##
**Some simple nonlinear PDE’s without solutions.**
*(English)*
Zbl 0907.35048

The authors establish with elementar tools very interesting nonexistence results for some classes of boundary value problems. Let \(\Omega\) be a smooth bounded domain in \({\mathbb R}^N\) (\(N\geq 3\)) which contains the origin. As a consequence of the inverse function theorem, the problem
\[
-\Delta u=\frac{u^2}{| x| ^a}+c\;\;\text{in} \Omega,\qquad u=0 \;\;\text{on} \partial\Omega
\]
has a unique small solution in \(W^{2,p}(\Omega)\) \((p>N/2)\), for any \(0<a<2\). The argument relies on the simple observation that the function \(| x| ^a\) belongs to \(L^p(\Omega)\). This fails if \(a=2\) and the authors study carefully this limiting case. More precisely, for any \(c<0\), the above considered problem has a weak solution \(u\leq 0\) which is the unique solution among nonpositive functions. If \(c=0\) the trivial solution is the only weak solution. In contrast with these results, for any \(c>0\) (no matter how small) the above problem has no solution, even in a very weak sense which is defined by the authors. These results are then extended to more general problems, such as
\[
-\Delta u=a(x)g(u)+b(x)\quad \text{in} \Omega,
\]
assuming only that \(g\) is nonnegative on the real axis, \(g\) is continuous and nondecreasing on \([0,\infty)\) and \(\int_0^\infty ds/g(s)<\infty\). The function \(a\in L^1_{\text{loc}}(\Omega)\) is assumed to be nonnegative and \(\int_\Omega (a(x)/| x| ^{N-2})dx=\infty\).

The authors also give some parabolic analogues of these results. They consider the problem \[ u_t-\Delta u=2(N-2)e^u\quad \text{in} B_1\times (0,T) \] with boundary condition \(u=0\) and initial condition \(u(x,0)=u_0\), where \(u_0\geq\ln (1/| x| ^2)\) and \(u_0\not\equiv \ln (1/| x| ^2)\). This problem has no solution \(u\geq \ln (1/| x| ^2)\) even for small time. Furthermore instantaneous and complete blow-up occurs.

The reviewer considers that this paper opens a large perspective in the study of elementary boundary value problems by using simple methods. We remark the accuracy of arguments and proofs, as well as the exciting comments made throughout the paper. We believe that this work will become soon a reference paper in its field.

The authors also give some parabolic analogues of these results. They consider the problem \[ u_t-\Delta u=2(N-2)e^u\quad \text{in} B_1\times (0,T) \] with boundary condition \(u=0\) and initial condition \(u(x,0)=u_0\), where \(u_0\geq\ln (1/| x| ^2)\) and \(u_0\not\equiv \ln (1/| x| ^2)\). This problem has no solution \(u\geq \ln (1/| x| ^2)\) even for small time. Furthermore instantaneous and complete blow-up occurs.

The reviewer considers that this paper opens a large perspective in the study of elementary boundary value problems by using simple methods. We remark the accuracy of arguments and proofs, as well as the exciting comments made throughout the paper. We believe that this work will become soon a reference paper in its field.

Reviewer: V.D.Rădulescu (Craiova)

### MSC:

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35K55 | Nonlinear parabolic equations |

35R05 | PDEs with low regular coefficients and/or low regular data |

47J25 | Iterative procedures involving nonlinear operators |