Infinitely many solutions of degenerate quasilinear Schrödinger equation with general potentials
2021
In this paper, we study the following quasilinear Schrodinger equation: $$ -\operatorname{div}\bigl(a(x,\nabla u)\bigr)+V(x) \vert x \vert ^{-\alpha p^{*}} \vert u \vert ^{p-2}u=K(x) \vert x \vert ^{- \alpha p^{*}}f(x,u) \quad \text{in } \mathbb{R}^{N}, $$
where $N\geq 3$
, $1< p< N$
, $-\infty <\alpha <\frac{N-p}{p}$
, $\alpha \leq e\leq \alpha +1$
, $d=1+\alpha -e$
, $p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}$
(critical Hardy–Sobolev exponent), V and K are nonnegative potentials, the function a satisfies suitable assumptions, and f is superlinear, which is weaker than the Ambrosetti–Rabinowitz-type condition. By using variational methods we obtain that the quasilinear Schrodinger equation has infinitely many nontrivial solutions.
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