Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness

We study the following Kirchhoff equation $$- \left(1 + b \int_{\mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u + V(x) u = f(x,u), \ x \in \mathbb{R}^3.$$ A special feature of this paper is that the nonlinearity $f$ and the potential $V$ are indefinite, hence sign-changing. Under some appropriate assumptions on $V$ and $f$, we prove the existence of two different solutions of the equation via the Ekeland variational principle and Mountain Pass Theorem

Nonlocal problems with critical Hardy nonlinearity

By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.Comment: 36 pages, revised versio

Triple positive solutions for semipositone fractional differential equations m-point boundary value problems with singularities and p–q-order derivatives

In this paper, by means of Leggett–Williams and Guo–Krasnosel'skii fixed point theorems, together with height functions of the nonlinearity on different bounded sets, triple positive solutions are obtained for some fractional differential equations with p–q-order derivatives involved in multi-point boundary value conditions. The nonlinearity may not only take negative infinity but also may permit singularities on both the time and the space variables