Multiplicity results for some nonlinear elliptic equations
The \(p\)-Laplacian problem under consideration is \[ - \Delta_p u= \lambda |u|^{a- 2} u+ |u|^{b- 2} u\quad \text{in} \quad \Omega,\quad u|_{\partial\Omega}= 0\tag{1} \] in a smooth bounded domain \(\Omega\subset \mathbb{R}^N\), for constants \(\lambda\), \(a\), \(b\), \(p\) with \(\lambda> 0\), \(1< a< p< b< p^*\), where \(p^*= Np/(N- p)\) if \(p< N\) and \(p^*= +\infty\) otherwise. If \(\Omega\) is a ball in \(\mathbb{R}^N\), one main theorem establishes the existence of a (finite) positive number \(\lambda^*\) such that (1) has at least two positive radial solutions for all \(\lambda\in (0, \lambda^*)\), and no positive solution for \(\lambda> \lambda^*\). This result sharpens theorems of the second author and \textit{I. Peral Alonso} [Trans. Am. Math. Soc. 323, No. 2, 877-895 (1991; Zbl 0729.35051); Ind. Univ. Math. J. 43, No. 3, 941-957 (1994; Zbl 0822.35048)]. The methods include a priori estimates and degree arguments. If \(p= 2\), the first author, \textit{H. Brezis} and \textit{G. Cerami} [J. Funct. Anal. 122, No. 2, 519-543 (1994; Zbl 0805.35028)] proved that (a generalization of) (1) has two pairs of one-signed solutions for all \(\lambda\in (0, \lambda^*)\). The present paper establishes the existence of an additional pair of solutions (which can change sign) for all \(\lambda\in (0, \lambda^{**})\), \(\lambda^{**}\leq \lambda^*\). The last section concerns bifurcation of positive solutions to a class of \(p\)-Laplacian eigenvalue problems in \(\Omega\).
scientific article; zbMATH DE number 871734