Superlinear Elliptic Equations with Variable Exponent via Perturbation Method.

We are concerned with the following p (x) -Laplacian equations in R N − △ p (x) u + | u | p (x) − 2 u = f (x , u) in R N. The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of this problem. To overcome this difficulty, by adding potential term and using mountain pass theorem, we get the weak solution u λ of perturbation equations. First, we prove that u λ ⇀ u as λ → 0 . Second, by using vanishing lemma, we get that u is a nontrivial solution of the original problem. [ABSTRACT FROM AUTHOR]