Nonlocal Kirchhoff Problems with Singular Exponential Nonlinearity

Xiang Mingqi,Binlin Zhang,Vicenţiu D Rădulescu

doi:10.1007/s00245-020-09666-3

Abstract

In this paper, we first develop the fractional Trudinger–Moser inequality in singular case and then we use it to study the existence and multiplicity of solutions for a class of perturbed fractional Kirchhoff type problems with singular exponential nonlinearity. Under some suitable assumptions, the existence of two nontrivial and nonnegative solutions is obtained by using the mountain pass theorem and Ekeland’s variational principle as the nonlinear term satisfies critical or subcritical exponential growth conditions. Moreover, the existence of ground state solutions for the aforementioned problems without perturbation and without the Ambrosetti–Rabinowitz condition is investigated.

Highlights

Introduction and Main ResultsLet N ≥ 2 and assume that ⊂ RN is a bounded domain with Lipschitz boundary and 0 ∈
Given s ∈ (0, 1), we study the following fractional Kirchhoff type problem with exponential growth:
Under suitable assumption on M and f, the authors obtained the existence of ground state solutions by using the mountain pass lemma combined with the fractional Trudinger–Moser inequality

Summary

Introduction and Main Results

Let N ≥ 2 and assume that ⊂ RN is a bounded domain with Lipschitz boundary and 0 ∈. Given s ∈ (0, 1), we study the following fractional Kirchhoff type problem with exponential growth:. Under suitable assumption on M and f , the authors obtained the existence of ground state solutions by using the mountain pass lemma combined with the fractional Trudinger–Moser inequality. Some new existence results of solutions for fractional non-degenerate Kirchhoff problems were given, for example, in [42,43,44,49]. To overcome the loss of compactness of the energy functional, we have to estimate the range of level value of energy functional This is the key point to obtain the existence of solutions for the critical case. 3, by using the mountain pass theorem and Ekeland’ variational principle, we obtain the existence of two nontrivial nonnegative solutions for problem (1.1) with subcritical exponential growth conditions as λ small.

Preliminary Results

The Subcritical Case

The Critical Case