Existence and multiplicity of solutions for fractional Schr\"odinger-Kirchhoff equations with Trudinger-Moser nonlinearity

Abstract

We study the existence and multiplicity of solutions for a class of fractional Schr\"{o}dinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider \begin{gather*} \begin{cases} M\big(\|u\|^{N/s}\big)\left[(-\Delta)^s_{N/s}u+V(x)|u|^{\frac{N}{s}-1}u\right]= f(x,u) +\lambda h(x)|u|^{p-2}u\, &{\rm in}\ \ \mathbb{R}^N,\\ \|u\|=\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy+\int_{\mathbb{R}^N}V(x)|u|^{N/s}dx\right)^{s/N}, \end{cases}\end{gather*} where $M:[0,\infty]\rightarrow [0,\infty)$ is a continuous function, $s\in (0,1)$, $N\geq2$, $\lambda>0$ is a parameter, $1<p<\infty$, $(-\Delta )^s_{N/s}$ is the fractional $N/s$--Laplacian, $V:\mathbb{R}^N\rightarrow(0,\infty)$ is a continuous function, $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R} $ is a continuous function, and $h:\mathbb{R}^N\rightarrow[0,\infty)$ is a measurable function. First, using the mountain pass theorem, a nonnegative solution is obtained when $f$ satisfies exponential growth conditions and $\lambda$ is large enough, and we prove that the solution converges to zero in $W_V^{s,N/s}(\mathbb{R}^N)$ as $\lambda\rightarrow\infty$. Then, using the Ekeland variational principle, a nonnegative nontrivial solution is obtained when $\lambda$ is small enough, and we show that the solution converges to zero in $W_V^{s,N/s}(\mathbb{R}^N)$ as $\lambda\rightarrow0$. Furthermore, using the genus theory, infinitely many solutions are obtained when $M$ is a special function and $\lambda$ is small enough. We note that our paper covers a novel feature of Kirchhoff problems, that is, the Kirchhoff function $M(0)=0$.