Pairs of Positive Solutions for a Carrier p ( x )-Laplacian Type Equation

Pasquale Candito (), Giuseppe Failla and Roberto Livrea
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Pasquale Candito: Department of Civil, Energy, Environmental and Material Engineering (DICEAM), University of Reggio Calabria, Via Zehender, Località Feo di Vito, 89122 Reggio Calabria, Italy
Giuseppe Failla: Department of Mathematics and Computer Sciences, Physical Sciences and Earth Sciences (MIFT), University of Messina, Viale Ferdinando Stagno d’Alcontres, 98166 Messina, Italy
Roberto Livrea: Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy

Mathematics, 2024, vol. 12, issue 16, 1-16

Abstract: The existence of multiple pairs of smooth positive solutions for a Carrier problem, driven by a p ( x ) -Laplacian operator, is studied. The approach adopted combines sub-super solutions, truncation, and variational techniques. In particular, after an explicit computation of a sub-solution, obtained combining a monotonicity type hypothesis on the reaction term and the Giacomoni–Takáč’s version of the celebrated Díaz–Saá’s inequality, we derive a multiplicity of solution by investigating an associated one-dimensional fixed point problem. The nonlocal term involved may be a sign-changing function and permit us to obtain the existence of multiple pairs of positive solutions, one for each “positive bump” of the nonlocal term. A new result, also for a constant exponent, is established and an illustrative example is proposed.

Keywords: p ( x )-Laplacian; variable exponent; multiple positive solutions; variational methods; sub-super solutions methods; fixed-point methods; truncation techniques (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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