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This paper discusses the existence and multiplicity of solutions for a class of $p (x)$ -Kirchhoff type problems with Dirichlet boundary data of the following form $\}\begin{matrix} - (a + b \int_{Ω} \frac{1}{p (x)} {| \nabla u |}^{p (x)} d x) {div (| \nabla u |}^{p (x) - 2} \nabla u) = f (x, u), & i n Ω \\ u = 0 & o n \partial Ω, \end{matrix}$ where $Ω$ is a smooth open subset of $ℝ^{N}$ and $p \in C (\bar{Ω})$ with $N < p^{-} = {inf}_{x \in Ω} p (x) \leq p^{+} = {sup}_{x \in Ω} p (x) < + \infty$ , $a$ , $b$ are positive constants and $f : \bar{Ω} \times ℝ \to ℝ$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

Existence and multiplicity of solutions for divergence type elliptic equations.

Zhao, Lin, Zhao, Peihao, Xie, Xiaoxia (2011)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence and multiplicity of solutions for the noncoercive Neumann $p$ -Laplacian.

Papageorgiou, Nikolaos S., Rocha, Eugenio M. (2010)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence and multiplicity of solutions to a $p$ -Laplacian equation with nonlinear boundary condition.

Pflüger, Klaus (1998)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence and multiplicity of solutions to elliptic problems with discontinuities and free boundary conditions.

Bensid, Sabri, Bouguima, Sidi Mohammed (2010)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Existence and multiplicity of weak solutions for a class of degenerate nonlinear elliptic equations.

Mihăilescu, Mihai (2006)

Boundary Value Problems [electronic only]

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), $x \in ℝ^{N}$ , $u \in H ¹ (ℝ^{N})$ , where λ is a positive parameter, a and b are positive functions, while $f : ℝ \to ℝ$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

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