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PAPER: Multiplicity theorems for superlinear elliptic problems

REF. DATE:8 March 2008 CREATED/MODIFIED: 14 January 2016 208 views No Comment

Multiplicity theorems for superlinear elliptic problems (with N.S. Papageorgiou, V. Staicu), Calculus of Variations and Partial Differential Equations 33:2(2008), 199-230.

ABSTRACT: Let \Omega\subseteq\bkR^N be a bounded domain with a C^2-boundary \partial \Omega. In this paper we study second order elliptic equations of the form

    \[\left\{ \begin{array}{ll}   -\mbox{div} \left(|Du|^{p-2}Du(x)\right)=f(x,u(x)) & \mbox{ a.e. on }\Omega,\\   \left. u\right|_{\partial \Omega}=0, & 1<p<\infty. \end{array}   \right.\]

driven by the Laplacian and p-Laplacian differential operators and a nonlinearity which is (p-)superlinear (it satisfies the Ambrosetti-Rabinowitz condition). For the p-Laplacian equations we prove the existence of five nontrivial smooth solutions, namely two positive, two negative and a nodal solution. For the semi linear problems using in addition Morse theory, we obtain six nontrivial solutions. We prove seven such multiplicity results. The first five concern problems driven by the p-Laplacian, while the last two deal with the particular case p=2 (semilinear problems). In all these theorems we also provide precise sign information about the solutions.

Download: Preprint version (RIA | download) / Publisher version (doi:10.1007/s00526-008-0172-7)
Remark: The paper follows Papageorgiou’s notation
Extra info: (P006/2008-03-08)

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