Existence and multiplicity of solutions for fractional \(p_{1}(x,\cdot )\& p_{2}(x,\cdot )\)-Laplacian Schrödinger-type equations with Robin boundary conditions

In this paper, we study fractional $p_1(x,\cdot )\mathrm{\&}{p}_2(x,\cdot )$ $p_{1}(x,\cdot )\& p_{2}(x,\cdot )$ -Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland’s variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in ${\mathbb{R}}^N\setminus \bar{\mathrm{\Omega }}$ $\mathbb{R}^{N}\setminus \overline{\Omega}$ for fractional order $p_1(x,\cdot )\mathrm{\&}{p}_2(x,\cdot )$ $p_{1}(x,\cdot )\& p_{2}(x,\cdot )$ -Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results.