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Zbl 1156.35046
Wu, Tsung-Fang
On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function.
(English)
[J] Commun. Pure Appl. Anal. 7, No. 2, 383-405 (2008). ISSN 1534-0392; ISSN 1553-5258/e

The author considers the problem $$-\Delta u = \lambda f(x)\vert u\vert ^{q-2}u + \vert u\vert ^{2^*-2}u~~\text {in}~~\Omega,~~~~u \in H_0^1(\Omega),$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N (N \geq 3)$, $1<q<2<2^*=2N/(N-2)$, $\lambda >0$ and $f:\overline{\Omega} \to \mathbb{R}$ is a continuous function with $f^+(x)=\max\{f(x),0\} \not \equiv 0$ in $\overline{\Omega}$. By using variational methods he proves that, for $\lambda>0$ small, the problem possesses at least two positive solutions. He also studies the asymptotic behavior of the obtained solutions as $\lambda \to 0$.
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35J20 Second order elliptic equations, variational methods
35B40 Asymptotic behavior of solutions of PDE

Keywords: concave-convex nonlinearities; critical Sobolev exponent; Nehari manifold

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