Existence of a Nontrivial Solution for a Class of Nonquadratic Elliptic Problems
Hou Yu, Chun‐Lei Tang · Journal of Southwest University · 2009
We study the existence of a nontrivial solution for the Dirichlet problem-Δu+a(x)u=f(x,u)u∈Ωu=0u∈Ω(1)where Ω is a smooth bounded domainin RN.Theorem 1 Assume that f(x,t)satisfies the following conditions:(f1)F(x,t)|t|2→+∞ as |t|→+∞ and F(x,t)|t|20 as |t|0 uniformly on Ω.(f2)There exist a10 and 1sN+2N-2 such that |f(x,t)|≤a1(1+|t|s)for all(x,t)∈Ω×R.(f3)There are constants β2NN+2s-1,a20 and L0 such that tf(x,t)-2F(x,t)≥a2 |t|β for all |t|≥L and x∈Ω.If 0 is an eigenvalue of-Δ+a(with Dirichlet boundary condition)and also satisfies the condition that:(f4)There exists δ0 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(1)has at least one nontrivial solution.