Existence and nonexistence of solutions for singular quadratic quasilinear equations

2025/08/08 by Arcoya, David, Carmona, José, Leonori, Tommaso +3 · 3 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.06375

Abstract

We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is \begincases -Δu + (|∇ u|²)/(u^γ) = f amp; in Ω,\newline \hfill u=0 \hfill amp; on ∂ Ω, \endcases where Ω is an open bounded subset of ℝ^N , γ> 0 and f is a function which is strictly positive on every compactly contained subset of Ω. As a consequence of our main results, we prove that the condition γ<2 is necessary and sufficient for the existence of solutions in H¹₀(Ω) for every sufficiently regular f as above.

Cited by

• Existence and non-existence phenomena for nonlinear elliptic equations with L¹ data and singular reactions
• A Class of Non-linear Anisotropic Elliptic problems with Unbounded Coefficients and Singular Quadratic Lower Order Terms
• Quasilinear elliptic equations with singular quadratic growth terms

Related

• Quasilinear elliptic equations with singular quadratic growth terms
• A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems
• A blow-up approach for singular elliptic problems with natural growth in the gradient
• Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions
• Multiplicity and Bifurcation Results for a Class of Quasilinear Elliptic Problems with Quadratic Growth on the Gradient