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Existence and multiplicity results for some nonlinear elliptic equations: A survey. (English) Zbl 1011.35049
This is a survey on existence, nonexistence and multiplicity results for the equation \(-\Delta u+au= \lambda h(x)|u|^{q-2}u+k(x) |u|^{\alpha-2} u\) either in a bounded smooth “nice” domain with homogeneous Dirichlet boundary conditions imposed on \(u\) or in \(\mathbb{R}^n\) with \(u\) vanishing at infinity. One section is also concerned with the \(p\)-Laplace operator as prinicipal part. Special emphasis is laid on their own contributions and in particular on the combination of convex and concave nonlinear terms, i.e. on \(1<q<2 <\alpha\), but also related relevant work of different authors is included. Most results deal with the subcritical and the critical case \(\alpha<2n/(n-2)\) and \(\alpha= 2n/(n- 2)\), respectively. The basic variational, functional analytic and analytic tools are briefly explained. The present paper gives a concise overview over a relevant part of the corresponding work of the past 20 years; about 50 original contributions are covered.

35J60 Nonlinear elliptic equations
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J50 Variational methods for elliptic systems
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35J65 Nonlinear boundary value problems for linear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs