Abstract
In the present paper we prove a multiplicity theorem for a quasi-linear elliptic problem with dependence on the gradient ensuring the existence of a positive solution and of a negative solution. In addition, we show the existence of the extremal constant-sign solutions: the smallest positive solution and the biggest negative solution. Our approach relies on extremal solutions for an auxiliary parametric problem. Other basic tools used in our paper are sub-supersolution techniques, Schaefer’s fixed point theorem, regularity results and strong maximum principle. In our hypotheses we only require a general growth condition with respect to the solution and its gradient, and an assumption near zero involving the first eigenvalue of the negative \(p\)-Laplacian operator.
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Acknowledgments
The second author was supported by “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA—INDAM). The work has been performed during the visit of the second author at the Department of Mathematics and Computer Sciences of the University of Catania.
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Communicated by P. Rabinowitz.
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Faraci, F., Motreanu, D. & Puglisi, D. Positive solutions of quasi-linear elliptic equations with dependence on the gradient. Calc. Var. 54, 525–538 (2015). https://doi.org/10.1007/s00526-014-0793-y
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DOI: https://doi.org/10.1007/s00526-014-0793-y