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.


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

Daomin Cao

Ezzat S. Noussair

Abstract

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