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Tohoku Mathematical Journal
SECOND SERIES VOL. 62, NO. 1
|Tohoku Math. J.|
62 (2010), 137-162
MULTIPLICITY OF SOLUTIONS FOR PARAMETRIC $P$-LAPLACIAN EQUATIONS WITH NONLINEARITY CONCAVE NEAR THE ORIGIN
Shouchuan Hu and Nikolaos S. Papageorgiou
(Received May 13, 2009, revised October 26, 2009)
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.
2000 Mathematics Subject Classification.
Primary 35J20; Secondary 35J60, 38J70.
Key words and phrases.
$p$-Laplacian, $p$-linear perturbation, $p$-superlinear perturbation, constant sign solutions, nodal solutions, multiple solutions, upper and lower solutions.
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