Nontrivial solution for the fractional \(p\)-Laplacian equations via perturbation methods
Summary: We study the existence of nontrivial solution of the following equation without compactness: \[ (- \Delta)_p^\alpha u + \left|u\right|^{p - 2} u = f(x, u),\quad x \in \mathbb{R}^N, \] where \(N, p \geq 2\), \(\alpha \in(0,1)\), \((- \Delta)_p^\alpha\) is the fractional \(p\)-Laplacian, and the subcritical \(p\)-superlinear term \(f \in C(\mathbb{R}^N \times \mathbb{R})\) is 1-periodic in \(x_i\) for \(i = 1,2, \ldots, N\). Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of fractional \(p\)-Laplacian type equation. To overcome this difficulty, by adding coercive potential term and using mountain pass theorem, we get the weak solution \(u_\lambda\) of perturbation equations. And we prove that \(u_\lambda \rightarrow u\) as \(\lambda \rightarrow 0\). Finally, by using vanishing lemma and periodic condition, we get that \(u\) is a nontrivial solution of fractional \(p\)-Laplacian equation.
scientific article; zbMATH DE number 6962641
scientific article