A strong comparison principle for positive solutions of degenerate elliptic equations
By M. Cuesta, Peter Takác. Differential and Integral Equations, 2000
110 citations recorded by the source corpus.
A strong comparison principle (SCP, for brevity) is obtained for nonnegative weak solutions u ∈ W 1,p 0 (Ω) of the following class of quasilinear elliptic boundary value problems, (P ) −div(a(x,∇u))− b(x, u) = f(x) in Ω; u = 0 on ∂Ω. Here, p ∈ (1,∞) is a given number, Ω is a bounded domain in IR with a connected C-boundary, a(x,∇u) and b(x, u) are slightly more general than the functions a0(x)|∇u|∇u and b0(x)|u|u, respectively, with a0 ≥ const > 0 and b0 ≥ 0 in L∞(Ω), and 0 ≤ f ∈ L∞(Ω). Validity of the SCP is investigated also in the case when b0 ≤ 0 depending upon whether p ≤ 2 or p > 2. The methods of proofs are new. AMS subject classifications: Primary 35B50, 35J70; Secondary 34C11, 47H05
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/cc28cdbf19b31986c45d80fc4e4b727410485574
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Citations in the corpus
- Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation (2007, methodology): “Theorem 2.3 holds if we replace the p-Laplacian operator by a more general quasilinear operator; see, for instance, conditions (3)-(7) in Cuesta and Takáč [14].”
- Liouville results for m-Laplace equations of Lane―Emden―Fowler type (2009, unknown): “Also, since div(|∇u|m−2∇u)(ui · xi) ∈ L1(Ω) by the regularity results in [7], and (|∇u|m−2∇u)ui · xi ∈ C0(Ω) by assumption, then we can exploit the divergence theorem (see [5]) obtaining ∫”
- Local "superlinearity"and "sublinearity" for the p-Laplacian (2009, unknown): “[11]).”
- A note on a non-linear Krein–Rutman theorem (2006, background): “ that L−1 µ be well defined on the positive cone and and the desired uniqueness of positive solutions follows from the weak comparison principle (WCP) proved for positive solutions of such operatorsin [5]. The sameWCPalsoshowsthat theoperatorT ismonotoneincreasingon the positive cone. Now, it is quit”