Eigenvalue Problems For The P-Laplacian On Unbounded Domains

In this paper, we consider the following nonlinear eigenvalue problems:where,â–³_pu:=div(|â–½u|^p-2â–½u) is p-Laplacian, 1＜p＜N,Ω:= R^N\ω,ωis a bounded do-main in R^N, and its boundary is smooth sufficiently, (?)u/(?)n denotes the outer normal derivativeof u with respect to (?)Ω. Denote g^±(x)=max{±g(x), 0},then g(x)=g⁺(x)-g^-(x).In this paper, we use a unified method to study the eigenvalue problems of (P₁)—(P⁴).The main results are the followings:Theorem A.1 Suppose 1＜p＜N, g(x)∈L_loc¹(Ω), g⁺(x)∈L^N/p(Ω)∩L^∞(Ω), g⁺(x)≠0,γ(x)∈L^∞((?)Ω) andγ(x)≥0 a.e. on (?)Ω, then problems (P₁)—(P₄) existαnoincreasingsequence of nonnegative eigenvalues, respectively. Theorem A.2 Suppose 1＜p＜N, g(x)∈L^∞(Ω), and (u,λ) is the weak solution ofproblems (P₁)—(P₄)respectively, then for any p^*≤Q≤∞, we have u∈L^Q(Ω), here,p^*=Np/(N-p).Theorem A.3 Ifλ₁ is the first eigenvalue of problems (P₁)—(P₄) respectively, thenλ₁ issimple and all eigenfunctions associated toλ₁ do not change sign.