Ground state and nodal solutions for fractional Orlicz problems with lack of regularity and without the Ambrosetti-Rabinowitz condition

TL;DR

This paper establishes the existence of ground state and nodal solutions for a fractional Orlicz problem on R^d, using Nehari manifold methods without relying on the Ambrosetti-Rabinowitz condition or differentiability of the nonlinearity.

Contribution

It introduces a novel approach to find solutions for fractional Orlicz problems without standard growth or smoothness assumptions.

Findings

Existence of a ground state solution with fixed sign.

Existence of a nodal (sign-changing) solution.

Solutions are obtained without the Ambrosetti-Rabinowitz condition.

Abstract

We consider a non-local Shr\"odinger problem driven by the fractional Orlicz g-Laplace operator as follows \begin{equation}\label{PP} (-\triangle_{g})^{\alpha}u+g(u)=K(x)f(x,u),\ \ \text{in}\ \mathbb{R}^{d},\tag{P} \end{equation} where $d\geq 3,\ (-\triangle_{g})^{\alpha}$ is the fractional Orlicz g-Laplace operator, $f:\mathbb{R}^d\times\mathbb{R}\rightarrow \mathbb{R}$ is a measurable function and $K$ is a positive continuous function. Employing the Nehari manifold method and without assuming the well-known Ambrosetti-Rabinowitz and differentiability conditions on the non-linear term $f$, we prove that the problem \eqref{PP} has a ground state of fixed sign and a nodal (or sign-changing) solutions.