Multiplicity of solutions for a nonhomogeneous problem involving a potential in Orlicz-Sobolev spaces

This paper is devoted to the study of the nonhomogeneous problem
$$$ -div (a(|\nabla u|)\nabla u)+a(| u|)u=\lambda V(x)|u|^{m(x)-2}u-\mu g(x,u) \mbox{ in} \ \Omega, \ u=0 \mbox{ on} \ \partial\Omega ,$$$ where $$$\Omega$$$ is a bounded smooth domain in $$$\mathbb{R}^N,\lambda, \mu$$$ are positive real numbers, $$$V(x)$$$ is a potential, $$$ m: \overline{ \Omega} \to (1, \infty)$$$ is a continuous function, $$$a$$$ is mapping such that $$$ \varphi(|t|)t$$$ is increasing homeomorphism from ℝ to ℝ and $$$g: \overline{\Omega}\times ℝ \to ℝ$$$ is a continuous function. We establish there main results with various assumptions, the first one asserts that any $$$\lambda$$$0> is an eigenvalue of our problem. The second Theorem states the existence of a constant $$$\lambda^{*}$$$ such that every $$$\lambda \in (0,\lambda^{*})$$$ is an eigenvalue of the problem. While the third Theorem claims the existence of a constant $$$\lambda^{**}$$$ such that every $$$\lambda \in [\lambda^{**},\infty)$$$ is an eigenvalue of the problem. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.

This paper is devoted to the study of the nonhomogeneous problem
$$$ -div (a(|\nabla u|)\nabla u)+a(| u|)u=\lambda V(x)|u|^{m(x)-2}u-\mu g(x,u) \mbox{ in} \ \Omega, \ u=0 \mbox{ on} \ \partial\Omega ,$$$ where $$$\Omega$$$ is a bounded smooth domain in $$$\mathbb{R}^N,\lambda, \mu$$$ are positive real numbers, $$$V(x)$$$ is a potential, $$$ m: \overline{ \Omega} \to (1, \infty)$$$ is a continuous function, $$$a$$$ is mapping such that $$$ \varphi(|t|)t$$$ is increasing homeomorphism from ℝ to ℝ and $$$g: \overline{\Omega}\times ℝ \to ℝ$$$ is a continuous function. We establish there main results with various assumptions, the first one asserts that any $$$\lambda$$$0> is an eigenvalue of our problem. The second Theorem states the existence of a constant $$$\lambda^{*}$$$ such that every $$$\lambda \in (0,\lambda^{*})$$$ is an eigenvalue of the problem. While the third Theorem claims the existence of a constant $$$\lambda^{**}$$$ such that every $$$\lambda \in [\lambda^{**},\infty)$$$ is an eigenvalue of the problem. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.

Mountain pass Theorem Ekeland’s variational principle Orlicz-Sobolev space.

Ekeland’s variational principle Mountain pass Theorem Orlicz-Sobolev space.