Infinitely many sign-changing solutions for a variable-exponent quasilinear Schrödinger equation at the quartic threshold

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Abstract

We prove the existence of infinitely many sign-changing solutions for a variable-exponent modified quasilinear Schrödinger equation on a smooth bounded domain in three dimensions. The exponent is allowed to attain the quartic threshold on a nonempty null set. Consequently, the transformed nonlinearity is not uniformly superquadratic. The classical dual change of variables v = h(u), where $h^{\prime}(s)=\left(1+s^{2}\right)^{1 / 2}$, converts the equation into a semilinear problem on $H_{0}^{1}(\Omega)$ with subcritical growth. An exact identity for h⁻¹ provides the scaling monotonicity used in the Cerami boundedness argument and a quantitative logarithmic nonquadraticity remainder at the threshold. We establish the Cerami condition, prove the strict invariance of small neighborhoods of the positive and negative cones, and verify the hypotheses of an invariant-set symmetric minimax theorem. These steps yield an unbounded sequence of nodal critical values. A bootstrap argument and an admissible transformed test function then convert the critical points into weak solutions of the original equation. The main contribution is a nonperturbative treatment of the variable-exponent regime in which the quartic threshold is attained on a null set. The dual substitution and the invariant-set theorem are used as established tools.

Keywords

variable exponent / quasilinear Schrödinger equation / dual transformation / quartic threshold / sign-changing solution / invariant cone

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