Existence of multiple solutions with precise sign information for superlinear Neumann problems

Abstract

We consider a nonlinear Neumann problem driven by the p-Laplacian differential operator and having a p-superlinear nonlinearity. Using truncation techniques combined with the method of upper-lower solutions and variational arguments based on critical point theory, we prove the existence of five nontrivial smooth solutions, two positive, two negative and one nodal. For the semilinear (i.e., p = 2) problem, using critical groups we produce a second nodal solution.

Anisotropic nonlinear Neumann problems

“…The study of the corresponding Neumann problem (for both the p-Laplacian and the p(z)-Laplacian) is in some sense lagging behind. We mention the works of Aizicovici-Papageorgiou-Staicu [4], Fan-Deng [16], Mihȃilescu [32]. In Aizicovici-Papageorgiou-Staicu [4] the authors deal with an equation driven by the p-Laplacian and having a potential F(z, ·) which is p-superlinear and satisfies the Ambrosetti-Rabinowitz condition.…”

“…We mention the works of Aizicovici-Papageorgiou-Staicu [4], Fan-Deng [16], Mihȃilescu [32]. In Aizicovici-Papageorgiou-Staicu [4] the authors deal with an equation driven by the p-Laplacian and having a potential F(z, ·) which is p-superlinear and satisfies the Ambrosetti-Rabinowitz condition. Fan-Deng [16] consider parametric problems driven by the p(z)-Laplacian.…”

Nodal Solutions for Neumann Problems with a Nonhomogeneous Differential Operator

“…Similarly, since L À is upward directed, i.e., if v; v 0 A L À , then we can find y A L À such that maxfv; v 0 g a y (see [2], Lemma 2), as above using the Kuratowski-Zorn lemma, we can find v À A ÀintC þ the biggest negative solution of (1.1). r…”

“…Since aðÁÞ is monotone, as in Aizicovici-Papageorgiou-Staicu [2] (Lemma 1), we show that L þ is downward directed, i.e., if u; u 0 A L þ , then we can find y A L þ such that y a minfu; u 0 g: So, if u A L þ , then we can find y A L þ such that y a minfu þ ; ug and so by the minimality of u þ , we infer that y ¼ u þ a u, hence u þ is the smallest nontrivial positive solution of (1.1).…”

“…To the best of our knowledge, our multiplicity theorem here is the first result in the literature, establishing the existence of nodal solutions for Neumann problems driven by a nonhomogeneous di¤erential operator. Note that the nonhomogeneity of the di¤erential operator is the source of serious technical di‰culties and the previous methods used to produce a nodal solution fail in this context (see, for example, the work of Aizicovici-Papageorgiou-Staicu [2]). So, we need new conditions and techniques in order to guarantee the existence of a nodal solution.…”