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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.

##### MSC:
 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
##### Keywords:
Laplacian; eigenvalue; critical exponent