Nonzero solutions for Dirichlet problems with the competing (p,q)-Laplacian
Abstract
The following equation is considered: −Δp u(x) + Δq u(x) = f(x, u(x)) for x ∈ Ω, u(x) = 0 for x∈ ∂ 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) ≥ 0 for u ≥ 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 for u > 0, all global minimizers of the action functional do not vanish in (−1,1) but fail to be of class C1.
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BibTeX
@misc{beldzinski2026nonzero,
title = {{Nonzero solutions for Dirichlet problems with the competing (p,q)-Laplacian}},
author = {Michał Bełdziński},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08624},
eprint = {2611.08624},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08624},
}
RIS
TY - UNPB
TI - Nonzero solutions for Dirichlet problems with the competing (p,q)-Laplacian
AU - Michał Bełdziński
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08624
UR - https://arxiv.org/abs/2611.08624
AB - The following equation is considered: −Δ_p u(x) + Δ_q u(x) = f(x, u(x)) for x ∈ Ω, u(x) = 0 for x∈ ∂ 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) ≥ 0 for u ≥ 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 for u > 0, all global minimizers of the action functional do not vanish in (−1,1) but fail to be of class C^1.
ER -