## On ground state solutions for singular and semi-linear problems including super-linear terms at infinity.(English)Zbl 1178.35169

Summary: We establish a result concerning the existence of entire, positive, classical and bounded solutions which converge to zero at infinity for the semi-linear equation $$-\Delta u=\lambda f(x,u),x\in \mathbb R^N$$, where $$f:\mathbb R^N\times (0,\infty)\to [0,\infty)$$ is a suitable function and $$\lambda >0$$ is a real parameter. This result completes the principal theorem of A. Mohammed [Nonlinear Anal., Theory Methods Appl. 71, No. 3–4 (A), 1276–1280 (2009; Zbl 1167.35371)] mainly because his result does not address the super-linear terms at infinity. Penalty arguments, lower-upper solutions and an approximation procedure are used.

### MSC:

 35J61 Semilinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35J20 Variational methods for second-order elliptic equations 35J67 Boundary values of solutions to elliptic equations and elliptic systems 35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian 35B40 Asymptotic behavior of solutions to PDEs 35B09 Positive solutions to PDEs

Zbl 1167.35371
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### References:

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