Project Details
Abstract
In this project, we will investigate the existence of weak solutions of the following problem:
K(j 1 (、dx)A p(x)u = If (x, u) + /dg(x, u) in Q;
Q P\X)
u = 0 on dQ,
where A, d > 0, Q c RN is a bounded domain with smooth boundary dQ, and
1 < p — = ess inf p(x) < p+ = ess sup p(x) < ⑴.
xeQ xeQ
Let K:[0, 00)-> R be a nondecreasing continuous function, and f,g: QxR — R be two Caratheodory functions with a subcritical growth.
Let W1,p(x) (Q) = {u g Lp(x) (Q) 11Vu| g Lp(x) (Q)} be the space with the norm
11 u 11 = lu 1 p( x )+ 1 Vu 1 p ( x )•
The variable exponent Lebesgue space is defined by
Lp(x)(Q) = {u is a measurable function | j |u(x)|p(x)dx < o}
Q
with the norm | ( )| p (x)
Hp(x)=匕取 > 0 1 j ~dx < 1}.
Q
The space W。1 ,p(x) (Q) is the closure of CV (Q) in w 1 , p(x) (Q).
A weak solution u of this problem is defined that u g W), p ()) (Q) satisfies
I V&|p( x)
K(j--- -dx)j |Vu(x)|p(x) 2Vu(x)Vv(x)dx = Aj f(x,u(x))v(x)dx + dj g(x,u(x))v(x)dx.
Q p ( x ) Q Q Q
for all v g W),p()) (Q).
Associated with this problem, we consider the functional Va
Ja (u) = (i K(s)ds)J -^|、dx - f (x, u(x))u(x)dx - dJQ g(x, u(x))u(x)dx.
Firstly, we need some preliminaries about W)1p(x) (Q).
Furthermore, we have to show that J( is still coercive and bounded from below on the Nehari manifold. The space W(,((x) (Q) is different from H((Q) orW(,((Q). For example, is Nehari manifold separated by m + uM- and MnMA =彡? Hence, we need some new
A A
inequalities to deal with the difficulties of this equation.
Besides, It is related to the stationary version of the Kirchhoff type equation
,|VHp( x) , ,p( x )-2
utt - K(^----dx)div(| Vu| Vu) = h(x, u) in Q.
Q p ( x )
Project IDs
Project ID:PA10507-0639
External Project ID:MOST105-2115-M182-001
External Project ID:MOST105-2115-M182-001
Status | Finished |
---|---|
Effective start/end date | 01/08/16 → 31/07/17 |
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