Positive solutions to a class of quasilinear elliptic equations on \(\mathbb{R}\)
The authors look for positive solutions \(u\in H^{1}({\mathbb R})\) to the equation \[ -u''+(1+\varepsilon a(x))u-k(1+\varepsilon b(x))(u^2)''u= (1+\varepsilon c(x))u^{p}, \] where \(\varepsilon\), \(k\in {\mathbb R}\), \(p>1\) and the real valued functions \(a\), \(b\), \(c\) are in the class \(S=\{h=h_{1}+h_{2}/ h_{1}\in L^{r}({\mathbb R})\cap L^{\infty}({\mathbb R})\) for some \(r>1\), \(h_{2}\in L^{\infty}({\mathbb R})\), \(\lim_{|x|\to\infty}h_{2}(x)=0\}\). The authors show that: Theorem. There exists \(k_{0}>0\) such that for \(k>-k_{0}\) and \(a,b,c\in S\) the problem admits a solution provided that \(|\varepsilon|\) is sufficiently small. In the proof the authors make use of a finite-dimensional reduction based on an abstract perturbation procedure already introduced in [\textit{A. Ambrosetti} and \textit{M. Badiale}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 15, 233-252 (1998; Zbl 1004.37043)]. The result extends and strengthens in particular, even in the perturbative setting, the results already obtained in [\textit{M. Poppenberg, K. Schmitt} and \textit{Z.-Q. Wang}, Calc. Var. Partial Differ. Equ. 14, 329-344 (2002; Zbl 1052.35060) and \textit{J. Liu, Y. Wang} and \textit{Z.-Q. Wang}, J. Differ. Equ. 187, 473-493 (2003; Zbl 1229.35268)] regarding the quasilinear Schrödinger equation \(-u''+V(x)u-k(u^2)''u= \lambda u^{p}\).
scientific article; zbMATH DE number 1890727