Infinitely many small solutions for a new class of sublinear-quasilinear Schrödinger equations

Safa Bridaa, Abderrazek Benhassine Hassine

Abstract


We prove the existence of infinitely many small solutions for the quasilinear Schrödinger equation
\[
-\Delta u + V(x)u + \tau \Delta\left(\sqrt{1+u^{2}}\right)\frac{u}{2\sqrt{1+u^{2}}} = W(x)f(x,u), \quad x \in \mathbb{R}^{N},
\]
where $N \geq 3$, $\tau \geq 2$. Under significantly weakened hypotheses requiring only local boundedness of potentials, spatially varying growth conditions, and oscillation along sequences rather than uniformly, we prove the existence of a sequence of weak solutions converging to zero in $L^\infty(\mathbb{R}^N)$. Our approach combines variational methods with $\mathbb{Z}_2$-genus theory and Moser iteration.


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DOI: https://doi.org/10.52846/ami.v53i1.2083