Maximal domains of the $$(\lambda ,\mu )$$-parameters to existence of entire positive solutions for singular quasilinear elliptic systems

Abstract

In this paper, we establish maximal domains on the real parameters $$\lambda ,\mu >0$$ to existence of $$C^{1}({\mathbb {R}}^{N})$$ -entire positive solutions for the quasilinear elliptic system $$\begin{aligned} \left\{ \begin{array}{l} -\Delta _p u = \eta a(x)f_1(u) + \lambda b(x)g_1(u)h_1(v)~in ~ {\mathbb {R}}^N,\\ -\Delta _p v = \theta c(x)f_2(v) + \mu d(x)g_2(v)h_2(u)~in~ {\mathbb {R}}^N,\\ u, v > 0~in~ {\mathbb {R}}^N,~~ u(x), v(x) {\mathop {\longrightarrow }\limits ^{|x|\rightarrow \infty }} 0, \end{array} \right. \end{aligned}$$ where $$\Delta _p$$ is the $$p-$$ Laplacian operator with $$1< p< N $$ ( $$3\le N$$ ); $$0<a, b, c, d\in C({\mathbb {R}}^{N})$$ ; either $$\eta =\theta =1$$ or $$\eta =\lambda ,\theta =\mu $$ and $$f_i, g_i, h_i~(i=1,2)$$ are positive continuous functions that satisfy some technical conditions, which allow $$f_{i}$$ to behave in a singular way at 0 and $$g_ih_i$$ as a $$(p-1)$$ -superlinear term at 0 and infinity. The main difficulties in approaching our problem come from its non-variational structure, building ordered sub-supersolutions and from the lack of a well-defined spectral theory. Using appropriate truncation, a generalization of the first eigenvalue in $${\mathbb {R}}^{N}$$ and a priori estimates, we are able to prove our principal results.