Abstract
We consider a nonlinear parametric Dirichlet problem driven by the anisotropic p-Laplacian with the combined effects of “concave” and “convex” terms. The “superlinear” nonlinearity need not satisfy the Ambrosetti-Rabinowitz condition. Using variational methods based on the critical point theory and the Ekeland variational principle, we show that for small values of the parameter, the problem has at least two nontrivial smooth positive solutions.
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This research has been partially supported by the Ministry of Science and Higher Education of Poland under Grant no. N201 542438.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Gasiński, L., Papageorgiou, N.S. A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities. J Glob Optim 56, 1347–1360 (2013). https://doi.org/10.1007/s10898-011-9841-8
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DOI: https://doi.org/10.1007/s10898-011-9841-8
Keywords
- Variable exponent
- Concave and convex terms
- Positive solutions
- Mountain pass theorem
- Maximum principle
- Ekeland variational principle