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Dhanya, R.; Pramanik, Sarbani; Harish, R.

On a class of infinite semipositone problems for \((p,q)\) Laplace operator. (English) Zbl 1534.35386

Asymptotic Anal. 136, No. 3-4, 291-307 (2024).
Summary: We analyze a non-linear elliptic boundary value problem that involves \((p,q)\) Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a singular, monotonically increasing continuous function in \((0, \infty)\) which is eventually positive. The novelty in proving the existence of a positive solution lies in the construction of a suitable subsolution. Our contribution marks an advancement in the theory of existence of positive solutions for infinite semipositone problems in arbitrary bounded domains, whereas the prevailing theory is limited to addressing similar problems only in symmetric domains. Additionally, using the ideas pertaining to the construction of subsolution, we establish the exact behavior of the solutions of “q-sublinear” problem involving \((p,q)\) Laplace operator when the parameter \(\lambda\) is very large. The parameter estimate that we derive is non-trivial due to the non-homogeneous nature of the operator and is of independent interest.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
78A02 Foundations in optics and electromagnetic theory
78A35 Motion of charged particles
83A05 Special relativity
35A21 Singularity in context of PDEs
35B09 Positive solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J75 Singular elliptic equations

Keywords:

\((p,q)\)-Laplacian; infinite semipositone problem; maximal solution; asymptotic estimate; singular problem
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References:

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