In the present paper we deal with a quasi-linear elliptic equation depending on a sublinear nonlinearity involving the gradient. We prove the existence of a nontrivial nodal solution employing the theory of invariant sets of descending flow together with sub-supersolution techniques, gradient regularity arguments, strong comparison principle for the $p$-Laplace operator. The same conclusion is obtained for an eigenvalue problem under a different set of assumptions.

We prove that the fractional Yamabe equation ${\rm {\cal L}}_\gamma u = \vert u \vert ^{((4\gamma )/(Q-2\gamma ))}u$ on the Heisenberg group ℍn has [n + 1/2] sequences of nodal (sign-changing) weak solutions whose elements have mutually different nodal properties, where ${\rm {\cal L}}_\gamma $ denotes the CR fractional sub-Laplacian operator on ℍn, Q = 2n + 2 is the homogeneous dimension of ℍn, and $\gamma \in \bigcup\nolimits_{k = 1}^n [k,((kQ)/Q-1)))$. Our argument is variational, based on a Ding-type conformal pulling-back transformation of the original problem into a problem on the CR sphere S2n + 1 combined with a suitable Hebey-Vaugon-type compactness result and group-theoretical constructions for special subgroups of the unitary group U(n + 1).

In this paper we establish the existence and multiplicity of multi-bump nodal solutions for the class of problems

where λ ∈ (0, ∞), f is a continuous function with exponential critical growth and V : ℝ2→ ℝ is a continuous function verifying some hypotheses.

We prove that the $p$-Laplacian problem $-\Delta_p u = f(x, u)$, with $u \in W^{1, p}_0 (\Omega)$ on a bounded domain $\Omega \subset R^N$, with $p > 1$ arbitrary, has a nodal solution provided that $f : \Omega \times R \to R$ is subcritical, and $f(x, t) / |t|^{p - 2}$ is superlinear. Infinitely many nodal solutions are obtained if, in addition, $f(x, -t) = -f(x, t)$.