Multiple solutions for some elliptic equations with a nonlinearity concave at the origin

Multiple Solutions to (p,q)-Laplacian Problems with Resonant Concave Nonlinearity

“…Such a type of growth rate has been widely studied, also combined with further conditions, provided p = and μ = , i.e., the equation is semilinear. As an example, besides the seminal paper [2], let us mention [8,16,21,22]. A similar comment holds true also when p ̸ = but μ = , in which case the literature looks to be daily increasing; see for instance the very recent papers [12,14,18,19] and, concerning the nonsmooth framework, [13,17].…”

Superlinear Elliptic Problems With Sign Changing Coefficients

“…When b(x) ≡ −1, in the already cited [2], it was proved that, along with a non-negative (and a non-positive) nontrivial solution, a further solution exists provided λ is small enough and a ≥ λ 1 ; this third solution is obtained via Linking Theorem.…”

“…On the other hand, if b(x) ≡ −1 and the superlinear term is subcritical, one non-negative solution for any a ∈ R and λ > 0 was found in [2], via Mountain Pass Theorem.…”

“…Our first purpose is to determine a class of functions f µ (x, u), for which the results in [2] are still valid, that is, for which we may obtain at least a non-negative solution and a non-positive one for any λ > 0 and a ∈ R, and a third solution for λ small and a ≥ λ 1 ; as observed above, in order to obtain this, one needs f µ (x, s) s < 0 in some region. However, it turns out that if f µ (x, s) s > 0 in some other region, which was not the case in [2], then one more non-positive (and one non-negative) solution arises. This will be shown in Theorems 2.1 and 2.2.…”

“…Our second purpose is to study this problem in the more delicate critical case, which was not considered in [2]: this result will be given in Theorems 2.4 and 2.5. Observe also that in [3] were considered, like here, unbounded coefficients, however, the critical case was not studied, and so our work complements, in some sense, also that result.…”