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Zbl 1153.35036
Wu, Tsung-Fang
On semilinear elliptic equations involving concave--convex nonlinearities and sign-changing weight function. (English)
[J] J. Math. Anal. Appl. 318, No. 1, 253-270 (2006). ISSN 0022-247X

The author studies the multiplicity results of positive solutions of the following elliptic equation: $$-\Delta u=u^p+\lambda f(x)u^q\text{ in } \Omega \quad 0\le u\in H_0^1(\Omega),\tag 1$$ where $\Omega$ is a bounded domain in $\bbfR^N$, $0<q<1<p<2^*$; $2^*=\frac{N+2}{N-2}$ if $N\ge 3$, $2^*=\infty$ if $N=2$, and $f$ is a given function, $\lambda>0$. The author under suitable assumptions on the data (1) proves that (1) possesses at least two positive solutions for $\lambda$ is sufficiently small.
[Messoud A. Efendiev (Berlin)]

MSC 2000:: *35J60 Nonlinear elliptic equations
35J20 Second order elliptic equations, variational methods
35J25 Second order elliptic equations, boundary value problems
47J30 Variational methods

Keywords: semilinear elliptic equations; Nehari manifold; concave-convex nonlinearities

Cited in: Zbl 1179.35150

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Zentralblatt MATH Berlin [Germany]