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On the Dirichlet problem for the generalized \(n\)-Laplacian: singular nonlinearity with the exponential and multiple exponential critical growth range. (English) Zbl 1273.35143

Let \(\Omega\subset\mathbb R^n\), \(n\geq 2\), be a bounded domain containing the origin, and let \(f:\Omega\times\mathbb R\rightarrow\mathbb R\) be a function uniformly continuous on \(\Omega\times [-t_0,t_0]\), for every \(t_0>0\), and such that \(f(x,0)=0\), \(tf(x,t)>0\) for every \(x\in\Omega\) and \(t\neq 0\). Finally, let \(\Phi:[0,\infty)\rightarrow [0,\infty)\) be a \(C^1\)-Young function (that is a convex increasing function such that \(\Phi(0)=0\) and \(\Phi(t)/t\rightarrow \infty\) as \(t\rightarrow \infty\)).
The author establishes the existence of at least a non-trivial weak solution for the following singular elliptic problem \[ -\mathrm{div}\left(\Phi'(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)=\frac{f(x,u)}{|x|^a} \text{ in } \Omega, u\in W_0L^{\Phi}(\Omega), \]
where \(a\in [0,n)\). The Young function \(\Phi\) is supposed to have the following behavior \[ \lim_{t\rightarrow\infty} \frac{\Phi(t)}{t^n\left(\prod_{j=1}^{l-1}\log_{[j]}^{n-1}(t)\right)\log_{[l]}^{\alpha}(t)}=1, \] where \(l\in \mathbb{N}\), \(\alpha<n-1\), and \(\log_{[j]}(t)=\log(\log_{[j-1]}(t))\), with \(\log_{[1]}(t)=\log(t)\). The nonlinearity \(f(x,t)\) is supposed, for \(t\) large, to have critical growth with respect to the Young function \(\Phi\).
Under some additional technical conditions on \(\Phi\) and \(f\), which include an Ambrosetti-Rabinowitz type growth condition on \(f\), the author proves the existence of a weak solution by using variational methods. In particular, it is proved that the energy functional associated to the problem has a Mountain Pass geometry. Then, choosing a Palais-Smale sequence \(\{u_k\}\), the author shows that \(u_k\) weakly converges in \(W_0L^{\Phi}(\Omega)\) to a function \(u\) and that \(\nabla u_k\) converges almost everywhere to \(\nabla u\). >From this, the author deduces that \(u\) is a weak nontrivial solution of the above problem.

MSC:

35J75 Singular elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D10 Inequalities involving derivatives and differential and integral operators
35D30 Weak solutions to PDEs
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