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

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

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