Search

This paper focuses on the study of ground states and nontrivial solutions for the following Hamiltonian elliptic system: { − Δ u + V ( x ) u = f 1 ( x , v ) , x ∈ R 2 , − Δ v + V ( x ) v = f 2 ( x , u ) , x ∈ R 2 , where V ∈ C ( R 2 , ( 0 , ∞ ) ) and f 1 , f 2 : R 2 × R → R have critical exponential growth. The strongly indefinite features together with the critical exponent bring some new difficulties in our analysis. In this paper, we develop a direct approach and use an approaching argument to seek Cerami sequences for the energy functional and estimate the minimax levels of such sequences. In particular, under some general assumptions imposed on the nonlinearity f i , we obtain the existence of ground states and nontrivial solutions for the above system as well as the following system in bounded domain, { − Δ u = f 1 ( v ) , x ∈ Ω , − Δ v = f 2 ( u ) , x ∈ Ω , u = 0 , v = 0 , x ∈ ∂ Ω . Our results improve and extend the related results of de Figueiredo-do O-Ruf (2004) [20] ; (2011) [21] , of Lam-Lu (2014) [30] , and of de Figueiredo-do O-Zhang (2020) [22] .

This paper is concerned with concentration and multiplicity properties of solutions to the following fractional problem with unbalanced growth and critical or supercritical reaction: { ( − Δ ) p s u + ( − Δ ) q s u + V ( e x ) ( | u | p − 2 u + | u | q − 2 u ) = h ( u ) + | u | r − 2 u in R N , u ∈ W s , p ( R N ) ∩ W s , q ( R N ) , u > 0 , in R N , } where e is a positive parameter, 0 s 1 , 2 ⩽ p q N / s , ( − Δ ) t s ( t ∈ { p , q } ) is the fractional t-Laplace operator, while V : R N ↦ R and h : R ↦ R are continuous functions. The analysis developed in this paper covers both critical and supercritical cases, that is, we assume that either r = q s ⁎ : = N q / ( N − s q ) or r > q s ⁎ . The main results establish the existence of multiple positive solutions as well as related concentration properties. In the first case, due to the strong influence of the critical term, the result holds true for “high perturbations” of the subcritical nonlinearity. In the second framework, the result holds true for “low perturbations” of the supercritical nonlinearity. The concentration properties are achieved by combining topological and variational methods, provided that e is small enough and in close relationship with the set where the potential V attains its minimum.

In the present paper, we study the following planar Choquard equation: { − Δ u + V ( x ) u = ( I α ⁎ F ( u ) ) f ( u ) , x ∈ R 2 , u ∈ H 1 ( R 2 ) , where V ( x ) is an 1-periodic function, I α : R 2 → R is the Riesz potential and f ( t ) behaves like ± e β t 2 as t → ± ∞ . A direct approach is developed in this paper to deal with the problems with both critical exponential growth and strongly indefinite features when 0 lies in a gap of the spectrum of the operator − △ + V . In particular, we find nontrivial solutions for the above equation with critical exponential growth, and establish the existence of ground states and geometrically distinct solutions for the equation when the nonlinearity has subcritical exponential growth. Our results complement and generalize the known ones in the literature concerning the positive potential V to the general sign-changing case, such as, the results of de Figueiredo-Miyagaki-Ruf (1995) [16] , of Alves-Cassani-Tarsi-Yang (2016) [4] , of Ackermann (2004) [1] , and of Alves-Germano (2018) [5] .

In this paper, we study the following non-autonomous Choquard-Pekar equation: { − Δ u + V ( x ) u = ( W ⁎ F ( u ) ) f ( u ) , x ∈ R N ( N ≥ 2 ) , u ∈ H 1 ( R N ) , where the potential V ( x ) is 1-periodic and 0 lies in a gap of the spectrum of the Schrodinger operator − Δ + V . Under some general assumptions on the potential W and the nonlinearity f, we show the existence of ground state solutions. We also construct infinitely many geometrically distinct solutions by using the variational method and deformation arguments.

This paper is concerned with the following planar Schrodinger-Poisson system { − Δ u + V ( x ) u + ϕ u = f ( x , u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where V ∈ C ( R 2 , [ 0 , ∞ ) ) is axially symmetric and f ∈ C ( R 2 × R , R ) is of subcritical or critical exponential growth in the sense of Trudinger-Moser. We obtain the existence of a nontrivial solution or a ground state solution of Nehari-type and infinitely many solutions to the above system under weak assumptions on V and f. Our theorems extend the results of Cingolani and Weth [Ann. Inst. H. Poincare Anal. Non Lineaire, 33 (2016) 169-197] and of Du and Weth [Nonlinearity, 30 (2017) 3492-3515] and Chen and Tang [J. Differential Equations, 268 (2020) 945-976], where f ( x , u ) has polynomial growth on u. In particular, some new tricks and approaches are introduced to overcome the double difficulties resulting from the appearance of both the convolution ϕ 2 , u ( x ) with sign-changing and unbounded logarithmic integral kernel and the critical growth nonlinearity f ( x , u ) .

Consider the semilinear Schrodinger equation { − △ u + V ( x ) u = f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where both V ( x ) and f ( x , u ) are periodic in x, 0 belongs to a spectral gap of − △ + V , and f ( x , u ) is subcritical and allowed to be super-linear at some x ∈ R N and asymptotically linear at the other x ∈ R N . In the existing works in the literature, it is commonly assumed that lim | u | → ∞ ⁡ ∫ 0 u f ( x , s ) d s u 2 = ∞ uniformly in x ∈ R N , to obtain the existence of ground state solutions or infinitely many geometrically distinct solutions. In this paper, for the first time, we prove the existence of ground state solutions and infinitely many geometrically distinct solutions under the weaker super-quadratic condition lim | u | → ∞ ⁡ ∫ 0 u f ( x , s ) d s u 2 = ∞ , a.e. x ∈ G just for some domain G ⊂ R N .

In this paper, we develop some new variational and analytic techniques to prove that the following planar Schrodinger-Poisson system { − Δ u + V ( x ) u + ϕ u = f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , admits a nontrivial solution and a ground state solution possessing the least energy in the axially symmetric functions space, where V ( x ) is axially symmetric. Our results improve and extend the ones in the case V = 1 and f ( u ) = | u | p − 2 u with 2 p 6 . In particular, we use the assumption that 2 V ( x ) + ∇ V ( x ) ⋅ x is bounded from below instead of the usually one that lim | x | → ∞ ⁡ V ( x ) = 1 . Moreover, V ( x ) is even admitted to be unbounded.

This paper is concerned with the following singularly perturbed problem in H 1 ( R 2 ) − e 2 Δ u + V ( x ) u + A 0 ( u ( x ) ) u + ∑ j = 1 2 A j 2 ( u ( x ) ) u = f ( u ) , e ( ∂ 1 A 2 ( u ( x ) ) − ∂ 2 A 1 ( u ( x ) ) ) = − 1 2 u 2 , ∂ 1 A 1 ( u ( x ) ) + ∂ 2 A 2 ( u ( x ) ) = 0 , e Δ A 0 ( u ) = ∂ 1 ( A 2 | u | 2 ) − ∂ 2 ( A 1 | u | 2 ) , where e is a small parameter, V ∈ C ( R 2 , R ) and f ∈ C ( R , R ) . By using some new variational and analytic techniques joined with the manifold of Pohozaev–Nehari type, we prove that there exists a constant e 0 > 0 determined by V and f such that for e ∈ ( 0 , e 0 ] , the above problem admits a semiclassical ground state solution v ˆ e with exponential decay at infinity. We also establish a new concentration behaviour of { v ˆ e } as e → 0 . In particular, our results are available to the nonlinearity f ( u ) ∼ | u | s − 2 u for s ∈ ( 4 , 6 ] , which extend the existing results concerning the case f ( u ) ∼ | u | s − 2 u for s > 6 .

In the present paper, we consider the following nonlinear Schrodinger–Poisson system with convolution nonlinearity: { − △ u + V ( x ) u + ϕ u = ( I α ⁎ F ( u ) ) f ( u ) , x ∈ R 3 , − △ ϕ = u 2 , x ∈ R 3 , where α ∈ ( 0 , 3 ) , I α : R 3 → R is the Riesz potential, V ∈ C ( R 3 , [ 0 , ∞ ) ) , f ∈ C ( R , R ) and F ( t ) = ∫ 0 t f ( s ) d s satisfies lim | t | → ∞ ⁡ F ( t ) / | t | σ = ∞ with σ = min ⁡ { 2 , ( 6 + α ) / 4 } . By using some new analytic techniques and new inequalities, we prove the above system admits a ground state solution under mild assumptions on V and f. In particular, our results cover and improve the existing ones for the Schrodinger–Poisson system which can be considered as the limited problem when α → 0 .

In this paper we consider the following Klein–Gorden–Maxwell system − Δ u + V 0 u − ( 2 w + ϕ ) ϕ u = λ | u | p − 2 u + | u | 4 u , x ∈ R 3 , Δ ϕ = ( w + ϕ ) u 2 , x ∈ R 3 , where λ , w and V 0 are positive real constants. By employing some analytical skills and using the variational method, we prove some results about the existence of ground state solutions for the system under a general condition imposed on V 0 . Our results improve and extend some related ones in the literature.

This paper is concerned with the following Klein–Gordon–Maxwell system − △ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) , x ∈ R 3 , △ ϕ = ( ω + ϕ ) u 2 , x ∈ R 3 , where ω > 0 is a constant, V and f are periodic with respect to x . By combining deformation type arguments, Lusternik–Schnirelmann theory and some new tricks, we prove that the above system admits infinitely many geometrically distinct solutions under weaker superlinear conditions instead of the common super-cubic conditions on f . Our result seems new and extends the previous results in the literature.

In this paper, we study the following doubly singularly perturbed fractional Schrodinger–Poissonsystem with critical Sobolev exponent e 2 α ( − Δ ) α u + V ( x ) u + ϕ u = | u | 2 α ∗ − 2 u + f ( u ) in R N , e θ ( − Δ ) s 2 ϕ = γ s u 2 in R N , where α ∈ ( 1 2 , 1 ) , N ∈ ( 2 α , 4 α ) , s ∈ ( N − 2 α , N ) , θ ∈ ( 0 , s ) , f is a subcritical nonlinearity, e is a small parameter, the positive potential V satisfies a local condition. By combining penalization techniques with Ljusternik–Schnirelmann theory, the number of positive solutions is estimated below by the topology of the set where the potential V attains its minimum.

This paper is concerned with the following Klein–Gordon–Maxwell system: − △ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = f ( x , u ) , x ∈ R 3 , △ ϕ = ( ω + ϕ ) u 2 , x ∈ R 3 , where ω > 0 is a constant, V ∈ C ( R 3 , R ) , f ∈ C ( R 3 × R , R ) , and f is superlinear at infinity. Using some weaker superlinear conditions instead of the common super-cubic conditions on f , we prove that the above system has (1) infinitely many solutions when V ( x ) is coercive and sign-changing; (2) a least energy solution when V ( x ) is positive periodic. These results improve the related ones in the literature.

In this paper, we study the following quasilinear Schrodinger equation − Δ u + u − Δ ( u 2 ) u = h ( u ) , x ∈ R N , where N ≥ 3 , 2 ∗ = 2 N N − 2 , h is a continuous function. By using a change of variable, we obtain the existence of ground state solutions. Unlike the condition lim | u | → ∞ ∫ 0 u h ( s ) d s | u | 4 = ∞ , we only need to assume that lim | u | → ∞ ∫ 0 u h ( s ) d s | u | 2 = ∞ .

In this paper, we study the following Hamiltonian elliptic system with gradient term − ϵ 2 Δ ψ + ϵ b → ⋅ ∇ ψ + ψ + V ( x ) φ = ∑ i = 1 I K i ( x ) | η | p i − 2 φ i n R N , − ϵ 2 Δ φ − ϵ b → ⋅ ∇ φ + φ + V ( x ) ψ = ∑ i = 1 I K i ( x ) | η | p i − 2 ψ i n R N , where η = ( ψ , φ ) : R N → R 2 , V , K i ∈ C ( R N , R ) , ϵ > 0 is a small parameter and b → is a constant vector. Suppose that V is sign-changing and has at least one global minimum, and K i has at least one global maximum. We prove that there are two families of semiclassical solutions, for sufficiently small ϵ , with the least energy, one concentrating on the set of minimal points of V and the other on the set of maximal points of K i . Moreover, the convergence and exponential decay of semiclassical solutions are also explored.

In this paper, we consider the following fractional Schrodinger–Poissonproblem ( − Δ ) s u + V ( x ) u + ϕ u = f ( u ) in R 3 , ( − Δ ) t ϕ = u 2 in R 3 , where 0 s ≤ t 1 and 2 s + 2 t > 3 , the potential V ( x ) is weakly differentiable and f ∈ C ( R , R ) . By introducing some new tricks, we prove that the problem admits a ground state solution of Nehari–Pohozaev type under mild assumptions on V and f . The results here extend the existing study.

Background and Objective: High-dimensional data generally contains more accurate information for medical image, e.g., computerized tomography (CT) data can depict the three dimensional structure of organs more precisely. However, the data in high-dimension often needs enormous computation and has high memory requirements in the deep learning convolution networks, while dimensional reduction usually leads to performance degradation. Methods: In this paper, a two-dimensional deep learning segmentation network was proposed for medical volume data based on multi-pinacoidal plane fusion to cover more information under the control of computation.This approach has conducive compatibility while using the model proposed to extract the global information between different inputs layers. Results: Our approach has worked in different backbone network. Using the approach, DeepUnet’s Dice coefficient (Dice) and Positive Predictive Value (PPV) are 0.883 and 0.982 showing the satisfied progress. Various backbones can enjoy the profit of the method. Conclusions: Through the comparison of different backbones, it can be found that the proposed network with multi-pinacoidal plane fusion can achieve better results both quantitively and qualitatively.

In this paper, we study the following generalized quasilinear Schrodinger equation − d i v ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = f ( x , u ) , x ∈ R N , where N ≥ 3 , 2 ∗ = 2 N N − 2 , g ∈ C 1 ( R , R + ) , V ( x ) and f ( x , u ) are 1-periodic on x . By using a change of variable, we obtain the existence of ground states solutions. Unlike the Nahari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the manifold by using the diagonal method.

This paper is concerned with existence of bound states for a coupled gradient system of nonautonomous Schrodinger equations which arises from studies of nonlinear optics and Bose–Einstein condensates. By using variational methods that are constrained to a Nehari–Pohozaev manifold, we obtain existence of nonnegative and positive bound states solutions. In Particular, we show non-existence result for a minimizing problem.

In this paper, we are concerned with the existence and multiplicity of radial and nonradial normalized solutions for the following quasilinear Schrodinger equation modeling the self-channeling of a high-power ultrashort laser in matter: { − Δ u + μ u − [ Δ ( 1 + u 2 ) 1 2 ] u 2 ( 1 + u 2 ) 1 2 = f ( u ) in R N , ∫ R N | u | 2 d x = m , u ∈ H 1 ( R N ) , where m > 0 is a given constant, μ ∈ R is a Lagrange multiplier, and f satisfies the well-known Berestycki–Lions condition.

In this paper, we study the following fractional Kirchhoff equation a + b ∫ R 3 | ( − Δ ) s 2 u | 2 d x ( − Δ ) s u + V ( x ) u = f ( x , u ) + λ | u | p − 2 u , in R 3 , where a , b > 0 , ( − Δ ) s is the fractional Laplace operator with p ≥ 2 s ∗ = 6 3 − 2 s and V , f satisfies suitable conditions. We show that the above equation has a ground state solution and a sign-changing solution via Moser iterative method and truncation technique.

This paper is dedicated to studying the following Schrodinger–Poisson system { − △ u + V ( x ) u + ϕ u = K ( x ) f ( u ) , x ∈ R 3 , − △ ϕ = u 2 , x ∈ R 3 , where V , K ∈ C ( R 3 , R ) and f ∈ C ( R , R ) . Under mild assumptions on the decay rate of V ( x ) , we establish the existence of one ground state sign-changing solution with precisely two nodal domains, by using a weaker growth condition lim ∣ t ∣ → ∞ ∫ 0 t f ( s ) d s ∣ t ∣ 3 = ∞ , instead of the usual super-cubic condition lim ∣ t ∣ → ∞ ∫ 0 t f ( s ) d s ∣ t ∣ 4 = ∞ . Our result unifies both asymptotically cubic and super-cubic cases, which generalizes and improves the corresponding ones in the literatures.

In this paper, we study the existence of ground state sign-changing solutions for the following generalized quasilinear Schrodinger-Maxwell system − d i v ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u + μ ϕ G ( u ) g ( u ) = K ( x ) f ( u ) , x ∈ R 3 , − Δ ϕ = G 2 ( u ) , x ∈ R 3 , where g ∈ C 1 ( R , R + ) , V ( x ) and K ( x ) are positive continuous functions and μ is a positive parameter. By making a change of variable as u = G − 1 ( v ) and G ( u ) = ∫ 0 u g ( t ) d t , we obtain one ground state sign-changing solution v μ = G − 1 ( u μ ) by using some new analytical skills and non-Nehari manifold method. Furthermore, the energy of v μ = G − 1 ( u μ ) is strictly larger than twice that of the ground state solutions of Nehari-type. We also establish the convergence property of v μ = G − 1 ( u μ ) as the parameter μ ↘ 0 . Our results improve and generalize some results obtained by Chen and Tang (2016), Zhu et al. (2016).

This paper is concerned with the following Kirchhoff type equation − a + b ∫ R 3 | ∇ u | 2 d x △ u + V ( x ) u = f ( x , u ) , x ∈ R 3 , u ( x ) → 0 , | x | → ∞ , where a > 0 , b ≥ 0 , V ∈ C ( R 3 , R ) , f ∈ C ( R 3 × R , R ) , V ( x ) and f ( x , u ) u are both allowed to be sign-changing. Using a weaker assumption lim | t | → ∞ ∫ 0 t f ( x , s ) d s | t | 3 = ∞ uniformly in x ∈ R 3 , instead of the common assumption lim | t | → ∞ ∫ 0 t f ( x , s ) d s | t | 4 = ∞ uniformly in x ∈ R 3 , we establish the existence of infinitely many high energy solutions of the above problem, where some new tricks are introduced to overcome the competing effect of nonlocal term. Our result unifies both asymptotically cubic or super-cubic cases, which generalizes and improves the existing ones.

In this paper, we prove the existence of positive solutions and negative solutions for the following modified SchrdingerKirchhoffPoisson type systems {(a+bR3u2)u+V(x)u+u12u(u2)=f(x,u),inR3,=u2,inR3, where a>0, b0, V and f are continuous functions and V(x) is allowed to be sign-changing. Under some certain assumptions on V and f, we prove the existence of nontrivial nonnegative solutions, nontrivial nonpositive solutions and sequence of high energy solutions via the perturbation method.

In the present paper, we consider the following modified quasilinear fourth-order elliptic equation (1.1) { Δ 2 u − Δ u + V ( x ) u − 1 2 Δ ( u 2 ) u = f ( x , u ) , for x ∈ R N , u ( x ) ∈ H 2 ( R N ) , where Δ 2 : = Δ ( Δ ) is the biharmonic operator, V ∈ C ( R N , R ) and f ∈ C ( R N × R , R ) , N ≤ 5 , are allowed to be sign-changing. Two main theorems on the existence of nontrivial solutions and infinitely many high energy solutions for Eq. (1.1) are obtained via variational methods.

In this paper, we study the following fractional Schrodinger–Poisson system ( − Δ ) s u + V ( x ) u + ϕ u = f ( x , u ) + λ | u | p − 2 u , in R 3 , ( − Δ ) t ϕ = u 2 , in R 3 , where s ∈ ( 3 4 , 1 ) , t ∈ ( 0 , 1 ) , λ > 0 is a real parameter, ( − Δ ) s is the fractional Laplace operator, p ≥ 2 s ∗ = 6 3 − 2 s . Under some suitable assumptions on V and f , by using Moser iterative method and truncation technique, we prove that the above equation has a ground state solution and a sign-changing solution.

The paper deals with the existence of nonnegative solutions for ( p , N ) systems in R N involving critical exponential growth nonlinearities. The constructed solution has both components nontrivial and different, that is it solves the actual system, which does not reduce into an equation. The main feature and novelty of the paper is the presence of a general coupled critical exponential term of the Trudinger–Moser type, set in R N .

In the present paper, we consider the existence of ground state sign-changing solutions for a class of Kirchhoff-type problems (0.1) { − ( a + b ∫ Ω | ∇ u | 2 d x ) △ u = f ( u ) , x ∈ Ω ; u = 0 , x ∈ ∂ Ω , where Ω ⊂ R N is a bounded domain with a smooth boundary ∂Ω, N = 1 , 2 , 3 , a > 0 , b > 0 and f ∈ C ( R , R ) . Under some weak assumptions on f , with the aid of some new analytical skills and Non-Nehari manifold method, we prove that (0.1) possesses one ground state sign-changing solution u b , and its energy is strictly larger than twice that of the ground state solutions of Nehari-type. Furthermore, we establish the convergence property of u b as the parameter b ↘ 0 . Our results improve and generalize some results obtained by W. Shuai (2015) [34] .

This paper is concerned with the following Kirchhoff type equation: { - ( a + b ? ? | ? v | 2 ) Δ v = f ( x , v ) in ? , v = 0 on ? ? . Assuming that the primitive of f is asymptotically 4-linear as | v | ? ∞ , a homeomorphism between the Nehari manifold and a subset of unit sphere is constructed based on some observations and new techniques. Inspired by recent work of Szulkin and Weth (2010), ground state solutions for the equation above are obtained, as well as infinitely many pairs of nontrivial solutions for case f ( x , v ) ?is odd with respect to v .

In this paper, we study the following fractional Schrodinger equation ( - Δ ) s u + V ( | x | ) u = f ( | x | , u ) , x ? R N , where ( - Δ ) s ( s ? ( 0 , 1 ) ) denotes the fractional Laplacian. By variational methods, we obtain the existence of a sequence of radial solutions for N ? 2 , a sequence of non-radial solutions for N = 4 or N ? 6 , and a non-radial solution for N = 5 .

In this paper, we study the following diffusion system { ? t u - Δ x u + b ( t , x ) ? ? x u + V ( x ) u = g ( t , x , v ) , - ? t v - Δ x v - b ( t , x ) ? ? x v + V ( x ) v = f ( t , x , u ) where b ? C 1 ( R × R N , R N ) and V ( x ) ? C ( R N , R ) . By using non-Nehari manifold method developed in Tang (2014), we investigate the existence of the ground state solutions of Nehari type for the above problem with periodic and asymptotically periodic nonlinearity.

This paper is concerned with the following Maxwell–Dirac system { − i ∑ k = 1 3 α k ∂ k u + a β u + M ( x ) u − K ( x ) ϕ u = F u ( x , u ) , − Δ ϕ = 4 π K ( x ) ∣ u ∣ 2 , in R 3 where M ( x ) is a external potential and F ( x , u ) is an asymptotically quadratic nonlinearity modeling various types of interactions. In view of the effects of the nonlocal term, we use some special techniques to deal with the nonlocal term. Moreover, existence and multiplicity of stationary solutions are obtained for system without any periodicity assumption via variational methods.

In this paper, we consider the following Kirchhoff problem: − ( 1 + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( u ) , x ∈ R 3 , u ∈ H 1 ( R 3 ) , where b is a positive constant. Assume that f ( u ) is an odd function of u . Under some appropriate assumptions on V but without radial symmetry or compactness hypotheses, we establish the existence of infinitely many solutions to the above problem by using an approximation method employed by Sato and Shibata (2018) and some new arguments.

This paper is concerned with two classes of critical Klein–Gordon–Maxwell systems as follows − Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = μ f ( u ) + u 5 , x ∈ R 3 , Δ ϕ = ( ω + ϕ ) u 2 , x ∈ R 3 and − e 2 Δ u + V ( x ) u − ( 2 ω + ϕ ) ϕ u = g ( x , u ) + K ( x ) u 5 , x ∈ R 3 , Δ ϕ = ( ω + ϕ ) u 2 , x ∈ R 3 , where μ > 0 , e > 0 , V , K ∈ C ( R 3 , R + ) , f ∈ C ( R , R ) and g ∈ C ( R 3 × R , R ) are super-linear at infinity. Instead of the expression “ μ sufficiently large” in the existing works, we give a certain range μ ≥ μ 0 under which the first system admits a ground state solution when V is positive and periodic. Under some mild conditions on V , K and g , we show that the second system has at least one nontrivial solution provided e ∈ ( 0 , e 0 ] , where the bound e 0 > 0 is expressed explicitly in terms of V , K and g .

This paper deals with the following nonlinear Schrodinger–Bopp–Podolsky system − Δ u + V ( x ) u + ϕ u = μ f ( u ) + u 5 − 2 Δ ϕ + a 2 Δ 2 ϕ = 4 π u 2 in R 3 , where a > 0 , V ∈ C ( R 3 , [ 0 , ∞ ) ) with V ∞ ≔ lim | y | → ∞ V ( y ) ≥ sup x ∈ R 3 V ( x ) > 0 , and f ∈ C ( R , R ) satisfying ∫ 0 t f ( s ) d s ≥ t p with p ∈ ( 2 , 6 ) for all t ≥ 0 . By using some new analytic techniques and new inequalities, we prove that the above system admits ground state solutions for all μ > 0 if p ∈ ( 4 , 6 ) ; for all μ > μ 0 if p ∈ ( 2 , 4 ] , where μ 0 is a positive constant determined by a , V ∞ and p . Our results extend and complement the existence results in d’Avenia and G. Siciliano (2019) which only dealt with the subcritical case.

This paper deals with the following variational problem with variable exponents { − div ( ( 1 + | ∇ u | p ( x ) 1 + | ∇ u | 2 p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) , in Ω , u = 0 , on ∂ Ω , where Ω ⊆ R N ( N ≥ 2 ) be a bounded domain with the smooth boundary ∂Ω, p : Ω ‾ → R is a Lipschitz continuous function with 1 p − : = ess inf x ∈ Ω ⁡ p ( x ) ≤ p ( x ) ≤ p + : = ess sup x ∈ Ω ⁡ p ( x ) N and f ∈ C ( Ω × R , R ) is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz type condition. Under three different superlinear conditions on f at infinity, we prove that the above equation has at least a nontrivial solution. Moreover, the existence of infinite many solutions is proved for odd nonlinearity. Especially, some new tricks are introduced to show the boundedness of Cerami sequences.

In this paper, we prove the existence of normalized solutions to the following Schrodinger-Poisson equation − Δ u + ( | x | − 1 ⁎ | u | 2 ) u − f ( u ) = λ u , x ∈ R 3 , λ ∈ R , where f ∈ C ( R , R ) satisfies more general conditions which cover the case f ( u ) ∼ | u | q − 2 u with q ∈ ( 3 , 10 3 ) ∪ ( 10 3 , 6 ) . Especially, some new analytical techniques are presented to overcome the difficulties due to the presence of three terms in the corresponding energy functional which scale differently.

In this paper, we study the following fractional Schrodinger–Poisson system ( − Δ ) s u + V ( x ) u + ϕ u = K ( x ) f ( u ) + | u | 2 s ∗ − 2 u , in R 3 , ( − Δ ) t ϕ = u 2 , in R 3 , where s ∈ ( 3 4 , 1 ) , t ∈ ( 0 , 1 ) are fixed constants, ( − Δ ) s is the fractional Laplace operator, 2 s ∗ = 6 3 − 2 s , V and K are positive functions and f is continuous, superlinear at infinity with quasicritical growth. We show that the above equation has a positive solution via the variational method.

This paper is concerned with the following Choquard equation − △ u + ( ω − β | x | ) u = ( I α ∗ | u | N + α N − 2 ) | u | N + α N − 2 − 2 u , x ∈ R N , where N ≥ 3 , α ∈ ( 0 , N ) , I α is the Riesz potential, ω and β are positive real numbers. For β = 0 , it is known that the above equation has no nontrivial solution. For β > 0 , − β ∕ | x | is called the external Coulomb potential. If 0 β c ( α , ω , N ) , we obtain a ground state solution of Pohozaev type for the above equation, where c ( α , ω , N ) is a constant that can be expressed explicitly via α , ω and N .

We consider the semilinear Schrodinger equation { − △ u + V ( x ) u = f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where V ( x ) is asymptotically periodic and sign-changing, f ( x , u ) is asymptotically periodic in x , and asymptotically linear as | u | → ∞ . Under some weak assumptions on V and f , we prove that the above problem has a nontrivial solution.

In this paper, we study the following diffusion system { ? t u - Δ x u + V ( x ) u = g ( t , x , v ) , - ? t v - Δ x v + V ( x ) v = f ( t , x , u ) where z : = ( u , v ) : R × R N ? R 2 , V ( x ) ? C ( R N , R ) is a general periodic function. Under weaker conditions on nonlinearity, we establish the existence of ground state solutions via variational methods.

In this paper, we study the following quasilinear Schrodinger equation of the form − Δ u + V ( x ) u − Δ ( u 2 ) u = g ( x , u ) , x ∈ R N , where the potential V ( x ) is allowed to be sign-changing, and the primitive of the nonlinearity g ( x , u ) is of superlinear growth at infinity in u and is also allowed to be sign-changing. We obtain the existence of infinitely many nontrivial solutions by using dual approach and Mountain Pass Theorem.

In this paper, we study the following quasilinear elliptic equation − Δ u − Δ ( u 2 ) u = g ( x , u ) , in Ω , u = 0 , on ∂ Ω , where Ω is a bounded smooth domain in R N . Using some special techniques, the existence of infinitely many solutions is obtained by a dual approach, which unifies and improves the recent result of Liu et al. (2013).

In this paper, we study the existence of infinitely many nontrivial solutions for a class of nonlinear Schrodinger–Kirchhoff type equation - a + b ∫ R N | ∇ u | 2 dx Δ u + V ( x ) u = f ( x , u ) in R N , u ( x ) → 0 , as | x | → ∞ , where constants a > 0 , b > 0 and the potential V ( x ) is allowed to be sign-changing. Under general superlinear assumption on nonlinearity f ( x , u ) , we establish the existence of infinitely many solutions via variational methods, which unifies and improves the recent results of Wu (2011) [9].

To improve lipid productivity, co-cultivation of Chlorella sp. U4341 and Monoraphidium sp. FXY-10 for lipid production was studied. Compared with mono-cultivations, co-cultivation of the two microalgae significantly increased the accumulation of total biomass and total lipid yield, and enhanced the lipid productivity (29.52 mg L(-1) d(-1)). Fatty acid compositions significantly varied in different cultivations. The content of C18 fatty acids in co-cultivation significantly increased, especially for oleic acid (32.45%) and linolenic acid (10.03%) compared with that in mono-cultivation. Moreover, high saturated and monounsaturated fatty acids (55.85%) were obtained in co-cultivation, which suggests their potential as a biodiesel feedstock.