On the Morse index of least energy nodal solutions for quasilinear elliptic problems (Q1984785)

The authors study the existence and properties of nodal solutions of the Dirichlet problem for the equation \(-\varepsilon^2 \Delta u -\Delta_p u = f(u)\), where \(\varepsilon \in \mathbb{R}\), \(p>2\), and the function \(f\) has a superlinear and subcritical growth. It is proved that the considered problem possesses a least energy nodal solution which is a minimizer of the energy functional \(J_\varepsilon\) over the nodal Nehari set \(\mathcal{M}_\varepsilon\), and each local minimizer of \(J_\varepsilon\) over \(\mathcal{M}_\varepsilon\) is also a critical point of \(J_\varepsilon\), and hence it is a nodal solution of the problem. Moreover, it is shown that in the case \(\varepsilon \neq 0\) any local minimizer of \(J_\varepsilon\) over \(\mathcal{M}_\varepsilon\) has the Morse index \(2\) and exactly two nodal domains. In order to establish the latter facts, the authors obtain fine regularity properties of solutions, which requires them to impose the assumption \(\varepsilon \neq 0\). On the other hand, using approximation arguments, it is shown that in the case \(\varepsilon=0\) the problem does possess a least energy nodal solution with the Morse index \(2\) and exactly two nodal domains. Moreover, each strict local minimizer of \(J_0\) over \(\mathcal{M}_0\) also obeys these properties.