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Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth. (English) Zbl 1448.35126

Summary: In this paper we study the nonhomogeneous semilinear fractional Schrödinger equation with critical growth \[\begin{cases}( - \Delta)^s u + u = {u^{2_s^* - 1}} + \lambda (f(x,u)) + h(x)),\quad & x \in\mathbb{R}^N, \\ u \in H^s(\mathbb{R}^N),\quad u(x) > 0,& x \in\mathbb{R}^N, \end{cases}\] where \(s \in (0, 1)\), \(N > 4s\), and \(\lambda > 0\) is a parameter, \(2_s^* = \frac{2N}{N - 2s}\) is the fractional critical Sobolev exponent, \(f\) and \(h\) are some given functions. We show that there exists \(0 < \lambda^* < + \infty\) such that the problem has exactly two positive solutions if \(\lambda \in (0, \lambda^* \)), no positive solutions for \(\lambda > \lambda^* \), a unique solution \(( \lambda^*,u_{ \lambda^*})\) if \(\lambda = \lambda^* \), which shows that \(( \lambda^*,u_{ \lambda^*})\) is a turning point in \(H^S( \mathbb{R}^N)\) for the problem. Our proofs are based on the variational methods and the principle of concentration-compactness.

MSC:

35J10 Schrödinger operator, Schrödinger equation
35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
35B33 Critical exponents in context of PDEs
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