2020/03/30 by Lauren Maria Mezzomo Bonaldo, Bonaldo, Lauren Maria Mezzomo, Olı́mpio H. Miyagaki +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2004.00416
openalex publication_date 2020/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations (\mathscrPλ) in a smooth bounded domain, driven by a nonlocal integrodifferential operator \mathscrLAK with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz type growth conditions. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for the problem (\mathscrPλ) and we show that the problem treated has at least one nontrivial solution for any parameter λ>0 small enough as well as that the solution blows up, in the fractional Sobolev norm, as λ→ 0. Moreover, for the case sublinear, by imposing some additional hypotheses on the nonlinearity f(x,⋅), by using a new version of the symmetric Mountain Pass Theorem due to Kajikiya [36], we obtain the existence of infinitely many weak solutions which tend to be zero, in the fractional Sobolev norm, for any parameter λ>0. As far as we know, the results of this paper are new in the literature.