We study Brezis-Nirenberg type theorems for the equationwhere Ω is a bounded domain in R N , g(x, ·) is increasing and f is a dissipative nonlinearity. We apply such theorems for studying existence and multiplicity of positive solutions for the equationwhere q > 0, p > 1 and λ > 0.

Dip. Me.Mo.Mat.via B o~i a m o 25b, Abstract We prove the existence of two nontrivial ~olutions for the fourth orderXk(Xk -c) where 2 5 k 5 i or 6 > X,(X, -c) and 6 is close to X, (X, -c) where j 2 i + 1. (Here (A,),?, is the sequence of the eige~ivalues of -A in H;(R)). Moreover if c > XI, c is close to XI , b > X,(Xj -c) and b is close to Xj(X, -c ) where j 2 2 we get three non trivial solutions. AMS: 35J40KEY WORDS: Multiple solutions, variational theorems of mixed type, aeroelastic oscillations.

We consider the existence of weak solutions for classical doubly resonant semilinear elliptic problems. We show how the main technical assumptions can be used to define appropriate metrics on the underlying function space, so that extensions of the results already known in the literature can be obtained using only basic facts from critical point theory for continuous functionals on complete metric spaces.

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We study the existence of multiple positive solutions of −∆u = λu −q +u p in Ω with homogeneous Dirichlet boundary condition, where Ω is a bounded domain in R N , λ > 0, and 0 < q ≤ 1 < p ≤ (N + 2)/(N − 2). We show by a variational method that if λ is less than some positive constant then the problem has at least two positive, weak solutions including the cases of q = 1 or p = (N + 2)/(N − 2). We also study the regularity of positive weak solutions.