## 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