<p>We consider semilinear elliptic problems of the form <Equation ID="Equ99"> <EquationSource Format="TEX">$ -\Delta u + \lambda u = f(x,u), \quad u\in H^1_0(A), $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>λ</mi> <mi>u</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>u</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A\subset \mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, is either a bounded or unbounded annulus, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We study a broad class of nonlinearities <i>f</i> with superlinear growth at infinity, including exponential- and power-type ones. Under suitable assumptions, we establish the existence of a positive nonradial solution via techniques in the spirit of Szulkin’s nonsmooth critical point theory, applied within a convex cone in Orlicz spaces. Notably, the Trudinger-Moser inequality fails in the whole Sobolev space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H^1_0(A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

An Orlicz space approach to exponential elliptic problems in higher dimensions

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2026年2月17日

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Alberto Boscaggin,
Francesca Colasuonno,
Benedetta Noris,
Federica Sani

摘要

We consider semilinear elliptic problems of the form $ -\Delta u + \lambda u = f(x,u), \quad u\in H^1_0(A), $ $- Δ u + λ u = f (x, u), u \in H_{0}^{1} (A),$ where $A\subset \mathbb {R}^N$ $A \subset R^{N}$ , $N\ge 3$ $N \geq 3$ , is either a bounded or unbounded annulus, and $\lambda \ge 0$ $λ \geq 0$ . We study a broad class of nonlinearities f with superlinear growth at infinity, including exponential- and power-type ones. Under suitable assumptions, we establish the existence of a positive nonradial solution via techniques in the spirit of Szulkin’s nonsmooth critical point theory, applied within a convex cone in Orlicz spaces. Notably, the Trudinger-Moser inequality fails in the whole Sobolev space $H^1_0(A)$ $H_{0}^{1} (A)$ .