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Existence of solution to a nonlocal biharmonic problem with dependence on gradient and Laplacian

Shilpa, Dwivedi Gaurav

Journal of Applied Analysis

2022

Vol. 28, nr 2

211--218

In this article, we prove the existence of a solution to a nonlocal biharmonic equation with nonlinearity depending on the gradient and the Laplacian.We employ an iterative technique based on the mountain pass theorem to prove our result.

Some recent results on singular p-Laplacian equations

Guarnotta Umberto, Livrea Roberto, Marano Salvatore A.

Demonstratio Mathematica

2022

Vol. 55, nr 1

416--428

A short account of some recent existence, multiplicity, and uniqueness results for singular p-Laplacian problems either in bounded domains or in the whole space is performed, with a special attention to the case of convective reactions. An extensive bibliography is also provided.

Dynamical systems approach of internal length in fractional calculus

Beda P. B.

Engineering Transactions

2017

Vol. 65, iss. 1

209--215

Conventionally, non-local properties are included in the constitutive equations in the form of strain gradient-dependent terms. In case of the second gradient dependence an internal material length can be obtained from the critical eigenmodes in instability problems. When non-locality is included by using fractional calculus, a generalized strain can be defined. Stability investigation can be also performed and internal length effects can be studied by analysing the critical eigenspace. Such an approach leads to classical results for second gradient, but new phenomena appear in the first gradient case.