Least energy solutions for a quasilinear Neumann problem

Abstract

We study the existence of nonconstant weak solutions and the qualitative behavior of the least energy level for a class of quasilinear elliptic problems with Neumann boundary conditions of the form −∑_i,j=1^N D_{x_j}(a_ij(x,u)D_{x_i}u) +1/2∑_i,j=1^N D_s a_ij(x,u)D_{x_i}uD_{x_j}u =g(u)−λu in Ω. Since the associated energy functional is neither of class C¹ nor locally Lipschitz, the classical Nehari manifold method cannot be applied directly. We therefore rely on variational techniques combined with nonsmooth critical point theory. For λ>0, we prove the existence of least energy solutions and show that, under suitable assumptions, they are nonconstant for large λ and constant for sufficiently small λ. Moreover, the least energy level tends to +∞ as λ→+∞. In the indefinite regime, a key ingredient is a spectral decomposition of H¹(Ω) associated with the linear operator ℒ₀u :=−∑_i,j=1^N D_{x_j}(a_ij(x,0)D_{x_i}u).

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BibTeX RIS

BibTeX

@misc{canino2026least,
  title = {{Least energy solutions for a quasilinear Neumann problem}},
  author = {Annamaria Canino and Simone Mauro},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24576},
  eprint = {2610.24576},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24576},
}

RIS

TY  - UNPB
TI  - Least energy solutions for a quasilinear Neumann problem
AU  - Annamaria Canino
AU  - Simone Mauro
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.24576
UR  - https://arxiv.org/abs/2610.24576
AB  - We study the existence of nonconstant weak solutions and the qualitative behavior of the least energy level for a class of quasilinear elliptic problems with Neumann boundary conditions of the form −∑_i,j=1^N D_x_j(a_ij(x,u)D_x_iu) +1/2∑_i,j=1^N D_s a_ij(x,u)D_x_iuD_x_ju =g(u)−λu in Ω. Since the associated energy functional is neither of class C^1 nor locally Lipschitz, the classical Nehari manifold method cannot be applied directly. We therefore rely on variational techniques combined with nonsmooth critical point theory. For λ>0, we prove the existence of least energy solutions and show that, under suitable assumptions, they are nonconstant for large λ and constant for sufficiently small λ. Moreover, the least energy level tends to +∞ as λ→+∞. In the indefinite regime, a key ingredient is a spectral decomposition of H^1(Ω) associated with the linear operator ℒ_0u :=−∑_i,j=1^N D_x_j(a_ij(x,0)D_x_iu).
ER  -

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