Sign-changing solutions for Kirchhoff weighted equations under double exponential nonlinearities growth

Abstract

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

Appendix

1.1 Appendix A.

Theorem A1

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:

$\textit{Step } 1.$:

For any $u \in E$ with $u^{\pm } \ne 0$, similar to Lemma 7, there is an unique maximum point pair $(s_u, t_u) \in {\mathbb {R}}_+\times {\mathbb {R}}_+$ of the function $J_p$ such that $s_u u^+ + t_u u^- \in {\mathcal {M}}_p.$

$\textit{Step } 2.$:

If $u \in E$ with $u^{\pm } \ne 0$, then $\langle J_p'(u),u^\pm \rangle \le 0$. Then, similar to Lemma 8, the unique maximum point pair $(s_u, t_u)$ in Step 1 satisfies $0<s_u, t_u\le 1$.

$\textit{Step } 3.$:

Similar to Lemma 10, for all $u \in {\mathcal {M}}_p$, there exists $\kappa > 0$ such that $\Vert u^+\Vert ,$ $\Vert u^-\Vert \ge \kappa .$

$\textit{Step } 4.$:

Now, let sequence $(w_n) \subset {\mathcal {M}}$ satisfy $\lim _{n \rightarrow +\infty } J(w_n) = m$. Similar to Lemma 13, we can show that, up to a subsequence, $w_{n}^\pm \rightharpoonup w^\pm ~~\text{ in }~~E.$ From Step 3, we show that $w^\pm \ne 0$. Using Steps 1, 2 and again similar to Lemma 13, $w \in {\mathcal {M}}_p$ such that $J_p(w) = m_p$, as desired.$\square$