈ Ω , u , v > 0 , x ∈ Ω , u = v = 0 , x ∈ ℝ N \ Ω. Under some assumptions of hi(x, u, v, ∇u, ∇v)(i = 1, 2), we get a priori bounds of the positive solutions to the problem (1.1) by the blow-up methods and rescaling argument. Based on these estimates and degree theory, we establish the existence of positive solutions to problem (1.1). [ABSTRACT FROM AUTHOR]

In this paper, we focus on the qualitative analysis of positive solutions for a specific class of static coupled nonlinear Hartree type systems. In the initial step, we convert these equations into integral systems that incorporate the Riesz potentials. Following that, we acquire integrability outcomes and decay characteristics for the integrable solutions of these integral systems, employing the regularity lifting lemma as our second approach. Lastly, as a third step, we establish Liouville-type results and conduct a comprehensive classification of positive solutions for the integral systems in R N using the moving plane method. To the best of our knowledge, this kind of Liouville type result is sharp and new even for Choquard equation. As an application, we deduce the sharp and new Liouville type results for the elliptic system with quadratic nonlinearity. [ABSTRACT FROM AUTHOR]

We establish a priori L ∞ -estimates for non-negative solutions of a semilinear nonlocal Neumann problem. As a consequence of these estimates, we get non-existence of non-constant solutions under suitable assumptions on the diffusion coefficient and on the nonlinearity. Moreover, we prove an existence result for radial, radially non-decreasing solutions in the case of a possible supercritical nonlinearity, extending to the case 0 < s ≤ 1 / 2 the analysis started in [7]. [ABSTRACT FROM AUTHOR]

We consider parabolic equation of the porous medium type model of which is u t - ▵ u m + V (x) f (u) = h (x) | ∇ u m | , u ≥ 0 , m > 1. Under the natural assumptions on the functions f, V and h we prove Harnack's type inequality with constant independent of the solution. [ABSTRACT FROM AUTHOR]

We investigate the existence, uniqueness, and long-time behavior of classical solutions to a coupled system of seven nonlinear parabolic equations. Four of them are determined in the interior of a region, and the other three are solved on a part of the boundary. In particular, such systems arise in modeling of surface reactions that involve the bulk diffusion of reactants toward and reaction products from the biocatalyst surface and surface diffusion of the intermediate reaction products. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of a nonnegative nontrivial weak solution for a fractional critical (p, q)-Laplacian problem with discontinuous nonlinearity. The approach is based on suitable variational methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the sign of solutions to Cauchy problems of linear and semilinear polyharmonic heat equations. Cauchy problems for higher order parabolic equations have no positivity preserving property in general, however, it is expected that solutions to these Cauchy problems are eventually globally positive if initial data decay slowly enough. We first show the existence of the threshold of the decay rate of initial datum which separates whether the corresponding solution to the Cauchy problem of the linear polyharmonic heat equation is eventually globally positive or not. Applying this result, we construct eventually globally positive solutions to the Cauchy problem of the semilinear polyharmonic heat equation under the super-Fujita condition. [ABSTRACT FROM AUTHOR]

Abstract approaches to maximum and anti-maximum principles for differential operators typically rely on the condition that all vectors in the domain of the operator are dominated by the leading eigenfunction of the operator. We study the necessity of this condition. In particular, we show that under a number of natural assumptions, so-called individual versions of both the maximum and the anti-maximum principle simultaneously hold if and only if the aforementioned domination condition is satisfied. Consequently, we are able to show that a variety of concrete differential operators do not satisfy an anti-maximum principle. [ABSTRACT FROM AUTHOR]

We consider the classical Holling–Tanner model extended on 1D space by introducing the diffusion term. Making a reasonable simplification, the diffusive Holling–Tanner system is studied by means of symmetry based methods. Lie and Q -conditional (nonclassical) symmetries are identified. The symmetries obtained are applied for finding a wide range of exact solutions, their properties are studied and a possible biological interpretation is proposed. 3D plots of the most interesting solutions are drown as well. [ABSTRACT FROM AUTHOR]

The article is concerned with the existence of positive solutions of a semi-linear elliptic system defined in a cylinder Ω = Ω ′ × (0 , a) ⊂ R n , where Ω ′ ⊂ R n - 1 is a bounded and smooth domain. The system couples a superlinear equation defined in the whole cylinder Ω with another superlinear (or linear) equation defined at the bottom of the cylinder Ω ′ × { 0 } . Possible applications for such systems are interacting substances (gas in the cylinder and fluid at the bottom) or competing species in a cylindrical habitat (insects in the air and plants on the ground). We provide a priori L ∞ bounds for all positive solutions of the system when the nonlinear terms satisfy certain growth conditions. It is interesting that due to the structure of the system our growth restrictions are weaker than those of the pioneering result by Brezis–Turner for a single equation. Using the a priori bounds and topological arguments, we prove the existence of positive solutions for these particular semi-linear elliptic systems. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of solutions for two classes of singular quasilinear elliptic problems in R N. For the first class of problems, we use a change of variable with global minimization argument and prove the existence and uniqueness of solution. For the second class of problems, we use a change of variable with local minimization argument and an application of a version of the mountain pass theorem combined with a truncation argument to obtain two solutions. [ABSTRACT FROM AUTHOR]