Tohoku Mathematical Journal

Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin

Shouchuan Hu and Nikolaos S. Papageorgiou

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We consider a nonlinear elliptic problem driven by the $p$-Laplacian and depending on a parameter. The right-hand side nonlinearity is concave, (i.e., $p$-sublinear) near the origin. For such problems we prove two multiplicity results, one when the right-hand side nonlinearity is $p$-linear near infinity and the other when it is $p$-superlinear. Both results show that there exists an open bounded interval such that the problem has five nontrivial solutions (two positive, two negative and one nodal), if the parameter is in that interval. We also consider the case when the parameter is in the right end of the interval.

Article information

Tohoku Math. J. (2), Volume 62, Number 1 (2010), 137-162.

First available in Project Euclid: 31 March 2010

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Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations 38J70

$p$-Laplacian $p$-linear perturbation $p$-superlinear perturbation constant sign solutions nodal solutions multiple solutions upper and lower solutions


Hu, Shouchuan; Papageorgiou, Nikolaos S. Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin. Tohoku Math. J. (2) 62 (2010), no. 1, 137--162. doi:10.2748/tmj/1270041030.

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