Superlinear Schrödinger operators
Copyright policy )Superlinear Schrödinger operators
Consider the following semilinear elliptic partial differential equation \[ \mathcal{A}u=f(x,u),\quad u\in D, \] in an unbounded domain. Note that the case of the Schrödinger operator \(\mathcal{A}=-\Delta+V(x)\) on \(D=H^1(\mathbb{R}^n)\) is included. The main objective is to find nontrivial solutions and, in particular, ground state solutions, that is, solutions minimizing the corresponding energy functional. This problem is studied under the assumptions that \(\mathcal{A}\) is self-adjoint and has a spectral gap, and the function \(f(x,u)\) is superlinear. Using the result from the author and \textit{W. Zou} [ESAIM, Control Optim. Calc. Var. 9, 601--619 (2003; Zbl 1173.35482)], the author finds nontrivial and ground state solutions for this equation under conditions on \(\mathcal{A}\) and \(f\) weaker than those previously assumed.
- University of California, Los Angeles United States
- University of California United States
- University of California, Berkely United States
- University of California United States
- University of California United States
Variational methods for second-order elliptic equations, Semilinear elliptic equations, variational methods, saddle point theory, Variational inequalities, ground state solutions, Variational methods, Schrödinger operator, Schrödinger equation, critical point theory, semilinear differential equations, Critical point theory, Saddle point theory, Analysis, Semilinear differential equations
Variational methods for second-order elliptic equations, Semilinear elliptic equations, variational methods, saddle point theory, Variational inequalities, ground state solutions, Variational methods, Schrödinger operator, Schrödinger equation, critical point theory, semilinear differential equations, Critical point theory, Saddle point theory, Analysis, Semilinear differential equations
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