Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations

In connection with the maximizing problem for the functional R(u) = ∥u∥_L^q ∥▽u∥_L^p in W₀ ^1,p(Ω)β{0}, we consider the equation -div(|▽u|^{p - 2} ▽u(x)) = |u|^{q - 2} u(x), x ε{lunate} Ω, 1 < p, q < ∞, p ≠ q, (E) u(x) = 0, x ε{lunate} ∂Ω. It is shown that for the case q < p^* (p^* = ∞ if p ≧ N, and p^* = Np (N - p) if p < N), (E) has always a nonnegative nontrivial solution belonging to W₀ ^1,p(Ω) ∩ L^∞(Ω), and for the case p < N and q > p^* (resp. q = p^*), (E) has no nontrivial (resp. nonnegative nontrivial) solution belonging to the class P = {u ε{lunate} W₀ ^1,p(Ω) ∩ L^q(Ω); x_i|u|^{q - 2}u ε{lunate} L ^{p (p - 1)}(Ω), i = 1, 2, ..., N} ⊂ W₀ ^1,p(Ω) ∩ ^∞(Ω), provided that Ω is star shaped. The crucial point of the proof of our result is to obtain an L^∞-estimate of weak solutions and to verify a certain "Pohozaev-type inequality" for weak solutions belonging to P.