Existence of weak solutions for quasilinear elliptic equations involving the \(p\)-Laplacian. (English) Zbl 1173.35483

Severo, Uberlandio

Electron. J. Differ. Equ. 2008, Paper No. 56, 16 p. (2008).

Summary: This paper shows the existence of nontrivial weak solutions for the quasilinear elliptic equation \[ -\big(\Delta_p u +\Delta_p (u^2)\big) +V(x)|u|^{p-2}u= h(u) \] in \(\mathbb{R}^N\). Here \(V\) is a positive continuous potential bounded away from zero and \(h(u)\) is a nonlinear term of subcritical type. Using minimax methods, we show the existence of a nontrivial solution in \(C^{1,\alpha}_{\text{loc}}(\mathbb{R}^N)\) and then show that it decays to zero at infinity when \(1<p<N\).

MSC:

35J60	Nonlinear elliptic equations
35J20	Variational methods for second-order elliptic equations
35Q55	NLS equations (nonlinear Schrödinger equations)
35D05	Existence of generalized solutions of PDE (MSC2000)
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

quasilinear Schrödinger equation; solitary waves; \(p\)-Laplacian; variational method; mountain-pass theorem