Strongly indefinite Choquard equation in ℝ2 with critical exponential growth

Abstract

In this paper, we show the existence of nontrivial solutions for the following planar Choquard equation: −normalΔu+Vfalse(xfalse)u=false(Iα∗Ffalse(ufalse)false)ffalse(ufalse),3.0235pt3.0235pt0.30emx∈ℝ2,u∈H1false(ℝ2false), where Iα:ℝ2→ℝ is the Riesz potential. We mainly consider the case that Vfalse(xfalse) is 1‐periodic, f:ℝ→ℝ has critical exponential growth at infinity and 0 lies in a gap of the spectrum of −△+V. There are few works in the literature concerned with such problem due to the compactness issue, appearance of the convolution term Iα∗Ffalse(ufalse), and the difficulties aroused by the strongly indefinite characteristic and the critical exponential growth of f. By employing an approximation scheme and some fine estimates of the mountain pass minimax levels in combination with related variational arguments, we show nontrivial solution for the above strongly indefinite problem.