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Ground state solutions for a semilinear problem with critical exponent. (English) Zbl 1240.35205
Summary: This work is devoted to the existence and qualitative properties of ground state solutions of the Dirchlet problem for the semilinear equation \(-\Delta u-\lambda u=| u| ^{2^{*}-2}u\) in a bounded domain. Here, \(2^{*}\) is the critical Sobolev exponent, and the term ground state refers to minimizers of the corresponding energy within the set of nontrivial solutions. We focus on the indefinite case where \(\lambda \) is larger than the first Dirichlet eigenvalue of the Laplacian, and we present a particularly simple approach to the study of ground states.

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35B33 Critical exponents in context of PDEs
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations