A Sign-Changing Solution For A Superlinear Dirichlet Problem With A Reaction Term Nonzero At Zero

In previous joint work with A. Castro and J. Cossio (See [5]), it was shown that a superlinear boundary value problem has at least 3 nontrivial solutions, one of which changes sign exactly-once. In this paper we provide an analogous result when the nonlinearity does not pass through the origin; this case includes the so-called semipositone case, i.e., where the nonlinearity is negative at zero. We find a small negative solution and a pair of larger solutions (negative and and of nontrivial positive part respectively), together with a fourth solution which changes sign. We briefly mention a gradient descent algorithm which follows our method of proof and can be used to obtain approximations to the four solutions (See [14]). 1 Introduction. Let\Omega be a smooth bounded region in R N , \Delta the Laplacian operator, and f 2 C 1 (R; R) such that f(0) = 0. We will take f to satisfy the assumptions below and consider the nonlinearity f(u) \Gamma ffl, where jfflj will be taken to be su..