Existence of solutions of the Dirichlet problem for \(\Delta u +\lambda u +h(u,\nabla u)\cdot\nabla u = f(x)\) (Q1817991)

We are interested in sufficient conditions for the existence of solutions of the boundary value problem \[ \Delta u+\lambda u+ h(u,\nabla u)\cdot\nabla u= f(x)\quad\text{in }\Omega;\quad u= 0\quad\text{on }\partial\Omega,\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with sufficiently smooth boundary \(\partial\Omega\), \(h\) is a continuous function and \(\lambda\) is a real number, not necessarily an eigenvalue of the linear problem. In what follows, let \(S_\Delta\) denote the spectrum of the Dirichlet problem for the Laplacian and fix an index \(p>n\). This last condition ensures the compact Sobolev embedding \(W^{2,p}(\Omega)\hookrightarrow C^1(\Omega)\). We note that due to regularity, \(S_\Delta\) does not depend on the choice of \(p\). Theorem. Suppose \(h:\mathbb{R}\times \mathbb{R}^n\to \mathbb{R}^n\) is continuous and \[ \lambda-\textstyle{{1\over 4}}|h(0,0)|^2\not\in S_\Delta. \] Then for sufficiently small \(\|f\|_{L^p}\), problem (1) has a solution in \(W^{2,p}(\Omega)\). If moreover \(h\) is Lipschitz in a neighborhood of the origin, then for sufficiently small \(R\), problem (1) has a unique solution in the ball \[ B_R= \{u\in C^1(\Omega):\|u\|_{C^1}\leq R\}. \]