Existence and multiplicity of solutions to equations of N -Laplacian type with critical exponential growth in RNRN

In this paper, we deal with the existence of solutions to the nonuniformly elliptic equation of the formequation(0.1)−div(a(x,∇u))+V(x)|u|N−2u=f(x,u)|x|β+εh(x) in RNRN when f:RN×R→Rf:RN×R→R behaves like exp(α|u|N/(N−1))exp(α|u|N/(N−1)) when |u|→∞|u|→∞ and satisfies the Ambrosetti–Rabinowitz condition. In particular, in the case of N  -Laplacian, i.e., a(x,∇u)=|∇u|N−2∇ua(x,∇u)=|∇u|N−2∇u, we obtain multiplicity of weak solutions of (0.1). Moreover, we can get the nontriviality of the solution in this case when ε=0ε=0. Finally, we show that the main results remain true if one replaces the Ambrosetti–Rabinowitz condition on the nonlinearity by weaker assumptions and thus we establish the existence and multiplicity results for a wider class of nonlinearity, see Section 7 for more details.