Multiple solutions for asymptotically q‐linear (p, q)‐Laplacian problems

Abstract

We investigate the existence and the multiplicity of solutions of the problemwhere Ω is a smooth, bounded domain of R N , 1 < p < q < ∞, and the nonlinearity g behaves as u q − 1 at infinity. We use variational methods and find multiple solutions as minimax critical points of the associated energy functional. Under suitable assumptions on the nonlinearity, we cover also the resonant case.

Multiplicity of positive solutions for (p,q)-Laplace equations with two parameters

“…We also refer to [11,38] for a more recent treatment of related problems in the framework of Orlicz-Sobolev spaces and for the discussion of some qualitative properties of the corresponding eigenvalues. The existence of abstract solutions of the problem (𝒟) with fixed parameters 𝛼, 𝛽 is investigated in [10,40,41]. In [10], as a particular case of a more general result, the author obtains multiplicity of solutions when the parameters satisfy certain relations with respect to the so-called quasi-eigenvalues of the 𝑝-and (𝑝, 𝑞)-Laplacians.…”

“…The existence of abstract solutions of the problem (𝒟) with fixed parameters 𝛼, 𝛽 is investigated in [10,40,41]. In [10], as a particular case of a more general result, the author obtains multiplicity of solutions when the parameters satisfy certain relations with respect to the so-called quasi-eigenvalues of the 𝑝-and (𝑝, 𝑞)-Laplacians. When either 𝛼 = 0 or 𝛽 = 0, multiplicity and bifurcation results when the parameter is nearby a variational eigenvalue of the 𝑞-or 𝑝-Laplacian, respectively, are studied in [40,41].…”

“…Our aim consists in investigating the multiplicity of solutions of the problem (𝒟) with fixed parameters 𝛼 and 𝛽 in ranges different from and extending those found in [10,40,41]. For this purpose, we employ several minimax variational methods (see, e.g., [9,31,33,34]) and determine three generally different regions on the (𝛼, 𝛽)-plane where the problem possesses a given number of distinct pairs of solutions with a prescribed sign of energy.…”

Corrigendum: Multiple solutions for asymptotically q‐linear (p,q)‐Laplacian problems

“…It was pointed out to the author by Professors Vladimir Bobkov and Mieko Tanaka that the case (H + ) in Theorem 1.1 never occurs. This is due to the fact that 𝜈 (1) k = +∞ for all k. On the other hand, under assumption (H − ), Theorem 1.1 remains valid in the form stated in [1]. We give below a modified version of Theorem 1.1 (H + ).…”

“…The proof of the present theorem can be done as in [1], relying on Theorem 2.2, and Lemmas 3.2 and 3.7 of [1], that continue to hold, and on the following adjustment of Lemma 3.6. Lemma 3.…”