In this paper, we consider the nonlinear eigenvalue problem:
$$\begin{array}{}
\displaystyle
\begin{cases}
{\it\Delta}(|{\it\Delta} u|^{p(x)-2}{\it\Delta} u)= \lambda \frac{|u|^{q(x)-2}u}{{\delta(x)}^{2q(x)}} \;\; \mbox{in}\;\; {\it\Omega}, \\
u\in W_0^{2,p(x)}({\it\Omega}),
\end{cases}
\end{array}$$
where Ω is a regular bounded domain of ℝN, δ(x) = dist(x, ∂Ω) the distance function from the boundary ∂Ω, λ is a positive real number, and functions p(⋅), q(⋅) are supposed to be continuous on Ω satisfying
$$\begin{array}{}
\displaystyle
1 \lt \min_{\overline{{\it\Omega} }}\,q\leq \max_{\overline{{\it\Omega}}}\,q \lt \min_{\overline{{\it\Omega} }}\,p \leq \max_{\overline{{\it\Omega}}}\,p \lt \frac{N}{2} \mbox{ and } \max_{\overline{{\it\Omega}}}\,q \lt p_2^*:= \frac{Np(x)}{N-2p(x)}
\end{array}$$
for any x ∈ Ω. We prove the existence of at least one non-decreasing sequence of positive eigenvalues. Moreover, we prove that sup Λ = +∞, where Λ is the spectrum of the problem. Furthermore, we give a proof of positivity of inf Λ > 0 provided that Hardy-Rellich inequality holds.
In the present paper, we study the nonlinear partial differential equation with the weighted $p$-Laplacian operator\begin{gather*}- \operatorname{div}(w(x)|\nabla u|^{p-2}\nabla u) = \frac{ f(x)}{(1-u)^{2}},\end{gather*}on a ball ${B}_{r}\subset \mathbb{R}^{N}(N\geq 2)$. Under some appropriate conditionson the functions $f, w$ and the nonlinearity $\frac{1}{(1-u)^{2}}$, we prove the existence and the uniqueness of solutions of the above problem. Our analysis mainly combines the variational method and critical point theory. Such solution is obtained as a minimizer for the energy functional associated with our problem in the setting of the weighted Sobolev spaces.
Wernicke’s encephalopathy is a rare disease, resulting from severe thiamine deficiency
(vitamin B1). Alcoholic subjects are most often affected, but it may occur outside
of this addiction.
Magnetic resonance imaging is the gold standard imaging modality to confirm the diagnosis. This pathology manifests itself in T2-weighted hyperintensity in the periaqueducal, thalami and mammillary bodies. A suitable treatment makes it possible to avoid heavy sequelae.
We report the case of a 23-year-old woman, pregnant with 12 weeks of amenorrhea, with diplopia, headache and abundant vomiting, reffering to Wernicke’s encephalopathy.
Key words: Wernicke’s encephalopathy, vitamin B1, Magnetic resonance imaging, hyperemesis gravidarum.
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