Multiplicity of Solutions for Quasilinear Elliptic Equations with Variable Exponents and Its Application

In this project, we will investigate the existence of at least one or multiple solutions of the following problem: 「— d/v(|Vu\P(x} 2 Vu) = Ag(x)|u|q(x) 2u + f(x)|—r(x) 2u in Q; \u = 0 on dQ where A ＞ 0 , Q d RN is a bounded domain with smooth boundary dQ. Let p，q，r，f，g be positive continuous functions in Q. and satisfy some suitable conditions. (1) 1＜q(x)＜p(x)＜r(x)＜p*(x)=Np(x)/(N-p(x)); (2) 1 ＜ q— ＜ q + ＜ p— = ess inf p(x) ＜ p + = ess supp(x) ＜ r— ＜ r + ＜ ⑴. Let The variable exponent Lebesgue space is defined by U(x)(Q) = {u is a measurable function 11 |u(x)|p(x) dx ＜ ⑴} Q with the norm p(x) lulp( x)=inf{£ ＞ oi \~ dx ＜1}. Q 6 The space W。1 ’ p(1 ) (Q) is the closure of CT (Q) in W1 , p(1) ( Q). Associated with this problem, we consider the functional JA，for u g W。1 ’ p(x) (Q) Ja(u)= f--- dx — Af g(x) |u|q(x)dx — f f(x) |u|r(x)dx. Q p(x) Jq q(x)n Jq r(x) 1 1 In fact, the weak solution u g wJ ;p (1 of this equation is the critical point of the functional ]入. Firstly, we need some preliminaries about W。1,p(x) (Q). Furthermore, we have to show that Ja is still coercive and bounded from below on the Nehari manifold. Because of the space W。1，)(x) (Q) we need some new inequalities to deal with the difficulties of this equation. For example, is Nehari manifold separated by MA u M1 and M1 n Ml 二 彡？ Besides, we also consider the following equation r|Vu|p (x) | ,p( x)—2 —K(^^1^dx)div(\Vu\pix) 2 Vu) = f (x,u) in Q; Q p ( x ) u = 0 on 5Q, It is related to the stationary version of the Kirchhoff type equation

Project ID:PA10406-1338
External Project ID:MOST104-2115-M182-001