<p>We study the existence of a positive solution for a class of nonlinear Schrödinger equations <Equation ID="Equ53"> <EquationSource Format="TEX">\(\begin{aligned} -\Delta u+V(x)u=f(u), \qquad u \in \mathcal {D}^{1,2}(\mathbb {R}^N), \ N\ge 3. \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>u</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="script">D</mi> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="4pt" /> <mi>N</mi> <mo>≥</mo> <mn>3</mn> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>Here the potential <i>V</i> is symmetric under a group action <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(G \subset O(N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>⊂</mo> <mi>O</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and decay to zero at infinity, and the nonlinearity <i>f</i>, under very mild hypotheses, is asymptotically linear or superlinear and subcritical at infinity, not satisfying any monotonicity condition.</p>