The nonlinear \((p,q)\)-Schrödinger equation with a general nonlinearity: existence and concentration. (English. French summary) Zbl 1522.35028
Summary: We investigate the following class of \((p,q)\)-Laplacian problems:
\[
\begin{cases}
-\varepsilon^p\Delta_pv-\varepsilon^q\Delta_qv+V(x)(|v|^{p-2}v+|v|^{q-2}v)=f(v)\text{ in }\mathbb{R}^N, \\
v\in W^{1,p}(\mathbb{R}^N)\cap W^{1,q}(\mathbb{R}^N),\, v>0\text{ in }\mathbb{R}^N, \end{cases}
\]
where \(\varepsilon>0\) is a small parameter, \(N\geq 3\), \(1<p<q<N\), \(\Delta_sv:=\operatorname{div}(|\nabla v|^{s-2}\nabla v)\), with \(s\in\{p,q\}\), is the \(s\)-Laplacian operator, \(V:\mathbb{R}^N\to\mathbb{R}\) is a continuous potential such that \(\inf_{\mathbb{R}^N}V>0\) and \(V_0:=\inf_{\Lambda}V<\min_{\partial\Lambda}V\) for some bounded open set \(\Lambda\subset\mathbb{R}^N\), and \(f:\mathbb{R}\to\mathbb{R}\) is a subcritical Berestycki-Lions type nonlinearity. Using variational arguments, we show the existence of a family of solutions \((v_{\varepsilon})\) which concentrates around \(\mathcal{M}:=\{x\in\Lambda:V(x)=V_0\}\) as \(\varepsilon\to 0\).
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
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