Ground state solutions for general Choquard equations with a variable potential and a local nonlinearity. (English) Zbl 1437.35209

Summary: This paper deals with the following Choquard equation with a local nonlinear perturbation: \[\begin{cases} - \Delta u+V(x)u=(I_{\alpha } \ast F(u))f(u)+g(u), \qquad x\in \mathbb{R}^N, \\ u\in H^1(\mathbb{R}^N), \end{cases}\] where \(I_{\alpha }: \mathbb{R}^N\rightarrow \mathbb{R}\) is the Riesz potential, \(N\ge 3\), \(\alpha \in (0, N)\), \(F(t)=\int_0^tf(s)\text{d}s\ge 0\) (\(\not \equiv 0\)), \(V\in{\mathcal{C}}^1(\mathbb{R}^N, [0, \infty ))\) and \(f, g\in{\mathcal{C}}(\mathbb{R}, \mathbb{R})\) satisfying the subcritical growth. Under some suitable conditions on \(V\), we prove that the above problem admits ground state solutions without super-linear conditions near infinity or monotonicity properties on \(f\) and \(g\). In particular, some new tricks are used to overcome the combined effects and the interaction of the nonlocal nonlinear term and the local nonlinear term. Our results improve and extend the previous related ones in the literature.


35J20 Variational methods for second-order elliptic equations
35J62 Quasilinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: DOI


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