Nonexistence of positive supersolutions to some nonlinear elliptic problems (Q390813)

The authors are concerned with the existence of positive supersolutions for the equation NEWLINE\[NEWLINE -\Delta u+|\nabla u|^q=\lambda f(u)\quad\text{ in }\mathbb R^N\setminus B_{R_0},\tag{1}NEWLINE\]NEWLINE where \(N\geq 3\), \(q>1\), \(R_0>0\) and \(f:(0,\infty)\to (0,\infty)\) is a continuous function. The main results of the paper are as follows.NEWLINENEWLINE Theorem 1. Assume \(f\) satisfies NEWLINE\[NEWLINE\gamma_1:=\liminf_{t\to 0}\frac{f(t)}{t^p}>0,\tag{2}NEWLINE\]NEWLINE for some \(p>0\). If NEWLINE\[NEWLINE 1<q<\frac{N}{N-1}\quad\text{ and }\quad p<\frac{q}{2-q}, NEWLINE\]NEWLINE or NEWLINE\[NEWLINE q\geq\frac{N}{N-1}\quad\text{ and }\quad p<\frac{N}{N-2}, NEWLINE\]NEWLINE then, there are no classical supersolutions of (1) which do not blow up at infinity.NEWLINENEWLINEThe second result of the paper provides some estimates on the critial value \(\lambda^*\) defined by NEWLINE\[NEWLINE \lambda^*:=\inf\{\lambda>0: \text{eq. (1) has no positive solutions which do not blow up at infinity}\}. NEWLINE\]NEWLINE Theorem 2. Assume \(f\) satisfies (2) and \(1<q<\frac{N}{N-1}\), \(p=\frac{q}{2-q}\). Then NEWLINE\[NEWLINE \lambda^*\leq \frac{2-q}{2\gamma_1}\left[\frac{q(2-q)}{2(N-q(N-1)) }\right]^{q/(2-q)}. NEWLINE\]NEWLINE If in addition NEWLINE\[NEWLINE \gamma_2:= \limsup_{t\to 0}\frac{f(t)}{t^p}<\infty, NEWLINE\]NEWLINE then NEWLINE\[NEWLINE \lambda^*\geq \frac{2-q}{2\gamma_2}\left[\frac{q(2-q)}{2(N-q(N-1)) } \right]^{q/(2-q)}, NEWLINE\]NEWLINE and for \(0<\lambda<\lambda^*\) there exist positive supersolutions of (1) which do not blow up at infinity.NEWLINENEWLINETheorem 3. Assume \(f\) is nondecreasing and satisfies (2). Then, for \(q>\frac{N}{N-1}\) and \(p=\frac{N}{N-2}\), the problem (1) has no positive supersolutions which do not blow up at infinity.NEWLINENEWLINETheorem 4. Assume \(f\) satisfies NEWLINE\[NEWLINE\liminf_{t\to \infty}\frac{f(t)}{t^p}>0, NEWLINE\]NEWLINE for some \(p>q\). Then, there are no positive classical solutions of (1) which blow up at infinity.NEWLINENEWLINEThe last part of the paper presents a study on the related problem NEWLINE\[NEWLINE -\Delta u-|\nabla u|^q=f(u)\quad\text{ in }\mathbb R^N\setminus B_{R_0},\tag{3}NEWLINE\]NEWLINE where \(N\geq 3\), \(q>1\), \(R_0>0\) and \(f:(0,\infty)\to (0,\infty)\) is a continuous function. The main result on equation (3) is the following.NEWLINENEWLINETheorem 5. Assume \(f\) satisfies (2) and that \(p>\frac{N}{N-2}\), \(1<q\leq\frac{N}{N-1}\). Then (3) has no positive classical supersolutions.