Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

TL;DR

This paper investigates the existence, uniqueness, and qualitative properties of positive solutions to a class of quasilinear elliptic equations, providing sharp criteria based on eigenvalues and analyzing solution behavior as parameters vary.

Contribution

It introduces new criteria for existence and nonexistence of positive solutions, including thresholds for parameter $eta$, and extends results even for the linear case $p=2$.

Findings

Established sharp eigenvalue-based criteria for solutions.

Identified threshold $eta^*$ for solution existence and nonexistence.

Analyzed limits of solutions as parameters approach critical values.

Abstract

We study the following quasilinear elliptic equation $$ -\Delta_p u + (\beta\Phi(x)-a(x)) u^{p-1} + b(x)g(u)=0 \quad \text{in } \mathbb{R}^N \quad \quad \quad (P_\beta) $$ where $p>1$, $\Phi \in L^\infty_{loc}(\mathbb{R}^N)$, $a,b \in L^\infty(\mathbb{R}^N)$, $\beta,b,g \geq 0$. We provide a sharp criterion in terms of generalized principal eigenvalue for existence/nonexistence of positive solutions of ($P_\beta$) in suitable classes of functions. We derive the uniqueness result of ($P_\beta$) in those classes. Under additional conditions on $\Phi$, we further show that : i) either for every $\beta \geq 0$ nonexistence phenomenon occurs, ii) or there exists a threshold value $\beta^*>0$ in the sense that for every $\beta \in [0,\beta^*)$ existence and uniqueness phenomenon occurs and for every $\beta \geq \beta^*$ nonexistence phenomenon occurs. In the latter case, we study the limits, as $\beta\to 0$ and $\beta\to\beta^*$, of the sequence of positive solutions of $(P_\beta)$. Our results are new even in the case $p=2$.