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In this paper, we are concerned with the nonlinear elliptic equations of the $p(x)$-Laplace type $$ \begin{cases} -\text{div}(a(x,\nabla u))+|u|^{p(x)-2}u=\lambda f(x,u) \quad &\textmd{in} \quad \Omega \\ a(x,\nabla u)\frac {\partial u}{\partial n} = \lambda\theta g(x,u) &\textmd{on} \quad \partial\Omega, \end{cases} $$ which is subject to nonlinear Neumann boundary condition. Here the function $a(x,v)$ is of type $|v|^{p(x)-2}v$ with continuous function $p: \overline{\Omega} \to (1,\infty)$ and the functions $f, g$ satisfy a Carath\'eodory condition. The main purpose of this paper is to establish the existence of at least three solutions for the above problem by applying three critical points theory due to Ricceri. Furthermore, we localize three critical points interval for the given problem as applications of the theorem introduced by Arcoya and Carmona.
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참고문헌 (31)
참고문헌 신청
A. Ambrosetti / 1973 / Dual variational methods in critical point theory and applications / J. Funct. Anal 14(4):349~381
S. Aouaoui / 2012 / On some degenerate quasilinear equations involving variable exponents / Nonlinear Anal 75(4):1843~1858
D. Arcoya / 2007 / A nondifferentiable extension of a theorem of Pucci and Serrin and applications / J. Differential Equations 235(2):683~700
J.-H. Bae / Multiple solutions for the equations involving nonhomogeneous operators of p(x)-Laplace type in RN / submitted
M. S. Berger / 1977 / Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis / Academic Press
S. Aouaoui / 2012 / On some degenerate quasilinear equations involving variable exponents / Nonlinear Anal 75(4):1843~1858
D. Arcoya / 2007 / A nondifferentiable extension of a theorem of Pucci and Serrin and applications / J. Differential Equations 235(2):683~700
J.-H. Bae / Multiple solutions for the equations involving nonhomogeneous operators of p(x)-Laplace type in RN / submitted
M. S. Berger / 1977 / Nonlinearity and Functional Analysis: Lectures on Nonlinear Problems in Mathematical Analysis / Academic Press
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