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This article is about the following Kirchhoff–Schrödinger–Poisson system with steep potential well 1.1 −(a+b∫ℝ3|∇u|2dx)△u+λV(x)u+μϕ(x)u=f(x,u)+h(x)|u|αinℝ3,−△ϕ=u2,inℝ3,$$ \left\{\begin{array}{ll}-\left(a&#x0002B;b{\int}_{{\mathrm{\mathbb{R}}}&#x0005E;3}{\left&#x0007C;\nabla u\right&#x0007C;}&#x0005E;2 dx\right)\triangle u&#x0002B;\lambda V(x)u&#x0002B;\mu \phi (x)u&#x0003D;f\left(x,u\right)&#x0002B;h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{\alpha }& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\\ {}-\triangle \phi &#x0003D;{u}&#x0005E;2,& \mathrm{in}\kern0.4em {\mathrm{\mathbb{R}}}&#x0005E;3,\end{array}\right. $$ where a,b,λ>0$$ a,b,\lambda >0 $$ are constants, μ>0$$ \mu >0 $$, and 0<α<1,f∈C(ℝN×ℝ,ℝ)$$ 0<\alpha <1,f\in C\left({\mathrm{\mathbb{R}}}&#x0005E;N\times \mathrm{\mathbb{R}},\mathrm{\mathbb{R}}\right) $$. By using the variational principle, we overcome the difficulties caused by Poisson's term and obtain system (1.1) that has two nontrivial solutions under certain assumptions. Moreover, we study the concentration of solutions and obtain new conclusions of system (1.1). Finally, we present the case where the solution to system (1.1) does not exist. [ABSTRACT FROM AUTHOR]

In this paper, we prove the multiplicity of positive solutions for the following singular problem involving the fractional p$p$‐Laplacian: (−Δ)psu+V(x)|u|p−2u=a(x)uκ−1+μb(x)uq−1,inRN$$\begin{equation*} \hspace*{4pc}(-\Delta)_p^su+V(x)|u|^{p-2}u=a(x)u^{\kappa -1}+\mu b(x)u^{q-1}, \mbox{ in } \mathbb {R}^N \end{equation*}$$u(x)>0,inRN.$$\begin{equation*} \hspace*{10.6pc}u(x)>0, \mbox{ in } \mathbb {R}^N. \end{equation*}$$Here, μ>0$\mu >0$, 0<κ<1$0<\kappa <1$, and p

In this paper, we consider a class of Kirchhoff type equation with Hartree nonlinearity: − (1 + a ∫ R N | ∇ u | 2 d x) Δ u + λ V (x) u = (I α ∗ Q | u | p) Q | u | p − 2 u i n R N , where N ≥ 3 , a , λ > 0 parameters, V ∈ C (R N , R +) , I α is the Riesz potential, Q (x) ≥ 0 in R N , and 1 + α N < p < N + α N − 2 . Under some suitable assumptions for potential V (x) and spatial dimensions N, by using the filtration of Nehari manifold and Ekeland variational principle, we prove that above nonlocal equation admits at least two positive solutions. The effect of the steep potential well and spatial dimensions on the number of positive solutions are completely showed. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence and multiplicity of r a d i a l sign-changing solutions for fractional Schrödinger–Poisson system: (℘) (− Δ) s u + u + ϕ u = f (u) , (− Δ) α ϕ = u 2 , in R 3 where s ∈ (3 4 , 1) , α ∈ (0 , 1) and f is a continuous function. Based on perturbation approach and the method of invariant sets of descending flow, we obtain the existence and multiplicity of r a d i a l sign-changing solutions of system ( P ). In addition, by applying the constrained variational method incorporated with Brouwer degree theory, we prove that system ( P ) possesses at least one r a d i a l ground state sign-changing solution. Furthermore, we show that the least energy of sign-changing solutions exceed twice than the least energy, and when f is odd, system ( P ) admits infinitely many nontrivial solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following fractional Schrödinger–Poisson system: { (− Δ) α u + V λ (x) u + μ ϕ u = f (x , u) + β (x) | u | ν − 2 u in R 3 , (− Δ) t ϕ = u 2 in R 3 , where μ > 0 is a parameter, α , t ∈ (0 , 1) , ν ∈ (1 , 2) , 2 α + 2 t > 3 , V λ is allowed to be sign-changing and f is an indefinite function. We require that V λ (x) = λ V + (x) − V − (x) with V + (x) having a potential well Ω whose depth is controlled by λ and V − (x) ≥ 0 for all x ∈ R 3 . Under some suitable assumptions on V λ (x) and f (x , u) , we prove that the above system has at least two nontrivial solutions. Moreover, the phenomenon of concentration of the solutions is also explored. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear fractional Schrödinger–Poisson system as follows: (- Δ) α u + λ V (x) u + G (x) ϕ u = f (x , u) , x ∈ R 3 , (- Δ) α ϕ = G (x) K α u 2 , x ∈ R 3 , where α ∈ (0 , 1) , K α = π - α Γ (α) π - (3 - 2 α) / 2 Γ ((3 - 2 α) / 2) , V ∈ C (R 3) is a potential well, G ∈ L ∞ (R 3) is nonnegative and f ∈ C (R 3 × R) satisfies the sublinear condition in R 3 . By using variational methods and the genus properties in critical point theory, we study the existence and multiplicity of negative energy solutions for the above system. Moreover, the concentration of solutions is also explored on the set V - (0) as λ → ∞ . [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the existence and multiplicity of nonnegative solutions for the following critical Kirchhoff equation involving the fractional p‐Laplace operator (−Δ)ps. More precisely, we consider M∬ℝ2N|u(x)−u(y)|p|x−y|N+psdxdy(−Δ)psu=λf(x)|u|q−2u+K(x)|u|ps*−2u,inΩ,u=0,inℝN\Ω,where Ω⊂ℝN is an open bounded domain with Lipschitz boundary ∂Ω,M(t)=a+btm−1 with m > 1, a > 0, b > 0, dimension N>sp,ps*=NpN−ps is the fractional critical Sobolev exponent, and the parameters λ > 0, 0 < s < 1 < q < p < ∞. Applying the Nehari manifold, fibering maps and Krasnoselskii genus theory, we investigate the existence and multiplicity of nonnegative solutions. [ABSTRACT FROM AUTHOR]

This paper studies the following fractional Kirchhoff equations with steep potential well: 0.1a+b∫ℝN(|(−△)α2u|2)dx(−△)αu+λV(x)u=f(x,u)+μg(x)uq,inℝN,u∈Hα(ℝN),N≥3,where a, b, λ > 0 are parameters, μ > 0, and 0

We are concerned with the existence and multiplicity of nontrivial solutions to the following double phase problems: - div (| ∇ u | p - 2 ∇ u + α (x) | ∇ u | q - 2 ∇ u) + V (x) | u | γ - 2 u = f (x , u) , in Ω , u = 0 , on ∂ Ω , applying the mountain pass theorem and fountain theorem. The Ambrosetti—Rabinowitz condition as well as the monotonicity of f (x , t) / | t | q - 1 are not assumed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions and concave-convex nonlinearities with subcritical or critical growth. Applying Nehari manifold, fibering maps and Ljusternik-Schnirelmann theory, we investigate a relationship between the number of positive solutions and the topology of the global maximum set of K. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following two-component system of Schrödinger equations with Hartree nonlinearity:where is a small parameter, , , and are constants. Under some suitable assumptions on the potentials and the coupled function , we prove the existence and multiplicity of positive solutions for the above system by using energy estimates, the Nehari manifold technique and the Lusternik-Schnirelmann theory. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored. [ABSTRACT FROM AUTHOR]

We consider the following parametric double-phase problem: − div ∇ u p − 2 ∇ u + a x ∇ u q − 2 ∇ u = λ f x , u in Ω , u = 0 , on ∂ Ω. . We do not impose any global growth conditions to the nonlinearity f x , u , which refer solely to its behavior in a neighborhood of u = 0. And we will show that they suffice for the multiplicity of signed and nodal solutions of the double-phase problem above when λ is large enough. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of standing waves with prescribed L2‐norm Schrödinger‐Poisson equations with general nonlinearities in R3: i∂tψ+Δψ−κ(|x|−1*|φ|2)ψ+f(ψ)=0,where κ>0 and f is superlinear and satisfies the monotonicity condition. To this end, we look for critical points of the following functional Eκ(u)=12∫R3|∇u|2+κ4∫R3(|x|−1*u2)u2−∫R3F(u)constrained on the L2‐spheres S(c)=u∈H1(R3):||u||22=c,c>0, where F(s):=∫0sf(t)dt. We consider the case where Eκ is unbounded from below on S(c) and establish the existence of critical points of Eκ on S(c) for c>0 sufficiently small and under some mild assumptions on f. In addition, we show that there are infinitely many radial critical points {unκ} of Eκ on S(c) when f is odd and present a convergence property of unκ as κ↘0. Our results generalize some recent results in the literature. [ABSTRACT FROM AUTHOR]

We consider the following double phase problem with variable exponents: − div ∇ u p x − 2 ∇ u + a x ∇ u q x − 2 ∇ u = λ f x , u in Ω , u = 0 , on ∂ Ω . By using the mountain pass theorem, we get the existence results of weak solutions for the aforementioned problem under some assumptions. Moreover, infinitely many pairs of solutions are provided by applying the Fountain Theorem, Dual Fountain Theorem, and Krasnoselskii's genus theory. [ABSTRACT FROM AUTHOR]

This paper is dedicated to studying the multiplicity of positive solutions for the following Schrödinger–Poisson problem - Δ u + u + ϕ u = λ Q (x) | u | q - 2 u + K (x) | u | 4 u , in R 3 , - Δ ϕ = u 2 , in R 3 , where 4 < q < 6 or q = 2 , λ > 0 is a parameter, K(x) and Q(x) satisfy some mild assumptions. With minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where K(x) attains its global maximum for small λ . [ABSTRACT FROM AUTHOR]

It is well known that a single nonlinear fractional Schrödinger equation with a potential V (x) may have a positive solution that is concentrated at the nondegenerate minimum point of V (x) when the positive parameter ε is sufficiently small [see G. Chen and Y. Zheng, Commun. Pure Appl. Anal. 13, 2359 (2014); J. Dávila et al., J. Differential Equations 256, 858 (2014); and M. M. Fall et al., Nonlinearity 28, 1937 (2015)]. In this work, we find multiple different positive solutions for a class of weakly coupled fractional Schrödinger systems with two potentials V1(x) and V2(x) that have some of the same minimum points; the positive solutions are concentrated at these minimum points. By using energy estimates, the Nehari manifold technique, and the Lusternik–Schnirelmann theory of critical points, we obtain some multiplicity results for a class of weakly coupled fractional Schrödinger systems. The existence and nonexistence of least energy positive solutions are also explored. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following semilinear elliptic systems: - Δ u + V (x) u = F u (x , u , v) , in R N , - Δ v + V (x) v = F v (x , u , v) , in R N , where V : R N → R , F u (x , u , v) and F v (x , u , v) are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of - ▵ + V . Under appropriate assumptions on F u (x , u , v) and F v (x , u , v) , we prove the above system has a ground-state solution by using the Nehari-type technique in a strongly indefinite setting. Furthermore, the existence of infinitely many geometrically distinct solutions is obtained via variational methods. Recent results from the literature are improved and extended. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of positive solutions for a class of nonhomogeneous Klein-Gordon-Maxwell equations { − Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) + h ( x ) , in R 3 , Δ ϕ = ( ω + ϕ ) u 2 , in R 3 , where ω is a positive constant. Under some suitable assumptions on V ( x ), f ( x, u ) and h ( x ), we prove the existence of two positive solutions by using the Ekeland’s variational principle and the Mountain Pass Theorem. These results improve the related ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of sublinear or superlinear p(x)‐Laplacian equations in RN. Some new criteria to guarantee that the existence of multiple solutions for the considered problem is established by using the Morse theory and minimax methods. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and multiplicity results of solutions with prescribed L 2 -norm for a class of Kirchhoff type problems − a + b ∫ R 3 | ∇ u | 2 d x Δ u − λ u = f ( u ) in R 3 , where a , b > 0 are constants, λ ∈ R and f ∈ C ( R , R ) . To obtain such solutions, we look into critical points of the energy functional E b ( u ) = a 2 ∫ R 3 | ∇ u | 2 + b 4 ∫ R 3 | ∇ u | 2 2 − ∫ R 3 F ( u ) constrained on the L 2 -spheres S ( c ) = u ∈ H 1 ( R 3 ) : | | u | | 2 2 = c . Here, c > 0 and F ( s ) ≔ ∫ 0 s f ( t ) d t . Under some mild assumptions on f , we show that critical points of E b unbounded from below on S ( c ) exist for c > 0 . In addition, we establish the existence of infinitely many radial critical points { u n b } of E b on S ( c ) provided that f is odd. Finally, the asymptotic behavior of u n b as b ↘ 0 is analyzed. These conclusions extend some known ones in previous papers. [ABSTRACT FROM AUTHOR]

This paper focuses on the following generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = f ( x , u ) , x ∈ R N , where N ≥ 3 , g ( s ) : R → R + is a nondecreasing function with respect to | s | . By using a change of variables and variational methods, we obtain the existence and multiplicity of nontrivial solutions for the above problem when the nonlinearity is superlinear but does not satisfy the Ambrosetti–Rabinowitz type condition. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following semilinear elliptic systems: where $V(x)$, $H(x)$ are nonnegative continuous functions. Under some appropriate assumptions on $V(x)$, $H(x)$, and $F(x, u, v)$, we prove the existence of infinitely many small negative-energy solutions by using the fountain theorem established by Zou. Recent results from the literature are extended. [ABSTRACT FROM AUTHOR]

In this paper we study a class of boundary value problems with impulsive effects. Under some natural assumptions, by using Morse theory in combination with the minimax arguments, we establish some new results on the existence of multiple nontrivial solutions for the problems. Recent results from the literature are improved and extended. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and multiplicity of nontrivial solutions for a nonlinear Schrödinger–Poisson system. Under some suitable conditions, some criteria on the existence of nontrivial solutions, including a ground state solution, two solutions and a sequence of nontrivial solutions { u k } with ‖ u k ‖ → 0 as k → + ∞ , is established by applying the Morse theory and variational methods. Some known results in the literatures are extended and improved. [ABSTRACT FROM AUTHOR]

In this paper, we study a 4-sublinear Schrödinger–Poisson system with sign-changing potential. Under some suitable assumptions, the existence of two nontrivial solutions are obtained by using the Morse theory. Our result improves the recent ones of Chen and Zhang (2014) [6] . [ABSTRACT FROM AUTHOR]

This paper deals with the existence of nontrivial solutions of the following Kirchhoff-type equation: $- (a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\, dx )\Delta u+V(x)u=f(x,u)$, in $\mathbb{R}^{N}$. Under more relaxed assumptions on V and f, some new criteria on the multiplicity of nontrivial solutions are established by combining Morse theory with local linking and Clark's theorem. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Schrödinger–Poisson system { − Δ u + V ( x ) u + λ ϕ ( x ) u = f ( x , u ) , in R 3 , − Δ ϕ = u 2 , in R 3 , where the nonlinearity f and the potential V are allowed to be sign-changing. Under some appropriate assumptions on V and f , we obtain the existence of two different solutions of the system via the Ekeland’s variational principle and the Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we investigate a boundary value problem for fourth-order differential systems. By using variational methods and a three-critical-point theorem, we establish sufficient conditions under which such a system possesses two solutions generated from a boundary condition. To illustrate the main results, an example is given. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of infinitely many classical solutions for a class of second-order impulsive differential equations. By using two new fountain theorems, we deal with two cases: that when the nonlinearity is superlinear and that when it is asymptotically linear at infinity. Some recent results are extended and improved. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the existence and multiplicity of classical solutions for a second-order impulsive differential equation with periodic boundary conditions. By using a variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution, two solutions and infinitely many solutions when the parameter pair lies in different intervals, respectively. Some examples are given in this paper to illustrate the main results. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of multiple solutions for a class of second-order impulsive Hamiltonian systems. We give some new criteria for guaranteeing that the impulsive Hamiltonian systems with a perturbed term have at least three solutions by using a variational method and some critical points theorems of B. Ricceri. We extend and improve on some recent results. Finally, some examples are presented to illustrate our main results. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence and multiplicity of solutions for two-point boundary value problems for even order ordinary differential equation. By using the critical point theory and the property of projective operator, we establish some conditions on the nonlinearity which guarantee that the nonlinear Hammerstein integral equation with quasi-positive definite kernel has at least one solution, two nontrivial solutions and finite pairs of solutions, respectively. As an application, we get the corresponding results for two-point boundary value problems for even order ordinary differential equation. Moreover, an example is presented to illustrate our main results. [Copyright &y& Elsevier]

In this paper, we consider the following fractional Choquard equation: ε 2 s (− Δ) s u + V (x) u = ε − α (I α ∗ F (u)) F ′ (u) , x ∈ R N , where N ⩾ 3 , s ∈ (0 , 1 and α ∈ (0 , N) , ε > 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. [ABSTRACT FROM AUTHOR]