Constant sign and nodal solutions for a class of nonlinear Dirichlet problems

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We consider a nonlinear Dirichlet problem with a Carathéodory reaction which has arbitrary growth from below. We show that the problem has at least three nontrivial smooth solutions, two of constant sign and the third nodal. In the semilinear case (i.e., p=2), with the reaction f(z,.) being C1 and with subcritical growth, we show that there is a second nodal solution, for a total of four nontrivial smooth solutions. Finally, when the reaction has concave terms and is subcritical and for the nonlinear problem (i.e., 1<p<∞) we show that again we can have the existence of three nontrivial smooth solutions, two of constant sign and a third nodal.