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Simplicity of the principal eigenvalue for indefinite quasilinear problems. (English) Zbl 1158.35069

Let \(p \in (1,\infty )\), \(\Omega \subseteq\mathbb R^N\) be a domain (bounded if \(p \geq N\), or arbitrary one if \(p \in (1,N))\). In the space \(D_0^{1,p} (\Omega )\) (the closure of \(C_0^\infty (\Omega )\) with respect to the norm \(L_p (\Omega)\)) the functional
\[ J(u) = {\int_\Omega A(x,\nabla u) }\left/ {\int_\Omega w(x)| u | q)^{ - p / q}},\right. \]
\(w \in L_{\text{loc}}^1 (\Omega )\), \(w^ + \not\equiv 0\) is considered, where 1) \(A:\Omega \times\mathbb R^N \to [0,\infty ]\), \(0 < A(x,\eta ) \leq C| \eta | ^p\), \(x \in \Omega\), \(\eta \in\mathbb R^N\backslash \{0\}\); 2) for some \(\eta \mapsto A(x,\eta )\) is convex for \(x \in \Omega \); 3) there exists \(\Phi \in W^ + \equiv \{\varphi \in W\mid \int_\Omega w| \varphi | ^q > 0 \}\), \(W=\{\varphi\in D_0^{1,p}(\Omega)\mid w| \varphi| ^p\in L^1(\Omega) \}\). The authors obtain sufficient conditions for the uniqueness up to a constant factor to the minimization problem \(\inf \{ J(u)\mid u \in D_0^{1,p} (\Omega ), 0 < \int_\Omega w(x)| u | ^p < \infty\} \).

MSC:

35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B50 Maximum principles in context of PDEs
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