WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 23, 2024
Nonexistence Results for the Elliptic Equations with Fractional Laplacian and Variable Exponents Nonlinearities
Authors: ,
Abstract: The main motivation behind this paper is to investigate the nonexistence of weak solution for the following Cauchy problem $$(−Δ)^{\frac{γ}{ 2}} w^{l} + Δw^{l} = |w|^{m(x)}, x ∈ IR^{n}$$, where $$γ ∈ (0, 2), l ≥ 1, m : IR^{n} → (1,+∞)$$ is a measurable function, and $$(−Δ)^{\frac{γ}{ 2}}$$ is the fractional Laplacian operator of order $${\frac{γ}{ 2}}$$. Then, this result is extended to the case of 2×2-system of the same type. The proof of our results is based on a contradiction argument by using the so-called test function method. The results obtained in this paper extend several contributions and we focus on new nonexistence result which is due to the presence of variable-exponents.
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Keywords: Variable-exponent nonlinearities, nonexistence, test functions, fractional Laplacian, elliptic
equations, blow-up
Pages: 494-501
DOI: 10.37394/23206.2024.23.52