Existence of weak non-negative solutions for a class of nonuniformly boundary value problem
Bull. Korean Math. Soc. 2012 Vol. 49, No. 4, 737-748
https://doi.org/10.4134/BKMS.2012.49.4.737
Published online July 1, 2012
Trinh Thi Minh Hang and Hoang Quoc Toan
Hanoi University of Civil Engineering, Hanoi University of Science
Abstract : 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
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