Radial and nonradial solutions for nonlinear scalar field equations and a new proof for certain Gagliardo-Nirenberg type inequality on the plane

A nonlinear scalar field equation is studied, which takes the form \begin{equation*} - \triangle{u}=g(u) \ \textrm{in} \ R^{N}, \quad \quad u \in E, \end{equation*} where $E$ is a suitable Banach space.. The nonlinear term $g(s)$ has subcritical growth at infinity. Near $s=0$, the nonlinear term $g(s)$ grows like $s^q$ for any $q>0$ if $N=2$ and for any $q \in (0,\ (N+2)/(N-2))$ if $N\ge 3$. Through the technique of adding one dimension of space, by mountain pass theorem and symmetric mountain theorem, we prove the multiplicity of radial solutions if $N\ge 2$ and the multiplicity of nonradial solutions if $N\ge 4$. Unlike [1] and [2], the proof presented here does not rely on any decomposition results. During the treatment for $N=2$, we provide a new proof for certain Gagliardo-Nirenberg type inequality on $R^2$, which is originally due to [3]. Based on this inequality we get a Trudinger-Moser type inequality involving gradient norm through Taylor expansion. Eventually we construct a special path to obtain the least energy level for solutions. [1] J. Mederski, Nonradial solutions of nonlinear scalar field equations. Nonlinearity 33 (2020), 6349--6380. [2] L. Jeanjean, S.S. Lu, Nonlinear scalar field equations with general nonlinearity. Nonlinear Anal. 190 (2020), 111604. [3] D.E. Edmunds, A.A.Ilyin, Asymptotically sharp multiplicative inequalities. Bull. London Math. Soc. 27 (1995) 71--74.