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Communications on Pure & Applied Analysis

May 2019 , Volume 18 , Issue 3

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Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator
Thi Tuyen Nguyen
2019, 18(3): 999-1021 doi: 10.3934/cpaa.2019049 +[Abstract](38) +[HTML](14) +[PDF](411.33KB)

We study the well-posedness of second order Hamilton-Jacobi equations with an Ornstein-Uhlenbeck operator in \begin{document}$\mathbb{R}^N$\end{document} and \begin{document}$\mathbb{R}^N× [0, +∞).$\end{document} As applications, we solve the associated ergodic problem associated to the stationary equation and obtain the large time behavior of the solutions of the evolution equation when it is nondegenerate. These results are some generalizations of the ones obtained by Fujita, Ishii & Loreti 2006 [19] by considering more general diffusion matrices or nonlocal operators of integro-differential type and general sublinear Hamiltonians. Our work uses as a key ingredient the a-priori Lipschitz estimates obtained in Chasseigne, Ley & Nguyen 2017 [10].

Uniqueness for Neumann problems for nonlinear elliptic equations
Maria Francesca Betta, Olivier Guibé and Anna Mercaldo
2019, 18(3): 1023-1048 doi: 10.3934/cpaa.2019050 +[Abstract](38) +[HTML](19) +[PDF](408.87KB)

In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is

where \begin{document}$Ω$\end{document} is a bounded domain of \begin{document}$\mathbb{R}^{N}$\end{document}, \begin{document}$N≥ 2$\end{document}, with Lipschitz boundary, \begin{document}$ 1 < p < N$\end{document}, \begin{document}$\underline n$\end{document} is the outer unit normal to \begin{document}$\partial Ω$\end{document}, the datum \begin{document}$f$\end{document} belongs to \begin{document}$L^{(p^{*})'}(Ω)$\end{document} or to \begin{document}$L^{1}(Ω)$\end{document} and satisfies the compatibility condition \begin{document}$∈t_Ω f \, dx = 0$\end{document}. Finally the coefficient \begin{document}$c(x)$\end{document} belongs to an appropriate Lebesgue space.

Entire solutions in nonlocal monostable equations: Asymmetric case
Yu-Juan Sun, Li Zhang, Wan-Tong Li and Zhi-Cheng Wang
2019, 18(3): 1049-1072 doi: 10.3934/cpaa.2019051 +[Abstract](25) +[HTML](17) +[PDF](416.88KB)

This paper is concerned with entire solutions of the monostable equation with nonlocal dispersal, i.e., \begin{document}$u_{t}=J*u-u+f(u)$\end{document}. Here the kernel \begin{document}$J$\end{document} is asymmetric. Unlike symmetric cases, this equation lacks symmetry between the nonincreasing and nondecreasing traveling wave solutions. We first give a relationship between the critical speeds \begin{document}$c^{*}$\end{document} and \begin{document}$\hat{c}^{*}$\end{document}, where \begin{document}$c^*$\end{document} and \begin{document}$\hat{c}^{*}$\end{document} are the minimal speeds of the nonincreasing and nondecreasing traveling wave solutions, respectively. Then we establish the existence and qualitative properties of entire solutions by combining two traveling wave solutions coming from both ends of real axis and some spatially independent solutions. Furthermore, when the kernel \begin{document}$J$\end{document} is symmetric, we prove that the entire solutions are 5-dimensional, 4-dimensional, and 3-dimensional manifolds, respectively.

The properties of positive solutions to semilinear equations involving the fractional Laplacian
Rongrong Yang and Zhongxue Lü
2019, 18(3): 1073-1089 doi: 10.3934/cpaa.2019052 +[Abstract](51) +[HTML](20) +[PDF](382.99KB)

Let \begin{document}$Ω$\end{document} be either a unit ball or a half space. Consider the following Dirichlet problem involving the fractional Laplacian

where \begin{document}$α$\end{document} is any real number between \begin{document}zhongwenzy$\end{document} and \begin{document}$\end{document}. Under some conditions on \begin{document}$f$\end{document}, we study the equivalent integral equation

here \begin{document}$G(x, y)$\end{document} is the Green's function associated with the fractional Laplacian in the domain \begin{document}$Ω$\end{document}. We apply the method of moving planes in integral forms to investigate the radial symmetry, monotonicity and regularity for positive solutions in the unit ball. Liouville type theorems-non-existence of positive solutions in the half space are also deduced.

A study of comparison, existence and regularity of Viscosity and weak solutions for quasilinear equations in the Heisenberg group
Pablo Ochoa and Julio Alejo Ruiz
2019, 18(3): 1091-1115 doi: 10.3934/cpaa.2019053 +[Abstract](24) +[HTML](20) +[PDF](424.17KB)

In this manuscript, we are interested in the study of existence, uniqueness and comparison of viscosity and weak solutions for quasilinear equations in the Heisenberg group. In particular, we highlight the limitation of applying the Euclidean theory of viscosity solutions to get comparison of solutions of sub-elliptic equations in the Heisenberg group. Moreover, we will be concerned with the equivalence of different notions of weak solutions under appropriate assumptions for the operators under analysis. We end the paper with an application to a Radó property.

The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise
T. Tachim Medjo
2019, 18(3): 1117-1138 doi: 10.3934/cpaa.2019054 +[Abstract](27) +[HTML](27) +[PDF](439.67KB)

In this article, we study the stability of weak solutions to a stochastic version of a coupled Cahn-Hilliard-Navier-Stokes model with multiplicative noise. The model consists of the Navier-Stokes equations for the velocity, coupled with an Cahn-Hilliard model for the order (phase) parameter. We prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions. We also prove a result related to the stabilization of these equations.

Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions
Eun Kyoung Lee, R. Shivaji, Inbo Sim and Byungjae Son
2019, 18(3): 1139-1154 doi: 10.3934/cpaa.2019055 +[Abstract](35) +[HTML](14) +[PDF](439.43KB)

We study positive solutions to (singular) boundary value problems of the form:

where \begin{document}$\varphi_p(u): = |u|^{p-2}u$\end{document} with \begin{document}$p>1$\end{document} is the \begin{document}$p$\end{document}-Laplacian operator of \begin{document}$u$\end{document}, \begin{document}$λ>0$\end{document}, \begin{document}$0≤α<1$\end{document}, \begin{document}$c:[0,∞)\rightarrow or \to(0,∞)$\end{document} is continuous and \begin{document}$h:(0,1)\rightarrow or \to(0,∞)$\end{document} is continuous and integrable. We assume that \begin{document}$f∈ C[0,∞)$\end{document} is such that \begin{document}$f(0)<0$\end{document}, \begin{document}$\lim_{s\rightarrow or \to∞}f(s) = ∞$\end{document} and \begin{document}$\frac{f(s)}{s^{α}}$\end{document} has a \begin{document}$p$\end{document}-sublinear growth at infinity, namely, \begin{document}$\lim_{s \rightarrow or \to ∞}\frac{f(s)}{s^{p-1+α}} = 0$\end{document}. We will discuss nonexistence results for \begin{document}$λ≈ 0$\end{document}, and existence and uniqueness results for \begin{document}$λ \gg 1$\end{document}. We establish the existence result by a method of sub-supersolutions and the uniqueness result by establishing growth estimates for solutions.

Backward compact and periodic random attractors for non-autonomous sine-Gordon equations with multiplicative noise
Yangrong Li and Shuang Yang
2019, 18(3): 1155-1175 doi: 10.3934/cpaa.2019056 +[Abstract](23) +[HTML](22) +[PDF](464.72KB)

A non-autonomous random attractor is called backward compact if its backward union is pre-compact. We show that such a backward compact random attractor exists if a non-autonomous random dynamical system is bounded dissipative and backward asymptotically compact. We also obtain both backward compact and periodic random attractor from a periodic and locally asymptotically compact system. The abstract results are applied to the sine-Gordon equation with multiplicative noise and a time-dependent force. If we assume that the density of noise is small and that the force is backward tempered and backward complement-small, then, we obtain a backward compact random attractor on the universe consisted of all backward tempered sets. Also, we obtain both backward compactness and periodicity of the attractor under the assumption of a periodic force.

The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations
Eddye Bustamante, José Jiménez Urrea and Jorge Mejía
2019, 18(3): 1177-1203 doi: 10.3934/cpaa.2019057 +[Abstract](32) +[HTML](17) +[PDF](447.86KB)

In this work we prove that the initial value problem (IVP) associated to the fractional two-dimensional Benjamin-Ono equation

where \begin{document}$0 < \alpha\leq1$, $D_x^{\alpha}$\end{document} denotes the operator defined through the Fourier transform by

and \begin{document}$\mathcal H$\end{document} denotes the Hilbert transform with respect to the variable x, is locally well posed in the Sobolev space \begin{document}$H^s(\mathbb R^2)$ with $s>\dfrac32+\dfrac14(1-\alpha)$\end{document}.

Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian
Ronghua Jiang and Jun Zhou
2019, 18(3): 1177-1203 doi: 10.3934/cpaa.2019058 +[Abstract](35) +[HTML](35) +[PDF](463.95KB)

We consider a nonlocal parabolic equation associated with the fractional p-laplace operator, which was studied by Gal and Warm in [On some degenerate non-local parabolic equation associated with the fractional p-Laplacian. Dyn. Partial Differ. Equ., 14(1): 47-77, 2017]. By exploiting the boundary condition and the variational structure of the equation, according to the size of the initial dada, we prove the finite time blow-up, global existence, vacuum isolating phenomenon of the solutions. Furthermore, the upper and lower bounds of the blow-up time for blow-up solutions are also studied. The results generalize the results got by Gal and Warm.

Reaction of the fluid flow on time-dependent boundary perturbation
Eduard Marušić-Paloka and Igor Pažanin
2019, 18(3): 1227-1246 doi: 10.3934/cpaa.2019059 +[Abstract](35) +[HTML](17) +[PDF](373.43KB)

The aim of this paper is to investigate the effects of time-dependent boundary perturbation on the flow of a viscous fluid via asymptotic analysis. We start from a simple rectangular domain and then perturb the upper part of its boundary by the product of a small parameter \begin{document}$\varepsilon$\end{document} and some smooth function \begin{document}$h(x, t)$\end{document}. The complete asymptotic expansion (in powers of \begin{document}$\varepsilon$\end{document}) of the solution of the evolutionary Stokes system has been constructed. The convergence of the expansion has been proved providing the rigorous justification of the formally derived asymptotic model.

Symmetry of solutions to a class of Monge-Ampère equations
Fan Cui and Huaiyu Jian
2019, 18(3): 1247-1259 doi: 10.3934/cpaa.2019060 +[Abstract](41) +[HTML](23) +[PDF](340.62KB)

We study the symmetry of solutions to a class of Monge-Ampère type equations from a few geometric problems. We use a new transform to analyze the asymptotic behavior of the solutions near the infinity. By this and a moving plane method, we prove the radially symmetry of the solutions.

Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials
Dongdong Qin, Xianhua Tang and Qingfang Wu
2019, 18(3): 1261-1280 doi: 10.3934/cpaa.2019061 +[Abstract](32) +[HTML](41) +[PDF](395.61KB)

In this paper we study a class of weakly coupled Schrödinger system arising in several branches of sciences, such as nonlinear optics and Bose-Einstein condensates. Instead of the well known super-quadratic condition that \begin{document} $\lim_{|z|\to∞}\frac{F(x,z)}{|z|^2} = ∞$ \end{document} uniformly in \begin{document} $x$ \end{document}, we introduce a new local super-quadratic condition that allows the nonlinearity \begin{document} $F$ \end{document} to be super-quadratic at some \begin{document} $x∈ \mathbb{R}^N$ \end{document} and asymptotically quadratic at other \begin{document} $x∈ \mathbb{R}^N$ \end{document}. Employing some analytical skills and using the variational method, we prove some results about the existence of ground states for the system with periodic or non-periodic potentials. In particular, any nontrivial solutions are continuous and decay to zero exponentially as \begin{document} $|x| \to ∞$ \end{document}. Our main results extend and improve some recent ones in the literature.

Stability and $ L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems
Yeping Li and Jie Liao
2019, 18(3): 1281-1302 doi: 10.3934/cpaa.2019062 +[Abstract](33) +[HTML](15) +[PDF](474.36KB)

In the paper, we consider a three-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler-Poisson with electric field and relaxation term added to the momentum equations. We first construct the planar diffusion waves. Next we show the global existence of smooth solutions for the initial value problem of three-dimensional bipolar Euler-Poisson systems when the initial data are near the planar diffusive waves. Finally, we also establish the \begin{document}$ L^p(p∈[2,+∞])$\end{document} convergence rates of the solutions toward the planar diffusion waves. A frequency decomposition, approximate Green function and delicate energy method are used to prove our results.

Solvability of nonlocal systems related to peridynamics
Moritz Kassmann, Tadele Mengesha and James Scott
2019, 18(3): 1303-1332 doi: 10.3934/cpaa.2019063 +[Abstract](22) +[HTML](14) +[PDF](625.24KB)

In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lamé system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces.

Some Remarks on Regularity Criteria of Axially Symmetric Navier-Stokes Equations
Zijin Li and Xinghong Pan
2019, 18(3): 1333-1350 doi: 10.3934/cpaa.2019064 +[Abstract](28) +[HTML](18) +[PDF](502.88KB)

Two main results will be presented in our paper. First, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a \begin{document} $log$ \end{document} supercritical assumption on the horizontally radial component \begin{document} $u^r$ \end{document} and vertical component \begin{document} $u^z$ \end{document}, accompanied by a \begin{document} $log$ \end{document} subcritical assumption on the horizontally angular component \begin{document} $u^θ$ \end{document} of the velocity. Second, the precise Green function for the operator \begin{document} $-(Δ-\frac{1}{r^2})$ \end{document} under the axially symmetric situation, where \begin{document} $r$ \end{document} is the distance to the symmetric axis, and some weighted \begin{document} $L^p$ \end{document} estimates of it will be given. This will serve as a tool for the study of axially symmetric Navier-Stokes equations. As an application, we will prove the regularity of solutions to axially symmetric Navier-Stokes equations under a critical (or a subcritical) assumption on the angular component \begin{document} $w^θ$ \end{document} of the vorticity.

Nonlinear wave equation with both strongly and weakly damped terms: supercritical initial energy finite time blow up
Yanbing Yang and Runzhang Xu
2019, 18(3): 1351-1358 doi: 10.3934/cpaa.2019065 +[Abstract](31) +[HTML](29) +[PDF](342.33KB)

By introducing a new increasing auxiliary function and employing the adapted concavity method, this paper presents a finite time blow up result of the solution for the initial boundary value problem of a class of nonlinear wave equations with both strongly and weakly damped terms at supercritical initial energy level.

A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach
Takeshi Wada
2019, 18(3): 1359-1374 doi: 10.3934/cpaa.2019066 +[Abstract](31) +[HTML](20) +[PDF](448.67KB)

We study the nonlinear Schrödinger equation (NLS)

in \begin{document}$\mathit{\boldsymbol{R}}^{1+n}$\end{document}, where \begin{document}$n\ge 3$\end{document}, \begin{document}$p>1$\end{document}, and \begin{document}$\lambda \in \mathit{\boldsymbol{C}}$\end{document}. We prove that (NLS) is locally well-posed in \begin{document}$H^s$\end{document} if \begin{document}$1<s<\min\{4;n/2\}$\end{document} and \begin{document}$\max\{1;s/2\}< p< 1+4/(n-2s)$\end{document}. To obtain a good lower bound for \begin{document}$p$\end{document}, we use fractional order Besov spaces for the time variable. The use of such spaces together with time cut-off makes it difficult to derive positive powers of time length from nonlinear estimates, so that it is difficult to apply the contraction mapping principle. For the proof we improve Pecher's inequality (1997), which is a modification of the Strichartz estimate, and apply this inequality to the nonlinear problem together with paraproduct formula.

A remark on norm inflation for nonlinear Schrödinger equations
Nobu Kishimoto
2019, 18(3): 1375-1402 doi: 10.3934/cpaa.2019067 +[Abstract](26) +[HTML](16) +[PDF](591.04KB)

We consider semilinear Schrödinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on \begin{document}$\mathbb{R}^d$\end{document} or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.

Perturbations of nonlinear eigenvalue problems
Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu and Dušan D. Repovš
2019, 18(3): 1403-1431 doi: 10.3934/cpaa.2019068 +[Abstract](30) +[HTML](45) +[PDF](430.79KB)

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive solutions changes as the real parameter \begin{document}$λ$\end{document} varies. We also show that there exists a minimal positive solution \begin{document}$\overline{u}_λ$\end{document} and determine the monotonicity and continuity properties of the map \begin{document}$λ\mapsto\overline{u}_λ$\end{document}. Special attention is given to the particular case of the \begin{document}$p$\end{document}-Laplacian.

Hilbert transforms along double variable fractional monomials
Haixia Yu
2019, 18(3): 1433-1446 doi: 10.3934/cpaa.2019069 +[Abstract](34) +[HTML](18) +[PDF](414.22KB)

In this paper, we obtain the \begin{document}$L^2(\mathbb{R}^2)$\end{document} boundedness and single annulus \begin{document}$L^p(\mathbb{R}^2)$\end{document} estimate for the Hilbert transform \begin{document}$H_{α,β}$\end{document} along double variable fractional monomial \begin{document}$u_1(x_1)[t]^α+u_2(x_1)[t]^β$\end{document}

with the bounds are independent of the measurable function \begin{document}$u_1$\end{document} and \begin{document}$u_2$\end{document}. At the same time, we also obtain the \begin{document}$L^p(\mathbb{R})$\end{document} boundedness of the corresponding Carleson operator

where \begin{document}$[t]^α$\end{document} stands for either \begin{document}$|t|^α$\end{document} or \begin{document}$\textrm{sgn}(t)|t|^α$\end{document}, \begin{document}$[t]^β$\end{document} stands for either \begin{document}$|t|^β$\end{document} or \begin{document}$\textrm{sgn}(t)|t|^β$\end{document} and \begin{document}$α,β,p∈ (1,∞)$\end{document}.

Stability of axially-symmetric solutions to incompressible magnetohydrodynamics with no azimuthal velocity and with only azimuthal magnetic field
Wojciech M. Zajączkowski
2019, 18(3): 1447-1482 doi: 10.3934/cpaa.2019070 +[Abstract](43) +[HTML](15) +[PDF](412.31KB)

Incompressible viscous axially-symmetric magnetohydrodynamics is considered in a bounded axially-symmetric cylinder. Vanishing of the normal components, azimuthal components and also azimuthal components of rotation of the velocity and the magnetic field is assumed on the boundary. First, global existence of regular special solutions, such that the velocity is without the swirl but the magnetic field has only the swirl component, is proved. Next, the existence of global regular axially-symmetric solutions which are initially close to the special solutions and remain close to them for all time is proved. The result is shown under sufficiently small differences of the external forces. All considerations are performed step by step in time and are made by the energy method. In view of complicated calculations estimates are only derived so existence should follow from the Faedo-Galerkin method.

Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models
Cheng-Hsiung Hsu and Jian-Jhong Lin
2019, 18(3): 1483-1508 doi: 10.3934/cpaa.2019071 +[Abstract](23) +[HTML](28) +[PDF](555.56KB)

This paper is concerned with the existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models. By using Schauder's fixed point theorem and the existence of contracting rectangles, we obtain the existence result. Then we investigate the asymptotic behavior of positive monotone traveling wave solutions by using the modified Ikehara's Theorem. With the help of their asymptotic behavior, we provide a sufficient condition which guarantee that all positive traveling wave solutions of the system are non-monotone. Furthermore, to illustrate our main results, the existence and non-monotonicity of traveling wave solutions of Lotka-Volterra predator-prey model and modified Leslie-Gower predator-prey models with different kinds of functional responses are also discussed.

Applications of generalized trigonometric functions with two parameters
Hiroyuki Kobayashi and Shingo Takeuchi
2019, 18(3): 1509-1521 doi: 10.3934/cpaa.2019072 +[Abstract](35) +[HTML](33) +[PDF](463.32KB)

Generalized trigonometric functions (GTFs) are simple generalization of the classical trigonometric functions. GTFs are deeply related to the \begin{document}$p$\end{document}-Laplacian, which is known as a typical nonlinear differential operator, and there are a lot of works on GTFs concerning the \begin{document}$p$\end{document}-Laplacian. However, few applications to differential equations unrelated to the $p$-Laplacian are known. We will apply GTFs with two parameters to nonlinear nonlocal boundary value problems without \begin{document}$p$\end{document}-Laplacian. Moreover, we will give integral formulas for the functions, e.g. Wallis-type formulas, and apply the formulas to the lemniscate function and the lemniscate constant.

Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term
Yu Zhang and Yanyan Zhang
2019, 18(3): 1523-1545 doi: 10.3934/cpaa.2019073 +[Abstract](49) +[HTML](19) +[PDF](455.36KB)

The Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term are studied. The Riemann solutions exactly include two kinds: delta-shock solutions and vacuum solutions. In order to see more clearly the influence of the source term on Riemann solutions, the generalized Rankine-Hugoniot relations of delta shock waves are derived in detail, and the position, propagation speed and strength of delta shock wave are given. It is also shown that, as the source term vanishes, the Riemann solutions converge to the corresponding ones of the homogeneous system, which is just the generalized zero-pressure flow model and contains the one-dimensional zero-pressure flow as a prototypical example. Furthermore, the generalized balance relations associated with the generalized mass and momentum transportation are established for the delta-shock solution. Finally, two typical examples are presented to illustrate the application of our results.

Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential
Xiaoyan Lin, Yubo He and Xianhua Tang
2019, 18(3): 1547-1565 doi: 10.3934/cpaa.2019074 +[Abstract](52) +[HTML](17) +[PDF](400.86KB)

We consider the following semilinear Schrödinger equation with inverse square potential

where $N≥ 3$, $f$ is asymptotically linear, $V$ is 1-periodic in each of $x_1, ..., x_N$ and $\sup[σ(-\triangle +V)\cap (-∞, 0)]<0<{\rm{inf}}[σ(-\triangle +V)\cap (0, ∞)]$. Under some mild assumptions on $V$ and $f$, we prove the existence and asymptotical behavior of ground state solutions of Nehari-Pankov type to the above problem.

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