A Sign-Changing Solution for a Superlinear Dirichlet Problem

We show that a superlinear boundary value problem has at least 3 nontrivial solutions. A pair are of one sign (positive and negative respectively), and the 3rd solution changes sign exactly once. The critical level of the sign-changing solution is bounded below by the sum of the 2 lesser levels of the one-sign solutions. If nondegenerate, the one sign solutions are of Morse index 1 and the sign-changing solution has Morse index 2. Our results extend and complement those of Z. Q. Wang ([12]). 1 Introduction. Let \Ω be a smooth bounded region in R N , \Δ the Laplacian operator, and f 2 C 1 (R; R) such that f(0) = 0. We assume that there exist constants A ? 0 and p 2 (1; N+2 N \Γ2 ) such that jf 0 (u)j A(juj p\Γ1 + 1) for all u 2 R. Hence f is subcritical, i.e., there exists B ? 0 such that jf(u)j B(juj p + 1). Also, we assume that there exists m 2 (0; 1) such that f(u)u \Γ 2F (u) muf(u); (1) where F (u) = R u 0 f(s) ds; for all u 2 R. Finally, we...