On a double resonant problem in \(\mathbb R^N\). (English) Zbl 1034.35024

Furtado, Marcelo F.; Maia, Liliane A.; Silva, Elves A. B.

Differ. Integral Equ. 15, No. 11, 1335-1344 (2002).

The authors establish via variational methods the existence and multiplicity of solutions for a class of semilinear elliptic equations in \(\mathbb R^n\): \[ -\Delta u+b(x)u=f(x,u),\quad x\in\mathbb R^n\,, \] where \(n\geq 3\) and the potential \(b\) is a continuous function satisfying \(b(x)\geq b_0>0\) for all \(x\in\mathbb R^n\). The main goal of the paper is to explore the compactness provided by the condition on the potential, to study a class of double resonant problems under a local nonquadraticity condition. For the existence of solution the nonlinearity may satisfy a critical growth condition. Related results can be found [{P. H. Rabinowitz}, Z. Angew. Math. Phys. 43, 270–291 (1992; Zbl 0763.35087)].

Reviewer: Nicolae Pop (Baia Mare)

MSC:

35J60	Nonlinear elliptic equations
35J20	Variational methods for second-order elliptic equations
35B33	Critical exponents in context of PDEs

Keywords:

boundary value problem for elliptic systems; variational methods; double resonant problems

Citations:

Zbl 0763.35087