We study existence and concentration of positive solutions for quasilinear elliptic equationswhere the potential V : R N → R has a positive infimum and inf ∂Ω V > inf Ω V for some bounded domain Ω in R N and h is a nonlinearity without some growth conditions such as AmbrosettiRabinowitz.
We consider ε-perturbed nonlinear Schrödinger equations of the form −ε 2 u + V (x)u = Q(x) f (u) in R 2 , where V and Q behave like (1 + |x|) −α with α ∈ (0, 2) and (1 + |x|) −β with β ∈ (α, +∞), respectively. When f has subcritical exponential growth-by means of a weighted Trudinger-Moser-type inequality and the mountain pass theorem in weighted Sobolev spaces-we prove the existence of nontrivial mountain pass solutions, for any ε > 0, and in the semi-classical limit, these solutions concentrate at a global minimum point of A = V /Q. Our existence result holds also when f has critical growth, for any ε > 0.
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