Nonzero solutions for Dirichlet problems with the competing (p,q)-Laplacian

Abstract

The following equation is considered: $$-Δ_p u(\boldsymbol{x}) + Δ_q u(\boldsymbol{x}) = f\big(\boldsymbol{x}, u(\boldsymbol{x})\big)\text{ for } \boldsymbol{x} \in Ω, \qquad u(\boldsymbol{x}) = 0 \text{ for } \boldsymbol{x}\in \partial Ω.$$ We start by showing that the energy functional associated with the considered problem no longer has the properties typical of its non-competing counterpart, which prevents a direct application of standard variational and monotonicity methods. Using Ekeland's variational principle combined with properties imposed on the nonlinear term~$f$, we prove the existence of a nonzero generalized solution to the problem. It is also shown that in the one-dimensional case $Ω= (-1,1)$, assuming that $f$ satisfies the symmetry conditions $$f(x,u) = f(-x,u) = -f(x,-u),$$ together with the sign condition $f(x,u) \geq 0$ for $u \geq 0$ and an appropriate growth bound on $f$, a convex relaxation of the action functional yields the existence of a positive weak solution. Furthermore, we prove that in this setting, and under a slightly stronger sign condition $$f(x,u) > 0\text{ for }u > 0,$$ all global minimizers of the action functional do not vanish in $(-1,1)$ but fail to be of class $C^1$.