Infinitely many solutions for problems in fractional Orlicz–Sobolev spaces

Abstract

We use a symmetric mountain pass lemma of Kajikiya to prove the existence of infinitely many weak solutions for the Schrödinger $\mathrm{\Phi }$-Laplace equation

$${\left(-\mathrm{\Delta }\right)}_{\mathrm{\Phi }}u+V\left(x\right)\phi \left(u\right)=\xi \left(x\right)f\left(u\right)\text{in }{ℝ}^d,$$

where $\mathrm{\Phi }\left(t\right)={\int }_0^t\phi \left(s\right)ds$ is an $N$-function, ${\mathrm{\Delta }}_{\mathrm{\Phi }}$ is the $\mathrm{\Phi }$-Laplacian operator, $V:{ℝ}^d\to ℝ$ is a continuous function, $\xi$ is a function with sign-changing on ${ℝ}^d$ and the nonlinearity $f$ is sublinear as $\left|u\right|\to \infty$. During the study of our problem, we deal with a new compact embedding theorem for the Orlicz–Sobolev spaces.

We also study the existence and multiplicity of solutions to the general fractional $\mathrm{\Phi }$-Laplacian equations of Kirchhoff type

$$\left\{\begin{array}{cc}M\left({\int }_{{ℝ}^{2d}}\mathrm{\Phi }\left(\frac{u\left(x\right)-u\left(y\right)}{K\left(|x-y|\right)}\right)\frac{dxdy}{N\left(|x-y|\right)}\right){\left(-\mathrm{\Delta }\right)}_{\mathrm{\Phi }}^{K,N}u=f\left(x,u\right)\hfill & \text{ in }\mathrm{\Omega },\hfill \\ & \\ u=0\hfill & \text{ in }{ℝ}^d\setminus \mathrm{\Omega },\hfill \end{array}\right.$$

where $\mathrm{\Omega }$ is an open bounded subset of ${ℝ}^d$ with smooth boundary $\partial \mathrm{\Omega }$, $d>2$, and $M:{ℝ}_0^{+}\to {ℝ}^{+}$ is a continuous function and $f:\mathrm{\Omega }\times ℝ\to ℝ$ is a Carathéodory function. The proofs rely essentially on the fountain theorem and the genus theory.