Multiple Solutions for Sublinear Double Phase Problems

Motivated by a wide interest in the current literature, in this paper, we study the existence of weak solutions for a wide class of double-phase Dirichlet problems depending on a positive real parameter. More precisely, under natural assumptions on the nonlinear term, we are able to show that the main problem admits only the trivial solution provided that the parameter is small, and at least two <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:msup> </mml:math> -bounded nonnegative weak solutions whenever the parameter is sufficiently large. The main approach, of a pure variational nature, is based on a fine analysis of the geometrical structure of the associated energy functional.