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

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 $p=2$.