Multiplicity and Stability of Normalized Solutions in Nonlocal Double Phase Problems

Abstract

In this paper, we deal with the following nonlocal double phase problem with general growth conditions

$$\begin{aligned} (-\Delta )_{p,a(\varepsilon x)}^\alpha v+(-\Delta )^\beta _{q}v=\lambda |v|^{q-2}v+|v|^{r-2}v+b(\varepsilon x)h(v)&\ \ \textrm{in} \ \mathbb {R}^N, \end{aligned}$$

where \(\alpha ,\beta \in (0,1)\), \(1<q\le p<N/\alpha \), \(\lambda \in \mathbb {R}\), \((-\Delta )_{p,a}^\alpha +(-\Delta )^\beta _q\) is the fractional (p, q)-Laplacian with weight \({a:\mathbb {R}^N\times \mathbb {R}^N}\rightarrow \mathbb {R}^+\), \(q<r<p+\frac{\alpha pq}{N}\), \(\varepsilon >0\) and \(b\in L^\infty (\mathbb {R}^N), h\in C(\mathbb {R})\). Such equations can be used to model anisotropic materials in which the geometric shape of composite materials made of two different materials is determined by the function a. Since the nonlinear term h may satisfy Sobolev critical or supercritical growth, we first consider a truncated problem and study the existence of normalized solutions by combining the fractional Gagliardo-Nirenberg inequality with variational methods. We show that any normalized solution of the truncated problem is also a solution of our problem. This is achieved by estimating the bound of solutions using the De Giorgi iteration technique. Then we reveal that the multiplicity of normalized ground state solutions may be caused by the geometric shape of composite materials. More precisely, we prove that the number of normalized ground state solutions is at least the number of intersections between the minimum points of function a and the maximum points of function b as \(\varepsilon \) is small enough. Moreover, we discuss the asymptotic behavior of normalized solutions as \(\varepsilon \rightarrow 0^+\). Finally, the orbital stability of the ground state set of the problem is investigated. The main features of this paper are that the operator \((-\Delta )_{p,a}^\alpha +(-\Delta )^\beta _{q}\) may generate double phase energy, and that the nonlinear term h may have Sobolev critical or supercritical growth at infinity. Our results are new even in the (p, q)-Laplacian case, i.e. when \(\alpha =\beta =1\).