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Multiplicity Results for Weak Solutions of a Semilinear Dirichlet Elliptic Problem with a Parametric Nonlinearity

This paper deals with the existence of weak solutions to a Dirichlet problem for a semilinear elliptic equation involving the difference of two main nonlinearities functions that depends on a real parameter 
 
 λ
 
 . According to the values of 
 
 λ
 
 , we give both nonexistence and multiplicity results by using variational methods. In particular, we first exhibit a critical positive value such that the problem admits at least a nontrivial non-negative weak solution if and only if 
 
 λ
 
 is greater than or equal to this critical value. Furthermore, for 
 
 λ
 
 greater than a second critical positive value, we show the existence of two independent nontrivial non-negative weak solutions to the problem.