Signed and Sign-Changing Solutions for a Kirchhoff-Type Problem Involving the Weighted N-Laplacian with Critical Double Exponential Growth

Abstract

In this paper, we are concerned with the existence of least energy sign-changing solutions for a nonlocal weighted Schrödinger–Kirchhoff problem, under boundary Dirichlet condition in the unit ball B of $\mathbb {R}^{N}$, $N>2$. The non-linearity of the equation is assumed to have double exponential growth in view of Trudinger–Moser type inequalities. The potential V is a continuous positive function and bounded away from zero in B. We use the constrained minimization in Nehari set, the quantitative deformation lemma and degree theory results.

A   Appendix

Theorem A.1

There exists $w \in \mathcal {M}_p$ such that $J_p(w) = m_p$ where $m_p = \inf _{\mathcal {M}_p} J_p(u)$.

Proof

The proof of this theorem is obtained by the following steps:

• (Step 1) For any $u \in E$ with $u^{\pm } \ne 0$, similar to Lemma 1, there is the unique maximum point pair $(s_u, t_u) \in \mathbb {R}_+\times \mathbb {R}_+$ of the function $J_p$ such that $s_u u^+ + t_u u^- \in \mathcal {M}_p$.

• (Step 2) If $u \in E$ with $u^{\pm } \ne 0$, such that $\langle J_p'(u),u^\pm \rangle \le 0$. Then, similar to Lemma 2, the unique maximum point pair $(s_u, t_u)$ in Step 1 satisfies $0<s_u, t_u\le 1$.

• (Step 3) Similar to Lemma 4, for all $u \in \mathcal {M}_p$, there exists $\kappa > 0$ such that $\Vert u^+\Vert$, $\Vert u^-\Vert \ge \kappa$.

• (Step 4) Now, let sequence $(w_n) \subset \mathcal {M}$ satisfies $\lim _{n \rightarrow +\infty } J(w_n) = m$. Similar to Lemma 7, we can to show that, up to a subsequence, $w_{n}^\pm \rightharpoonup w^\pm$ in E. From Step 3, we show that $w^\pm \ne 0$. Using the Steps 1, 2 and again similar to Lemma 7, $w \in \mathcal {M}_p$ such that $J_p(w) = m_p$, as desired.$\square$