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

This article concerns the existence and multiplicity of solutions for the following class of non-linear elliptic equations with variable exponents {-Delta u+lambda u=f(x,u),inR(N), u is an element of H-1(RN), where lambda > 0, N >= 2 and f:R(N)xR -> R is a function of the following types: Type 1: The subcritical case: f(x,t)=|t|(p(epsilon x)-2)t,for all(x,t)is an element of R(N)xR,N >= 3, Type 2: The critical case: f(x,t)=mu|t|(p(epsilon x)-2)t+|t|(2 & lowast;-2)t,for all(x,t)is an element of R(N)xR,N >= 3, Type 3: The exponential subcritical growth case: f(x,t)=mu|t|(p(epsilon x)-2)te(alpha|t|beta),for all(x,t)is an element of R(2)xR, where parameter epsilon >0, alpha >0, beta is an element of(0,2), 2 & lowast;=2N/N-2 if N >= 3 and 2(& lowast;)=+infinity if N = 2. Related to the function p:RN -> R, we assume that it is a continuous function with p max, p min is an element of(2,2 & lowast;), where p max=max x is an element of R(N)p(x) and p min=min x is an element of R(N)p(x). We show that for each lambda > 0 the number of solutions is associated with the number of global maximum or global minimum points of p when epsilon 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 R-N. Ann. Inst. Henri Poincare, Anal. Non Lineaire. 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).