Boundedness of the extremal solution of some -Laplacian problems

Abstract

In this article we consider the p-Laplace equation − p u = λ f (u) on a smooth bounded domain of R N with zero Dirichlet boundary conditions. Under adequate assumptions on f we prove that the extremal solution of this problem is in the energy class W 1, p 0 (Ω) independently of the domain. We also obtain L q and W 1,q estimates for such a solution. Moreover, we prove its boundedness for some range of dimensions depending on the nonlinearity f .

Boundedness of the extremal solution for some p-Laplacian problems

“…In this case, we assume in addition that γ ≤ 1, then N * 1 ≥ N (p), here N (p) is given in [18]. Hence, our result extends partially the main results in [18]. On the other hand, for p = 2, we find that N * 2 = 6 + 4 √ γ, this has been given in [22].…”

“…In this case, we assume in addition that γ ≤ 1, then N * 1 ≥ N (p), here N (p) is given in [18]. Hence, our result extends partially the main results in [18].…”

“…if the coefficients a i satisfy some constraints such that f (u) satisfies (2). So our assumption (7) and the condition (21) in [18] are not covered each other. Next, we assume that f (u) satisfies the following condition:…”

On the Connection Between Two Quasilinear Elliptic Problems With Source Terms of Order 0 or 1

“…The same conclusions hold when the function g behaves like an exponential or a power, see [67], [19], [63], and [27]. Up to our knowledge, the gap between N 0 = pp ′ and N 2 remains for general g, excepted in the radial case, see [18].…”

Boundedness of Stable Solutions to Semilinear Elliptic Equations: A Survey

“…Part (a) of Theorem 1 was proven by Crandall and Rabinowitz [10] and by Mignot and Puel [17], not only for the exponential nonlinearity but also for some others. The improved result under assumption (1.5) is due to Sanchón [19]. The proofs use the equation in (1.3) together with the stability condition (1.2) applied to…”