Weighted Sobolev Type Embedded In The Application Of Quasilinear Elliptic Equations

In this paper, by using weighted Sobolev embedding and Variational methods in radially symmetric function spaces, we study the existence of nontrivial solutions of a class of quasilinear elliptic equation with unbounded or decaying radial potentials. We consider the following quasilinear elliptic equationwhere -△_pu = (?), 1＜p＜N,λ≥0, and the non-negative radial potentials V,Q,h are continuous functions and verify(V) There exist real numbers a and a₀, such that(Q) Q(r) > 0 there exist real numbers b and b₀, such that(H) h(r) > 0, there exist real numbers c and c₀, such thatThe function f satisfies the following conditions:(f₁) J∈C(R, R).(f₂) There exist C＞0 and s_*＜q₁＜q₂＜s^* (respectively, max{s_*,s_**}＜q₁≤q₂＜∞) such that(f'₂) If c＜max{a,-p},c₀＞min{-p,a₀} hold, |f(u)|≤C|u|^p-1 + |u|^s-1) and (?)|u|^-pF(u) = 0 also hold, where C＞0 and s verifies p＜s＜s^* and F(u) = (?) f(v)dv. (f₃)μ＞s_* (respectively.μ＞max{s_* , s_**}) ,0＜μF(u)≤uf(u). for any u in R except u = 0 is right.(f'₃) 0＜sF(u)＜uf(u), for any u in R except u = 0 is right.Where the indices s_*,s^*,s_** are given below. Under these conditions, we get the following existence theorem by applying weighted Sobolev embedding([1]) and Mountain Pass Theorem([17]):Theorem A Let 1＜p＜N. Assume that the functions V, Q, h verify the conditions (V),(Q),(H) respectively and the function f verify (f₁), (f₂), (f₃) or (f₁), (f'₂), (f'₃). If q_* < q < q^* ( respectively, max{q_*. q_**}＜q＜∞), then the problem (P) has at least one nontrivial radial solution.Theorem B Let 1＜p＜N. Assume that the functions V, Q, h satisfy the conditions (V),(Q),(H) respectively and f verify (f₁), (f₂), (f₃) or (f₁), (f'₂), (f'₃)- If b＜max{a, -p}, b₀＞min{- p, a₀} hold, and 0≤λC_H^p＜1, where C_H is the constant of the embedding W_r^1.p(R^N; V) (?) L^p(R^N: Q). Then for q = p, the problem (P) has at least one nontrivial radial solution.