Extremal and sign-changing solutions of supercritical logistic-type equations in $\mathbb {R}^N$

Abstract

We prove the existence of constant-sign and sign-changing (weak) solutions of the following logistic-type equation in $\mathbb {R}^N$, $N\ge 3$,

$$\begin{aligned} u\in \mathcal {D}^{1,2}(\mathbb {R}^N):-\Delta u = \lambda a(x) u - b(x) g(u). \end{aligned}$$

The problem under consideration is treated in a rather weak setting regarding the regularity assumptions on the coefficients a, b and the growth condition on the nonlinear function g on the one hand, as well as the solution space $\mathcal {D}^{1,2}(\mathbb {R}^N)$ on the other hand. The nonlinearity g we are dealing with may have supercritical growth which does not allow for an immediate variational approach and which makes the difference to the existing literature. Instead we combine truncation and differential inequality techniques with variational methods and rather involved topological tools to achieve our goal. A sub-supersolution principle for the nonlinear equation in question has been developed as a tool to prove the existence of minimal positive and maximal negative solutions which are used to prove the existence of sign-changing solutions via truncation and variational methods.