Existence of blow-up solutions for a class of elliptic systems. (Q364243)

The authors study existence of solutions to the problem NEWLINE\[NEWLINE \begin{aligned} \Delta u = F_u(x,u,v) \;\;\text{in}\;\;\Omega , \\ \Delta v = F_v(x,u,v) \;\;\text{in}\;\;\Omega , \tag{P}\\ u,v> 0 \;\;\text{in}\;\;\Omega . \end{aligned} NEWLINE\]NEWLINE Here \(\Omega \) is a smoothly bounded domain in \(\mathbb R^N \) or \(\Omega = \mathbb R^N\) and \(F\: \Omega \times \mathbb R^{+} \times \mathbb R^{+} \to \mathbb R^{+} \) is continuously differentiable. In case of a bounded domain, three types of boundary value problems are considered:NEWLINENEWLINE(F) the finite case: for \(\alpha , \beta \in (0, \infty)\) it holds \(u = \alpha\), \(v = \beta \) on \(\partial \Omega \),NEWLINENEWLINE(I) the infinite case: \(u(x) \to +\infty\), \(v(x) \to +\infty \) as dist\((x,\partial \Omega) \to 0\),NEWLINENEWLINEthe semifinite cases:NEWLINENEWLINE(SF1) \(u(x) \to +\infty \) as \(\mathrm{dist}(x,\partial \Omega) \to 0\), \(v(x) = \beta \) on \(\partial \Omega \),NEWLINENEWLINE(SF2) \(u(x) = \alpha \) on \(\partial \Omega \), \(v(x) \to +\infty \) as \(\mathrm{dist}(x,\partial \Omega) \to 0\).NEWLINENEWLINETo formulate the main theorem for the case of a bounded domain, the authors define a class \(\mathcal {F}\) as follows:NEWLINENEWLINEA function \(h\: [0, \infty) \to [0, \infty)\) belongs to \(\mathcal {F}\) if \(h \in C^1([0, \infty))\), \(h(0)= 0\), \(h'(t)\geq 0\) in \([0, \infty)\), \(h(t)> 0\) on \((0, \infty) \) and NEWLINE\[NEWLINE \int_1^{\infty }\frac {1}{H(t)^{1/2}}dt < +\infty , NEWLINE\]NEWLINE where \(H(t)= \int_0^th(s) ds\). The authors prove the following:NEWLINENEWLINETheorem 1.1: Assume that \(\Omega \) is bounded and that there exist functions \(f_1, f_2, g \in \mathcal {F}\) satisfying NEWLINE\[NEWLINE \begin{aligned} & F_t(x,t,s) \geq f_1(t)\;\;\forall x \in \overline {\Omega },\quad t,s > 0 \\ & F_s(x,t,s) \geq f_2(s)\;\;\forall x \in \overline {\Omega },\quad t,s > 0 \tag{C1}\\ & g(t) \geq \max \{F_t(x,t,t),F_s(x,t,t) \}\;\;\forall x \in \overline {\Omega },\quad t > 0. \end{aligned} NEWLINE\]NEWLINE Then system (P) with one of the boundary conditions (F), (I), (SF1) or (SF2) admits a positive solution.NEWLINENEWLINEThe second part of the paper is devoted to the system NEWLINE\[NEWLINE \begin{aligned} \Delta u = p(x) F_u(x,u,v) & \text{ in } \mathbb R^N, \\ \Delta v = q(x) F_v(x,u,v) &\text{ in } \mathbb R^N, \tag{LS}\\ u,v> 0 & \text{ in } \mathbb R^N. \end{aligned} NEWLINE\]NEWLINE The functions \(p\), \(q\) are supposed to satisfy the conditions NEWLINE\[NEWLINE \begin{aligned} & p,q > 0 \text{ on } \mathbb R^N, \quad p,q \in C^{0,\nu }_{\mathrm{loc}}(\mathbb R^N),\\ & \int_0^{\infty }r\Phi_p(r) dr < \infty , \text{ where }\Phi_p(r)= \max \{p(x); | x| = r\}, \tag{C2}\\ & \int_0^{\infty }r\Phi_q(r) dr < \infty , \text{ where } \Phi_q(r)= \max \{q(x); | x| = r\}. \end{aligned} NEWLINE\]NEWLINE The obtained result in this case reads:NEWLINENEWLINETheorem 1.2: Assume that \(\Omega = \mathbb R^N \) and conditions (C1), (C2) are satisfied. Then the system (LS) admits an entire large solution, i.e., a solution \((u,v)\) satisfying \(u(x) \to \infty\), \(v(x) \to \infty \) as \(| x| \to \infty \).NEWLINENEWLINEIn the proofs both variational methods and method of sub and supersolutions are combined.