What question did this study set out to answer?

To explore the existence and multiplicity of real solutions for the nonlinear Helmholtz equation.

April 16, 2026Demonstratio Mathematica0 citationsOpen Access

Existence and multiplicity of real solutions to the Helmholtz equation with concave and convex nonlinearities

SCShaoXiong ChenYunnan Normal University BLBiao LiuHarbin University of Science and Technology RQRuowen QiuYunnan Normal University

Key Points

To explore the existence and multiplicity of real solutions for the nonlinear Helmholtz equation.
Analyzed the nonlinear Helmholtz equation within the context of asymptotic conditions.
Considered cases with convex and concave nonlinearities.
Applied mathematical techniques to establish conditions for solutions.
Identified conditions under which multiple real solutions exist.
Showed the dependence of solutions on the nature of the nonlinearity.
Established asymptotic behavior of solutions as x approaches infinity.

Abstract

Abstract In this paper, we study real solutions of the nonlinear Helmholtz equation − Δ u − k 2 u = γ M (x) | u | q − 2 u + f (x, u) for x ∈ R N, -u-k^2u= M (x) u ^q-2u+f (x, u) \, for\, x R^N, subject to the asymptotic conditions u (x) = O | x | 1 − N 2 and ∂ 2 u ∂ r 2 (x) + k 2 u (x) = o | x | 1 − N 2, as r = | x | → ∞, u (x) =O (x ^1-N{2}) and ^{2u} {r^2} (x) +k^2u (x) =o (x ^1-N{2}), as\, r= x, where N ≥ 1, k > 0, q ∈ (1, 2). Assuming that f (x, t</jats: i

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