Existence of space-time localized rogue wave solutions is shown for the semilinear wave equation with variable coefficients via variational methods under sufficient conditions on coefficients, operator, and p>1.
Willem.Minimax theorems, volume 24 ofProgress in Nonlinear Differential Equations and their Applications
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For the mixed local-nonlocal problem with concave-critical nonlinearity, a threshold Lambda_epsilon exists such that positive solutions occur for lambda below it, with at least two solutions for small lambda when epsilon is small enough.
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Quasilinear problems with mixed local-nonlocal operator and concave-critical nonlinearities: Multiplicity of positive solutions
For the mixed local-nonlocal problem with concave-critical nonlinearity, a threshold Lambda_epsilon exists such that positive solutions occur for lambda below it, with at least two solutions for small lambda when epsilon is small enough.