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SOLVABILITY OF NONLINEAR ELLIPTIC TYPE EQUATION WITH TWO UNRELATED NON STANDARD GROWTHS
Sert, Ugur,Soltanov, Kamal Korean Mathematical Society 2018 대한수학회지 Vol.55 No.6
In this paper, we study the solvability of the nonlinear Dirichlet problem with sum of the operators of independent non standard growths $$-div\({\mid}{\nabla}u{\mid}^{p_1(x)-2}{\nabla}u\)-\sum\limits^n_{i=1}D_i\({\mid}u{\mid}^{p_0(x)-2}D_iu\)+c(x,u)=h(x),\;{\in}{\Omega}$$ 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.
Solvability of nonlinear elliptic type equation with two unrelated non standard growths
Ugur Sert,Kamal Soltanov 대한수학회 2018 대한수학회지 Vol.55 No.6
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.