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In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths \begin{equation*} -div\left(\left\vert\nabla u\right\vert^{p_{1}\left( x\right)-2}\nabla u\right)-\sum\limits_{i=1}^{n}D_{i}\left( \left\vert u\right\vert ^{p_{0}\left( x\right)-2}D_{i}u\right)+c\left( x,u\right) \!=\!h\left( x\right) ,~x\!\in\! \Omega \end{equation*} in a bounded domain $\Omega \subset \mathbb{R}^{n}$. Here, one of the operators in the sum is monotone and the other is weakly compact. We obtain sufficient conditions and show the existence of weak solutions of the considered problem by using monotonicity and compactness methods together.
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참고문헌 (14)
참고문헌 신청
E. Acerbi / 2002 / Regularity results for stationary electro-rheological uids / Arch. Ration. Mech. Anal 164(3):213~259
S. N. Antontsev / 2006 / On stationary thermo-rheological viscous ows / Ann. Univ. Ferrara Sez. VII Sci. Mat 52(1):19~36
S. N. Antontsev / 2005 / A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions / Nonlinear Anal 60(3):515~545
S. N. Antontsev / 2005 / On the localization of solutions of elliptic equations with nonhomogeneous anisotropic degeneration / Siberian Math. J 46(5):765~782
S. N. Antontsev / 2006 / Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions / Nonlinear Anal 65(4):728~761
S. N. Antontsev / 2006 / On stationary thermo-rheological viscous ows / Ann. Univ. Ferrara Sez. VII Sci. Mat 52(1):19~36
S. N. Antontsev / 2005 / A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions / Nonlinear Anal 60(3):515~545
S. N. Antontsev / 2005 / On the localization of solutions of elliptic equations with nonhomogeneous anisotropic degeneration / Siberian Math. J 46(5):765~782
S. N. Antontsev / 2006 / Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions / Nonlinear Anal 65(4):728~761
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