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Existence of solution for a class of quasilinear elliptic problem without \(\Delta_2\)-condition. (English) Zbl 1421.35126

Summary: The existence and multiplicity of solutions for a class of quasilinear elliptic problems are established for the type \[\begin{cases} -\Delta_{\Phi} u = f(u) \quad &\text{in } \Omega, \\ u = 0 \quad &\text{on } \partial \Omega, \end{cases}\] where \(\Omega \subset \mathbb{R}^N\), \(N \geq 2\), is a smooth bounded domain. The nonlinear term \(f : \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function which is superlinear at the origin and infinity. The function \(\Phi : \mathbb{R} \rightarrow \mathbb{R}\) is an \(N\)-function where the well-known \(\Delta_2\)-condition is not assumed. Then the Orlicz-Sobolev space \(W_0^{1, \Phi}(\Omega)\) may be non-reflexive. As a main model, we have the function \(\Phi(t) = (e^{t^2} - 1) / 2\), \(t \geq 0\). Here, we consider some situations where it is possible to work with global minimization, local minimization and mountain pass theorem. However, some estimates employed here are not standard for this type of problem taking into account the modular given by the \(N\)-function \(\Phi\).

MSC:

35J62 Quasilinear elliptic equations
35A15 Variational methods applied to PDEs
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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