Sequences of weak solutions for fractional equations

By Giovanni Molica Bisci. 2013

59 citations recorded by the source corpus.

This work is devoted to study the existence of infinitely many weak solutions to nonlocal equations involving a general integrodifferential operator of fractional type. These equations have a variational structure and we find a sequence of nontrivial weak solutions for them exploiting the ${\mathbb{Z}}_2$-symmetric version of the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. As a particular case, we derive an existence theorem for the fractional Laplacian, finding nontrivial solutions of the equation $$ \left\{\begin{array}{ll} (-\Delta)^s u=f(x,u) & {\mbox{in}} \Omega\\ u=0 & {\mbox{in}} \erre^n\setminus \Omega. \end{array} \right. $$ As far as we know, all these results are new and represent a fractional version of classical theorems obtained working with Laplacian equations.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/b2e6444adcc852b7adf697d9511993cabe440f87

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

On a fractional degenerate Kirchhoff-type problem (2017, background): “The nonlocal analysis that we perform here is quite general and exploited for other goals in several recent contributions; see [26, 27, 28, 31, 32].”
Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem (2015, methodology): “In [27], applying the symmetric Mountain Pass Theorem, Molica Bisci obtained the existence of infinite solutions for problem (1.”
On doubly nonlocal fractional elliptic equations (2015, methodology): “Indeed, the nonlocal analysis which we perform in this paper in order to use the Mountain Pass Theorem is quite general and it was successfully exploited for other goals in several recent contributions (see [16, 19, 20, 21, 23, 28, 30, 31, 32] and [15] for various properties on the fractional Sobole”
Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian (2015, background): “We refer also to [13,14,29,30,32,40,43,44] for further details on the functional framework and its applications to the existence of solutions for problem (1.”
Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians (2017, background): “In the constant exponent case it is known as the fractional p-Laplacian, see [2, 3, 5, 7, 8, 9, 11, 12, 15, 16, 17] and references therein.”
Ground state solutions of scalar field fractional Schrödinger equations (2015, methodology): “Remark 7 For completenesswe justmention here that, by using a variational approach similar to the one adopted here, in [36–38] the authors proved the existence and the multiplicity of weak solutions for nonlocal problems involving regional fractional Laplacian operators in a suitable abstract settin”