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In this paper, the existence and nonexistence of energy minimizer of the Kirchhoff–Schrodinger energy function with prescribed L 2 -norm in dimension four are considered. The energy infimum values are completely classified in terms of coefficient and exponent of the nonlinearity. The sharp existence results of global constraint minimizers for both the subcritical and critical exponent cases are obtained, and the criticality is in the sense of both Sobolev embedding and Gagliardo–Nirenberg inequality. Our results also show the delicate difference between the case without a trapping potential function and the one with potential function.

In this paper, we prove the existence and multiplicity of nontrivial solutions for the following fractional Choquard equation with critical exponent ( − Δ ) s u = λ f ( x ) | u | q − 2 u + g ( x ) ∫ Ω | u | 2 μ , s ∗ | x − y | μ d y | u | 2 μ , s ∗ − 2 u in Ω u = 0 , in R n ∖ Ω where Ω is a bounded domain in R n with smooth boundary, s ∈ ( 0 , 1 ) , 0 μ n , n > 2 s , 1 q 2 and 2 μ , s ∗ = ( 2 n − μ ) ∕ ( n − 2 s ) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. λ > 0 is a parameter and f , g : Ω → R are continuous functions but may change sign on Ω .

We consider the Cauchy problem for semilinear wave equations with variable coefficients and time-dependent scattering damping in R n , where n ≥ 2 . It is expected that the critical exponent will be Strauss’ number p 0 ( n ) , which is also the one for semilinear wave equations without damping terms. Lai and Takamura (2018) have obtained the blow-up part, together with the upper bound of lifespan, in the sub-critical case p p 0 ( n ) . In this paper, we extend their results to the critical case p = p 0 ( n ) . The proof is based on Wakasa and Yordanov (0000), which concerns the blow-up and upper bound of lifespan for critical semilinear wave equations with variable coefficients.

In this paper, we study the symmetry of components of positive solutions for the fractional elliptic systems in the whole space R N . As an application, we prove the uniqueness of positive solutions for fractional Schrodinger type systems with critical exponents. We also prove the symmetry of components and some Liouville type theorems for a class of fractional noncooperative systems, which contains the systems arising in Bose–Einstein condensation and in biological models of Lotka–Volterra type.

In this paper we study the asymptotic and qualitative properties of least energy radial sign-changing solutions of the fractional Brezis--Nirenberg problem ruled by the s-laplacian, in a ball of $\mathbb{R}^n$, when $s \in (0,1)$ and $n > 6s$. As usual, $\lambda$ is the (positive) parameter in the linear part in $u$, and we consider $\lambda$ close to zero. We prove that if such solutions vanish at the center of the ball then they vanish everywhere, we establish a bound on the number of sign-changes and, when $s$ is close to $1$, for a suitable value of the parameter $\lambda$ such solutions change sign exactly once. Moreover, for any $s \in (0,1)$ and $\lambda$ sufficiently small we prove that the number of connected components of the complement of the nodal set corresponds to the number of sign-changes plus one. In addition, for any $s \in (\frac{1}{2},1)$, we prove that least energy nodal solutions which change sign exactly once have the limit profile of a "tower of bubbles", as $\lambda \to 0^+$, i.e. the positive and negative parts concentrate at the same point (which is the center of the ball) with different concentration speeds.

In this paper we study the Henon type equation − Δ u + a ( x ) u = | x | α u p α − e in R N , where p α = N + 2 + 2 α N − 2 , α > 0 , e > 0 , N ≥ 7 . We construct, for e small enough, a family of positive solutions concentrating at the origin under certain conditions on the potential a ( x ) .

The paper is concerned with existence, multiplicity and asymptotic behavior of nonnegative solutions for a fractional Schrodinger–Poisson–Kirchhoff type system. As a consequence, the results can be applied to the special case (a+b‖u‖2)[(−Δ)su+V(x)u]+ϕk(x)|u|p−2u=λh(x)|u|q−2u+|u|2s∗−2uinR3,(−Δ)tϕ=k(x)|u|pinR3,‖u‖=∫R3∫R3|u(x)−u(y)|2|x−y|3+2sdxdy+∫R3V(x)|u|2dx1∕2, where a,b≥0 are two numbers, with a+b>0, 1 0 is a parameter, s,t∈(0,1), (−Δ)s is the fractional Laplacian, k∈L63+2t−p(3−2s)(R3) may change sign, V:R3→[0,∞) is a potential function, 2s∗=6∕(3−2s) is the critical Sobolev exponent, 1

The paper investigates the well-posedness and longtime dynamics of Boussinesq type equations with fractional damping: u t t + Δ 2 u + ( − Δ ) α u t − Δ f ( u ) = g ( x ) , with α ∈ ( 1 , 2 ) . The main results focus on the relations among the dissipative exponent α , the growth exponent p of nonlinearity f ( u ) and the well-posedness and the longtime dynamics of the equations. We find a new critical exponent p α ≡ N + 2 ( 2 α − 1 ) ( N − 2 ( 2 α − 1 ) ) + rather than p ∗ ≡ N + 2 N − 2 ( N ≥ 3 ) as known before and show that when 1 ≤ p p α : (i) The equations are like parabolic, that is, not only the IBVP of the equations are well-posedness, but also their weak solutions are of higher global regularity as t > 0 . (ii) The related solution semigroup has a global attractor A α in natural energy space, and also has an exponential attractor A e x p α in the sense of partially strong topology. In particular, when 1 ≤ p p α ′ ≡ N + 2 α ( N − 2 α ) + ( p α ) , the partially strong topology becomes the strong one. (iii) For any α 0 ∈ [ 1 , 2 ) , the family of global attractors A α is upper semicontinuous at the point α 0 .

In the present paper, we are interested in the following Choquard type equation − Δ u + ( λ V ( x ) − μ ) u = ( I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ − 2 u + | u | p − 2 u in R 3 , where p ∈ ( 4 , 6 ) , λ ∈ R + , μ ∈ R is a constant such that the operator L λ ≔ − Δ + λ V ( x ) − μ is non-degenerate, I α is a Riesz potential of order α ∈ ( 0 , 3 ) , 2 α ∗ = 3 + α is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality, the function V ∈ C ( R 3 , R ) is nonnegative and has a potential well Ω ≔ int V − 1 ( 0 ) , additionally, the operator − Δ has a sequence of Dirichlet eigenvalues in H 0 1 ( Ω ) expressed as 0 μ 1 μ 2 ⋯ μ n ⟶ n + ∞ . If μ μ 1 , via the Nehari manifold techniques and the Ekeland variational principle, we prove the existence and concentration of positive ground state solutions for sufficiently large λ . If μ > μ 1 and μ ≠ μ j for all j ∈ N + , employing the Nehari–Pankov manifold methods and the constrained minimization arguments, we obtain the existence of ground state solution for λ large enough and verify the asymptotic behaviour of ground state solutions as λ → + ∞ .

This paper is devoted to the study of a semilinear equation of the form: − Δ u − μ V ( x ) u = | u | 2 ∗ − 2 u + θ h ( x ) , u ∈ H 0 1 ( Ω ) , where Ω ⊂ R N ( N ≥ 3 ) is an open bounded domain with smooth boundary ∂ Ω , 0 μ μ = ( N − 2 2 ) 2 , 2 ∗ = 2 N N − 2 , h ( x ) > 0 and V ( x ) has prescribed finitely many singular points. Using variational methods, we establish some existence and multiplicity of positive solutions for the problem.

In this paper, we consider the following fractional Choquard equation: e 2 s ( − Δ ) s u + V ( x ) u = e − α ( I α ∗ F ( u ) ) F ′ ( u ) , x ∈ R N , where N ⩾ 3 , s ∈ ( 0 , 1 ] and α ∈ ( 0 , N ) , e > 0 is a parameter, I α is the Riesz potential, and F ( u ) ≔ 1 2 α ♯ | u | 2 α ♯ + 1 2 α ∗ | u | 2 α ∗ , where 2 α ♯ = N + α N and 2 α ∗ = N + α N − 2 s are lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. Firstly, by the refined Sobolev inequality with Morrey norm, we show a generalization of Lions type theorem. Secondly, combining this theorem with variational methods, we show the multiplicity and concentration of positive solutions for above equation. Moreover, the multiplicity and concentration results are obtained in the case where F ( u ) has the lower critical exponent 2 α ♯ , which remains unsolved in the existing literature.

In this paper, we study following Choquard equation with lower critical exponent: − Δ u + V ( x ) u = ( I α ∗ | u | α N + 1 ) | u | α N − 1 u + f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where N ≥ 1 , I α is the Riesz potential, and V : R N → R allows to be sign-changing. Under some mild assumptions imposed on the nonlinearity f , we prove that above equation has a ground state solution for the periodic case and asymptotically periodic case, respectively. The characterization of the ground states is also investigated by a direct approach that are constrained to the Nehari manifold. Moreover, we show a non-existence result for the equation via a generalized Pohožaev identity established for the non-autonomous nonlinearity f . The results extend and improve some recent ones in the literature.

This paper deals with the following Choquard equation with a local nonlinear perturbation: − Δ u + u = I α ∗ | u | α N + 1 | u | α N − 1 u + f ( u ) , x ∈ R N ; u ∈ H 1 ( R N ) , where I α : R N → R is the Riesz potential, N ≥ 3 , α ∈ ( 0 , N ) , the exponent α N + 1 is critical with respect to the Hardy–Littlewood–Sobolev inequality, and the nonlinear perturbation f is only required to satisfy some weak assumptions near 0 and ∞ . Our results improve the previous related ones in the literature.

This paper is devoted to the fractional Laplacian system with critical exponents. We use the method of moving sphere to derive a Liouville Theorem, and then prove the solutions in R^n\{0} are radially symmetric and monotonically decreasing radially. Together with blow up analysis and the Pohozaev integral, we get the upper and lower bound of the local solutions in B_1\{0}. Our results is an extension of the classical work by Caffarelli et al [6, 7], Chen et al[16], arXiv admin note: text overlap with arXiv:1309.2021, arXiv:1404.1756 by other authors