Uma solução que muda de sinal para um problema superlinear de Dirichlet

Abstract

In this work, we have studied the existence of nodal solution for nonlinear elliptic problems with Diriichlet boundary conditions "Formula disponibilizado no texto completo" where onde ΩcRN(N ≥3) it is a sufficiently smooth limited region. The present work is based on the studies carried out by Castro, Cossio and Neuberger in [10] and [11]. We will prove that a superlinear elliptic problem has at least three nontrivial solutions. Two of these solutions, w1 and w2, are of definite signal (positive and negative, respectively). And a third solution w3, called the nodal solution, we will prove that this solution changes signal exactly once. In addition, we will prove that w1 and w2 have Morse index 1 and that solution w3 has Morse index 2. Thus, this result extends and complements the results of Wang in [35]. We will prove that if is isolated, then its index of Leray-Schauder of any solution given by the principle of min-max is +1. Finally combining the results of [10] with the theoretical results of grade, of Castro and Cossio in [9]. In the case where non-linearity is asymptotically linear, we shall provide sufficient conditions to: (i) the existence of at least four solutions, one of these signal changes signal exactly once, (ii) the existence of at least five solutions, two of which change sign. In addition, one of these two signal-changing solutions changes exactly once.