Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $ℝ^N$

Abstract

We are concerned with the multiplicity of positive and nodal solutions of \begin{cases} −\mathrm{\Delta }u + \mu u = Q(x)|u|^{p−2}u &\text{in }\: ℝ^{N} \\ u \in H^{1}(\mathbf{ℝ}^{N}) \end{cases} where 2 < p < \frac{2N}{N−2} , N ≥ 3 , μ > 0 , Q ∈ C(ℝ^N) and Q(x) ≥ 0 for x ∈ ℝ^N . We show how the “shape” of the graph of Q(x) affects the number of both positive and nodal solutions.