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

Abstract

We study the following quasilinear elliptic equation $$-{\Delta }_{p}u+(\beta \Phi (x)-a(x))u^{p-1}+b(x)g(u)=0\text{in }{\mathbb{R}}^N(P_{\beta })$$ where $p>1$, $\Phi \in L_{loc}^{\infty }({\mathbb{R}}^N)$, $a,b\in L^{\infty }({\mathbb{R}}^N)$, $\beta ,b,g\ge 0$. 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 $\beta \ge 0$ nonexistence phenomenon occurs, ii) or there exists a threshold value ${\beta }^{\ast }>0$ in the sense that for every $\beta \in [0,{\beta }^{\ast })$ existence and uniqueness phenomenon occurs and for every $\beta \ge {\beta }^{\ast }$ nonexistence phenomenon occurs. In the latter case, we study the limits, as $\beta \to 0$ and $\beta \to {\beta }^{\ast }$, of the sequence of positive solutions of $(P_{\beta })$. Our results are new even in the case $p=2$.

We have amended the title and made some minor corrections. The paper will appear in Journal of Functional Analysis (2015)

References in corpus (1)

• Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

Cited by in corpus (1)

• On the generalized principal eigenvalue of quasilinear operators