| Contents |
| We consider the question of the existence of sign-changing solutions for elliptic problems of the following type
\begin{equation}
\tag{$\mathcal{D}$}
\begin{cases}
-\Delta_p u = f_\lambda(x,u), \quad x \in \Omega, \\
\, ~~u|_{\partial \Omega} = 0,
\end{cases}
\end{equation}
where $f_\lambda(x, u)$ has two forms: 1) indefinite nonlinearity $f(x, u) = \lambda |u|^{p-2} u + f(x) |u|^{\gamma-2} u$, where $f \in L^\infty(\Omega)$ changes the sign and $p < \gamma < p^*$; 2) convex-concave nonlinearity $f_\lambda(x, u) = \lambda |u|^{q-2} u + |u|^{\gamma-2} u$, where $1 < q < p < \gamma < p^*$.
Using the variational and topological approaches we obtain the continuous branch of sign-changing solutions to $(\mathcal{D})$ on non-local spectral interval $(-\infty, \lambda^*)$. The obtained solutions have precisely 2 nodal domains and least energy property. |
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