Quasilinear Elliptic Differential Equations of Critical Nonlinear Growth

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We studied a Dirichlet boundary value problem of a degenerate quasilinear elliptic equation which has the φ-Laplace operator in the principal part. The main result is the existence theorem of nonnegative nontrivial solutions via variational methods in Orlicz-Sobolev space settings. It can be applied to a wide class of elliptic equations even if the principal parts have non power-like nonlinearities. In the following let φ(t)t = Φ'(t)(1) The quasilinear elliptic problem of subcritical growth: An existence theorem of multiple nonnegative nontrivial solutions is proved. In a previous paper we have discussed the problem under the hypothesis φ(t)t = o(f(x, t)) at t = 0 and ∞, which corresponds to classical results about a semilinear elliptic equation with a concave-convex lower term. In this time we consider the case f(x, t) = ο(φ(t)t) at t = 0 and ∞ contrary to the problem treated above. Further we also consider an equation with more general principal part.(2) The quasilinear elliptic problem of critical Orlicz-Sobolev growth : We make some modification of the standard concentration-compactness principle and obtain an existence theorem of a nonnegative nontrivial solution. For example it can be applied to the the case Φ(t) = t^p log(1 + t), p > 1.(3) Minimax problem of a nonsmooth functional : The variational problem for a functional with slowly growing principal part and involving critical Orlicz-Sobolev lower term (with respect to the principal part) is discussed. The functional is not Frechet differentiable, although it Gateaux differentiable. A nonnegative nontrivial solution for the Euler equation is given. For example the result can be applied to the case Φ(t) = t log(1 + t).