Solutions to Semilinear Elliptic Problems with Combined Nonlinearities

Abstract

We are concerned with the following nonlinear Dirichlet problem:

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].…”

On the Neumann problem with combined nonlinearities

“…The proof is similar to that of Proposition 4.1 in [8] and is omitted. We point out here that Proposition 4.1 in [8] has been proved for {u m } ∈ H 1 0 (Ω) (for the Dirichlet problem) but this result can be extended easily to the situation considered in this paper.…”

“…However, we do not impose the usual Ambrosetti-Rabinowitz condition on f . To overcome this difficulty we use some ideas from the papers [8], [13] and [14]. If h ≡ 0, then under our assumptions on f problem (1.1) does not have a positive solution.…”

“…We follow the ideas of [8]. It follows from the comments in the paragraph preceding Theorem 3.2 that the functional J + has a mountainpass structure.…”

Superlinear Elliptic Problems With Sign Changing Coefficients

“…In [10], Perera considers the case a > λ 1 when the coefficient b ≡ −1, but with different hypotheses on the behavior of the nonlinearity at infinity which impose a interaction with the first eigenvalue; in this case, a key role is played by the coercivity of the functional and he obtains up to five nontrivial solutions (of which two non positive and two non-negative). In [11] and [12], the coefficient b is assumed in L ∞ but is allowed to change sign and the nonlinearity is asymptotically linear: in [11] a nontrivial solution is obtained for a slightly above λ 1 , while in [12] the case a < λ 1 is considered and two non-negative solutions are obtained.…”