Some results of nontrivial solutions for a nonlinear PDE in Sobolev space

S. Benmehidi, Badji Mokhtar University, Annaba, Algeria
B. Khodja, Badji Mokhtar University, Annaba, Algeria

E. J. Qualitative Theory of Diff. Equ., No. 44. (2009), pp. 1-14.

Abstract: In this study, we investigate the question of nonexistence of nontrivial solutions of the Robin problem
\begin{equation}
\left\vert\begin{array}{l}
-\dfrac{\partial ^{2}u}{\partial x^{2}}-\sum\limits_{s=1}^{n}\dfrac{\partial}{\partial y_{s}}a_{s}(y,\frac{\partial u}{\partial y_{s}})+f(y,u)=0\text{in }\Omega =\mathbb{R}\times D, \\ \\
u+\varepsilon \dfrac{\partial u}{\partial n}=0\text{ on }\mathbb{R}\times \partial D.
\end{array}\right. \tag*{$\left( P\right) $}
\end{equation}
where $a_{s}:D\times \mathbb{R}\rightarrow \mathbb{R}$ are $H^{1}$-functions with constant sign such that
\begin{equation}\begin{array}{c}
2\int\limits_{0}^{\xi _{s}}a_{s}(y,t_{s})dt_{s}-\xi _{s}a_{s}(y,\xi_{s})\leq 0,s=1,...,n
\end{array}\tag*{$\left( H_{1}\right) $}\end{equation}
and $f:D\times \mathbb{R}\rightarrow \mathbb{R}$ is a real continuous locally Liptschitz function such that
\begin{equation} 2F(y,u)-uf(y,u)\leq 0, \tag*{$\left( H_{2}\right) $} \end{equation}.
We show that the function \begin{equation*} E(x)=\int\limits_{D}\left\vert u(x,y)\right\vert ^{2}dy \end{equation*} is convex on $\mathbb{R}$ . Our proof is based on energy (integral) identities.

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