Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in RN (original) (raw)

In this paper, we deal with the existence of solutions to the nonuniformly elliptic equation of the form (0.1) − div (a (x, ∇u)) + V (x) |u| N −2 u = f (x, u) |x| β + εh(x) in R N where 0 ≤ β < N , V : R N → R is a continuous potential satisfying V (x) ≥ V 0 > 0 in R N and V −1 ∈ L 1 (R N) or |{x ∈ R N : V (x) ≤ M }| < ∞ for every M > 0, f : R N ×R → R behaves like exp α |u| N/(N −1) when |u| → ∞ and satisfies the Ambrosetti-Rabinowitz condition, h ∈ W 1,N R N * , h = 0 and ε is a positive parameter. In particular, in the case of N −Laplacian, i.e, (0.2) − ∆ N u + V (x) |u| N −2 u = f (x, u) |x| β + εh(x) using the minimization and the Ekeland variational principle, we obtain multiplicity of weak solutions of (0.2). Moreover, we prove that it is not necessary to have the small nonzero perturbation εh(x) to get the nontriviality of the solution to the N −Laplacian equation (0.3) − ∆ N u + V (x) |u| N −2 u = f (x, u) |x| β Finally, we will prove the above results when our nonlinearity f doesn't satisfy the wellknown Ambrosetti-Rabinowitz condition and thus derive the existence and multiplicity of solutions for a wider class of nonlinear terms f .