We prove the Lazer-McKenna conjecture for the superlinear elliptic problem of the Ambrosetti-Prodi type with a non-homogeneous non-linearity by constructing solutions with sharp peaks. We also compute the critical groups provided the critical points are isolated.

We prove that $\Delta u+au=b(x)f(u)$ possesses a unique positive solution such that $\lim_{{\rm dist}(x,\partial\Omega)\to 0}u(x)=\infty$, where $\Omega$ is a smooth bounded domain in ${{\mathbb R} }^N$ and $a\in {{\mathbb R} }$. Here $b$ is a smooth function on $\overline \Omega$ which is positive in $\Omega$ and may vanish on $\partial{\Omega}$ (possibly at a very degenerate rate such as $\exp(-[{\rm dist}(x,\partial\Omega)]^q)$ with $q < 0$). We assume that $f$ is locally Lipschitz continuous on $[0,\infty)$ with $f(u)/u$ increasing for $u>0$ and $f(u)$ grows at $\infty$ faster than any power $u^p$ ($p>1$). As a distinct feature of this study appears the asymptotic behaviour of the boundary blow-up solution, which breaks up depending on how $b(x)$ vanishes on $\partial\Omega$ and how fast $f$ grows at $\infty$.

This text is devoted to maximal regularity results for second-order parabolic systems on Lipschitz domains of space dimension $n \ge 3$ with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial-value problems generates isomorphisms between two scales of Sobolev--Morrey spaces for solutions and right-hand sides introduced in the first part [12] of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are Hölder continuous in time and space up to the boundary for a certain range of Morrey exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower-order coefficients.

A part of the proof of Theorem 1 after formula (2.6) should be changed in accordance to page 288 in the book by S. Yakubov and Ya. Yakubov, "Differential-Operator Equations. Ordinary and Partial Differential Equations", Chapman and Hall/CRC, Boca Raton, 2000. Then, one has to change $\frac {m_k}{2}$ to $m_k$ in estimates of all theorems.