Search

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 investigate the existence and the asymptotic behavior of positive continuous solutions for a class of nonlinear polyharmonic boundary value problems in the punctured unit ball of $ \mathbb {R}^{n} (n \geqslant 3) $ R n (n ⩾ 3). Our arguments are based on potential theory associated to the polyharmonic operator $ (-\Delta)^{m} $ (− Δ) m , where m is a positive integer less than $ \frac {n}{2} $ n 2 , properties of functions in the Kato class $ K_{m,n} $ K m , n , Karamata regular variation theory tools and the Schauder fixed point theorem. [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]

We consider the nonlinear elliptic system $$\begin{align*} \left\{ \begin{array}{ll} u \in W_{0}^{\frac{N}{N-1}}(\Omega): & -\textrm{div}(M(x)\nabla u) + u = -\textrm{div}(u\,M(x)\,\nabla\psi) + f(x),\\ \displaystyle \psi \in W_0^{1,2}(\Omega): & -\textrm{div}(M(x)\nabla \psi) + \psi = R(u) + E(x)\,\nabla\psi, \end{array} \right. \end{align*}$$ { u ∈ W 0 N N − 1 (Ω) : − div (M (x) ∇u) + u = − div (u M (x) ∇ψ) + f (x) , ψ ∈ W 0 1 , 2 (Ω) : − div (M (x) ∇ψ) + ψ = R (u) + E (x) ∇ψ , where Ω is a bounded, open subset of $ \mathbb {R}^N $ R N , $ N \geq ~3 $ N ≥ 3 ; $ M(x) $ M (x) is a coercive, symmetric matrix with $ L^{\infty }(\Omega) $ L ∞ (Ω) coefficients; $ f(x) $ f (x) and $ E(x) $ E (x) belong to some Lebesgue space, and $ R(s) $ R (s) is a continuous function such that $$\begin{align*} 0 \leq R(s)\leq |s|^{\theta}, \quad \mbox{for}\ \theta { \lt } \frac{2}{N}. \end{align*}$$ 0 ≤ R (s) ≤ | s | θ , for θ < 2 N. Using a duality technique, we prove existence of at least a weak solution $ (u,\psi) $ (u , ψ). Moreover, if N=3 or N=4, we prove under stronger assumptions on $ f(x) $ f (x) and $ E(x) $ E (x) that the solution u belongs to $ W_0^{1,2}(\Omega) $ W 0 1 , 2 (Ω). [ABSTRACT FROM AUTHOR]

This article considers existence of solution for a class of fractional Schrödinger-Poisson system. By using the Nehari method and the variational method, we obtain a uniqueness result for positive solutions.

In this article, we study the following nonlinear kk-Hessian system with singular weights Sk1k(σ(D2u1))=λb(∣x∣)f(−u1,−u2),inΩ,Sk1k(σ(D2u2))=λh(∣x∣)g(−u1,−u2),inΩ,u1=u2=0,on∂Ω,\left\{\begin{array}{ll}{S}_{k}^{\frac{1}{k}}(\sigma ({D}^{2}{u}_{1}))=\lambda b\left(| x| )f\left(-{u}_{1},-{u}_{2}),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {S}_{k}^{\frac{1}{k}}(\sigma ({D}^{2}{u}_{2}))=\lambda h\left(| x| )g\left(-{u}_{1},-{u}_{2}),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {u}_{1}={u}_{2}=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where λ>0\lambda \gt 0, 1≤k≤N1\le k\le N is an integer, Ω\Omega stands for the open unit ball in RN{{\mathbb{R}}}^{N}, and Sk(σ(D2u)){S}_{k}(\sigma ({D}^{2}u)) is the kk-Hessian operator of uu. By using the fixed point index theory, we prove the existence and nonexistence of negative kk-convex radial solutions. Furthermore, we establish the multiplicity result of negative kk-convex radial solutions based on a priori estimate achieved. More precisely, there exists a constant λ∗>0{\lambda }^{\ast }\gt 0 such that the system admits at least two negative kk-convex radial solutions for λ∈(0,λ∗)\lambda \in \left(0,{\lambda }^{\ast }).

This article is concerned with the following Kirchhoff equation: −a+b∫R3∣∇u∣2dxΔu=g(u)+h(x)inR3,-\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u=g\left(u)+h\left(x)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where aa and bb are positive constants and h≠0h\ne 0. Under the Berestycki-Lions type conditions on gg, we prove that the equation has at least two positive solutions by using variational methods. Furthermore, we obtain the existence of ground state solutions.

For the following quasilinear Choquard-type equation: −Δu−Δ(u2)u+V(x)u=(Iμ*∣u∣p)∣u∣p−2u,x∈RN,-\Delta u-\Delta \left({u}^{2})u+V\left(x)u=\left({I}_{\mu }* {| u| }^{p}){| u| }^{p-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N≥3,0

We study the gradient flow of an energy with mixed homogeneity, which is at the interface of Finsler and sub-Riemannian geometry.

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 study the existence of ground state solution for the following Kirchhoff equation with combined nonlinearities \[ -\left(a+b\displaystyle\int_{\mathbb{R}^N} |\nabla u|^2\right)\Delta u=\lambda u+|u|^{p-2}u+\mu |u|^{q-2}u \ \hbox{in}\ \mathbb{R}^N, \quad 1\leq N\leq 3 \] − (a + b ∫ R N | ∇u | 2) Δu = λu + | u | p − 2 u + μ | u | q − 2 u in R N , 1 ≤ N ≤ 3 with prescribed mass $ \int _{\mathbb {R}^N} u^2=c^2 $ ∫ R N u 2 = c 2 , where $ a{ \gt }0,b{ \gt }0,c{ \gt }0 $ a > 0 , b > 0 , c > 0 , $ \mu { \lt }0 $ μ < 0 , $ 2{ \lt }q\leq 2+\frac {8}{N}{ \lt }p{ \lt }2^{*} $ 2 < q ≤ 2 + 8 N < p < 2 ∗ . Under certain assumptions on the parameter μ, we obtain the existence of normalized solution $ (\tilde {u},\lambda _c)\in S_c \times \mathbb {R} $ (u ~ , λ c) ∈ S c × R , where $ S_c=\{ u\in H^1(\mathbb {R}^N): \int _{\mathbb {R}^N} u^2=c^2 \} $ S c = { u ∈ H 1 (R N) : ∫ R N u 2 = c 2 }. [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 article, we are first concerned with the existence of multiple normalized solutions to the following fractional p-Laplace problem: { (- Δ) p s ⁢ v + 풱 ⁢ (ξ ⁢ x) ⁢ | v | p - 2 ⁢ v = λ ⁢ | v | p - 2 ⁢ v + f ⁢ (v) in ⁢ ℝ N , ∫ ℝ N | v | p ⁢ 푑 x = a p , where a , ξ > 0 , p ≥ 2 , λ ∈ ℝ is an unknown parameter that appears as a Lagrange multiplier, 풱 : ℝ N → [ 0 , ∞) is a continuous function, and f is a continuous function with L p -subcritical growth. We prove that there exists the multiplicity of solutions by using the Lusternik–Schnirelmann category. Next, we show that the number of normalized solutions is at least the number of global minimum points of 풱 , as ξ is small enough via Ekeland's variational principle. [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]

AbstractIn this paper, we consider a class of nonlinear p-k-Hessian systems with the gradient termwhere γ > 0, p > 1, and 1 ≤ k ≤ N is an integer. We show some existence, boundedness and blow-up results of the entire positive p-k-convex radial solutions based on the monotone iterative method. Finally, we present an example to illustrate our result. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of positive solutions of an equation with p-Laplacian where n ≥ 3, 1 < p < n, q > 0, Δpu = div(|∇u|p−2∇u), and K(x) is a double bounded function. We will prove the following results. When 0 < q < p − 1, the equation has no solution satisfying inf Rnu = 0 for any double bounded function K(x). When q = p − 1, the equation has a positive radial singular solution for K(x) ≡ Const.. In addition, q = p − 1 is a necessary condition such that this equation (with K(x) ≡ 1) has finite energy solutions. In addition, the existence of positive solutions are also derived for the system where u, v > 0 in Rn \ {0}, n ≥ 3, 1 < p1, p2 < n, q1, q2 > 0, and K1(x), K2(x) are double bounded functions. [ABSTRACT FROM AUTHOR]

In this article, we consider the double-phase problem with nonlocal reaction. For the autonomous case, we introduce the methods of the Pohozaev manifold, Hardy-Littlewood Sobolev subcritical approximation, adding the mass term to prove the existence and nonexistence of normalized solutions to this problem. For the nonautonomous case, we show the existence of normalized solutions to the double-phase problem by using the Pohozaev restrict method and describing the relationship between the energy of this problem and its limit problem. Moreover, we study the existence of normalized solutions to the double-phase problem involving double Hardy-Littlewood-Sobolev critical exponents.

In this paper, we consider the nonlinear equation involving the nonlocal pseudo-relativistic operators \[ (-\Delta +m^2)^s u(x) = f(x,u(x)) , \] (− Δ + m 2) s u (x) = f (x , u (x)) , where 00. The nonlocal pseudo-relativistic operator includes the pseudo-relativistic Schrödinger operator $ \sqrt {-\Delta +m^2} $ − Δ + m 2 . When $ m\rightarrow 0^+ $ m → 0 + , the nonlocal pseudo-relativistic operator $ (-\Delta +m^2)^s $ (− Δ + m 2) s is also closely related to the fractional Laplacian operator $ (-\Delta)^s $ (− Δ) s . But these two operators are quite different. We first establish a Hopf type lemma for anti-symmetric functions to nonlocal pseudo-relativistic operators, which play a key role in the method of moving planes. The main difficulty is to construct a suitable sub-solution to nonlocal pseudo-relativistic operators. Then we prove a pointwise estimate to nonlocal pseudo-relativistic operators. As an application, combined with the Hopf type lemma and the pointwise estimate, we obtain the radial symmetry and monotonicity of positive solutions to the above nonlinear nonlocal pseudo-relativistic equation in the whole space. We believe that the Hopf type lemma will become a powerful tool in applying the method of moving planes on nonlocal pseudo-relativistic equations to obtain qualitative properties of solutions. [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]

The paper demonstrates two approaches to constructing monotonic difference schemes for the transfer equation in divergent form from the family of semi-Lagrangian methods: Eulerian–Lagrangian and Lagrangian–Eulerian. Within each approach, a monotonic conservative difference scheme is proposed. It is shown that within the framework of the Lagrangian–Eulerian approach, based on the use of curvilinear grids formed by the characteristics of the approximated transfer operator, it is possible to construct monotonic difference schemes of second order accuracy. [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 article is concerned with the structure of solutions to the elliptic problem for a Hénon-type equation with a forcing term: −Δu=α(x)up+κμ,inRN,u>0,inRN,(Pκ)\hspace{11.3em}-\Delta u=\alpha \left(x){u}^{p}+\kappa \mu ,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}u\gt 0,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{13.0em}\left({{\rm{P}}}_{\kappa }) where N≥3N\ge 3, p>1p\gt 1, κ>0\kappa \gt 0, and α\alpha is a positive continuous function in RN\{0}{{\mathbb{R}}}^{N}\setminus \left\{0\right\}, and μ\mu is a nonnegative Radon measure in RN{{\mathbb{R}}}^{N}. Under suitable assumptions on the exponent pp, the coefficient α\alpha , and the forcing term μ\mu , we give a complete classification of the existence/nonexistence of solutions to problem (Pκ{{\rm{P}}}_{\kappa }) with respect to κ\kappa .

The existence of L 2–normalized solutions is studied for the equation −Δu+μu=f(x,u) inRN,∫RNu2dx=m. $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R}}^{N},\quad {\int }_{{\mathbf{R}}^{N}}{u}^{2} \mathrm{d}x=m.$ Here m > 0 and f(x, s) are given, f(x, s) has the L 2-subcritical growth and (μ, u) ∈ R × H 1(R N) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.

In this paper, we are concerned with the following fractional relativistic Schrödinger equation with critical growth: (−Δ+m2)su+V(εx)u=f(u)+u2s*−1inRN,u∈Hs(RN),u>0inRN,\left\{\begin{array}{ll}{\left(-\Delta +{m}^{2})}^{s}u+V\left(\varepsilon x)u=f\left(u)+{u}^{{2}_{s}^{* }-1}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{s}\left({{\mathbb{R}}}^{N}),\hspace{1.0em}u\gt 0& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where ε>0\varepsilon \gt 0 is a small parameter, s∈(0,1)s\in \left(0,1), m>0m\gt 0, N>2sN\gt 2s, 2s*=2NN−2s{2}_{s}^{* }=\frac{2N}{N-2s} is the fractional critical exponent, (−Δ+m2)s{\left(-\Delta +{m}^{2})}^{s} is the fractional relativistic Schrödinger operator, V:RN→RV:{{\mathbb{R}}}^{N}\to {\mathbb{R}} is a continuous potential, and f:R→Rf:{\mathbb{R}}\to {\mathbb{R}} is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential VV, we construct a family of positive solutions uε∈Hs(RN){u}_{\varepsilon }\in {H}^{s}\left({{\mathbb{R}}}^{N}), with exponential decay, which concentrates around a local minimum of VV as ε→0\varepsilon \to 0.

In this article, we are concerned about the existence, uniqueness, and nonexistence of the positive solution for: −Δu−12(x⋅∇u)=μh(x)uq−1+λu−up,x∈RN,u(x)→0,as∣x∣→+∞,\left\{\begin{array}{l}-\Delta u-\frac{1}{2}\left(x\cdot \nabla u)=\mu h\left(x){u}^{q-1}+\lambda u-{u}^{p},\hspace{1.0em}x\in {{\mathbb{R}}}^{N},\\ u\left(x)\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| x| \to +\infty ,\end{array}\right. where N⩾3N\geqslant 3, 00\lambda \gt 0, p>1p\gt 1, μ>0\mu \gt 0 is a parameter and the function h(x)h\left(x) satisfies certain conditions. To start with, based on the variational argument and perturbation method, we obtain the existence and uniqueness of the positive solution for the aforementioned singular elliptic differential equation as λ>N2\lambda \gt \frac{N}{2}. In addition, there is no solution as λ⩽N2\lambda \leqslant \frac{N}{2}. Later, from an experimental point of view, we give the numerical solution of the aforementioned singular elliptic differential equation by means of a neural network in some special cases, which enrich the theoretical results. Our conclusions partially extend the results corresponding to the nonsingular case.

We show that the classical strong maximum principle, concerning positive supersolutions of linear elliptic equations vanishing on the boundary of the domain can be extended, under suitable conditions, to the case in which the forcing term is sign-changing. In addition, for the case of solutions, the normal derivative on the boundary may also vanish on the boundary (definition of flat solution). This leads to examples in which the unique continuation property fails. As a first application, we show the existence of positive solutions for a sublinear semilinear elliptic problem of indefinite sign. A second application, concerning the positivity of solutions of the linear heat equation, for some large values of time, with forcing and/or initial datum changing sign, is also given.