Pairs of positive radial solutions for a Minkowski-curvature Neumann problem with indefinite weight

Abstract

We prove the existence of a pair of positive radial solutions for the Neumann boundary value problemwhere B is a ball centered at the origin, a(|x|) is a radial sign-changing function with B a(|x|) dx < 0, p > 1 and λ > 0 is a large parameter. The proof is based on the Leray-Schauder degree theory and extends to a larger class of nonlinearities.

Positive solutions for a Minkowski-curvature equation with indefinite weight and super-exponential nonlinearity

“…where p > 1 and a(t) satisfies (besides some technical assumptions) the meanvalue condition T 0 a(t) dt < 0, has at least two positive T -periodic solutions if the parameter λ is positive and large enough (say, λ > λ * ) and no positive T -periodic solutions if λ is positive and sufficiently small (say, λ ∈ (0, λ * ), with λ * ≤ λ * ). An analogous result for the Neumann boundary value problem was established in [9]. Incidentally, let us recall that the condition T 0 a(t) dt < 0 is actually necessary for the existence of a positive solution of (1.2) with periodic or Neumann boundary conditions, as it can be easily checked by dividing the equation by u p and integrating by parts.…”

“…Incidentally, let us also observe that the existence of a family of positive solutions u λ (t) for λ large was already proved in [10, Theorem 3.3], dealing with the periodic problem (the Neumann case can be treated as well, using the approach in [9]). By [10,Theorem 4.1], it holds that u λ ∞ → 0 as λ → +∞.…”

“…On the other hand, the specific choice for the nonlinear term made in equation (1.1) casts it into the family of nonlinear problems with indefinite weight, a quite popular topic in Nonlinear Analysis since the pioneering papers [1,3,7,13,20]. In the case of linear differential operators, the bibliography about indefinite weight problems is nowadays very wide (we refer to [18,22,30] for an extensive list of references); more recently, the case of mean curvature equations, both in Euclidean and Minkowski space, has been considered as well (see, among others, [6,10,9,23,24,29]).…”

Uniqueness of positive solutions for boundary value problems associated with indefinite $ϕ$-Laplacian type equations

“…Recent works for the Neumann and periodic problems associated with (4.4) show that for g(u) = u γ with γ > 1 multiple positive solutions do exist also in the case of a weight with a single change of sign as in (1.4) (see [8,9]). On the other hand, numerical simulations suggest the possibility of uniqueness results when g(u) is a strictly increasing function with "super-exponential" growth at infinity (cf.…”