Normalized Solutions for a Quasilinear Schrodinger Choquard Equation with Exponential Critical Growth in \(\mathbb{R^2}\)
SiRu Wen *
School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China.
*Author to whom correspondence should be addressed.
Abstract
In this paper, we are concerned with normalized solutions to the following quasilinear Schrodinger Choquard equation
\(\begin{gathered}
-\Delta u-u \Delta u^2+\lambda u=\left(I_\alpha * F(u)\right) f(u), \quad \text { in } \mathbb{R}^2, \\
\int_{\mathbb{R}^2}|u|^2 d x=a^2,
\end{gathered}\)
where a > 0, λ ∈ R, α ∈ (0, 2), Iα denotes the Riesz potential, ∗ denotes the convolution opertor, and the nonlinearity f has an exponential critical growth in the sense of Trudinger-Moser inequality. Using Perturbation method and variational methods with Pohozaev manifold, we can avoid the nondifferentiability of the quasilinear term uΔu2 and prove the existence of normalized solutions with some further assumption.
Keywords: Normalized solutions, quasilinear schr¨odinger equation, choquard equation, exponential critical growth