Pairs of Positive Solutions for a Carrier <i>p</i>(<i>x</i>)-Laplacian Type Equation

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

• <i>p</i>(<i>x</i>)-Laplacian
• variable exponent
• multiple positive solutions
• variational methods
• sub-super solutions methods
• fixed-point methods