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Positive solutions of superlinear indefinite prescribed mean curvature problems. (English) Zbl 1464.35128

Let \(\Omega\) be a bounded domain in \(\mathbb R^N, N\geq 2\) with boundary of class \(C^2\). The following problem is considered: \[ -\operatorname{div}\big(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\big)=\lambda f(x,u,\nabla u)+g(x,u,\nabla u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega,\tag{1} \] where \(\lambda\) is a parameter, and \(f,g:\bar{\Omega}\times \mathbb R\times \mathbb R^N \to \mathbb R\) are continuous. Set \[ \mathcal S=\{(u,\lambda)\in W^{2,q}(\Omega)\times ]0,\infty[: u \text{ is a stricly positive solution to (1) for some }\lambda >0\}, \] where \(q>N\) is given. \(\mathcal S\) is endowed with the topology \(C^{1,\gamma}(\bar{\Omega}),\) for some fixed \(\gamma\in ]0,1-N/q[\). The authors present several strong additional assumptions on the data, which imply the existence of positive, regular solutions when \(\lambda\) is sufficiently large, and the existence of positive bounded variation solutions for \(\lambda\) small. When \(\lambda\) is large, they show the existence of \(\lambda^* \geq 0\) and a connected component \(\mathcal C\) of \(\mathcal S\) such that \(\text{proj}_{\mathbb R}(\mathcal C)=]\lambda^*,\infty[\) and \[ \lim_{\lambda\to\infty} \max\{ \|u\|_{W^{2,q}}:(u,\lambda)\in\mathcal C\}=0 \] (Theorem 2.2). This result is valid, in particular, when \(f(x,u)=a(x)h(u)\) and \(g=0\) (Proposition 1.1).

MSC:

35J62 Quasilinear elliptic equations
35J93 Quasilinear elliptic equations with mean curvature operator
35J25 Boundary value problems for second-order elliptic equations
35B09 Positive solutions to PDEs
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