Multiple solutions for some classes of non-linear elliptic equations with variable exponents

Abstract This article concerns the existence and multiplicity of solutions for the following class of non-linear elliptic equations with variable exponents \begin{equation*} \left\{ \begin{array}{l} -\Delta u+\lambda u=f(x,u), \quad \text{in} \quad \mathbb{R}^N, \\ \,u\in H^{1}(\mathbb{R}^N),\\ \end{array} \right. \end{equation*} where λ > 0, $N\geq 2$ and $f:\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ is a function of the following types: Type 1: The subcritical case: \begin{equation*}f(x,t)=|t|^{p(\varepsilon x)-2}t, \quad \forall (x,t) \in \mathbb{R}^N\times \mathbb{R}, N\geq3, \end{equation*} Type 2: The critical case: \begin{equation*}f(x,t)=\mu|t|^{p(\varepsilon x)-2}t+|t|^{2^*-2}t, \quad \forall (x,t) \in \mathbb{R}^N\times \mathbb{R}, N\geq3, \end{equation*} Type 3: The exponential subcritical growth case: \begin{equation*} f(x,t)=\mu|t|^{p(\varepsilon x)-2}te^{\alpha|t|^\beta}, \quad \forall (x,t) \in \mathbb{R}^2\times \mathbb{R}, \end{equation*} where parameter ɛ > 0, α > 0, $\beta \in (0,2)$ , $2^*=\frac{2N}{N-2}$ if $N \geq 3$ and $2^*=+\infty$ if N = 2. Related to the function $p:\mathbb{R}^{N}\rightarrow \mathbb{R}$ , we assume that it is a continuous function with $p_{\max}, p_{\min} \in (2,2^*)$ , where $p_{\max}=\displaystyle \max_{x \in \mathbb{R}^N}p(x)$ and $p_{\min}=\displaystyle \min_{x \in \mathbb{R}^N}p(x)$ . We show that for each λ > 0 the number of solutions is associated with the number of global maximum or global minimum points of p when ɛ is small enough. The proof of is based on the variational methods, Ekeland’s variational principle, Trundiger–Moser inequality, and the monotonicity of the ground state energy with respect to p . Our results extend those of Cao and Noussair (Multiplicity of positive and nodal solutions for nonlinear elliptic problem in $\mathbb{R}^{N}$ . Ann. Inst. Henri Poincaré, Anal. Non Linéaire . 13 (1996), 567–588) and Ji, Wang and Wu (A monotone property of the ground state energy to the scalar field equation and applications. J. Lond. Math. Soc., II. Ser. 100 (2019), 804–824).