A numerical algorithm for finding positive solutions for classes of p-Laplacian equations

Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms

“…, we can prove that, if ν is sufficiently small, then 1). Indeed, let (u ε , v ε ) be a solution of (2.24).…”

“…Now, all the pairs (u ε , v ε ) such that (1, u ε , v ε ) ∈ S ε are T -periodic solutions of (2.1) with (u ε , v ε ) = (0, 0) and, hence, satisfy (2.12). Since the L 2 -norm is continuous with respect to the L ∞ -norm and S ε is a continuum, for every ν > 0 there is 1]. Observe that, if (σ, u ε , v ε ) ∈ S ε for σ < 1, then u ε and v ε are positive solutions of (2.24).…”

“…Here Ω is an open bounded domain of R N with smooth boundary ∂Ω, satisfying the property of positive geometric density, see [39], Q T := Ω × (0, T ), T > 0, τ i ∈ (0, +∞), the functions K i , a, and b belong to L ∞ (Q T ). The exponents p and q belong to the interval (1,2), m > p, n > q and s m = |s| m−1 s. Setting Au := div(|∇u m | p−2 ∇u m ) and l := (m − 1)(p − 1), the operator Au becomes m p−1 div(|u| l |∇u| p−2 ∇u), which is the operator considered by Ivanov in [32], [33] and [34]. According to the classification proposed in these papers, we say that the first equation in (1.1) is of…”

Nontrivial, nonnegative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

“…In terms of the biological interpretations, system (1.1) is a model of the interactions of two biological species, with density u and v respectively, disliking crowding, i.e. m, n > 1, see [27], [28] and [50], and whose diffusion involves, as in [1], [2], [51] and [52], the singular p-Laplacian, i.e. 1 < p < 2.…”

“…On the other hand, some previous biological models found in the literature, see e.g. [1], [2], [51], [52] involve the p-Laplacian with p > 1 (and m = 1). Furthermore, we observe that the equations of the system we consider treat all the possible types of diffusion: slow, normal and fast, while in [25], [60] and [26] only the slow and normal diffusions were presented.…”