Fractional Kirchhoff-type problems with exponential growth without the Ambrosetti–Rabinowitz condition

The main aim of this paper is to investigate the existence of nontrivial solutions for a class of fractional Kirchhoff-type problems with right-hand side nonlinearity which is subcritical or critical exponential growth (subcritical polynomial growth) at infinity. However, it need not satisfy the Ambrosetti–Rabinowitz (AR) condition. Some existence results of nontrivial solutions are established via Mountain Pass Theorem combined with the fractional Moser–Trudinger inequality.

On solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity

In this paper, we prove the existence of at least three nontrivial solutions for the following class of fractional Kirchhoff-type problems: [Formula: see text] where [Formula: see text] is a constant, [Formula: see text] is a bounded open interval, [Formula: see text] is a continuous potential, the nonlinear term [Formula: see text] has exponential growth of Trudinger–Moser type, [Formula: see text] and [Formula: see text] denotes the standard Gagliardo seminorm of the fractional Sobolev space [Formula: see text]. More precisely, by exploring a minimization argument and the quantitative deformation lemma, we establish the existence of a nodal (or sign-changing) solution and by means of the Mountain Pass Theorem, we get one nonpositive and one nonnegative ground state solution. Moreover, we show that the energy of the nodal solution is strictly larger than twice the ground state level. When we regard [Formula: see text] as a positive parameter, we study the behavior of the nodal solutions as [Formula: see text].

Supercritical fractional Kirchhoff type problems

In this paper we deal with the following fractional Kirchhoff problem $$\begin{array}{} \displaystyle \left\{ {\begin{array}{l} \left[M\left(\displaystyle \iint_{\mathbb R^n\times \mathbb R^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} dx dy\right)\right]^{p-1}(-\Delta)^{s}_{p}u = f(x, u)+\lambda |u|^{r-2}u \\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \mbox{ in } \, \Omega, \\ \\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad u=0 \, \, ~\mbox{ in } \, \mathbb R^n\setminus \Omega. \end{array}} \right. \end{array}$$ Here Ω ⊂ ℝn is a smooth bounded open set with continuous boundary ∂Ω, p ∈ (1, +∞), s ∈ (0, 1), n > sp, $\begin{array}{} (-\Delta)^{s}_{p} \end{array}$ is the fractional p-Laplacian, M is a Kirchhoff function, f is a continuous function with subcritical growth, λ is a nonnegative parameter and r > $\begin{array}{} p^*_s \end{array}$, where $\begin{array}{} p^*_s=\frac{np}{n-sp} \end{array}$ is the fractional critical Sobolev exponent. By combining variational techniques and a truncation argument, we prove two existence results for this problem, provided that the parameter λ is sufficiently small.