Multi-bump solutions for a class of quasilinear equations on \(\mathbb R\)
The authors consider the following quasilinear elliptic equation \[ -v''+(\lambda V(x)+1)v-\frac{1}{2}(|v|^2)''v=|v|^{q-1}v,\;x\in \mathbb{R}, \] where \(q>3\), \(\lambda\) is a positive parameter and \(V\) verifies the following condition. (V): \(V\geq 0\) is a continuous function such that \(\Omega :=\operatorname{int} V^{-1}(\{0\})\) is a non-empty bounded set and has \(m\) connected components \(\Omega_j, j\in\{1,\dotsc,m\}\), satisfying \(d(\Omega_j,\Omega_i)>0\) for \(i\neq j\). The main result which is proved by variational methods reads as follows. Suppose \(q>3\) and (V) holds. Then, for any non-empty subset \(\Gamma\subset\{1,\dotsc,m\}\), there exists \(\lambda^*>0\) such that, for \(\lambda\geq\lambda^*\) the above equation has a family \(\{u_\lambda\}\) of solutions such that, for any sequence \(\lambda_n\rightarrow +\infty\), there is a subsequence \(\lambda_{n_i}\) such that \(u_{\lambda_{n_i}}\) converges strongly in \(H^1(\mathbb{R})\) to a function \(u\) which satisfies \(u(x)=0\) for \(x\not\in \Omega_\Gamma :=\bigcup_{j\in\Gamma}\Omega_j\) and the restriction \(u|_{\Omega_j}\), for \(j\in \Gamma\), is a least energy solution of \[ -u''+u-\frac{1}{2}(|u|^2)''u=u^q,\;x\in \Omega_j;\;u=0\text{ on }\partial\Omega_j. \]
scientific article; zbMATH DE number 6159036