Asymmetric superlinear problems under strong resonance conditions




Recova, Leandro L.
Rumbos, Adolfo J.

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Texas State University, Department of Mathematics


We study the existence and multiplicity of solutions of the problem -∆u = -λ1u‾ + g(x, u), in Ω; u = 0, on ∂Ω, where Ω is a smooth bounded domain in ℝN (N ≥ 2), u‾ denotes the negative part of u : Ω → ℝ, λ1 is the first eigenvalue of the N-dimensional Laplacian with Dirichlet boundary conditions in Ω, and g : Ω x ℝ → ℝ is a continuous function with g(x, 0) = 0 for all x ∈ Ω. We assume that the nonlinearity g(x, s) has a strong resonant behavior for large negative values of s and is superlinear, but subcritical, for large positive values of s. Because of the lack of compactness in this kind of problem, we establish conditions under which the associated energy functional satisfies the Palais-Smale condition. We prove the existence of three nontrivial solutions of problem (1) as a consequence of Ekeland's Variational Principle and a variant of the mountain pass theorem due to Pucci and Serrin [14].



Strong resonance, Palais-Smale condition, Ekeland's principle


Recova, L., & Rumbos, A. (2017). Asymmetric superlinear problems under strong resonance conditions. <i>Electronic Journal of Differential Equations, 2017</i>(149), pp. 1-27.


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