| In this paper, firstly, we consider the following Dirichlet problem:whereΩis a smooth bounded domain in Rn, f(x, t) is odd with respect to t. In our discussion, we do not suppose f(x,t) satisfies (AR) condition, infinitely many solutions are obtained by fountain theorem.The main result is the following theorem.Theorem 1 If f(x,t) satisfies the following conditions:(f1) f(x, -t) = -f(x, t) for all x∈Ω. and t∈R;(f2) There exist a > 0 and 1 < r < (N + 2)/(N-2) such that |f(x,t)|≤a(1 + |t|r), x∈(?), t∈R if N≥3; lim|t|→∞ln(|f(x,t)| +1)×|t|-2 = 0 uniformly for x∈Ωif N = 2; and no assumption if N = 1;(f3) F(x,t)/|t|2→+∞as |t|→+∞uniformly onΩ. F(x,t)/|t|2→0 as |t|→0 uniformlyonΩ, where F(x,t) = (?)f(x,s)ds; (f4) there are constants (?)(r - 1) <μ< 2*, a > 0, L > 0 such that tf(x, t) - 2F(x, t)≥a|t|μfor all |t|≥L and x∈Ω. Then for any n∈N, there exists infinitely many solutions {un} for promblem (1) such thatwhen n→∞.Next we consider the following Dilichlet problem:whereΩis a smooth bounded domain in Rn. In our discussion, we do not suppose f(x,t) satisfies (AR) condition and a nontrivial solution is obtained by variational methods. The main result is the following theorem.Theorem 2 Assume that f(x,t) satisfies the following conditions:(f5) F{x,t)/|t|2→∞as |t|→+∞and F(x,t)/|t|2→0 as |t|→0 uniformly onΩ, where F(x,t) =(?)f(x,s)ds;(f6) There exist a1 > 0 and 1 < s < (N + 2)/(N - 2) such that |f(x,t)|≤a1(l + |t|s). for all (x, t)∈Ω×R;(f7) There are constantsβ> (?)s - 1, a2 > 0 and L > 0 such that tf(x, t) - 2F(x, t)≥a2|t|βfor all |t|≥L and x∈Ω. If 0 is an eigenvalue of -△+ a (with Dirichlet boundary condition) assume also the condition that:(f8) There existsδ> 0 such that(i) F(x, t)≥0, for all |t|≤δ,x∈Ω:or(ii) F(x,t)≤0, for all |t|≤δ,x∈Ω. Then problem (2) has at least one nontrivial solution. At last.we consider the following p-Laplacian equation with Dirichlet boundary value condition:whereΩis a smooth bounded domain in Rn. In our discussion, we do not suppose f(x,t) satisfies (AR) condition and a nontrivial solution is obtained by Mountain Pass theorem. The main result is the following theorem.Theorem 3 Assume that f(x, t) satisfies the following conditions:(f9)f∈C((?)×R,R), when t∈R, x∈(?), f(x, t)≥0;(f10)limt→0+ (?) = 0,limt→+∞(?) = +∞uniformly on x∈(?)(f11)There exist q∈(p, (?) - 1) such that limt→+∞ (?) = 0 uniformly on x∈(?);(f12)There are constantsμ> (?)(q + l - p),a> 0,L > 0 such that tf(x, t) - pF(x, t)≥a|t|μfor all |t|≥L and x∈Ω. Then problem (3) has a positive solution for everyλ> 0.
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