Existence of Weak Solutions for a Class of Elliptic Equations Involving P(X)-Laplacian Operator

Project: National Science and Technology CouncilNational Science and Technology Council Academic Grants

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
StatusFinished
Effective start/end date01/08/1631/07/17

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