Existence and multiplicity of solutions for a nonvariational elliptic problem
The paper is concerned with the Dirichlet problem \[ \Delta u + b \cdot \nabla u + \lambda f(x,u) = 0, \quad u > 0 \quad \text{in } \Omega, \quad u |_{\partial \Omega} = 0. \tag{*} \] Beside some smoothness assumptions on \(b,f, \Omega\), a convexity assumption on \(\partial \Omega\) and structure conditions on \(f\) and \(b\) near \(\partial \Omega\), the authors impose on \(f\) the following sign and growth conditions \[ f(x,0) > 0, \quad f_ u (x,u) > 0, \quad {1 \over C} u^ s \leq f(x,u) \leq C(1 + u^ s) \] for \(x \in \overline \Omega\), \(u \geq 0\), where \(1 < s < n/(n - 2)\). The existence of a number \(\lambda^*>0\) is shown such that for \(0 < \lambda < \lambda^*\) the Dirichlet problem \((*)\) has at least two solutions, \(\lambda = \lambda^*\), \((*)\) has at least one solution, \(\lambda > \lambda^*\), \((*)\) has no solution. Moreover, if \(f\) is strictly convex in \(u\), the authors show uniqueness for the case \(\lambda = \lambda^*\). If \(b \not \equiv 0\), variational methods are not suitable to treat the Dirichlet problem \((*)\). The ``minimal'' solution \(u^{\min}_ \lambda\) is constructed by means of sub- and supersolution and a monotone iteration method. The second solution is obtained with help of \(C^{2, \alpha}\)-a-priori-estimates and Leray's degree of mapping. For this purpose the strict ordering of the minimal solutions \(0< \lambda_ 1 < \lambda_ 2 < \lambda^* \Rightarrow u^{\min}_{\lambda_ 1} < u^{\min}_{\lambda_ 2}\) is essential.
scientific article; zbMATH DE number 567008