Sign-changing solutions of boundary value problems for semilinear ${\Delta }_{\gamma }$-Laplace equations

Luyen, Duong Trong

Rendiconti del Seminario Matematico della Università di Padova, Tome 143 (2020), pp. 113-134.

In this article, we study the multiplicity of weak solutions to the boundary value problem

$$\left\{\begin{array}{ccc}-G_{\alpha }u=g(x,y,u)+f(x,y,u)\hfill & \text{in}\Omega ,u=0\hfill & \text{on}\partial \Omega ,\end{array}\right.$$

where $\Omega$ is a bounded domain with smooth boundary in ${ℝ}^N(N\ge 2),\alpha \in \mathbb{N},g(x,y,\xi ),f(x,y,\xi )$ are Carathéodory functions and $G_{\alpha }$ is the Grushin operator. We use the lower bounds of eigenvalues and an abstract theory on sign-changing solutions.

Classification : 35, 00
Mots clés : $\Delta_{\gamma}$-Laplace equations, critical point theorem, sign-changing solutions, boundary value problems, Grushin operator