Bulletin of the
Korean Mathematical Society BKMS

The goal of this paper is to study the existence of non-trivial non-negative weak solution for the nonlinear elliptic equation: $$ -\text{div}(h(x)\nabla u)= f(x,u)\text{ in } \Omega$$ with Dirichlet boundary condition in a bounded domain $\Omega \subset \mathbb R^N, N\geq 3$, where $ h(x) \in L^1_{loc}(\Omega), f(x,s)$ has asymptotically linear behavior. The solutions will be obtained in a subspace of the space $H^1_0(\Omega)$ and the proofs rely essentially on a variation of the mountain pass theorem in [12].

Keywords: mountain pass theorem, the weakly continuously differentiable functional

MSC numbers: 35J20, 35J65