Existence and multiplicity for an elliptic problem with critical growth\n in the gradient and sign-changing coefficients

2019/09/11 by Colette De Coster, De Coster, Colette, Antonio José Campesino Fernández +1
Computer Science · Mathematics · #35J20 #35J25 #35J62 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1909.04962

openalex publication_date 2019/09/11 · openalex created_date 2022/09/27 · openalex updated_date 2026/07/28

Abstract

Let \Ω \⊂ \ℝ^N, N \≥ 2, be a smooth bounded domain. We\nconsider the boundary value problem n
labelPlambda-Abstract-ch3
tagP
_lambda -
Delta u = c
_lambda(x) u +\n
mu |
nabla u|² + h(x)
,,
quad u
in H₀¹(
Omega)
cap L
^infty(
Omega)
,,\n where c_\λ and h belong to L^q(\Ω) for some q\n> N/2, \μ belongs to \ℝ \∖ 0 and we write\nc_\λ under the form c_\λ:= \λ c₊ - c_- with c₊\n gneqq 0, c_- \≥ 0, c₊ c_- \≡ 0 and \λ \∈ \ℝ.\nHere c_\λ and h are both allowed to change sign. As a first main\nresult we give a necessary and sufficient condition which guarantees the\nexistence of a unique solution to eqrefPlambda-Abstract-ch3 when \λ\n\≤ 0. Then, assuming that (P₀) has a solution, we prove existence and\nmultiplicity results for \λ > 0. Our proofs rely on a suitable change of\nvariable of type v = F(u) and the combination of variational methods with\nlower and upper solution techniques.\n

Related

• MULTIPLICITY RESULTS IN THE NON-COERCIVE CASE FOR AN ELLIPTIC PROBLEM WITH CRITICAL GROWTH IN THE GRADIENT
• Existence and Multiplicity for elliptic p-Laplacian problems with\n critical growth in the gradient
• Continuum of solutions for an elliptic problem with critical growth in the gradient
• Multiplicity results for fully nonlinear elliptic equations with natural gradient growth
• A priori bounds and multiplicity of solutions for an indefinite elliptic problem with critical growth in the gradient