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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 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]

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]

In this paper, we focus on the dynamical properties of a diffusive Lotka–Volterra system with fear effects subjected to the Neuman boundary condition in a bounded domain. The nonexistence of nonconstant solutions is obtained when diffusion rate is sufficiently large. We also establish the existence and multiplicity of the nonhomogeneous steady-state solutions bifurcating from the constant solution of the system by means of the Lyapunov–Schmidt reduction method. In addition, we consider the influence of the intrinsic growth rate on the population dynamics of the model and show that not only can the population density of both predator and prey change by changing the intrinsic growth rate, but also the coexistence equilibrium can possibly be destabilized. At the same time, we also consider the fear effect on the stability, we find that there is no obvious impact on the stability when the fear effect k is small. However, the results on stability will obviously change when the fear effect is large, which are very different from the ODE system (Wang et al. in J Math Biol 73:1179-1204, 2016) without the fear effect. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of multiple normalized solutions to the following class of elliptic problems - Δ u = λ u + h (ϵ x) f (u) , in R N , ∫ R N | u | 2 d x = a 2 , where a , ϵ > 0 , λ ∈ R is an unknown parameter that appears as a Lagrange multiplier, h : R N → [ 0 , ∞) is a continuous function, and f is continuous function with L 2 -subcritical growth. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ϵ is small enough. [ABSTRACT FROM AUTHOR]

We study the following Kirchhoff equation involving fractional Laplacian in R N where N ≥ 2 , a ≥ 0 , b , μ > 0 , 0 < s < 1 , and (- Δ) s is the fractional Laplacian with order s. By reducing (K) to an equivalent system, we obtain the existence of a positive solution of (K) with general nonlinearities. The positive solution is unique if g (u) = | u | p - 1 u , 1 < p < N + 2 s N - 2 s . Moreover, if the function g is odd, the existence of infinitely many (sign-changing) solutions is concluded. As we shall see, for the case where 0 < s ≤ N 4 , a necessary condition of existence of nontrivial solutions of (K) is that b is small. Our method works well for the so-called degenerate case a = 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity and concentration of positive solutions for the following (p, q)-Laplacian problem: - Δ p u - Δ q u + V (ε x) | u | p - 2 u + | u | q - 2 u = f (u) in R N , u ∈ W 1 , p (R N) ∩ W 1 , q (R N) , u > 0 in R N , where ε > 0 is a small parameter, 1 < p < q < N , Δ r u = div (| ∇ u | r - 2 ∇ u) , with r ∈ { p , q } , is the r-Laplacian operator, V : R N → R is a continuous function satisfying the global Rabinowitz condition, and f : R → R is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik–Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small ε . [ABSTRACT FROM AUTHOR]

We consider positive solutions to the weighted elliptic problem - div (| x | θ ∇ u) = | x | ℓ u p in R N \ B ¯ , u = 0 on ∂ B , where B is the standard unit ball of R N . We give a complete answer for the existence question for N ′ : = N + θ > 2 and p > 0 . In particular, for N ′ > 2 and τ : = ℓ - θ > - 2 , it is shown that for 0 < p ≤ p s : = N ′ + 2 + 2 τ N ′ - 2 , the only nonnegative solution to the problem is u ≡ 0 . This nonexistence result is new, even for the classical case θ = ℓ = 0 and N N - 2 < p ≤ N + 2 N - 2 , N ≥ 3 . The interesting feature here is that we do not require any behavior at infinity or any symmetry assumption. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the qualitative properties of positive solutions for the following two coupled elliptic systems in the punctured space: - Δ u = μ 1 u 2 q + 1 + β u q v q + 1 - Δ v = μ 2 v 2 q + 1 + β v q u q + 1 in R n \ { 0 } , where μ 1 , μ 2 and β are all positive constants, n ≥ 3 . This system is related to two coupled nonlinear Schrödinger equations from nonlinear optics and Bose–Einstein condensates. We establish a monotonicity formula that completely characterizes the singularity of positive solutions. We prove a sharp global estimate for both components of positive solutions. We also prove the nonexistence of positive semi-singular solutions, which means that one component is bounded near the singularity and the other component is unbounded near the singularity. [ABSTRACT FROM AUTHOR]

This paper is concerned with the concentration of positive ground states solutions for a modified Schrödinger equation where $$40$$ is a parameter and $$2^{*}:=\frac{2N}{N-2}(N\ge 3)$$ is the critical Sobolev exponent. We prove the existence of a positive ground state solution $$v_{\varepsilon }$$ and $$\varepsilon $$ sufficiently small under some suitable conditions on the nonnegative functions V( x) and K( x). Moreover, $$v_{\varepsilon }$$ concentrates around a global minimum point of V as $$\varepsilon \rightarrow 0$$ . The proof of the main result is based on minimax theorems and concentration compact theory. [ABSTRACT FROM AUTHOR]

In this paper, we study local bifurcations of an indefinite elliptic system with multiple components: Here $${\Omega \subset\mathbb{R}^N}$$ is a smooth and bounded domain, $${n \geq 3, a < - \Lambda_1 \,\,{\rm where}\,\, \Lambda_1}$$ is the principal eigenvalue of $${(-\Delta, H_{0}^1(\Omega)); \mu_{j} \,\,{\rm and}\,\, \beta}$$ are real constants. Using the positive and nondegenerate solution of the scalar equation $${-\Delta\omega - \omega = -\omega^3, \omega \in H_{0}^1(\Omega)}$$ , we construct a synchronized solution branch $${\mathcal{T}_\omega}$$ . Then we find a sequence of local bifurcations with respect to $${\mathcal{T}_\omega}$$ , and we find global bifurcation branches of partially synchronized solutions. .. [ABSTRACT FROM AUTHOR]

In this manuscript, we deal with an equation involving a combination of quasi-linear elliptic operators of local and non-local nature with p-structure, and concave–convex nonlinearities. The prototypical model is given by - Δ p u + (- Δ) p s u = λ p u q (x) + u r (x) in Ω , u (x) > 0 in Ω , u (x) = 0 on R n \ Ω , where Ω ⊂ R n is a bounded and smooth domain, s ∈ (0 , 1) , 2 ≤ p < ∞ , 0 < q (p) < p - 1 < r (p) < ∞ and 0 < λ p < ∞ , being Δ p and (- Δ) p s the p-Laplace and fractional p-Laplace operators, respectively. We study existence and global uniform and explicit boundedness results to weak solutions. Then, we perform an asymptotic analysis for the limit of a family of weak solutions { u p } p ≥ 2 as p → ∞ , which converges, up to a subsequence (under suitable assumptions on the problem data), to a non-trivial profile with uniform and explicit bounds, enjoying of a universal Lipschitz modulus of continuity, and verifying a nonlinear limiting PDE in the viscosity sense, which exhibits both local/non-local character. [ABSTRACT FROM AUTHOR]

A Kirchhoff-type fractional elliptic system with Hartree-type nonlinearity is proposed to provide a unified framework for well-known nonlinear Schrödinger equations, Kirchhoff equations and Schrödinger-Poisson systems. The existence of nontrivial nonnegative ground state solutions to the system is proved when the coefficient of the potential function is larger than a threshold value, and a precise estimate of the threshold value is given for a prototypical example. It is also shown that the ground state solution concentrates on the zero set of the potential function when the coefficient tends to infinity. [ABSTRACT FROM AUTHOR]

This article concerns with the existence and concentration of positive solutions for the following logarithmic elliptic equation -ϵ2Δu+V(x)u=ulogu2,inRN,u∈H1(RN),where ϵ>0,N≥3 and V is a continuous function with a global minimum. Using variational method developed by Szulkin (Ann Inst H Poincaré Anal Non Linéaire 3:77-109, 1986) for functionals which are sum of a C1 functional with a convex lower semicontinuous functional, we prove, for small enough ϵ>0, the existence of positive solutions and concentration around of a minimum point of V, when ϵ approaches zero. We also study the cases when V is periodic or asymptotically periodic. [ABSTRACT FROM AUTHOR]

Consider the following elliptic system: -ε2Δu1+V1(x)u1=μ1u13+α1u1p-1+βu22u1inR4,-ε2Δu2+V2(x)u2=μ2u23+α2u2p-1+βu12u2inR4,u1,u2>0inR4,u1,u2∈H1(R4),where Vi(x) are trapping potentials, μi,αi>0(i=1,2) and β<0 are constants, ε>0 is a small parameter, and 2. By using the variational method, we obtain a solution to this system for ε>0 small enough. The concentration behaviors of this solution as ε→0+, involving the location of the spikes, are also studied by combining the uniformly elliptic estimates and local energy estimates. To the best of our knowledge, this is the first result devoted to the spikes in the Bose-Einstein condensate with trapping potentials in R4 for the repulsive case. [ABSTRACT FROM AUTHOR]