Existência e multiplicidade de soluções e existência de ground state para uma classe de problemas elípticos em RN com uma não linearidade geral

Abstract

In this work, the existence of infinitely many radially symmetric solutions is proved for the problem $$(-\Delta_p)^s u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, \ \ u\in W^{s,p}(\mathbb{R}^N),$$ where $s\in (0,1], \ 2 \leq p < \infty, \ sp \leq N,\ 2 \leq N \in \mathbb{N}$ and $(-\Delta_p)^s$ is the (fractional if $0<s<1$) $p$-Laplacian operator. Both the cases were handled $sp=N$ and $sp<N.$ The nonlinearity $g$ was a function of Berestycki-Lions type with critical exponential growth if $sp=N$ and critical polynomial growth if $sp<N$. After that, decomposing the space $\mathbb{R}^N$ in the form $\mathbb{R}^M\times \mathbb{R}^M\times \mathbb{R}^{N-2M}$, we prove the existence of infinitely many nonradially symmetric solutions in the cases $N=4 $ or $ N\geq 6$ and $N-2M \neq 1$, on that $M>0$ is integer and $0\leq M\leq N/2$. We also prove the existence of a ground state solution for the same problem. Next, we consider the problem $$\Delta^2 u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, \ \ u\in H^2(\mathbb{R}^N),$$ where $N = 4$ and $\Delta^2$ is the bilaplacian operator. We obtain the same results stated above. Finally, we consider the problem $$(-\Delta)^s u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, \ \ u\in W^{s,2}(\mathbb{R}^N),$$ where $s\in (1,2), \quad N\geq 3 $ and $(-\Delta)^s$ is the higher order fractional $2$-Laplacian operator and, once more, obtain the same results described in the case of the $p$-Laplacian operator.