Zhongwei Tang

Solutions for the Problems Involving Fractional Laplacian and Indefinite Potentials

Abstract In this paper, we consider a class of Schrödinger equations involving fractional Laplacian and indefinite potentials. By modifying the definition of the Nehari–Pankov manifold, we prove the existence and asymptotic behavior of least energy solutions. As the fractional Laplacian is nonlocal, when the bottom of the potentials contains more than one isolated components, the least energy solutions may localize near all the isolated components simultaneously. This phenomenon is different from the Laplacian.

On the least energy solutions for semilinear Schrödinger equation with electromagnetic fields involving critical growth and indefinite potentials

Abstract In this paper, we are concerned with the following semilinear Schrödinger equation with electromagnetic fields and critical growth for sufficiently large , where , and its zero set is not empty, is the critical Sobolev exponent, is a constant such that the operator might be indefinite but is non-degenerate. Using variational method and modified Nehari–Pankov method, we prove the equation admits a least energy solution which localizes near the potential well . The results we obtain here extend the corresponding results for the Schrödinger equation which involves critical growth but does not involve electromagnetic fields.

Least energy solutions for semilinear Schrödinger systems with electromagnetic fields and critical growth

In the paper, we study the following semilinear Schrödinger systems with electromagnetic field and critical growth, where , , are real-valued magnetic vector potentials and . and such that , here is the critical Sobolev exponent. are constants such that the operators and are positively definite. We prove the existence of least energy solutions which localize near the common potential well for large enough.

Global existence and asymptotic stability of equilibria to reaction-diffusion systems

In this paper, we study weakly coupled reaction-diffusion systems in unbounded domains of or , where the reaction terms are sums of quasimonotone nondecreasing and nonincreasing functions. Such systems are more complicated than those in many previous publications and little is known about them. A comparison principle and global existence, and boundedness theorems for solutions to these systems are established. Sufficient conditions on the nonlinearities, ensuring the positively Ljapunov stability of the zero solution with respect to H2-perturbations, are also obtained. As samples of applications, these results are applied to an autocatalytic chemical model and a concrete problem, whose nonlinearities are nonquasimonotone. Our results are novel. In particular, we present a solution to an open problem posed by Escher and Yin (2005 J. Nonlinear Anal. Theory Methods Appl. 60 1065–84).

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS FOR A NONLINEAR ELLIPTIC EQUATION ARISING IN ASTROPHYSICS

The author first analyzes the existence of ground state solutions and cylindrically symmetric solutions and then the asymptotic behavior of the ground state solution of the equation −Δu = o(r)up-1, u > 0 in ℝN, u ∈ D1,2(ℝN), where N ≥ 3,x = (x′, z) ∈ ℝK × ℝN−K,2 ≤ K ≤ N,r = ∣x′∣. It is proved that for 2(N − s)/(N − 2) < p < 2* = 2N/(N-2), 0 < s < 2, the above equation has a ground state solution and a cylindrically symmetric solution. For p = 2*, the above equation does not have a ground state solution but a cylindrically symmetric solution, and when p close to 2*, the ground state solutions are not cylindrically symmetric. On the other hand, it is proved that as p close to 2*, the ground state solution up has a unique maximum point xp = (x′p, zp) and as p → 2*, ∣x′p∣ → r0 which attains the maximum of o on ℝN. The asymptotic behavior of ground state solution up is also given, which also deduces that the ground state solution is not cylindrically symmetric as p goes to 2*.