Existence and multiplicity of solutions for fractional Schrödinger-Kirchhoff equations with Trudinger-Moser nonlinearity

Abstract

We study the existence and multiplicity of solutions for a class of fractional Schrödinger-Kirchhoff type equations with the Trudinger-Moser nonlinearity. More precisely, we consider \begin{gather*} \begin{cases} M\big(\|u\|^{N/s}\big)\left[(-Δ)^s_{N/s}u+V(x)|u|^{\frac{N}{s}-1}u\right]= f(x,u) +λh(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 $M:[0,\infty ]\to [0,\infty )$ is a continuous function, $s\in (0,1)$, $N\ge 2$, $\lambda >0$ is a parameter, $1<p<\infty$, $(-\Delta {)}_{N/s}^s$ is the fractional $N/s$--Laplacian, $V:{\mathbb{R}}^N\to (0,\infty )$ is a continuous function, $f:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is a continuous function, and $h:{\mathbb{R}}^N\to [0,\infty )$ 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 $\lambda \to \infty$. 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 $\lambda \to 0$. 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 $M(0)=0$.