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Article Dans Une Revue Calculus of Variations and Partial Differential Equations Année : 2015

Superlinear critical resonant problems with small forcing term

Résumé

We prove the existence of solutions of a class of quasilinear elliptic problems with Dirichlet boundary conditions of the following form $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} Lu= g(u) -f &{} \hbox { in } \quad \Omega ,\\ u\in X, \end{array} \right. \end{aligned}$$ where \(\Omega \subset \mathbb R^N\) is a bounded domain, \(N\ge 2\), the differential operator is \(Lu= -\hbox {div}( |\nabla u|^{p-2}\nabla u )-\lambda _1 |u|^{p-2}u\) with \(X=W^{1,p}_0(\Omega )\) or \(Lu= \Delta ^2u -\lambda _1 u\) with \(X=H^2_0(\Omega )\), the nonlinearity is given by \(g(u)=(u^+ )^q\) or \(g(u)=|u|^{q-1}u\) i.e. it is a superlinear, at most critical, term and \(f\) is a small reaction term. We give an abstract formulation for which solutions are found by minimization on an appropriate subset of the Nehari manifold associated to our problem. Our method can be also applied to other related problems involving indefinite weights.
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Dates et versions

hal-03137590 , version 1 (10-02-2021)

Identifiants

Citer

Mabel Cuesta, Colette De Coster. Superlinear critical resonant problems with small forcing term. Calculus of Variations and Partial Differential Equations, 2015, 54 (1), pp.349-363. ⟨10.1007/s00526-014-0788-8⟩. ⟨hal-03137590⟩
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