An asymptotically periodic and asymptotically linear Schrödinger equation with indefinite linear part

Abstract

a b s t r a c tWe consider the semilinear Schrödinger equationwhere 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.

Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part

“…Moreover, suppose f is asymptotically periodic in x in the following sense, and there exists a continuous function f p ( x , u ) on , which is 1‐periodic in each component x j (1 ≤ j ≤ N) of x , such that is strictly increasing on ( − ∞ ,0) and (0, ∞ ), | f ( x , u ) | ≥ | f p ( x , u ) | , , | f ( x , u ) − f p ( x , u ) | ≤ | h ( x ) | ( | u | + | u | q ), , , where is the class of functions such that, for every ϵ > 0 the set has finite Lebesgue measure. Theorem If (H 1 )–(H 8 ) hold, then the problem (NLS) has a ground state solution. Remark In, f ( x , u ) is asymptotically periodic in x if there is a periodic function f p such that …”

“…0 in L qC1 .R N /, then for preceding , there exists J 2 N such that a ju n j qC1 qC1 < C , for n > J. Then by (15) for n > J, we obtain thatˇZ f .x, u n /u n dxˇ< C .…”

Ground state solutions for a diffusion system

“…The following generalized linking theorem plays an important role in proving our main result (see [12,40] …”

Multiple solutions for a semilinear elliptic system in

“…The functionalˆis strongly indefinite; such type of functional appeared extensively when one study differential equations via critical point theory; see for example [19,[23][24][25][26] and the references therein. Suppose that f .x, t/, g.x, t/ satisfy the assumptions .f 1 /-.f 3 / or .f 1 /, .f 2 /, .f 6 /, then it is easy to see that for any > 0, there is a C > 0 such that for t 2 R 1 , we have…”