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EXISTENCE OF WEAK NON-NEGATIVE SOLUTIONS FOR A CLASS OF NONUNIFORMLY BOUNDARY VALUE PROBLEM

  • Hang, Trinh Thi Minh ;
  • Toan, Hoang Quoc
  • Received : 2011.03.26
  • Published : 2012.07.31

Abstract

The goal of this paper is to study the existence of non-trivial non-negative weak solution for the nonlinear elliptic equation: $$-div(h(x){\nabla}u)=f(x,u)\;in\;{\Omega}$$ 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

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Acknowledgement

Supported by : National Foundation for Science and Technology Development of Vietnam (NAFOSTED)