Opuscula Math. 32, no. 4 (2012), 731-750
http://dx.doi.org/10.7494/OpMath.2012.32.4.731

Opuscula Mathematica

# On a class of nonhomogenous quasilinear problems in Orlicz-Sobolev spaces

Asma Karoui Souayah

Abstract. We study the nonlinear boundary value problem $$-div ((a_1(|\nabla u(x)|)+a_2(|\nabla u(x)|))\nabla u(x))=\lambda |u|^{q(x)-2}u-\mu |u|^{\alpha(x)-2}u$$ in $$\Omega$$, $$u = 0$$ on $$\partial \Omega$$ , where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$ with smooth boundary, $$\lambda$$, $$\mu$$ are positive real numbers, $$q$$ and $$\alpha$$ are continuous functions and $$a_1$$, $$a_2$$ are two mappings such that $$a_1(|t|)t$$, $$a_2(|t|)t$$ are increasing homeomorphisms from $$\mathbb{R}$$ to $$\mathbb{R}$$. The problem is analyzed in the context of Orlicz-Soboev spaces. First we show the existence of infinitely many weak solutions for any $$\lambda,\mu \gt 0$$. Second we prove that for any $$\mu \gt 0$$, there exists $$\lambda_*$$ sufficiently small, and $$\lambda ^*$$ large enough such that for any $$\lambda \in (0,\lambda_*)\cup(\lambda^*,\infty)$$, the above nonhomogeneous quasilinear problem has a non-trivial weak solution.

Keywords: variable exponent Lebesgue space, Orlicz-Sobolev space, critical point, weak solution.

Mathematics Subject Classification: 35D05, 35J60, 35J70, 58E05, 68T40, 76A02.

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• Asma Karoui Souayah
• Institut Préparatoire aux Etudes d'ingénieurs de Gafsa, Campus Universitaire Sidi Ahmed Zarrouk - 2112 Gafsa, Tunisia