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On the Dirichlet problem for the \(n,\alpha \)-Laplacian with the nonlinearity in the critical growth range. (English) Zbl 1225.35062

Summary: Let \(\Omega\subset\mathbb R^n\), \(n\geq 2\), be a bounded domain. Applying the mountain pass theorem we prove the existence of a non-trivial weak solution to the Dirichlet problem
\[ -\text{div}\bigg(\Phi'(|\nabla u|) \frac{\nabla u}{|\nabla u|}\bigg)= f(x,u) \quad\text{in }\Omega, \]
where \(u\) is in the Orlicz-Sobolev space \(W_0^1L^\Phi(\Omega)\) with a Young function of the type \(\Phi (t)\approx t^n\log^\alpha(t)\), \(\alpha<n-1\), and \(|f(x,t)|\approx \exp(\beta|t|^{\frac{n}{n-1-\alpha}})\), \(\beta>0\).

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J40 Boundary value problems for higher-order elliptic equations
35D30 Weak solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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.)
35A15 Variational methods applied to PDEs
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References:

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