Existence of solutions for \(p(x)\)-solitons type equations in several space dimensions (Q2799640)

Let \(a,b>0\), let \(p:\mathbb{R}^n\rightarrow \mathbb{R}\) be a continuous bounded function such that \(\lim_{|x|\rightarrow \infty}p(x)=\inf_{\mathbb{R}^n} p>n\), and let \(V:\mathbb{R}^{n+1}\setminus \{\eta\}\rightarrow \mathbb{R}\), where \(\eta=(1,0,...,0)\), be a \(C^1\) function satisfying the following conditions: NEWLINE{\parindent=6mm NEWLINE\begin{itemize} \item[{\((V_1)\)}] \(V(\xi)> V(0)=0\), for all \(\xi \in \mathbb{R}^{n+1}\setminus\{0,\eta\}\),\ and \(\displaystyle{\liminf_{|\xi| \rightarrow \infty}V(\xi)>0}\).NEWLINE\item [{\((V_2)\)}] \(V\) is twice differentiable at \(0\), and \(V''(0)\) is nondegenerate. NEWLINE\item [{\((V_3)\)}] there exist \(c,\rho>0\) such that: NEWLINENEWLINE\end{itemize}} NEWLINE\(V(\eta+\xi)\geq c(|\xi|^{-q^+}+|\xi|^{-q^-})\), for all \(\xi \in \mathbb{R}^{n+1}\setminus\{0\}\), with \(|\xi|<\rho\), where NEWLINE\[NEWLINE{\frac{1}{q^+}=\frac{1}{n}-\frac{1}{\sup_{\mathbb{R}^n} p}}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE{\frac{1}{q^-}=\frac{1}{n}-\frac{1}{\inf_{\mathbb{R}^n} p}}.NEWLINE\]NEWLINE Finally, let \(E_a\) be the completion of \(C_0^\infty(\mathbb{R}^n,\mathbb{R}^{n+1})\) with respect to the norm NEWLINE\[NEWLINE\|u\|_a:=a\|\nabla u\|_{L^2}+\|\nabla u\|_{L^{p(\cdot)}}+\|u\|_{L^2},NEWLINE\]NEWLINE where \(\|\cdot\|_{L^{p(\cdot)}}\) is the standard Luxemburg-norm.NEWLINENEWLINEUnder the above assumptions, the authors prove the existence of a solution \(u\in E_a\) to the equation \(\displaystyle{-a\Delta u -\frac{b}{2}\Delta_{p(\cdot)}+V'(u)=0}\) in \(\mathbb{R}^n\).NEWLINENEWLINEThe solution \(u\) is found by minimizing the energy functional on the set \(\Gamma_a:=\{v\in E_a: v(x)\neq \eta, \;\text{for all} \;x\in \mathbb{R}^n\}\). The authors use the solution \(u\) to construct solitons for Euler-Lagrange equations associated to a class Lagrangian densities involving variable exponents.