Existence of Normalized Positive Solutions for a Class of Nonhomogeneous Elliptic Equations

Abstract

In this paper, we prove the existence of normalized positive solutions to the following Schrödinger equation with a perturbation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + \lambda u= |u|^{p-2}u+ h(x) \quad &{} \hbox {in}\;{\mathbb {R}}^N, \\ \int _{{\mathbb {R}}^N}u^2 {\text {d}}x =a, \ \ u \in H^1({\mathbb {R}}^N). \\ \end{array}\right. } \end{aligned}$$

We treat two cases. First, for the mass-subcritical case $2<p < 2+ \frac{4}{N}$, we show that there exists a global minimizer with negative energy for arbitrarily positive perturbation. Secondly, for the mass-supercritical case $2+ \frac{4}{N}<p <2^*$ where $2^*=\frac{2N}{N-2}$ if $N\geqslant 3$ and $2^*= + \infty$ if $N=1,2$, when h is small radial positive function, we prove the existence of a mountain pass solution with positive energy. It seems this is the first contribution to the normalized solution for such a perturbed equation.