Weighted Function Spaces Of Sobolev Type Embedding And Quasilinear Elliptic Equations Of Radial Solutions

In this paper, we study weighted Sobolev embedding in radially systemtric spaces and obtain nontrivial solutions of quasilinear elliptic equation with unbounded or dedaying potentials. We consider the following quasilinear elliptic equation(P)where-â–³_pu = -div(|â–½u|^p-2â–½u),1 < p, q < N and the non-negative radial potentials V and Q are continuous functions and verify(V) There exist real number a and a₀, such that(Q) Q(r) > 0, there exist real number b and b₀, such thatLet C_0,r^∞(R^N) denote the function space consists of radially C^∞smooth functions on R^N with supporting set. Let D_r^1,p(R^N) be the completion of C_0,r^∞(R^N) with respect to the following normalDefine for q > 1, s≥1L^q(R^N; V) := {u : R^Nâ†'R| u is measurable,∫_RN V(|x|)|u|^q dx <∞}, L^s(R^N; Q) := {u : R^Nâ†'R| u is measurable,∫_RN Q(|x|)|u|^sdx <∞}.Then we define which is a Banach space under the normAccording to the relations between p, q, N and a, b or a₀,b₀, we define indexes s_* and s^* such that s_* < s^*. One main result of this paper is the following theorem.Theorem A. Let 1 < p, q < N. Potentials V and Q verify condition (V) and (Q) respectively. If s_* < s < s^*, thenis compact embedding.Assume the assumptions of theorem A hold. If s_* < s < s^*, then functionalis well defined and I∈C¹(X_r,R). The Frechet derivative of I isThe weak solution of problem (P) is the critical point of functional I. With Mountain Pass Theorem, we can draw the main existence result as follows.Theorem B. Let 1 < p,q< N. Potentials V(x) and Q(x) verify conditions (V) and (Q) respectively. If s_* < s < s^* and s > max{p,q}, then problem (P) has one nontrivial radial solution.In this paper, we mainly generalize the results in [29, 30] and [25, 26].