## Abstract and Applied Analysis

### On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers

#### Abstract

We prove that the semilinear elliptic equation $-\Delta u=f(u)$, in $\Omega$, $u=0$, on $\partial \Omega$ has a positive solution when the nonlinearity $f$ belongs to a class which satisfies $\mu {t}^{q}\leq f(t)\leq C{t}^{p}$ at infinity and behaves like ${t}^{q}$ near the origin, where $1< q < (N+2)/(N-2)$ if $N\geq 3$ and $1< q< +\infty$ if $N=1,2$. In our approach, we do not need the Ambrosetti-Rabinowitz condition, and the nonlinearity does not satisfy any hypotheses such those required by the blowup method. Furthermore, we do not impose any restriction on the growth of $p$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 578417, 6 pages.

Dates
First available in Project Euclid: 9 September 2008

https://projecteuclid.org/euclid.aaa/1220969173

Digital Object Identifier
doi:10.1155/2008/578417

Mathematical Reviews number (MathSciNet)
MR2411040

Zentralblatt MATH identifier
1187.35064

#### Citation

Alves, Claudianor O.; Souto, Marco A. S. On Existence of Solution for a Class of Semilinear Elliptic Equations with Nonlinearities That Lies between Different Powers. Abstr. Appl. Anal. 2008 (2008), Article ID 578417, 6 pages. doi:10.1155/2008/578417. https://projecteuclid.org/euclid.aaa/1220969173

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