Differential and Integral Equations

Ground state solutions for a semilinear problem with critical exponent

Andrzej Szulkin, Tobias Weth, and Michel Willem

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This work is devoted to the existence and qualitative properties of ground state solutions of the Dirchlet problem for the semilinear equation $-\Delta u-\lambda u=\vert u\vert^{2^*-2}u$ 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.

Article information

Differential Integral Equations, Volume 22, Number 9/10 (2009), 913-926.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian
Secondary: 35B33: Critical exponents 35J20: Variational methods for second-order elliptic equations


Szulkin, Andrzej; Weth, Tobias; Willem, Michel. Ground state solutions for a semilinear problem with critical exponent. Differential Integral Equations 22 (2009), no. 9/10, 913--926. https://projecteuclid.org/euclid.die/1356019515

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