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Multiplicity results for some nonlinear elliptic equations. (English) Zbl 0852.35045
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 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, H. Brezis and 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 $$\Omega$$.

MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B32 Bifurcations in context of PDEs
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