Existence and concentration of positive solutions for \(p\)-fractional Schrödinger equations. (English) Zbl 1431.35222
Summary: We deal with the existence and concentration of positive solutions for the following \(p\)-fractional Schrödinger equation \[\varepsilon^{sp}(-\Delta )_p^su+V(x)|u|^{p-2}u=f(u)+\gamma |u|^{p^*_s-2}u \] in \(\mathbb{R}^N\), where \(\varepsilon >0\) is a small parameter, \(s\in (0, 1), p\in (1, \infty )\), \(N>sp, \gamma \in \{0, 1\}\), \(p^*_s=\frac{Np}{N-sp}\) is the fractional critical Sobolev exponent, \((-\Delta )_p^s\) is the fractional \(p\)-Laplacian operator, \(V\) is a continuous positive potential having a local minimum and \(f\) is a superlinear continuous function with subcritical growth. The main results are obtained by using penalization techniques and suitable variational arguments.
MSC:
| 35R11 | Fractional partial differential equations |
| 35J60 | Nonlinear elliptic equations |
| 35A15 | Variational methods applied to PDEs |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
Keywords:
fractional Schrödinger equation; fractional \(p\)-Laplacian operator; variational methods; critical exponentReferences:
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